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FLUID MECHANICS FIFTH EDITION Authors

The fifth edition of this established text provides an excellent and comprehensive treatment of fluid mechanics that is concisely written and supported by good worked examples. This revision of a classic text presents relevant material for mechanical and civil engineers, as well as energy and environmental services engineers. It recognises the evolution of the subject and provides thorough coverage of both established theory and emerging topics.

Excellent coverage • All the latest developments and applications, including emerging specialisms • Strong coverage of the principles of fluid flow – fundamentals emphasized early in the text Emphasis on understanding • Good, clear explanations, together with extensive worked examples and tutorials – brought together to reinforce the reader’s understanding of all the key principles Helpful resources on accompanying website • With solutions to tutorials and simulations of fluid mechanics, presented through some 20 programs, all fully discussed in the text

Building on the success of previous editions, this fifth edition introduces the following new features: • New chapters on the impact of environmental change and the fluid mechanics required to both understand the consequences of climate change and develop renewable energy technologies • Updated chapter on dimensional analysis • Updated solutions manual • Extended website to include enhanced and additional simulations

Dr Janusz Gasiorek, formerly of South Bank University, London where he led the Fluid Mechanics group in Mechanical Engineering, with specialist research interest in rotodynamic machinery and fan engineering. Professor John Swaffield, Heriot–Watt University, has taught fluid mechanics for 30 years with specialist research in pressure transients, free surface unsteady flows and water conservation. Dr Lynne Jack, Heriot–Watt University, senior lecturer in energy systems and associated environmental impacts, with research interests in unsteady flow modelling and the implications for the built environment of climate change.

FIFTH EDITION

FLUID MECHANICS

Fluid Mechanics has become a textbook of choice with both students and lecturers, due to its:

Dr John Douglas, formerly of South Bank University, London.

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DOUGLAS GASIOREK SWAFFIELD JACK

Fluid Mechanics is ideal for use throughout a first degree course in all engineering disciplines where a good understanding of the subject is required. It is also suitable for conversion MSc courses requiring a fundamental treatment of fluid mechanics and will be a valuable resource for specialist Continuing Professional Development courses, including those offered by Distance Learning.

John F. Douglas Janusz M. Gasiorek John A. Swaffield www.pearson-books.com

Additional student support at www.pearsoned.co.uk/douglas

Lynne B. Jack

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Fluid Mechanics Visit the Fluid Mechanics, fifth edition Companion Website at www.pearsoned.co.uk/douglas to find valuable student learning material including: l

Simulations and computer programs for students, with clear instructions on how to use them to enhance study

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We work with leading authors to develop the strongest educational materials in engineering, bringing cutting-edge thinking and best learning practice to a global market. Under a range of well-known imprints, including Prentice Hall, we craft high quality print and electronic publications which help readers to understand and apply their content, whether studying or at work. To find out more about the complete range of our publishing please visit us on the World Wide Web at: www.pearsoned.co.uk

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Fluid Mechanics Fifth edition JOHN F. DOUGLAS M.Sc., Ph.D., A.C.G.I., D.I.C., C.Eng., M.I.C.E., M.I.Mech.E. Formerly of London South Bank University

JANUSZ M. GASIOREK B.Sc., Ph.D., C.Eng., M.I.Mech.E., M.C.I.B.S.E. Formerly of London South Bank University

JOHN A. SWAFFIELD F.R.S.E., B.Sc., M.Phil., Ph.D., C.Eng., M.R.Ae.S., F.C.I.W.E.M., F.C.I.B.S.E. William Watson Professor of Building Engineering and Head of the School of the Built Environment, Heriot-Watt University, Edinburgh

LYNNE B. JACK B.Eng., Ph.D., M.I.L.T. Senior Lecturer in Environmental Engineering, School of the Built Environment, Heriot-Watt University, Edinburgh

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Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk First published by Pitman Publishing Limited 1979 Second Edition 1985 Third Edition published under the Longman imprint 1995 Fourth Edition 2001 Fifth Edition 2005 © J. F. Douglas, J. M. Gasiorek and J. A. Swaffield 1979, 2001 © J. F. Douglas, J. M. Gasiorek, J. A. Swaffield and Lynne B. Jack 2005 The rights of J. F. Douglas, J. M. Gasiorek, J. A. Swaffield and Lynne B. Jack to be identified as authors of this Work have been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP. ISBN-13: 978-0-13-129293-2 ISBN-10: 0-13-129293-5 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data Fluid mechanics / John F. Douglas ... [et al.].— 5th ed. p. cm. Includes bibliographical references and index. 1. Fluid mechanics. I. Douglas, John F. TA357.D68 2006 620.1’06—dc22 2005054617 10 9 8 7 6 5 4 3 2 10 09 08 07 06 Set by 35 in 10/12pt Times Roman Printed and bound by Ashford Colour Press Ltd, Gosport

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for Janusz M. Gasiorek 1927–2003 friend and guide

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Contents Preface to the Fifth Edition xix Preface to the Fourth Edition xxi Preface to the Third Edition xxiv Preface to the Second Edition xxvi Preface to the First Edition xxvii Acknowledgements xxviii List of Computer Programs xxix List of Symbols xxxi

PART I ELEMENTS OF FLUID MECHANICS Chapter 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20

Fluids and their Properties 2

Fluids 4 Shear stress in a moving fluid 4 Differences between solids and fluids 5 Newtonian and non-Newtonian fluids 6 Liquids and gases 7 Molecular structure of materials 7 The continuum concept of a fluid 9 Density 10 Viscosity 11 Causes of viscosity in gases 12 Causes of viscosity in a liquid 13 Surface tension 14 Capillarity 15 Vapour pressure 16 Cavitation 17 Compressibility and the bulk modulus 17 Equation of state of a perfect gas 19 The universal gas constant 19 Specific heats of a gas 19 Expansion of a gas 20 Concluding remarks 22 Summary of important equations and concepts 22

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Chapter 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20

Statics of fluid systems 26 Pressure 27 Pascal’s law for pressure at a point 28 Variation of pressure vertically in a fluid under gravity 29 Equality of pressure at the same level in a static fluid 30 General equation for the variation of pressure due to gravity from point to point in a static fluid 32 Variation of pressure with altitude in a fluid of constant density 33 Variation of pressure with altitude in a gas at constant temperature 34 Variation of pressure with altitude in a gas under adiabatic conditions 35 Variation of pressure and density with altitude for a constant temperature gradient 38 Variation of temperature and pressure in the atmosphere 39 Stability of the atmosphere 41 Pressure and head 43 The hydrostatic paradox 44 Pressure measurement by manometer 45 Relative equilibrium 51 Pressure distribution in a liquid subject to horizontal acceleration 51 Effect of vertical acceleration 52 General expression for the pressure in a fluid in relative equilibrium 52 Forced vortex 56 Concluding remarks 57 Summary of important equations and concepts 57 Problems 57

Chapter 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13

Pressure and Head 24

Static Forces on Surfaces. Buoyancy 60

Action of fluid pressure on a surface 62 Resultant force and centre of pressure on a plane surface under uniform pressure 62 Resultant force and centre of pressure on a plane surface immersed in a liquid 63 Pressure diagrams 68 Force on a curved surface due to hydrostatic pressure 71 Buoyancy 73 Equilibrium of floating bodies 76 Stability of a submerged body 76 Stability of floating bodies 77 Determination of the metacentric height 78 Determination of the position of the metacentre relative to the centre of buoyancy 78 Periodic time of oscillation 81 Stability of a vessel carrying liquid in tanks with a free surface 82 Concluding remarks 85 Summary of important equations and concepts 85 Problems 85

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PART II

CONCEPTS OF FLUID FLOW 88

Chapter 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14

Fluid flow 92 Uniform flow and steady flow 93 Frames of reference 93 Real and ideal fluids 94 Compressible and incompressible flow 94 One-, two- and three-dimensional flow 95 Analyzing fluid flow 96 Motion of a fluid particle 96 Acceleration of a fluid particle 98 Laminar and turbulent flow 100 Discharge and mean velocity 102 Continuity of flow 104 Continuity equations for three-dimensional flow using Cartesian coordinates 107 Continuity equation for cylindrical coordinates 109 Concluding remarks 109 Summary of important equations and concepts 110 Problems 110

Chapter 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17

Motion of Fluid Particles and Streams 90

The Momentum Equation and its Applications 112

Momentum and fluid flow 114 Momentum equation for two- and three-dimensional flow along a streamline 115 Momentum correction factor 116 Gradual acceleration of a fluid in a pipeline neglecting elasticity 119 Force exerted by a jet striking a flat plate 120 Force due to the deflection of a jet by a curved vane 123 Force exerted when a jet is deflected by a moving curved vane 124 Force exerted on pipe bends and closed conduits 126 Reaction of a jet 129 Drag exerted when a fluid flows over a flat plate 136 Angular motion 138 Euler’s equation of motion along a streamline 141 Pressure waves and the velocity of sound in a fluid 143 Velocity of propagation of a small surface wave 146 Differential form of the continuity and momentum equations 148 Computational treatment of the differential forms of the continuity and momentum equations 151 Comparison of CFD methodologies 155 Concluding remarks 162 Summary of important equations and concepts 162 Further reading 163 Problems 163

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Chapter 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19

Mechanical energy of a flowing fluid 168 Steady flow energy equation 172 Kinetic energy correction factor 174 Applications of the steady flow energy equation 175 Representation of energy changes in a fluid system 178 The Pitot tube 180 Determination of volumetric flow rate via Pitot tube 181 Computer program VOLFLO 183 Changes of pressure in a tapering pipe 183 Principle of the venturi meter 185 Pipe orifices 187 Limitation on the velocity of flow in a pipeline 188 Theory of small orifices discharging to atmosphere 188 Theory of large orifices 192 Elementary theory of notches and weirs 193 The power of a stream of fluid 197 Radial flow 198 Flow in a curved path. Pressure gradient and change of total energy across the streamlines 199 Vortex motion 202 Concluding remarks 208 Summary of important equations and concepts 209 Problems 209

Chapter 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11

The Energy Equation and its Applications 166

Two-dimensional Ideal Flow 212

Rotational and irrotational flow 214 Circulation and vorticity 216 Streamlines and the stream function 218 Velocity potential and potential flow 220 Relationship between stream function and velocity potential. Flow nets 224 Straight line flows and their combinations 228 Combined source and sink flows. Doublet 236 Flow past a cylinder 241 Curved flows and their combinations 244 Flow past a cylinder with circulation. Kutta–Joukowsky’s law 249 Computer program ROTCYL 252 Concluding remarks 253 Summary of important equations and concepts 253 Problems 254

PART III DIMENSIONAL ANALYSIS AND SIMILARITY 256 Chapter 8 8.1 8.2

Dimensional Analysis 258

Dimensional analysis 260 Dimensions and units 260

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8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11

Dimensional reasoning, hom*ogeneity and dimensionless groups 260 Fundamental and derived units and dimensions 261 Additional fundamental dimensions 263 Dimensions of derivatives and integrals 265 Units of derived quantities 266 Conversion between systems of units, including the treatment of dimensional constants 266 Dimensional analysis by the indicial method 269 Dimensional analysis by the group method 271 The significance of dimensionless groups 279 Concluding remarks 280 Summary of important equations and concepts 280 Further reading 280 Problems 281

Chapter 9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12

Similarity 282

Geometric similarity 286 Dynamic similarity 286 Model studies for flows without a free surface. Introduction to approximate similitude at high Reynolds numbers 291 Zone of dependence of Mach number 293 Significance of the pressure coefficient 294 Model studies in cases involving free surface flow 295 Similarity applied to rotodynamic machines 297 River and harbour models 299 Groundwater and seepage models 305 Computer program GROUND, the simulation of groundwater seepage 310 Pollution dispersion modelling, outfall effluent and stack plumes 311 Pollutant dispersion in one-dimensional steady uniform flow 314 Concluding remarks 319 Summary of important equations and concepts 319 Further reading 320 References 320 Problems 321

PART IV BEHAVIOUR OF REAL FLUIDS 322 Chapter 10 Laminar and Turbulent Flows in Bounded Systems 324 10.1 10.2 10.3

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Incompressible, steady and uniform laminar flow between parallel plates 326 Incompressible, steady and uniform laminar flow in circular cross-section pipes 331 Incompressible, steady and uniform turbulent flow in bounded conduits 335

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10.4

Incompressible, steady and uniform turbulent flow in circular cross-section pipes 338 10.5 Steady and uniform turbulent flow in open channels 342 10.6 Velocity distribution in turbulent, fully developed pipe flow 343 10.7 Velocity distribution in fully developed, turbulent flow in open channels 352 10.8 Separation losses in pipe flow 352 10.9 Significance of the Colebrook–White equation in pipe and duct design 359 10.10 Computer program CBW 362 Concluding remarks 362 Summary of important equations and concepts 363 Further reading 363 Problems 364

Chapter 11

Boundary Layer 366

11.1 11.2 11.3

Qualitative description of the boundary layer 368 Dependence of pipe flow on boundary layer development at entry 370 Factors affecting transition from laminar to turbulent flow regimes 371 11.4 Discussion of flow patterns and regions within the turbulent boundary layer 372 11.5 Prandtl mixing length theory 374 11.6 Definitions of boundary layer thicknesses 377 11.7 Application of the momentum equation to a general section of boundary layer 378 11.8 Properties of the laminar boundary layer formed over a flat plate in the absence of a pressure gradient in the flow direction 379 11.9 Properties of the turbulent boundary layer over a flat plate in the absence of a pressure gradient in the flow direction 384 11.10 Effect of surface roughness on turbulent boundary layer development and skin friction coefficients 388 11.11 Effect of pressure gradient on boundary layer development 388 Concluding remarks 391 Summary of important equations and concepts 391 Further reading 392 Problems 392

Chapter 12 12.1 12.2 12.3 12.4 12.5 12.6 12.7

Incompressible Flow around a Body 394

Regimes of external flow 396 Drag 397 Drag coefficient and similarity considerations 401 Resistance of ships 403 Flow past a cylinder 407 Flow past a sphere 411 Flow past an infinitely long aerofoil 418

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12.8 Flow past an aerofoil of finite length 426 12.9 Wakes and drag 430 12.10 Computer program WAKE 435 Concluding remarks 436 Summary of important equations and concepts 436 Problems 436

Chapter 13 13.1 13.2 13.3 13.4 13.5

Compressible Flow around a Body 438

Effects of compressibility 440 Shock waves 445 Oblique shock waves 455 Supersonic expansion and compression 457 Computer program NORSH 459 Concluding remarks 459 Summary of important equations and concepts 460 Problems 460

PART V STEADY FLOW IN PIPES, DUCTS AND OPEN CHANNELS 462 Chapter 14 Steady Incompressible Flow in Pipe and Duct Systems 464 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 14.11 14.12 14.13 14.14

General approach 466 Incompressible flow through ducts and pipes 467 Computer program SIPHON 470 Incompressible flow through pipes in series 471 Incompressible flow through pipes in parallel 473 Incompressible flow through branching pipes. The three-reservoir problem 475 Incompressible steady flow in duct networks 478 Resistance coefficients for pipelines in series and in parallel 486 Incompressible flow in a pipeline with uniform draw-off 490 Incompressible flow through a pipe network 490 Head balance method for pipe networks 491 Computer program HARDYC 492 The quantity balance method for pipe networks 494 Quasi-steady flow 497 Concluding remarks 503 Summary of important equations and concepts 503 Further reading 504 Problems 504

Chapter 15 15.1 15.2

Uniform Flow in Open Channels 508

Flow with a free surface in pipes and open channels 510 Resistance formulae for steady uniform flow in open channels 512

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15.3 15.4

Optimum shape of cross-section for uniform flow in open channels 517 Optimum depth for flow with a free surface in covered channels 521 Concluding remarks 524 Summary of important equations and concepts 525 Further reading 525 Problems 526

Chapter 16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10 16.11 16.12 16.13 16.14

Specific energy and alternative depths of flow 530 Critical depth in non-rectangular channels 532 Computer program CRITNOR 534 Non-dimensional specific energy curves 535 Occurrence of critical flow conditions 536 Flow over a broad-crested weir 537 Effect of lateral contraction of a channel 538 Non-uniform steady flow in channels 541 Equations for gradually varied flow 542 Classification of water surface profiles 544 The hydraulic jump 547 Location of a hydraulic jump 549 Computer program CHANNEL 550 Annular water flow considerations 551 Concluding remarks 556 Summary of important equations and concepts 556 Further reading 557 Problems 558

Chapter 17 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9

Non-uniform Flow in Open Channels 528

Compressible Flow in Pipes 560

Compressible flow. The basic equations 562 Steady isentropic flow in non-parallel-sided ducts neglecting friction 563 Mass flow through a venturi meter 564 Mass flow from a reservoir through an orifice or convergent–divergent nozzle 567 Conditions for maximum discharge from a reservoir through a convergent–divergent duct or orifice 568 The Laval nozzle 569 Normal shock wave in a diffuser 573 Compressible flow in a duct with friction under adiabatic conditions. Fanno flow 578 Isothermal flow of a compressible fluid in a pipeline 582 Concluding remarks 585 Summary of important equations and concepts 586 Problems 586

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PART VI FLUID MECHANICS FOR ENVIRONMENTAL CHANGE 588 Chapter 18 Environmental Change and Renewable Energy Technologies 590 18.1 18.2 18.3 18.4

Environmental change 592 The application of wind turbines to electrical power generation 602 Wave energy conversion for electrical power generation 616 Tidal power 631 Concluding remarks 632 Summary of important concepts 633 Further reading 634 References 635

Chapter 19 Environmental Change and Rainfall Runoff Flow Modelling 636 19.1 19.2 19.3 19.4 19.5 19.6 19.7

Gradually varied unsteady free surface flow 638 Computer program UNSCHAN 646 Implicit four-point scheme 648 Flood routeing 650 The prediction of flood behaviour 652 Time-dependent urban stormwater routeing 657 Combined free surface and pressure surge analysis. Siphonic rainwater systems 660 Concluding remarks 669 Summary of important equations and concepts 669 Further reading 670 References 670

PART VII UNSTEADY FLOW IN BOUNDED SYSTEMS 672 Chapter 20 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8

Pressure Transient Theory and Surge Control 674

Wave propagation velocity and its dependence on pipe and fluid parameters and free gas 682 Computer program WAVESPD 688 Simplification of the basic pressure transient equations 690 Application of the simplified equations to explain pressure transient oscillations 690 Surge control 695 Control of surge following valve closure, with pump running and surge tank applications 697 Computer program SHAFT 704 Control of surge following pump shutdown 706 Concluding remarks 711 Summary of important equations and concepts 711

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Further reading 712 Problems 714

Chapter 21 Simulation of Unsteady Flow Phenomena in Pipe, Channel and Duct Systems 716 21.1 21.2 21.3 21.4 21.5

21.6 21.7 21.8

Development of the St Venant equations of continuity and motion 718 The method of characteristics 724 Network simulation 737 Computer program FM5SURG. The simulation of waterhammer 739 Computer programs FM5WAVE and FM5GUTT. The simulation of open-channel free surface and partially filled pipe flow, with and without lateral inflow 749 Simulation of low-amplitude air pressure transient propagation 755 Computer program FM5AIR. The simulation of unsteady air flow in pipe and duct networks 756 Entrained air flow analysis review 760 Concluding remarks 763 Summary of important equations and concepts 764 Further reading 764 References 765

PART VIII FLUID MACHINERY. THEORY, PERFORMANCE AND APPLICATION 766 Chapter 22 22.1 22.2 22.3 22.4 22.5

Introduction 770 One-dimensional theory 772 Isolated blade and cascade considerations 780 Departures from Euler’s theory and losses 788 Compressible flow through rotodynamic machines 794 Concluding remarks 798 Summary of important equations and concepts 798 Further reading 798 Problems 799

Chapter 23 23.1 23.2 23.3 23.4 23.5 23.6 23.7

Theory of Rotodynamic Machines 768

Performance of Rotodynamic Machines 800

The concept of performance characteristics 802 Losses and efficiencies 803 Dimensionless coefficients and similarity laws 809 Computer program SIMPUMP 815 Scale effects 816 Type number 817 Centrifugal pumps and fans 820

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23.8 23.9 23.10 23.11 23.12 23.13 23.14

Axial flow pumps and fans 822 Mixed flow pumps and fans 825 Water turbines 826 The Pelton wheel 827 Francis turbines 831 Axial flow turbines 836 Hydraulic transmissions 839 Concluding remarks 846 Summary of important equations and concepts 847 Problems 848

Chapter 24 24.1 24.2 24.3 24.4 24.5 24.6

Reciprocating pumps 852 Rotary pumps 863 Rotary gear pumps 864 Rotary vane pumps 865 Rotary piston pumps 866 Hydraulic motors 868 Concluding remarks 868 Summary of important equations and concepts 869 Problems 870

Chapter 25 25.1 25.2 25.3 25.4 25.5 25.6 25.7 25.8 25.9 25.10 25.11 25.12 25.13 25.14

Positive Displacement Machines 850

Machine–Network Interactions 872

Fans, pumps and fluid networks 874 Parallel and series pump operation 881 Fans in series and parallel 883 Fan and system matching. An application of the steady flow energy equation 888 Change in the pump speed and the system 892 Change in the pump size and the system 895 Changes in fan speed, diameter and air density 897 Jet fans 900 Computer program MATCH 908 Cavitation in pumps and turbines 909 Fan and pump selection 914 Fan suitability 918 Ventilation and airborne contamination as a criterion for fan selection 921 Computer program CONTAM 929 Concluding remarks 931 Summary of important equations and concepts 932 Further reading 933 Problems 933

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Appendix 1

Some Properties of Common Fluids 938

A1.1 Variation of some properties of water with temperature 938 A1.2 Variation of bulk modulus of elasticity of water with temperature and pressure 939 A1.3 Variation of some properties of air with temperature at atmospheric pressure 939 A1.4 Some properties of common liquids 939 A1.5 Some properties of common gases (at p = 1 atm, T = 273 K) 940 A1.6 International Standard Atmosphere 940 A1.7 Solubility of air in pure water at various temperatures 941 A1.8 Absolute viscosity of some common fluids 941

Appendix 2 Values of Drag Coefficient CD for Various Body Shapes 942 Index 943

Supporting resources Visit www.pearsoned.co.uk/douglas to find valuable online resources Companion Website for students l Simulations and computer programs for students, with clear instructions on how to use them to enhance study For instructors l Complete, downloadable Solutions Manual For more information please contact your local Pearson Education sales representative or visit www.pearsoned.co.uk/douglas

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Preface to the Fifth Edition Fluid mechanics remains a core component of engineering education. While its applications may have changed from the preoccupations of earlier engineers, and while our ability to apply its principles has been transformed by modern computing capabilities, those principles remain unchanged. Indeed it is possible that early engineers, perhaps two millennia in the past, concerned with the efficient delivery of a water supply dependent upon open channel flows, would recognize some of our current concerns as to the representation of flood routing. This fifth edition of Fluid Mechanics therefore recognizes evolution in the application of fluid mechanics as well as the necessity to reinforce and underpin the student’s understanding of its fundamental precepts. The fifth edition retains its emphasis on fundamentals in the early Parts of the text. As in previous editions, fundamentals are reinforced with both ample worked examples and tutorial examples whose solutions are available on the supporting website. Similarly computing support is provided on the website with some 20 simulations that the student or lecturer may use to extend the scope of the material provided by allowing a wide range of applications to be modelled, ranging from contamination decay in a ventilated space to pressure surge in pipe and duct flow or unsteady free surface flows in long channels. Later Parts introduce more specialist topics, including as in previous editions rotodynamic machinery and unsteady flow. The authors believe that a continuation of this presentation is both appropriate and essential. However, at the start of the twenty-first century the text cannot avoid the issues of global climate change that now increasingly appear to be corroborated by environmental research. If that research is fully substantiated then engineers will have two main roles to play, namely providing alternative energy sources and power generation to allow a continuation of supply without exacerbating environmental change, and a role in mitigating the environmental consequences of climate change, through for example the management of flood risk. The fifth edition of Fluid Mechanics addresses these concerns in a major new Part consisting of two chapters that deal with environment change, the application of fluid mechanics to energy generation from renewable sources, including wind and wave power, the fundamentals of flow simulation necessary to support flood modelling, and the development of improved techniques for controlling and attenuating rainfall runoff. The emphasis is firmly on identifying the fluid mechanics principles that future engineers will require to deploy to contribute to our response to climate change. Thus the fifth edition continues the ethos that the fundamental principles of the subject must be fully understood to allow the later introduction of specialized content, and therefore will continue to be attractive across the range of courses that include fluid mechanics.

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As in previous editions the authors wish to thank all their colleagues, both at Heriot Watt and internationally, who have contributed to the development of the text by their comments and suggestions following earlier editions. As in previous editions we have attempted to recognize and incorporate all the helpful suggestions we have received; however, any errors of understanding are ours. In particular at Heriot Watt we are grateful to Dr Ian McDougall, for again ensuring that our computing simulations are in a form suitable for dissemination via the website, and to Dr David Campbell for providing the worked solutions manual. Professor Julian Wolfram is to be thanked for his support in the development of the wave energy device sections, along with Professor Garry Pender, who contributed a review of flood modelling, and Dr Grant Wright for making available his work on siphonic roof drainage and the application of the McCormack technique, both in Chapter 19. Dr Steve Wallis and Dr Sylvain Neelz are thanked for their contribution to the pollution dispersion in channel flow content. Pauline Gillett, Assistant Editor for Science and Engineering and our main contact at Pearsons, is to be thanked for continuing the tradition of pressure tempered with infinite patience we have encountered in all our dealings with staff at Pearson, Longman and initially Pitman over the 30 year development of the text through five editions. However, in one essential respect this edition is different to any of its predecessors. Janusz Gasiorek, a founder of this series of texts, died in the summer of 2003. He is deeply missed, and his guidance during the four preceding editions will be impossible to replace. John, as he was known to all his friends and colleagues at London South Bank University, was committed to the educational approach represented by this text. He firmly believed in the importance of emphasising the fundamentals of the subject. Twenty-five years ago he recognized the central importance of computer-aided material and the need to include simulations to broaden the text. His initial work used the leading edge computing of its time and led directly to the emphasis in this edition on simulations that would have been impossible in 1980. We hope that this fifth edition, dedicated to his memory, will continue and reinforce that ethos. John A. Swaffield Lynne B. Jack Edinburgh, February 2005

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Preface to the Fourth Edition The study of fluid mechanics remains within the core of engineering education. Advances in the media available for the delivery of that process provide exciting challenges to the academic. The availability of fast, powerful and inexpensive computing and the multi-various opportunities presented by the web and Internet have the potential to transform fluid mechanics education. Always an experimental subject that traditionally relied heavily on laboratory demonstrations and experience, the opportunities offered by validated simulations are particularly appropriate, extending the student’s experience far beyond the constraints of laboratory space or equipment. While these changes are to be welcomed it remains essential that any fluid mechanics course or supporting text provides the fundamental underpinning that will allow the student, and later the practitioner, to recognize when a flow simulation, however ‘sophisticated’ the package, is less than accurate. Advances in media and particularly computing have therefore provided both the challenges and the solutions necessary to ensure that fluid mechanics remains at the centre of engineering education. However, the objectives of that educational process have also changed, particularly in the UK where fundamental reassessments of the academic and practice levels necessary for professional recognition have introduced differentiated courses. Current Engineering Council regulations will progressively reduce the percentage of graduates reaching Chartered Engineer status, while the introduction of BEng and MEng course requirements, incorporating Matching Sections to allow those unable to progress directly to an MEng qualification the opportunity to reach chartered status, will inevitably determine the partition of fluid mechanics into fundamental principles required by all and a range of more specialist topics that may be covered in greater depth. The Matching Section approach will also inevitably lead to post-university courses taken in many instances by part-time or distance learning routes, again offering both tremendous challenges to the course provider and the opportunity to fully utilize the advantages of the computing and delivery media advances already mentioned. The original aims of this text, dating back to the first edition and explicit by the third edition, clearly meet the needs of this changed educational landscape. The text has consistently emphasized the importance of a fundamental understanding of the principles of fluid mechanics, while at the same time providing specialist topics to be covered in greater depth, whether in the area of rotodynamic machinery or unsteady flow. The fundamental material may be seen as crossing the boundaries of the engineering disciplines committed to a coverage of fluid mechanics and it may be argued that the specialist areas chosen also meet this criterion, although in a more selective sense. The second and third editions experimented with the provision of computing simulations; however, the support infrastructure is only now fully available to allow the maximum benefit to be drawn from the provision, in this fourth edition, of a wider range of computing applications. In many ways the development

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Preface to the Fourth Edition

of this provision, from a separately purchased BBC Basic floppy disk in 1984 to the opportunity in the current edition to utilize simulations of a complexity unavailable in the early 1980s, is an allegory for the development of computing over this period. Clearly the lifetime of this edition will see continued exponential change in delivery systems so that it is probably not practical to predict the format of the fifth edition. While the third edition provided a hardcopy solutions manual, this will now be available for downloading, making the ‘Problem’ sections provided more attractive as a basis for student tutorial activity or distance learning education; applications already available to students at Heriot-Watt University through the extensive distance learning Masters course provision from the Department of Building Engineering. Thus the fourth edition retains the educational aims and objectives of earlier editions, while continuing to make full use of the available computing infrastructure. The content has been revised and extended – in the treatment of air and gas distribution networks to include time dependency, including the provision of simulations to extend any laboratory provision in this area, the inclusion of modelling and simulation considerations for both water and airborne pollution and groundwater seepage flow and the introduction of wind turbine coverage aimed at power generation. The impact of computing through computational fluid dynamics (CFD) is recognized with the emphasis placed firmly on the development of the fundamental principles, including the essential recognition that boundary equation definition, if not based on a full understanding of the flow condition, can lead to worthless predictions. Similarly the computational constraints defining stability have been reinforced. While all these are defined in terms of the exciting field of CFD, the text emphasizes that these considerations were always present in the simulations provided in this and earlier editions and may be found at the root of the cases described in the reworked coverage of unsteady flows across the whole spectrum of conditions from free surface wave attenuation to low-amplitude air pressure transient propagation and traditional waterhammer. As in previous editions the text emphasizes the linkage between theory and practice; engineering is still fundamentally about changing the rules and making things work. Examples throughout the text illustrate the application of theory. All the computing is presented in terms of a description of the calculation or simulation, followed by an example and an invitation to consider further several linked problems. The programs provided with the text fall into a number of natural categories, firstly relatively simple calculations, for example friction factor, lift coefficient or free surface flow depths; then calculations designed to provide solutions for steady state system operation, for example fan or pump operating points, free surface gradually varied flow surface profiles or groundwater seepage flow nets beneath dams; and finally unsteady flow simulations, whether for air distribution and recirculation networks or for waterhammer in response to changes in system operating conditions. This content both extends that provided in previous editions and enhances its presentation. The authors would again like to thank all their colleagues, both at Heriot-Watt University and at many other universities worldwide, who have contributed to the development of this series of editions directly or through informed comment, and all those students who have used the texts. In particular the authors wish to thank Dr Ruth Thomas and Dr Nils Tomes at Heriot-Watt University for the development of several of the third edition computer programs into a form suitable for both the Heriot-Watt distance learning MSc programme and for this text. From the Department of Building Engineering and Surveying Dr Ian McDougall contributed to the translation of the authors’ Fortran-based simulations into a form suitable for

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Preface to the Fourth Edition

xxiii

the twenty-first century; Dr David Campbell is again to be thanked for the quality of the additional original artwork provided with the text and for collating the solutions manual into a form suitable for electronic transmission; and Dr Fan Wang provided the background to the CFD descriptions included in this fourth edition. Through their subtle but effective insistence Anna Faherty and Karen Sutherland, Commissioning Editors at Pearson Education, are ultimately responsible for the manuscript being produced almost to schedule and for this and the support of all our other colleagues, past and present, at Pitman, Longman and Pearson Education since 1974, we are grateful. Nevertheless any errors, factual or of understanding, remain the authors’ responsibility. Fluid mechanics is the most fascinating and exciting of the engineering disciplines and one that impinges on all our lives in a multitude of ways both recognized and taken for granted. The authors hope that this text will communicate some of that excitement to the reader. Finally it was with deep sadness that we learnt of the death in 1998 of John Douglas who initiated this series of editions in 1974. John was always committed to the educational concepts embodied in this text. His commitment to engineering education and his ability to introduce the fundamentals of fluids to students was exceptional, as evidenced by the parallel and highly successful ‘Solving Problems in Fluid Mechanics’ series. This text will we hope continue that commitment and is dedicated to his memory. J. M. Gasiorek J. A. Swaffield Edinburgh, January 2000

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Preface to the Third Edition This third edition of Fluid Mechanics has retained the aims of the original text in providing a broad-based approach to the study of fluid flow together with a detailed and more advanced treatment of specialist topics that find wide application within the design and analysis of flow systems. The text repeats the previous mix of exposition and example shown to be successful by earlier editions. The study of fluid flow is one of the few areas within engineering that truly crosses the boundaries between the various engineering disciplines. It is of equal importance to mechanical, civil, chemical and process, aeronautical and environmental and building services engineers and is to be found as a fundamental building block in the education and formation of these engineers. While this has remained true, the techniques available to enable students to achieve an understanding of fluid mechanics have been revolutionized by the readily available access to computing facilities; facilities that have already advanced immeasurably since the second edition of this text was published. The use of the computer to aid understanding through the provision of interactive simulations, including the use of multi-media packages, will undoubtedly advance even more rapidly during the lifetime of this edition. Whereas the second edition was accompanied by an optional floppy disk containing the programs presented in the text, this is no longer appropriate. Instead the program listings, including a number of new or enhanced programs, have been presented in a format that will make them readily scannable and so usable on a wide range of machines. While the third edition text retains the philosophy and methodology introduced with the earlier editions, the content has been refined to both extend and, in the authors’ view, improve the presentation of existing material. In particular the text has been reordered to present earlier the fundamentals of dimensional analysis and the laws of similarity. The treatment given to the steady flow energy equation has been extended, together with a general enhancement of the analysis of air and gas flow networks. In this context the coverage of fans within rotodynamic machinery has been strengthened and new material covering the use of fans in ventilation, and the ventilation of tunnels by jet fans in particular, has been presented. A new chapter dealing with the mechanisms of mechanical and natural ventilation has been added to provide both a treatment of this important topic and a background to one of the most common applications for fan technology. As in previous editions current research has been utilized in the treatment of specialist topics, such as the jet fan tunnel ventilation and the unsteady flow analysis presentations. In the latter case the treatment presented in this edition seeks to emphasize the commonality of a range of unsteady flow analyses, from classical waterhammer to free surface waves and low-amplitude transient propagation in gas flows, by demonstrating the general development of the defining equations and the

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Preface to the Third Edition

xxv

identical solution by finite difference techniques, allied to computer simulation, once the appropriate terms have been identified for each application. Once again the authors would like to thank all their colleagues in the many universities in the UK and overseas who have contributed to this text by their support for, and comments on, the earlier editions. The authors are grateful to the staff at Longman, particularly Ian Francis and Chris Leeding, who have both supported us in completing this edition and shown considerable patience with the process. Nevertheless, any errors, factual or of understanding, remain ours. We have found fluid mechanics, in all its multi-disciplinary manifestations, to be the most stimulating of engineering areas; we hope that this text will communicate some of that experience and enthusiasm to students of this most demanding of engineering disciplines. J. F. Douglas J. M. Gasiorek J. A. Swaffield Edinburgh, December 1993

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Preface to the Second Edition In the preparation of this second edition we have retained the aims of the original text, namely to provide a broad-based treatment of the essentials of fluid mechanics, while at the same time demonstrating the application of the subject, particularly to the study and solution of higher level problems in selected areas. In retaining this ‘applications’ approach we are both aware and pleased that this technique currently features in the UK Engineering Council statements on the training, education and ‘formation’ of engineers, strengthening our view that this is one of the most efficient and relevant methods of helping students in general to understand our subject. We believe that such an approach should also include the use of improved computer-based numerical solutions as these will become part of the engineer’s everyday activities. In the five years since the first edition was published there has been a significant change in the availability of and access to micro and other computers for both the student and the practising engineer. Computers and programs are of course not ends in themselves but rather they are powerful tools that we can utilize to dispense with many tedious and repetitive calculations, thereby allowing the study, within an educational framework, of problems of greater complexity and relevance, including time-dependent phenomena that were previously beyond our capability without recourse to simplifying assumptions. This second edition therefore includes a series of computer programs chosen to illustrate these aspects of computer application and to be of direct use to both student and practising engineer alike. While the programs have been written in BBC Basic they may be transferred with little difficulty to Apple, Commodore or Sinclair machines. A program cassette tape will also be available to support the text. None of this of course removes the necessity to provide a thorough basis for the subject and this remains one of the text’s main objectives. We have included new material in areas that have been found particularly interesting by our readers, as well as updating and refining the existing text. The treatment of incompressible flows around a body has been extended to include the study of wakes, while the coverage of fluid machinery has been strengthened by the inclusion of a major new chapter on positive displacement machines. The existing treatment of unsteady flow has been extended to allow the application of numerical modelling techniques to unsteady open channel or partially filled pipe flows. Taken together with the introduction of computing methods we view these additions as supporting and reaffirming the aims and objectives of the original text. Once again we would like to thank all our colleagues in many universities and polytechnics in the UK and overseas who have encouraged us by their positive response to and constructive comments on the first edition. All have helped us to formulate this new edition which we hope will fulfil a useful role for both the student and the practising engineer. J. F. Douglas J. M. Gasiorek J. A. Swaffield London, May 1984

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Preface to the First Edition This is a textbook for all manner of engineers. Whether the reader is concerned with Civil, Mechanical or Chemical Engineering, Buiding Services or Environmental Engineering, the principles of fluid mechanics remain the same. Drawing on our joint experience of teaching students in all these disciplines, we have tried to set out these principles simply and clearly and to illustrate their application by examples drawn from the various branches of engineering. In the planning of this book we are indebted to our colleagues in other colleges, polytechnics and universities for the opportunity to study their syllabuses and examination papers which has enabled us to cover the general requirements of the Honours Degree and Professional examinations. We have also deliberately dealt with the elementary aspects of the subject very fully and so the book will meet the requirements of those studying for the Higher National Diploma or for the Higher Diploma or Higher Certificate of the Business and Technician Education Council (B.T.E.C.). For ease of reference the contents has been divided into Parts which are substantially self-contained and we hope that they will provide a convenient source of information for the practising engineer in his day to day activities. J. F. Douglas J. M. Gasiorek J. A. Swaffield

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Acknowledgements We are grateful to the following for permission to reproduce copyright material: Figures 5.20 and 5.21 reproduced with permission of FLUENT Inc.; Figure 5.22 reproduced by permission of Building Research Establishment Ltd.; Figure 5.23 reproduced with permission of Computational Dynamics Ltd.; Figures 6.11 (a) and (b), 25.6, 25.12, 25.17, 25.28 and 25.35 reproduced with permission of Woods Air Movement Ltd.; Figures 9.3 (a) and (b) reproduced with permission of Dr Stephen Huntingdon, HR Wallingford Group Ltd; Figures 18.4 and 18.5 reproduced with permission of UK Climate Impacts Programme; Figure 18.7 reproduced with permission of British Wind Energy Association; Figure 18.24 reproduced with permission of Ocean Power Delivery Ltd; Figures 18.29 (a) and (b) reproduced with permission of Marine Current Turbines Ltd; Figures 19.7 (a) and (b) reproduced with permission of Prof. Garry Pender, Heriot-Watt University; Figures 19.9 and 19.10 reproduced with permission of the Belgian Building Research Institute, Brussels; Figure 19.11 reproduced with permission of Fullfow Group Ltd, © 2000 UV-system; Figure 20.12 (a) reproduced with permission of Thames Water; Figures 21.3, 21.4 and 21.5 reproduced from Pressure Surges in Pipe and Duct Systems by J.A. Swaffield and A.P. Boldy, with kind permission from Ashgate Publishing Group and Adrian P. Boldy. Chapters 8 and 9 photographs reproduced with permission of Dr Carl Trygve Stansberg, Marintek, Trondheim, Norway; Chapter 18 photograph reproduced courtesy of Windcluster 2000 Ltd; Chapter 19 photographs reproduced courtesy of Scottish Water and City of York Council. Parts I, V and VII photographs reproduced with permission of Thames Water; Part II photograph reproduced with permission of NEG Micon AS © NEG Micon; Part V photograph reproduced with permission of Scottish and Southern Energy plc; Part VI photograph courtesy of NASA Earth Observatory, http://earthobservatory.nasa.gov; Part VII photograph © Crown CopyrightMOD. Reproduced with permission of Her Majesty’s Stationery Office; Part VIII photograph reproduced with permission of Woods Air Movement Ltd. Whilst every effort has been made to trace the owners of copyright material, in a few cases this has proved impossible and we take this opportunity to offer our apologies to any copyright holders whose rights we may have unwittingly infringed.

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List of Computer Programs SECTION NO. 6.8

PROGRAM

DESCRIPTION

VOLFLO

7.11

ROTCYL

9.10 10.10

GROUND CBW

12.10

WAKE

13.5

NORSH

14.3

SIPHON

14.12

HARDYC

16.3

CRITNOR

16.13

CHANNEL

19.2

UNSCHAN

20.2

WAVESPD

20.7

SHAFT

21.4

FM5SURG

Flow summation for a circular or rectangular duct cross-section based on a velocity or pitot pressure traverse Stagnation points and lift coefficient calculation for a rotating cylinder Simulation of groundwater seepage Friction factor calculation based on the Colebrook–White equation for a circular section pipe or duct Drag on a body calculated from a traverse across its wake Parameter of state calculations across a normal shock Flow between reservoirs based on the steady flow energy equation, including a high point between reservoirs Hardy–Cross method applied to determine flow distribution in a network Normal and critical depth calculations for free surface flows in rectangular section open channels or partially filled circular section pipes Gradually varied flow profile calculations for free surface flows in rectangular section channels or partially filled circular section pipe flows Unsteady gradually varied flow prediction in long free surface channels using the McCormack method Pressure transient propagation velocity calculations, including fluid and pipe wall properties and free gas Surge tank surface oscillation predictions following turbine load rejection Pressure transient prediction in a three-pipe network, including boundary conditions representing valve closure, column separation and gas release

PAGE

183 252 310

362 435 459

470 492

534

550

646

688 703

739

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xxx

List of Computer Programs

SECTION NO. 21.5

PROGRAM

DESCRIPTION

FM5WAVE

21.5

FM5GUTT

21.7

FM5AIR

23.4

SIMPUMP

25.9

MATCH

25.14

CONTAM

Free surface wave attenuation in open channels or partially filled pipe flows, including circular, parabolic or rectangular cross sections Free surface profile prediction for unsteady flow in rainwater gutters of circular, parabolic, rectangular or trapezoidal cross-section Unsteady airflow prediction in circular or rectangular section distribution ductwork as a result of fan speed or control valve setting changes Application of the similarity laws for fans or pumps System operating point determination for fans and pumps, utilizing either pressure-flow data or non-dimensional pressure-flow coefficients Contamination decay in a ventilated space for one or more non-reacting contaminants and series alterations in ventilation rate, contamination generation or occupation parameters

PAGE

749

749

756 815

908

929

In each case the program background theory is presented, together with an application example and output and a series of suggested further investigations. In addition four further program listings are provided as solutions to end of chapter problems, namely CHAPTER Chapter 9

PROBLEM 18

Chapter 14

19 and 20

Chapter 20

19

DESCRIPTION Finite difference representation of seepage flow beneath a dam Quasi-steady discharge from a tank and fluid transfer between two reservoirs using a finite difference approach Surge shaft oscillation as an application of a finite difference approach

PAGE 321

506 714

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List of Symbols a A b B c cp cv C Cc Cd CD Cf CL Cp Cr Cv d D e e E f f( ) F F( ) g h h H i I k K l L m M n N

acceleration, area, amplitude area, constant width, breadth, channel width width, breadth, constant chord length, velocity of sound, wave speed specific heat at constant pressure specific heat at constant volume constant, contaminant concentration coefficient of contraction coefficient of discharge coefficient of drag coefficient of friction coefficient of lift power coefficient Courant number coefficient of velocity diameter, depth drag, diameter, depth, diffusion coefficient base of natural logarithms error, internal energy per unit mass modulus of elasticity, energy friction factor, function or variable, frequency, force reflected pressure wave force, stress pressure wave gravitational acceleration vertical height, depth head loss head, enthalpy, building height, wave height hydraulic gradient, node identifier moment of inertia constant, radius of gyration, pipe wall roughness, concentration dependent rate coefficient, wave number bulk modulus, channel conveyance, buoyancy factor length lift, channel length mass, area ratio, doublet strength, hydraulic mean depth molecular weight, mass number of, polytropic index, Mannings channel roughness coefficient rotational speed

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xxxii

List of Symbols

p P q Q r R R s S S0 Sox,y Sfx,y t T u U v, V vf vr vx vy vz vr vθ V w W x, y, z y Z

pressure force, power, wetted perimeter, contaminant flux flow rate per unit width or unit depth, lateral channel inflow volumetric flow rate radius, radial distance radius, reaction force, hydraulic radius, combined damping coefficient gas constant slope, distance, arbitrary coordinate within Cartesian system, slip surface, entropy, channel and friction slope longitudinal channel slope bed slope in x and y directions bed friction slope in x and y directions time, annular film thickness temperature, torque surface width, flow surface width, wave period velocity, peripheral blade velocity internal energy, velocity, wind velocity velocity velocity of flow relative velocity velocity component in x direction velocity component in y direction velocity component in z direction radial velocity tangential velocity volume, volume storage specific weight, Priessmann slot width weight, work orthogonal coordinates gas content (per cent), variable potential head, depth

α β γ Γ δ Δ Δx, Δy ε ζ η θ λ μ ν ρ σ τ φ Φ

angle, angular acceleration angle adiabatic index (cp cv), turbine damping coefficient, angle of yaw circulation difference, increment change in cell dimensions absolute roughness, eddy viscosity vorticity efficiency, free surface amplitude (from datum) angle multiplier in methods of characteristics, wavelength coefficient of dynamic viscosity, radiation damping coefficient coefficient of kinematic viscosity, Poisson’s ratio mass density relative density (specific gravity), surface tension, temporal multiplier shear stress shear strain, angle velocity potential

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List of Symbols

Ψ ω

stream function angular (rotational) velocity, stage variable

Fr Ma Re Str We

Froude number Mach number Reynolds number Strouhal number Weber number

L M T Θ

Dimensions of Dimensions of Dimensions of Dimensions of

length mass time temperature

xxxiii

FM5_C01a.fm Page xxxiv Tuesday, September 20, 2005 1:21 PM

Part I

Elements of Fluid Mechanics 1 Fluids and their Properties 2 2 Pressure and Head 24 3 Static Forces on Surfaces. Buoyancy 60

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Fluid mechanics, as the name indicates, is that branch of applied mechanics that is concerned with the statics and dynamics of liquids and gases. The analysis of the behaviour of fluids is based upon the fundamental laws of applied mechanics that relate to the conservation of mass–energy and the force– momentum equation, together with other concepts and equations with which the student who has already studied solid-body mechanics will be familiar. There are, however, two major aspects of fluid mechanics which differ from solid-body mechanics. The first is the nature and properties of the fluid itself, which are very different from those of a solid. The second is that, instead of dealing with individual bodies or elements of known mass, we are frequently concerned with the behaviour of a continuous stream of fluid, without beginning or end.

Opposite: Water effects, image courtesy of Thames Water

A further problem is that it can be extremely difficult to specify either the precise movement of a stream of fluid or that of individual particles within it. It is, therefore, often necessary – for the purpose of theoretical analysis – to assume ideal, simplified conditions and patterns of flow. The results so obtained may then be modified by introducing appropriate coefficients and factors, determined experimentally, to provide a basis for the design of fluid systems. This approach has proved to be reasonably satisfactory – in so far as the theoretical analysis usually establishes the form of the relationship between the variables; the experimental investigation corrects for the factors omitted from the theoretical model and establishes a quantitative relationship.

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Chapter 1

Fluids and their Properties Fluids Shear stress in a moving fluid Differences between solids and fluids 1.4 Newtonian and non-Newtonian fluids 1.5 Liquids and gases 1.6 Molecular structure of materials 1.7 The continuum concept of a fluid 1.8 Density 1.9 Viscosity 1.10 Causes of viscosity in gases 1.1 1.2 1.3

1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20

Causes of viscosity in a liquid Surface tension Capillarity Vapour pressure Cavitation Compressibility and the bulk modulus Equation of state of a perfect gas The universal gas constant Specific heats of a gas Expansion of a gas

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This chapter will define the nature of fluids, stressing both the commonality with concepts of applied mechanics applied to solid-body systems and the fundamental differences that arise from the nature of fluids. The appropriate physical properties that define these differences and allow the

differentiation of fluids into gases and liquids, Newtonian and non-Newtonian, compressible and incompressible, will be identified. The application of the equation of state for perfect gases will be introduced. l l l

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4

Chapter 1

Fluids and their Properties

1.1

FLUIDS

In everyday life, we recognize three states of matter: solid, liquid and gas. Although different in many respects, liquids and gases have a common characteristic in which they differ from solids: they are fluids, lacking the ability of solids to offer permanent resistance to a deforming force. Fluids flow under the action of such forces, deforming continuously for as long as the force is applied. A fluid is unable to retain any unsupported shape; it flows under its own weight and takes the shape of any solid body with which it comes into contact. Deformation is caused by shearing forces, i.e. forces such as F (Fig. 1.1), which act tangentially to the surfaces to which they are applied and cause the material originally occupying the space ABCD to deform to AB′C′D. This leads to the definition:

FIGURE 1.1 Deformation caused by shearing forces

A fluid is a substance which deforms continuously under the action of shearing forces, however small they may be. Conversely, it follows that: If a fluid is at rest, there can be no shearing forces acting and, therefore, all forces in the fluid must be perpendicular to the planes upon which they act.

1.2

SHEAR STRESS IN A MOVING FLUID

Although there can be no shear stress in a fluid at rest, shear stresses are developed when the fluid is in motion, if the particles of the fluid move relative to each other so that they have different velocities, causing the original shape of the fluid to become distorted. If, on the other hand, the velocity of the fluid is the same at every point, no shear stresses will be produced, since the fluid particles are at rest relative to each other. Usually, we are concerned with flow past a solid boundary. The fluid in contact with the boundary adheres to it and will, therefore, have the same velocity as the boundary. Considering successive layers parallel to the boundary (Fig. 1.2), the velocity of the fluid will vary from layer to layer as y increases. If ABCD (Fig. 1.1) represents an element in a fluid with thickness s perpendicular to the diagram, then the force F will act over an area A equal to BC × s. The force per

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1.3

Differences between solids and fluids

5

FIGURE 1.2 Variation of velocity with distance from a solid boundary

unit area FA is the shear stress τ and the deformation, measured by the angle φ (the shear strain), will be proportional to the shear stress. In a solid, φ will be a fixed quantity for a given value of τ, since a solid can resist shear stress permanently. In a fluid, the shear strain φ will continue to increase with time and the fluid will flow. It is found experimentally that, in a true fluid, the rate of shear strain (or shear strain per unit time) is directly proportional to the shear stress. Suppose that in time t a particle at E (Fig. 1.1) moves through a distance x. If E is a distance y from AD then, for small angles, Shear strain, φ = xy, Rate of shear strain = xyt = (xt)y = uy, where u = xt is the velocity of the particle at E. Assuming the experimental result that shear stress is proportional to shear strain, then

τ = constant × uy.

(1.1)

The term uy is the change of velocity with y and may be written in the differential form dudy. The constant of proportionality is known as the dynamic viscosity µ of the fluid. Substituting into equation (1.1), du τ = µ −−− , dy

(1.2)

which is Newton’s law of viscosity. The value of µ depends upon the fluid under consideration.

1.3

DIFFERENCES BETWEEN SOLIDS AND FLUIDS

To summarize, the differences between the behaviours of solids and fluids under an applied force are as follows: 1.

2.

For a solid, the strain is a function of the applied stress, provided that the elastic limit is not exceeded. For a fluid, the rate of strain is proportional to the applied stress. The strain in a solid is independent of the time over which the force is applied and, if the elastic limit is not exceeded, the deformation disappears when the force is removed. A fluid continues to flow for as long as the force is applied and will not recover its original form when the force is removed.

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6

Chapter 1

Fluids and their Properties

In most cases, substances can be classified easily as either solids or fluids. However, certain cases (e.g. pitch, glass) appear to be solids because their rate of deformation under their own weight is very small. Pitch is actually a fluid which will flow and spread out over a surface under its own weight – but it will take days to do so rather than milliseconds! Similarly, solids will flow and become plastic when subjected to forces sufficiently large to produce a stress in the material which exceeds the elastic limit. They will also ‘creep’ under sustained loading, so that the deformation increases with time. A plastic substance does not meet the definition of a true fluid, since the shear stress must exceed a certain minimum value before flow commences.

1.4

NEWTONIAN AND NON-NEWTONIAN FLUIDS

Even among substances commonly accepted as fluids, there is a wide variation in behaviour under stress. Fluids obeying Newton’s law of viscosity (equation (1.2)) and for which µ has a constant value are known as Newtonian fluids. Most common fluids fall into this category, for which shear stress is linearly related to velocity gradient (Fig. 1.3). Fluids which do not obey Newton’s law of viscosity are known as nonNewtonian and fall into one of the following groups:

FIGURE 1.3 Variation of shear stress with velocity gradient

1.

Plastic, for which the shear stress must reach a certain minimum value before flow commences. Thereafter, shear stress increases with the rate of shear according to the relationship n

du τ = A + B ⎛⎝ −−−⎞⎠ , dy where A, B and n are constants. If n = 1, the material is known as a Bingham plastic (e.g. sewage sludge).

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1.6

2. 3. 4. 5. 6.

Molecular structure of materials

7

Pseudo-plastic, for which dynamic viscosity decreases as the rate of shear increases (e.g. colloidal solutions, clay, milk, cement). Dilatant substances, in which dynamic viscosity increases as the rate of shear increases (e.g. quicksand). Thixotropic substances, for which the dynamic viscosity decreases with the time for which shearing forces are applied (e.g. thixotropic jelly paints). Rheopectic materials, for which the dynamic viscosity increases with the time for which shearing forces are applied. Viscoelastic materials, which behave in a manner similar to Newtonian fluids under time-invariant conditions but, if the shear stress changes suddenly, behave as if plastic.

The above is a classification of actual fluids. In analyzing some of the problems arising in fluid mechanics we shall have cause to consider the behaviour of an ideal fluid, which is assumed to have no viscosity. Theoretical solutions obtained for such a fluid often give valuable insight into the problems involved, and can, where necessary, be related to real conditions by experimental investigation.

1.5

FIGURE 1.4 Behaviour of a fluid in a container

LIQUIDS AND GASES

Although liquids and gases both share the common characteristics of fluids, they have many distinctive characteristics of their own. A liquid is difficult to compress and, for many purposes, may be regarded as incompressible. A given mass of liquid occupies a fixed volume, irrespective of the size or shape of its container, and a free surface is formed (Fig. 1.4(a)) if the volume of the container is greater than that of the liquid. A gas is comparatively easy to compress. Changes of volume with pressure are large, cannot normally be neglected and are related to changes of temperature. A given mass of a gas has no fixed volume and will expand continuously unless restrained by a containing vessel. It will completely fill any vessel in which it is placed and, therefore, does not form a free surface (Fig. 1.4(b)).

1.6

MOLECULAR STRUCTURE OF MATERIALS

Solids, liquids and gases are all composed of molecules in continuous motion. However, the arrangement of these molecules, and the spaces between them, differ, giving rise to the characteristic properties of the three different states of matter. In solids, the molecules are densely and regularly packed and movement is slight, each molecule being restrained by its neighbours. In liquids, the structure is looser; individual molecules have greater freedom of movement and, although restrained to some degree by the surrounding molecules, can break away from this restraint, causing a change of structure. In gases, there is no formal structure, the spaces between molecules are large and the molecules can move freely. The molecules of a substance exert forces on each other which vary with their intermolecular distance. Consider, for simplicity, a monatomic substance in which each molecule consists of a single atom. An idea of the nature of the forces acting may be formed from observing the behaviour of such a substance on a macroscopic scale.

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8

Chapter 1

Fluids and their Properties

FIGURE 1.5 (a) Variation of force with separation. (b) Variation of potential energy with separation

1.

2.

3.

If two pieces of the same material are far apart, there is no detectable force exerted between them. Thus, the forces between molecules are negligible when widely separated and tend to zero as the separation tends towards infinity. Two pieces of the same material can be made to weld together if they are forced into very close contact. Under these conditions, the forces between the molecules are attractive when the separation is very small. Very large forces are required to compress solids or liquids, indicating that a repulsive force between the molecules must be overcome to reduce the spacing between them.

It appears from these observations that interatomic forces vary with the distance of separation (Fig. 1.5(a)) and that there are two types of force, one attractive and the other repulsive. At small separations, the repulsive force is dominant; at larger separations, it becomes insignificant by comparison with the attractive force. These conclusions can also be expressed in terms of the potential energy, defined as the energy required to bring one atom from infinity to a distance r from the second atom. The potential energy is zero if the atoms are infinitely far apart and is positive if external energy is required to move the first atom towards the second. Since Fig. 1.5(a) is the graph of the force F between the atom vs. the distance of separation, the potential energy curve (Fig. 1.5(b)) will be the integral of this curve from ∞ to r, which is the shaded area in Fig. 1.5(a). At r0, there is a condition of minimum energy, corresponding to F = 0 and representing a position of stable equilibrium, accounting for the inherent stability of solids and liquids in which the molecules are sufficiently densely packed for this condition to exist. Figure 1.5(b) also indicates that a pair of atoms can be separated

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1.7

The continuum concept of a fluid

9

completely, so that r = ∞, by the application of a finite amount of energy ∆E, which is called the dissociation or binding energy. Considering a large number of particles of a substance, each particle will have kinetic energy −12 mu2, where m is the mass of the particle and u its velocity. If a particle collides with a pair of particles, it will only cause them to separate if it can transfer to the pair energy in excess of ∆E. Thus, the possibility of stable pairs forming will depend on the average value of −12 mu2 in relation to ∆E. 1.

If the average value of −12 mu2 ∆E, no stable pairs can form. The system will behave as a gas, consisting of individual particles moving rapidly with no apparent tendency to aggregate or occupy a fixed space.

2.

If the average values of −12 mu2 ∆E, no dissociation of pairs is possible and the colliding particle may be captured by the pair. The system has the properties of a solid, forming a stable conglomeration of particles which can only be dissociated by supplying energy from outside (e.g. by heating to produce melting and, subsequently, boiling).

3.

If the average value of −12 mu2 ∆E, we have a system intermediate between (1) and (2), corresponding to the liquid state, since some particles will have values of −1 mu2 ∆ E, causing dissociation, while others will have values of −1 mu2 ∆E 2 2 and will aggregate.

Summing up, in a solid, the individual molecules are close packed and their movement is restricted to vibrations of small amplitude. The kinetic energy is small compared with the dissociation energy, so that the molecules do not become separated but retain the same relative conditions. In a liquid, the molecules are still close packed, but their movement is greater. Certain of the molecules will have sufficient kinetic energy to break through the surrounding molecules, so that the relative positions of the molecules can change from time to time. The material will cease to be rigid and can flow under the action of applied forces. However, the attraction between molecules is still sufficient to ensure that a given mass of liquid has a fixed volume and that a free surface will be formed. In a gas, the spacing between molecules is some ten times as great as in a liquid. The kinetic energy is far greater than the dissociation energy. The attractive forces between molecules are very weak and intermolecular effects are negligible, so that molecules are free to travel until stopped by a solid or a liquid boundary. A gas will, therefore, expand to fill a container completely, irrespective of volume.

1.7

THE CONTINUUM CONCEPT OF A FLUID

Although the properties of a fluid arise from its molecular structure, engineering problems are usually concerned with the bulk behaviour of fluids. The number of molecules involved is immense, and the separation between them is normally negligible by comparison with the distances involved in the practical situation being studied. Under these conditions, it is usual to consider a fluid as a continuum – a hypothetical continuous substance – and the conditions at a point as the average of a very large number of molecules surrounding that point within a distance which is large compared with the mean intermolecular distance (although very small in absolute

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Fluids and their Properties

terms). Quantities such as velocity and pressure can then be considered to be constant at any point, and changes due to molecular motion may be ignored. Variations in such quantities can also be assumed to take place smoothly, from point to point. This assumption breaks down in the case of rarefied gases, for which the ratio of the mean free path of the molecules to the physical dimensions of the problem is very much larger. In this book, fluids will be assumed to be continuous substances and, when the behaviour of a small element or particle of fluid is studied, it will be assumed that it contains so many molecules that it can be treated as part of this continuum. Properties of fluids The following properties of fluids are of general importance to the study of fluid mechanics. For convenience, a fuller list of the values of these properties for common fluids is given in Appendix 1, but typical values, SI units and dimensions in the MLT system (see Chapter 8) are given here.

1.8

DENSITY

The density of a substance is that quantity of matter contained in unit volume of the substance. It can be expressed in three different ways, which must be clearly distinguished.

1.8.1 Mass density Mass density ρ is defined as the mass of the substance per unit volume. As mentioned above, we are concerned, in considering this and other properties, with the substance as a continuum and not with the properties of individual molecules. The mass density at a point is determined by considering the mass δ m of a very small volume δV surrounding the point. In order to preserve the concept of the continuum, δV cannot be made smaller than x3, where x is a linear dimension which is large compared with the mean distance between molecules. The density at a point is the limiting value as δV tends to x3:

δm ρ = δlim −−−− . V→x δ V 3

Units: kilograms per cubic metre (kg m−3). Dimensions: ML−3. Typical values at p = 1.013 × 105 N m−2, T = 288.15 K: water, 1000 kg m−3; air, 1.23 kg m−3.

1.8.2 Specific weight Specific weight w is defined as the weight per unit volume. Since weight is dependent on gravitational attraction, the specific weight will vary from point to point, according to the local value of gravitational acceleration g. The relationship between w and ρ can be deduced from Newton’s second law, since

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1.9

Viscosity

11

Weight per unit volume = Mass per unit volume × g w = ρg. Units: newtons per cubic metre (N m−3). Dimensions: ML−2 T −2. Typical values: water, 9.81 × 103 N m−3; air, 12.07 N m−3.

1.8.3 Relative density Relative density (or specific gravity) σ is defined as the ratio of the mass density of a substance to some standard mass density. For solids and liquids, the standard mass density chosen is the maximum density of water (which occurs at 4 °C at atmospheric pressure):

σ = ρsubstanceρH O at 4 °C . 2

For gases, the standard density may be that of air or of hydrogen at a specified temperature and pressure, but the term is not used frequently. Units: since relative density is a ratio of two quantities of the same kind, it is a pure number having no units. Dimensions: as a pure number, its dimensions are M0L0 T 0 = 1. Typical values: water, 1.0; oil, 0.9.

1.8.4 Specific volume In addition to these measures of density, the quantity specific volume is sometimes used, being defined as the reciprocal of mass density, i.e. it is used to mean volume per unit mass.

1.9

VISCOSITY

A fluid at rest cannot resist shearing forces, and, if such forces act on a fluid which is in contact with a solid boundary (Fig. 1.2), the fluid will flow over the boundary in such a way that the particles immediately in contact with the boundary have the same velocity as the boundary, while successive layers of fluid parallel to the boundary move with increasing velocities. Shear stresses opposing the relative motion of these layers are set up, their magnitude depending on the velocity gradient from layer to layer. For fluids obeying Newton’s law of viscosity, taking the direction of motion as the x direction and vx as the velocity of the fluid in the x direction at a distance y from the boundary, the shear stress in the x direction is given by dv τ x = µ −−−−x . dy

(1.3)

1.9.1 Coefficient of dynamic viscosity The coefficient of dynamic viscosity µ can be defined as the shear force per unit area (or shear stress τ) required to drag one layer of fluid with unit velocity past another layer a unit distance away from it in the fluid. Rearranging equation (1.3),

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Chapter 1

Fluids and their Properties

dv Force Velocity Force × Time µ = τ −−− = −−−−−−−− −−−−−−−−−−−−− = −−−−−−−−−−−−−−−−−−−− dy Area Distance Area

or

Mass −−−−−−−−−−−−−−−−−−−−−−−. Length × Time

Units: newton seconds per square metre (N s m−2) or kilograms per metre per second (kg m−1 s−1). (But note that the coefficient of viscosity is often measured in poise (P); 10 P = 1 kg m−1 s−1.) Dimensions: ML−1T−1. Typical values: water, 1.14 × 10−3 kg m−1 s−1; air, 1.78 × 10−5 kg m−1 s−1.

1.9.2 Kinematic viscosity The kinematic viscosity ν is defined as the ratio of dynamic viscosity to mass density:

ν = µ ρ. Units: square metres per second (m2 s−1). (But note that kinematic viscosity is often measured in stokes (St); 104 St = 1 m2 s−1.) Dimensions: L2 T −1. Typical values: water, 1.14 × 10 −6 m2 s−1; air, 1.46 × 10 −5 m2 s−1.

1.10 CAUSES OF VISCOSITY IN GASES When a gas flows over a solid boundary, the velocity of flow in the x direction, parallel to the boundary, will change with the distance y, measured perpendicular to the boundary. In Fig. 1.6, the velocity in the x direction is vx at a distance y from the boundary and vx + δ vx at a distance y + δ y. As the molecules of gas are not rigidly constrained, and cohesive forces are small, there will be a continuous interchange of molecules between adjacent layers which are travelling at different velocities. Molecules moving from the slower layer will exert a drag on the faster, while those moving from the faster layer will exert an accelerating force on the slower. Assuming that the mass interchange per unit time is proportional to the area A under consideration, and inversely proportional to the distance δ y between them, Mass interchange per unit time = kAδy, where k is a constant of proportionality;

FIGURE 1.6

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1.11

Causes of viscosity in a liquid

13

Change of velocity = δ vx ; Force exerted by one layer on the other = Rate of change of momentum = Mass interchange per unit time × Change of velocity

δv F = kA −−−−x ; δy δv Viscous shear stress, τ = FA = k −−−−x . δy Thus, from consideration of molecular mass interchange occurring in a gas, we arrive at Newton’s law of viscosity. If the temperature of a gas increases, the molecular interchange will increase. The viscosity of a gas will, therefore, increase as the temperature increases. According to the kinetic theory of gases, viscosity should be proportional to the square root of the absolute temperature; in practice, it increases more rapidly. Over the normal range of pressures, the viscosity of a gas is found to be independent of pressure, but it is affected by very high pressures.

1.11 CAUSES OF VISCOSITY IN A LIQUID While there will be shear stresses due to molecular interchange similar to those developed in a gas, there are substantial attractive, cohesive forces between the molecules of a liquid (which are very much closer together than those of a gas). Both molecular interchange and cohesion contribute to viscous shear stress in liquids. The effect of increasing the temperature of a fluid is to reduce the cohesive forces while simultaneously increasing the rate of molecular interchange. The former effect tends to cause a decrease of shear stress, while the latter causes it to increase. The net result is that liquids show a reduction in viscosity with increasing temperature which is of the form

µT = µ0 (1 + A1T + B1T 2 ),

(1.4)

where µT is the viscosity at T °C, µ0 is the viscosity at 0 °C and A1 and B1 are constants depending upon the liquid. For water, µ 0 = 0.0179 P, A1 = 0.033 68 and B1 = 0.000 221. When plotted, equation (1.4) gives a hyperbola, viscosity tending to zero as temperature tends to infinity. An alternative relationship is

µµ0 = A2 exp[B2(1T′ − 1T0)],

(1.5)

where A2 and B2 are constants and T′ is the absolute temperature. High pressures also affect the viscosity of a liquid. The energy required for the relative movement of the molecules is increased and, therefore, the viscosity increases with increasing pressure. The relationship depends on the nature of the liquid and is exponential, having the form

µp = µ 0 exp[C( p − p0)],

(1.6)

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Fluids and their Properties

where C is a constant for the liquid and µp is the viscosity at pressure p. For oils of the type used in oil hydraulic machinery, the increase in viscosity is of the order of 10 to 15 per cent for a pressure increase of 70 atm. Water, however, behaves rather differently from other fluids, since its viscosity only doubles for an increase in pressure from 1 to 1000 atm.

1.12 SURFACE TENSION Although all molecules are in constant motion, a molecule within the body of the liquid is, on average, attracted equally in all directions by the other molecules surrounding it, but, at the surface between liquid and air, or the interface between one substance and another, the upward and downward attractions are unbalanced, the surface molecules being pulled inward towards the bulk of the liquid. This effect causes the liquid surface to behave as if it were an elastic membrane under tension. The surface tension σ is measured as the force acting across the unit length of a line drawn in the surface. It acts in the plane of the surface, normal to any line in the surface, and is the same at all points. Surface tension is constant at any given temperature for the surface of separation of two particular substances, but it decreases with increasing temperature. The effect of surface tension is to reduce the surface of a free body of liquid to a minimum, since to expand the surface area molecules have to be brought to the surface from the bulk of the liquid against the unbalanced attraction pulling the surface molecules inwards. For this reason, drops of liquid tend to take a spherical shape in order to minimize surface area. For such a small droplet, surface tension will cause an increase of internal pressure p in order to balance the surface force. Considering the forces acting on a diametral plane through a spherical drop of radius r, the force due to internal pressure = p × π r 2, and the force due to surface tension around the perimeter = 2π r × σ. For equilibrium, pπ r 2 = 2π rσ or p = 2σr. Surface tension will also increase the internal pressure in a cylindrical jet of fluid, for which p = σr. In either case, if r is very small, the value of p becomes very large. For small bubbles in a liquid, if this pressure is greater than the pressure of vapour or gas in a bubble, the bubble will collapse. In many of the problems with which engineers are concerned, the magnitude of surface tension forces is very small compared with the other forces acting on the fluid and may, therefore, be neglected. However, these forces can cause serious errors in hydraulic scale models and through capillary effects. Surface tension forces can be reduced by the addition of detergents.

EXAMPLE 1.1

Air is introduced through a nozzle into a tank of water to form a stream of bubbles. If the bubbles are intended to have a diameter of 2 mm, calculate by how much the pressure of the air at the nozzle must exceed that of the surrounding water. Assume that σ = 72.7 × 10−3 N m−1.

Solution Excess pressure, p = 2σr.

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1.13

Capillarity

15

Putting r = 1 mm = 10 −3 m, σ = 72.7 × 10 −3 N m−1. Excess pressure, p = (2 × 72.7 × 10−3)(1 × 10−3) = 145.4 N m−2.

1.13 CAPILLARITY If a fine tube, open at both ends, is lowered vertically into a liquid which wets the tube, the level of the liquid will rise in the tube (Fig. 1.7(a)). If the liquid does not wet the tube, the level of liquid in the tube will be depressed below the level of the free surface outside (Fig. 1.7(b)). If θ is the angle of contact between liquid and solid and d is the tube diameter (Fig. 1.7(a)),

FIGURE 1.7 Capillarity

Upward pull Component of Perimeter due to surface = surface tension × of = σ cos θ × π d. tension acting upwards tube

(1.7)

The atmospheric pressure is the same inside and outside the tube, and, therefore, the only force opposing this upward pull is the weight of the vertical-sided column of liquid of height H, since, by definition, there are no shear stresses in a liquid at rest. Therefore, in Fig. 1.7, there will be no shear stress on the vertical sides of the column of liquid under consideration. Weight of column raised = ρ g(π4)d 2H,

(1.8)

where ρ is the mass density of the liquid. Equating the upward pull to the weight of the column, from equations (1.7) and (1.8),

σ cos θ × π d = ρg(π4)d 2H, Capillary rise, H = 4σ cos θρgd.

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Chapter 1

Fluids and their Properties

FIGURE 1.8 Capillary rise in glass tubes of circular crosssection

Capillary action is a serious source of error in reading liquid levels in fine-gauge tubes, particularly as the degree of wetting and, therefore, the contact angle θ are affected by the cleanness of the surfaces in contact. For water in a tube of 5 mm diameter, the capillary rise will be approximately 4.5 mm, while for mercury the corresponding figure would be −1.4 mm (Fig. 1.8). Gauge glasses for reading the level of liquids should have as large a diameter as is conveniently possible, to minimize errors due to capillarity.

1.14 VAPOUR PRESSURE Since the molecules of a liquid are in constant agitation, some of the molecules in the surface layer will have sufficient energy to escape from the attraction of the surrounding molecules into the space above the free surface. Some of these molecules will return and condense, but others will take their place. If the space above the liquid is confined, an equilibrium will be reached so that the number of molecules of liquid in the space above the free surface is constant. These molecules produce a partial pressure known as the vapour pressure in the space. The degree of molecular activity increases with increasing temperature, and, therefore, the vapour pressure will also increase. Boiling will occur when the vapour pressure is equal to the pressure above the liquid. By reducing the pressure, boiling can be made to occur at temperatures well below the boiling point at atmospheric

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1.16

Compressibility and the bulk modulus

17

pressure: for example, if the pressure is reduced to 0.2 bar (0.2 atm), water will boil at a temperature of 60 °C.

1.15 CAVITATION Under certain conditions, areas of low pressure can occur locally in a flowing fluid. If the pressure in such areas falls below the vapour pressure, there will be local boiling and a cloud of vapour bubbles will form. This phenomenon is known as cavitation and can cause serious problems, since the flow of liquid can sweep this cloud of bubbles on into an area of higher pressure where the bubbles will collapse suddenly. If this should occur in contact with a solid surface, very serious damage can result due to the very large force with which the liquid hits the surface. Cavitation can affect the performance of hydraulic machinery such as pumps, turbines and propellers, and the impact of collapsing bubbles can cause local erosion of metal surfaces. Cavitation can also occur if a liquid contains dissolved air or other gases, since the solubility of gases in a liquid decreases as the pressure is reduced. Gas or air bubbles will be released in the same way as vapour bubbles, with the same damaging effects. Usually, this release occurs at higher pressures and, therefore, before vapour cavitation commences.

1.16 COMPRESSIBILITY AND THE BULK MODULUS All materials, whether solids, liquids or gases, are compressible, i.e. the volume V of a given mass will be reduced to V – δV when a force is exerted uniformly all over its surface. If the force per unit area of surface increases from p to p + δ p, the relationship between change of pressure and change of volume depends on the bulk modulus of the material: Bulk modulus = Change in pressureVolumetric strain. Volumetric strain is the change in volume divided by the original volume; therefore, Change in volume Change in pressure −−−−−−−−−−−−−−−−−−−−−−−−−−− = −−−−−−−−−−−−−−−−−−−−−−−−−−−−− , Original volume Bulk modulus −δVV = δpK, the minus sign indicating that the volume decreases as pressure increases. In the limit, as δp → 0, dp K = – V −−−− . dV

(1.9)

Considering unit mass of a substance, V = 1ρ.

(1.10)

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Differentiating, V dρ + ρ dV = 0 dV = −(Vρ) dρ. Substituting for V from equation (1.10), dV = −(1ρ 2) dρ.

(1.11)

Putting the values of V and dV obtained from equations (1.10) and (1.11) in equation (1.9),

dp K = ρ −−− . dρ

(1.12)

The value of K is shown by equation (1.12) to be dependent on the relationship between pressure and density and, since density is also affected by temperature, it will depend on how the temperature changes during compression. If the temperature is constant, conditions are said to be isothermal, while, if no heat is allowed to enter or leave during compression, conditions are adiabatic. The ratio of the adiabatic bulk modulus to the isothermal bulk modulus is equal to γ , the ratio of the specific heat of a fluid at constant pressure to that at constant volume. For liquids, γ is approximately unity and the two conditions need not be distinguished; for gases, the difference is substantial (for air, γ = 1.4). The concept of the bulk modulus is mainly applied to liquids, since for gases the compressibility is so great that the value of K is not a constant, but proportional to pressure and changes very rapidly. The relationship between pressure and mass density is more conveniently found from the characteristic equation of a gas (1.13). For liquids, the value of K is high and changes of density with pressure are small, but increasing pressure does bring the molecules of the liquid closer together, increasing the value of K. For water, the value of K will double if the pressure is increased from 1 to 3500 atm. An increase of temperature will cause the value of K to fall. For liquids, the changes in pressure occurring in many fluid mechanics problems are not sufficiently great to cause appreciable changes in density. It is, therefore, usual to ignore such changes and to treat liquids as incompressible. Where, however, sudden changes of velocity generate large inertial forces, high pressures can occur and compressibility effects cannot be disregarded in liquids (see Chapter 20). Gases may also be treated as incompressible if the pressure changes are very small, but, usually, compressibility cannot be ignored. In general, compressibility becomes important when the velocity of the fluid exceeds about one-fifth of the velocity of a pressure wave (e.g. the velocity of sound) in the fluid. Units: since volumetric strain is the ratio of two volumes, the units of bulk modulus will be the same as those of pressure, newtons per square metre (N m−2). Dimensions: ML−1T−2. Typical values: water, 2.05 × 109 N m−2; oil, 1.62 × 109 N m−2.

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1.19

Specific heats of a gas

19

1.17 EQUATION OF STATE OF A PERFECT GAS The mass density of a gas varies with its absolute pressure p and absolute temperature T. For a perfect gas, p = ρRT,

(1.13)

where R is the gas constant for the gas concerned. Most gases at pressures and temperatures well removed from liquefaction follow this characteristic equation closely, but it does not apply to vapours. Units: the gas contant is measured in joules per kilogram per kelvin (J kg−1 K−1). Dimensions: L 2 T −2Θ −1. Typical values: air, 287 J kg−1 K−1; hydrogen, 4110 J kg−1 K−1.

1.18 THE UNIVERSAL GAS CONSTANT From equation (1.13) ρR is constant for a given value of pressure p and temperature T. By Avogadro’s hypothesis, all pure gases have the same number of molecules per unit volume at the same temperature and pressure, so that ρ is proportional to the molar mass M (kg kmol−1). Therefore, the quantity MR will be constant for all perfect gases, and is known as the universal gas constant R0. R0 = MR = 8.314 kJ kmol−1 K−1.

1.19 SPECIFIC HEATS OF A GAS Since pressure, temperature and density of a gas are interrelated, the amount of heat energy H required to raise the temperature of a gas from T1 to T2 will depend upon whether the gas is allowed to expand during the process, so that some of the energy supplied is used in doing work instead of raising the temperature of the gas. Two different specific heats are, therefore, given for a gas, corresponding to the two extreme conditions of constant volume and constant pressure. 1.

Specific heat at constant volume cv. For a temperature change from T1 to T2 at constant volume, Heat supplied per unit mass, H = cv(T2 − T1).

2.

Since there is no change in volume, no external work is done, so that the increase of internal energy per unit mass of gas is cv (T2 − T1) heat units. Specific heat at constant pressure cp. If the pressure is kept constant, the gas will expand as the temperature changes from T1 to T2: Heat supplied per unit mass = cp(T2 − T1) heat units.

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Only part of this energy is used to raise the temperature of the gas; the rest goes to external work. Thus, cp cv: cp(T2 − T1 ) = cv(T2 − T1) + External work (in heat units). It can be shown that R = (cp − cv), where R, cp and cv have the same units. Units: specific heat is measured in joules per kilogram per kelvin, as is R. Dimensions: L2 T −2 Θ−1. Typical values: air, cp = 1.005 kJ kg−1 K−1, cv = 0.718 kJ kg−1 K−1.

1.20 EXPANSION OF A GAS When a gas expands, the amount of work done will depend upon the relationship between pressure and volume, which, in turn, depends upon whether the gas receives or loses heat during the process. If a unit mass of a gas has a volume V1 at pressure p1 and volume V2 at pressure p2, as shown in Fig. 1.9, then, Work done Area under p–V = = during expansion curve between V1 and V2

V2

p dV.

V1

FIGURE 1.9 Expansion of a gas

1. If the expansion is isothermal, the absolute temperature T (in kelvin) of the gas remains unchanged and the characteristic equation p = ρRT becomes pρ = constant; or, putting V = volume of unit mass = 1ρ, pV = constant = p1V1 = RT, p = p1V1(1V ).

(1.14)

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1.20

Expansion of a gas

21

From equation (1.14),

Work done per unit mass = p 1 V 1

V2

V1

dV −−− V

= p1V1 loge(V2 V1 ) = RT log e(V2 V1 ). 2. For any known relationship between pressure and mass density of the form pρ n = constant, putting V = 1ρ, pV n = p 1V n1 = constant.

(1.15)

Therefore, p = p 1V 1 V . n

–n

Work done by gas per unit mass =

V2

p dV

V1

= p1 V 1 n

V2

V

–n

dV

V1 (1 – n)

= [ p 1V 1 ( 1 – n ) ] [V 2 n

= ( 1 – n ) [ p 1V V –1

n 1

(1 – n) 2

(1 – n)

–V 1

]

– p 1 V 1 ],

or, since p 1V 1 = p 2V 2 , n

n

Work done by gas per unit mass = (p2V2 − p1V1)(1 − n) = ( p1V1 − p2V2)(n − 1) = R(T1 − T2)(n − 1).

(1.16)

3. If the compression is carried out adiabatically, no heat enters or leaves the system. Now, for any mode of compression, considering unit mass, Heat supplied = Change of internal energy + Work done (in heat units). Change of internal energy = cv(T2 − T1 ). Mechanical work done = ( p2V2 − p1V1)(1 − n). Thus, in general, if H is the heat supplied, H = cv(T2 − T1) + ( p2V2 − p1V1)(1 − n). Now,

R = (cp − cv) or cv = R(cpcv − 1).

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Also

R(T2 − T1) = ( p2V2 − p1V1).

Thus,

H = ( p2V2 − p1V1)(cpcv − 1) + ( p2V2 − p1V1)(1 − n).

For an adiabatic change, H = 0, so that ( p2V2 − p1V1)(cpcv − 1) = −( p2V2 − p1V1)(1 − n) = ( p2V2 − p1V1)(n − 1), and, therefore, n = cp cv = γ. Thus, for an adiabatic change, the relationship between pressure and density is given by pV γ = pρ γ = constant,

(1.17)

and, from (2), Work done by gas per unit mass = ( p1V1 − p2V2)(γ − 1) = R(T1 − T2)(γ − 1).

(1.18)

Concluding remarks The material presented in this chapter will be utilized in all sections of the text; in particular the influence of fluid viscosity will be of the utmost importance. The equation of state and the definition of compressible flows will also be used to differentiate flow conditions.

Summary of important equations and concepts 1.

2.

3.

The relationship between shear stress, viscosity and velocity gradient, equation (1.3), will recur throughout the text. While this text will be concerned with Newtonian fluids, as defined in Section 1.4, the reader should be familiar with the differentiation between these and other non-Newtonian fluid types. The fundamental fluid properties introduced must be understood and their dependence on temperature and pressure appreciated. In particular the defining differences between liquids and gases become essential in dealing with concepts of compressibility and time dependency, Section 1.5. A range of properties are introduced in this chapter whose importance will be returned to later under particular flow conditions: for example surface tension and its effects on capillary action; vapour pressure and its role in pressure surge analysis; and cavitation as a limit to pump operation and propellerturbine blade design.

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Summary of important equations and concepts

4.

5.

The concept of compressibility, the differences between gas and liquid compressibility and the conditions under which flows may be considered incompressible must be understood, Section 1.16. The gas laws are essential to the later development of the concepts of gaseous fluid flow, Sections 1.17 to 1.20.

23

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Chapter 2

Pressure and Head 2.1 2.2 2.3 2.4 2.5 2.6

2.7 2.8 2.9

Statics of fluid systems Pressure Pascal’s law for pressure at a point Variation of pressure vertically in a fluid under gravity Equality of pressure at the same level in a static fluid General equation for the variation of pressure due to gravity from point to point in a static fluid Variation of pressure with altitude in a fluid of constant density Variation of pressure with altitude in a gas at constant temperature Variation of pressure with altitude in a gas under adiabatic conditions

2.10 Variation of pressure and density with altitude for a constant temperature gradient 2.11 Variation of temperature and pressure in the atmosphere 2.12 Stability of the atmosphere 2.13 Pressure and head 2.14 The hydrostatic paradox 2.15 Pressure measurement by manometer 2.16 Relative equilibrium 2.17 Pressure distribution in a liquid subject to horizontal acceleration 2.18 Effect of vertical acceleration 2.19 General expression for the pressure in a fluid in relative equilibrium 2.20 Forced vortex

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This chapter will consider and introduce the forces acting on, or generated by, fluids at rest. In particular the concept of pressure will be introduced, including its variation with depth of submergence, via the hydrostatic equation, its unique value at any particular depth in a continuous fluid and direction of application at that depth. The concept that the atmosphere dictates that all activities on the Earth’s

surface are effectively carried out submerged in a fluid will be stressed and the pressure variations within, and stability of, the atmosphere will be treated. This understanding of pressure will be used to introduce methods of pressure measurement that will be essential to the treatment, in later chapters, of fluids in motion. l l l

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Pressure and Head

2.1

STATICS OF FLUID SYSTEMS

The general rules of statics apply to fluids at rest, but, from the definition of a fluid (Section 1.1), there will be no shearing forces acting and, therefore, all forces (such as F in Fig. 2.1(a)) exerted between the fluid and a solid boundary must act at right angles to the boundary. If the boundary is curved (Fig. 2.1(b)), it can be considered to be composed of a series of chords on each of which a force F1, F2, . . . , Fn acts perpendicular to the surface at the section concerned. Similarly, considering any plane drawn through a body of fluid at rest (Fig. 2.1(c)), the force exerted by one portion of the fluid on the other acts at right angles to this plane.

FIGURE 2.1 Forces in a fluid at rest

Shear stresses due to viscosity are only generated when there is relative motion between elements of the fluid. The principles of fluid statics can, therefore, be extended to cases in which the fluid is moving as a whole but all parts are stationary relative to each other. In the analysis of a problem it is usual to consider an element of the fluid defined by solid boundaries or imaginary planes. A free body diagram can be drawn for this element, showing the forces acting on it due to the solid boundaries or surrounding fluid. Since the fluid is at rest, the element will be in equilibrium, and the sum of the component forces acting in any direction must be zero. Similarly, the sum of the moments of the forces about any point must be zero. It is usual to test equilibrium by resolving along three mutually perpendicular axes and, also, by taking moments in three mutually perpendicular planes. Although a body or element may be in equilibrium, it can also be of interest to know what will happen if it is displaced from its equilibrium position. For example, in the case of a ship it is of the utmost importance to know whether it will overturn when it pitches or rolls or whether it will tend to right itself and return to its original position. There are three possible conditions of equilibrium:

1.

Stable equilibrium. A small displacement from the equilibrium position generates a force producing a righting moment tending to restore the body to its equilibrium position.

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2.2

2. 3.

Pressure

27

Unstable equilibrium. A small displacement produces an overturning moment tending to displace the body further from its equilibrium position. Neutral equilibrium. The body remains at rest in any position to which it is displaced.

These conditions are typified by the three positions of a cone on a horizontal surface shown in Fig. 2.2.

FIGURE 2.2 Types of equilibrium

2.2

PRESSURE

A fluid will exert a force normal to a solid boundary or any plane drawn through the fluid. Since problems may involve bodies of fluids of indefinite extent and, in many cases, the magnitude of the force exerted on a small area of the boundary or plane may vary from place to place, it is convenient to work in terms of the pressure p of the fluid, defined as the force exerted per unit area. If the force exerted on each unit area of a boundary is the same, the pressure is said to be uniform: Force exerted Pressure = −−−−−−−−−−−−−−−−−−−−−−−−−−− Area of boundary

F or p = −−. A

If, as is more commonly the case, the pressure changes from point to point, we consider the element of force δF normal to a small area δA surrounding the point under consideration:

δF Mean pressure, p = −−−−. δA In the limit, as δA → 0 (but remains large enough to preserve the concept of the fluid as a continuum),

δ F dF Pressure at a point, p = lim −−−− = −−−−. dA δ A→ 0 δ A Units: newtons per square metre (N m−2). (Note that an alternative metric unit is the bar; 1 bar = 105 N m−2.) Dimensions: ML−1T −2.

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2.3

PASCAL’S LAW FOR PRESSURE AT A POINT

By considering the equilibrium of a small fluid element in the form of a triangular prism surrounding a point in the fluid (Fig. 2.3), a relationship can be established between the pressures px in the x direction, py in the y direction and ps normal to any plane inclined at any angle θ to the horizontal at this point. FIGURE 2.3 Equality of pressure in all directions at a point

If the fluid is at rest, px will act at right angles to the plane ABFE, py at right angles to CDEF and ps at right angles to ABCD. Since the fluid is at rest, there will be no shearing forces on the faces of the element and the element will not be accelerating. The sum of the forces in any direction must, therefore, be zero. Considering the x direction: Force due to px = px × Area ABFE = pxδ yδ z; Component of force due to ps = − ( ps × Area ABCD) sin θ

δy = – p s δ s δ z −−− = −psδ yδ z δs (since sin θ = δyδ s). As py has no compound in the x direction, the element will be in equilibrium if pxδ yδ z + (−psδ yδ z) = 0, px = ps.

(2.1)

Similarly, in the y direction, Force due to py = py × Area CDEF = pyδ xδ z; Component of force due to ps = −( ps × Area ABCD) cos θ

δx = – p s δ s δ z −−− = −psδ xδ z δs (since cos θ = δxδs).

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2.4

Variation of pressure vertically in a fluid under gravity

29

Weight of element = −Specific weight × Volume = −ρg × −12 δ xδ yδ z. As px has no component in the y direction, the element will be in equilibrium if pyδ xδ z + (−psδ xδ z) + (−ρg × −12 δ xδ yδ z) = 0. Since δx, δy and δz are all very small quantities, δxδyδz is negligible in comparison with the other two terms, and the equation reduces to py = ps .

(2.2)

Thus, from equations (2.1) and (2.2), ps = px = py.

(2.3)

Now ps is the pressure on a plane inclined at any angle θ ; the x, y and z axes have not been chosen with any particular orientation, and the element is so small that it can be considered to be a point. This proof may be extended to the z axis. Equation (2.3), therefore, indicates that the pressure at a point is the same in all directions. This is known as Pascal’s law and applies to a fluid at rest. If the fluid is flowing, shear stresses will be set up as a result of relative motion between the particles of the fluid. The pressure at a point is then considered to be the mean of the normal forces per unit area (stresses) on three mutually perpendicular planes. Since these normal stresses are usually large compared with shear stresses it is generally assumed that Pascal’s law still applies.

2.4

VARIATION OF PRESSURE VERTICALLY IN A FLUID UNDER GRAVITY

Figure 2.4 shows an element of fluid consisting of a vertical column of constant crosssectional area A and totally surrounded by the same fluid of mass density ρ. Suppose that the pressure is p1 on the underside at level z1 and p2 on the top at level z2. Since the fluid is at rest the element must be in equilibrium and the sum of all the vertical forces must be zero. The forces acting are: Force due to p1 on area A acting up = p1 A, Force due to p2 on area A acting down = p2 A, Force due to the weight of the element = mg = Mass density × g × Volume = ρgA(z2 − z1). Since the fluid is at rest, there can be no shear forces and, therefore, no vertical forces act on the side of the element due to the surrounding fluid. Taking upward forces as positive and equating the algebraic sum of the forces acting to zero, p1A − p2A − ρgA(z2 − z1) = 0,

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FIGURE 2.4 Vertical variation of pressure

p2 − p1 = −ρg(z2 − z1).

(2.4)

Thus, in any fluid under gravitational attraction, pressure decreases with increase of height z.

EXAMPLE 2.1

A diver descends from the surface of the sea to a depth of 30 m. What would be the pressure under which the diver would be working above that at the surface assuming that the density of sea water is 1025 kg m−3 and remains constant?

Solution In equation (2.4), taking sea level as datum, z1 = 0. Since z2 is lower then z1 the value of z2 is −30 m. Substituting these values and putting ρ = 1025 kg m−3: Increase of pressure = p2 − p1 = −1025 × 9.81(−30 − 0) = 301.7 × 103 N m−2.

2.5

EQUALITY OF PRESSURE AT THE SAME LEVEL IN A STATIC FLUID

If P and Q are two points at the same level in a fluid at rest (Fig. 2.5), a horizontal prism of fluid of constant cross-sectional area A will be in equilibrium. The forces acting on this element horizontally are p1A at P and p2A at Q. Since the fluid is at rest, there will be no horizontal shear stresses on the sides of the element. For static equilibrium the sum of the horizontal forces must be zero: p1A = p2A,

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2.5

Equality of pressure at the same level in a static fluid

31

FIGURE 2.5 Equality of pressures at the same level

p1 = p2. Thus, the pressure at any two points at the same level in a body of fluid at rest will be the same. In mathematical terms, if (x, y) is the horizontal plane,

∂p −−− = 0 ∂x

and

∂p −−− = 0; ∂y

partial derivatives are used because pressure p could vary in three directions. Pressures at the same level will be equal even though there is no direct horizontal path between P and Q, provided that P and Q are in the same continuous body of fluid. Thus, in Fig. 2.6, P and Q are connected by a horizontal pipe, R and S being two points at the same level at the entrance and exit to the pipe. If the pressure is pP at P, pQ at Q, pR at R and pS at S, then, since R and S are at the same level, pR = pS; also

pR = pP + ρgz and pS = pQ + ρgz.

Substituting in equation (2.5), pP + ρgz = pQ + ρgz, pP = pQ.

FIGURE 2.6 Equality of pressures in a continuous body of fluid

(2.5)

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2.6

GENERAL EQUATION FOR THE VARIATION OF PRESSURE DUE TO GRAVITY FROM POINT TO POINT IN A STATIC FLUID

Let p be the pressure acting on the end P of an element of fluid of constant crosssectional area A and p + δp be the pressure at the other end Q (Fig. 2.7). FIGURE 2.7 Variation of pressure in a stationary fluid

The axis of the element is inclined at an angle θ to the vertical, the height of P above a horizontal datum is z and that of Q is z + δz. The forces acting on the element are: pA acting at right angles to the end face at P along the axis of the element, ( p + δp)A acting at Q along the axis in the opposite direction; mg the weight of the element, due to gravity, acting vertically down = Mass density × Volume × Gravitational acceleration = ρ × Aδ s × g. There are also forces due to the surrounding fluid acting normal to the sides of the element, since the fluid is at rest, and, therefore, perpendicular to its axis PQ. For equilibrium of the element PQ, the resultant of these forces in any direction must be zero. Resolving along the axis PQ, pA − ( p + δp)A − ρgAδs cos θ = 0, δp = −ρgδs cos θ, or, in differential form, dp −−− = – ρ g cos θ . ds In the general three-dimensional case, s is a vector with components in the x, y and z directions. Taking the (x, y) plane as horizontal, if the axis of the element is also horizontal, θ = 90° and dp⎞ ∂p ∂p ⎛− = −−− = −−− = 0, −− ⎝ ds ⎠ θ = 90° ∂ x ∂ y

(2.6)

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2.7

Variation of pressure with altitude in a fluid of constant density

33

confirming the results of Section 2.5 that, in a static fluid, pressure is constant everywhere in a horizontal plane. It is for this reason that the free surface of a liquid is horizontal. If the axis of the element is in the vertical z direction, θ = 0° and dp⎞ ∂p ⎛− −− = −−− = – ρ g, ⎝ ds ⎠ θ = 0° ∂ z and, since ∂p∂x = ∂p∂y = 0, the partial derivative ∂p∂z can be replaced by the total differential dpdz, giving dp −−− = – ρ g, dz

(2.7)

which corresponds to the result obtained in Section 2.4. Also, considering any two horizontal planes a vertical distance z apart, Pressure at all points on lower plane = p,

∂p Pressure at all points on upper plane = p + z −−− , ∂z ∂p Difference of pressure = z −−−. ∂z Since the planes are horizontal, the pressure must be constant over each plane; therefore, ∂p∂z cannot vary horizontally. From equation (2.7), this implies that ρg shall be constant and, therefore, for equilibrium, the density ρ must be constant over any horizontal plane. Thus, the conditions for equilibrium under gravity are: 1. 2. 3.

The pressure at all points on a horizontal plane must be the same. The density at all points on a horizontal plane must be the same. The change of pressure with elevation is given by dpdz = −ρg.

The actual pressure variation with elevation is found by integrating equation (2.7):

dp = – ρ g dz

or

p2 – p1 = –

z2

ρ g dz,

(2.8)

z1

but this cannot be done unless the relationship between ρ and p is known.

2.7

VARIATION OF PRESSURE WITH ALTITUDE IN A FLUID OF CONSTANT DENSITY

For most problems involving liquids it is usual to assume that the density ρ is constant, and the same assumption can also be made for a gas if pressure differences are very small. Equation (2.8) can then be written

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p = –ρ g

dz = –ρgz + constant,

or, for any two points at altitude z1 and z2 above datum, p2 − p1 = −ρg(z2 − z1 ).

VARIATION OF PRESSURE WITH ALTITUDE IN A GAS AT CONSTANT TEMPERATURE

2.8

The relation between pressure, density and temperature for a perfect gas is given by the equation pρ = RT. If conditions are assumed to be isothermal, so that temperature does not vary with altitude, ρ can be expressed in terms of p and the result substituted in equation (2.7): p ρ = −−−− , RT and, from equation (2.7), dp pg −−− = – ρ g = – −−−− , dz RT dp g −−− = – −−−− dz. p RT Integrating from p = p1 when z = z1, to p = p2 when z = z2, log e( p2p1) = −(gRT )(z2 − z1), p2 p1 = exp[−( gRT )(z2 − z1) ].

EXAMPLE 2.2

At an altitude z, of 11 000 m, the atmospheric temperature T is −56.6 °C and the pressure p is 22.4 kN m−2. Assuming that the temperature remains the same at higher altitudes, calculate the density of the air at an altitude of z2 of 15 000 m. Assume R = 287 J kg−1 K−1.

Solution Let p2 be the absolute pressure at z2. Since the temperature is constant, p2 p1 = exp [−( gRT )(z2 − z1) ].

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2.9

Variation of pressure with altitude in a gas under adiabatic conditions

35

Putting p 1 = 22.4 kN m −2 = 22.4 × 10 3 N m −2, z 1 = 11 000 m, z 2 = 15 000 m, R = 287 J kg−1 K−1, T = −56.6 °C = 216.4 K: 9.81 ( 15 000 – 11 000 ) 3 p 2 = 22.4 × 10 × exp – −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 287 × 216.4 = 22.4 × 103 × exp(−0.632) = 11.91 × 103 N m−2. Also, from the equation of state for a perfect gas (see equation (1.13)), p2 = ρ2RT and so Density of air at 15 000 m, ρ2 = p2 RT = (11.91 × 103)(287 × 216.4) = 0.192 kg m−3.

2.9

VARIATION OF PRESSURE WITH ALTITUDE IN A GAS UNDER ADIABATIC CONDITIONS

If conditions are adiabatic, the relationship between pressure and density is given by pργ = constant = p1 ρ γ1 , so that

ρ = ρ1( pp1)1γ. Substituting in equation (2.7),

ρ 1 g 1γ dp −−− = – −−− −γ p , p 1 dz 1 γ p 1 1 ⎞ – 1γ − p dp. dz = – ⎛ −−− ⎝ ρ1 g ⎠

Integrating from p = p1 when z = z1, to p = p2 when z = z2 , γ p ( γ –1 )γ p 1 1 − −−−−−−−−−−−−− z 2 – z 1 = – −−− ρ 1 g ( γ – 1 )γ

p2

p1

γ γ p 1 1 − ( p (2γ –1 )γ – p (1γ –1 )γ ) = – ⎛ −−−−−− ⎞ −−− ⎝ γ – 1⎠ ρ 1 g

p 1 ⎛ p 2⎞ ( γ –1 )γ γ −− –1 , = – ⎛ −−−−−− ⎞ −−−− ⎝ γ – 1⎠ ρ 1 g ⎝ p 1⎠ or, since p1ρ1 = RT, for any gas,

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γ RT ⎛ p−2⎞ ( γ –1 )γ – 1 , z 2 – z 1 = – ⎛ −−−−−− ⎞ −−−−−−1 ⎝ − p 1⎠ ⎝ γ – 1⎠ g ⎛ p−−2⎞ ⎝ p 1⎠

( γ –1 )γ

g ( z 2 – z 1 ) ⎛ γ – 1⎞ = −−−−−− −−−−−−− −−−−−− + 1, RT 1 ⎝ γ ⎠

g ( z2 – z1 ) γ – 1 p2 −− = 1 – −−−−−−−−−−−−− ⎛⎝ −−−−−−⎞⎠ γ RT 1 p1

γ ( γ –1 )

.

(2.9)

This can be extended to any isentropic process for which pρ n = constant, to give g ( z2 – z1 ) n – 1 p2 −− = 1 – −−−−−−−−−−−−− ⎛⎝ −−−−−−⎞⎠ n RT 1 p1

n( n –1 )

.

(2.10)

The rate of change of temperature with altitude – the temperature lapse rate – can also be found for adiabatic conditions. From the characteristic equation, ρ = pRT and, since, from equation (2.7), dz = −dpρg, substituting for ρ, dz = −(RTgp) dp. For adiabatic conditions, γ

pργ = p1ρ 1 , or, since pρ = RT, p = p1(T1T )γ (1−γ ) and, differentiating, dp = −[γ (1 − γ )] p1 T γ1(1 – γ ) T −1(1−γ ) dT. Substituting these values of p and dp in equation (2.11): RT { – [ γ ( 1 – γ ) ]p T γ ( 1 – γ ) T –1 ( 1 – γ ) dT } dz = – −−−− −−−−−−−−−−−−−−−−−−−−−γ−1−(−1−–−−1γ−)−−−−–−− −−−−−−−−−−−−−−−−−−−−− p1 T 1 T γ (1 – γ ) g = [γ (1 − γ )](Rg) dT.

(2.11)

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2.9

Variation of pressure with altitude in a gas under adiabatic conditions

37

Temperature gradient, dT −−−− = – [ ( γ – 1 )γ ] ( gR ). dz

EXAMPLE 2.3

(2.12)

Calculate the pressure, temperature and density of the atmosphere at an altitude of 1200 m if at zero altitude the temperature is 15 °C and the pressure 101 kN m−2. Assume that conditions are adiabatic (γ = 1.4) and R = 287 J kg−1 K−1.

Solution From equation (2.9), g ( z2 – z1 ) γ – 1 p 2 = p 1 1 – −−−−−−−−−−−−− ⎛⎝ −−−−−−⎞⎠ RT 1 γ

γ ( γ –1 )

.

Putting p1 = 101 × 103 N m−2, z1 = 0, z2 = 1200 m, T1 = 15 °C = 288 K, γ = 1.4, R = 287 J kg−1 K−1: 9.81 × 1200 0.4 3 p 2 = 101 × 10 1 – −−−−−−−−−−−−−−−− ⎛⎝ −−−−⎞⎠ 287 × 288 1.4

1.40.4

Nm

–2

= 87.33 × 103 N m−2. From equation (2.12), Temperature gradient, dT −−−− = – [ ( γ – 1 )γ ] ( gR ) = −(0.41.4) × (9.81287) dz = −9.76 × 10−3 K m−1, dT T2 = T 1 + −−− (z 2 – z 1 ) dz = 288 − (9.76 × 10−3 × 1200) = 276.3 K = 3.3 °C. From the equation of state, Density at 1200 m,

ρ2 = p2 RT2 = (87.33 × 103)(287 × 276.3) = 1.101 kg m−3.

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2.10 VARIATION OF PRESSURE AND DENSITY WITH ALTITUDE FOR A CONSTANT TEMPERATURE GRADIENT Assuming that there is a constant temperature lapse rate (i.e. dTdz = constant) with elevation in a gas, so that its temperature falls by an amount δT for a unit change of elevation, then, if T1 = temperature at level z1, T = temperature at level z, T = T1 − δ T(z − z1).

(2.13)

From equation (2.7), dpdz = −ρg and, since pρ = RT, putting ρ = pRT, dp g −−− = – p −−−− , dz RT dp g −−− = – −−−− dz. p RT Substituting for T from equation (2.13), dpp = −{gR[T1 − δ T(z − z1) ]}dz. Integrating between the limits p1 and p2 and z1 and z2, log e (p2 p1) = ( gRδT ) log e{[T1 − δ T(z2 − z1)]T1}, p2 p1 = [1 − (δ TT1)(z2 − z1)] gRδ T.

(2.14)

Comparing this with the result obtained in Section 2.9, and putting dT n–1 g δ T = – −−− = ⎛⎝ −−−−−−⎞⎠ ⎛⎝ −−⎞⎠ , dz n R we have gRδT = n(n − 1). Substituting in equation (2.14), p2 p1 = [1 − (gRT1)(z2 − z1)(n − 1)n]n(n−1) which agrees with equation (2.10). To find the corresponding change of density ρ, since pρ = RT,

ρ2 p2 T1 p2 T −− = −− × −−− = −− × −−−−−−−−−−−−−1−−−−−−−−−− ρ1 p1 T2 p1 T1 – δ T ( z2 – z1 ) and, substituting from equation (2.14) for p2p1,

ρ2 ρ1 = [1 − (δ TT1)(z2 − z1)] gRδT[1 − (δTT1)(z2 − z1)]−1 = [1 − (δ TT1)(z2 − z1)](gRδ T)−1.

(2.15)

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2.11

EXAMPLE 2.4

Variation of temperature and pressure in the atmosphere

39

Assuming that the temperature of the atmosphere diminishes with increasing altitude at the rate of 6.5 °C per 1000 m, find the pressure and density at a height of 7000 m if the corresponding values at sea level are 101 kN m−2 and 1.235 kg m−3 when the temperature is 15 °C. Take R = 287 J kg−1 K−1.

Solution From equation (2.14), p2 = p1[1 − (δTT1)(z2 − z1)] gRδ T. Putting p1 = 101 × 103 N m−2, δT = 6.5 °C per 1000 m = 0.0065 K m−1, T1 = 15 °C = 288 K, (z2 − z1) = 7000 m, R = 287 J kg−1 K−1: p2 = 101 × 103[1 − (0.0065288) × 7000]9.81(287×0.0065) = 40.89 × 103 N m−2. From the equation of state, Density, ρ2 = p2 RT2 = p2 R[T1 − δ T(z2 − z1)] = 40.89 × 103287(288 − 0.0065 × 7000) = 0.588 kg m−3.

2.11 VARIATION OF TEMPERATURE AND PRESSURE IN THE ATMOSPHERE A body of fluid which is of importance to the engineer is the atmosphere. In practice, it is never in perfect equilibrium and is subject to large incalculable disturbances. In order to provide a basis for the design of aircraft an International Standard Atmosphere has been adopted which represents the average conditions in Western Europe; the relations between altitude, temperature and density have been tabulated (Table 2.1). Essentially, the standard atmosphere comprises the troposphere – extending from sea level to 11 000 m – in which the temperature lapse rate is constant at approximately 0.0065 K m−1 and the pressure–density relationship is pρ n = constant, where n = 1.238. Above 11 000 m lies the stratosphere, in which conditions are assumed to be isothermal, with the temperature constant at −56 °C. Figure 2.8 shows the variation of pressure with altitude in the International Standard Atmosphere. The atmospheric pressure at sea level is assumed to be equivalent to 760 mm of mercury, the temperature 15 °C and the density 1.225 kg m−3. In the real atmosphere, the troposphere extends to an average of 11 000 m, but can vary from 7600 m at the poles to 18 000 m at the equator. While the temperature, in general, falls steadily with altitude, meteorological conditions can arise in the lower layers which produce temperature inversion – the temperature increasing with altitude. In the stratosphere, the temperature remains substantially constant up to approximately 32 000 m; it then rises to about 70 °C before falling again. Figure 2.9 shows typical values for pressure and temperature.

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TABLE 2.1 International Standard Atmosphere

FIGURE 2.8 Variation of temperature with altitude in the International Standard Atmosphere

Altitude above sea level (m)

Absolute pressure (bar)

Absolute temperature (K)

Mass density (kg m−3)

Kinematic viscosity (m2 s−1 × 10−5)

0 1 000 2 000 4 000 6 000 8 000 10 000 11 500 12 000 14 000 16 000 18 000 20 000 22 000 24 000 26 000 28 000 30 000 32 000

1.013 25 0.898 8 0.795 0 0.616 6 0.472 2 0.356 5 0.265 0 0.209 8 0.194 0 0.141 7 0.103 5 0.075 65 0.055 29 0.040 47 0.029 72 0.021 88 0.016 16 0.011 97 0.008 89

288.15 281.7 275.2 262.2 249.2 236.2 223.3 216.7 216.7 216.7 216.7 216.7 216.7 218.6 220.6 222.5 224.5 226.5 228.5

1.225 0 1.111 7 1.006 6 0.819 4 0.660 2 0.525 8 0.413 6 0.337 5 0.311 9 0.227 9 0.166 5 0.121 6 0.088 91 0.064 51 0.046 94 0.034 26 0.025 08 0.018 41 0.013 56

1.461 1.581 1.715 2.028 2.416 2.904 3.525 4.213 4.557 6.239 8.540 11.686 15.989 22.201 30.743 42.439 58.405 80.134 109.620

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2.12

Stability of the atmosphere

41

FIGURE 2.9 Variation of temperature and pressure in the real atmosphere

Because of vertical currents, the composition of the air remains practically constant in both the troposphere and the stratosphere, except that there is a negligible amount of water vapour in the stratosphere and a slight reduction in the ratio of oxygen to nitrogen above an altitude of 20 km. Nine-tenths of the mass of the atmosphere is contained below 20 km and 99 per cent below 60 km.

2.12 STABILITY OF THE ATMOSPHERE We have seen that there are variations of density from point to point in the atmosphere when it is at rest. In practice, there are local disturbances due to air currents. There are also changes of density as a result of local thermal effects, which cause the movement of elements of air into regions where they are surrounded by air of slightly different density and temperature. If the density of the surrounding air is less than that of the newly arrived element, there is a tendency for the element to return to its original position – since the net upward force exerted by the surrounding fluid is less than the weight of the element. In Fig. 2.10, if ρ1 is the density of the surrounding air and ρ2 is the density of the air in the displaced element, Weight of element, mg = ρ2 gAδ z. Upward force due to surrounding fluid = δp × A = ρ1gδzA, and there is, therefore, a net downward force of ( ρ2 − ρ1)gδzA. As an element of fluid rises in the atmosphere, its pressure and temperature fall. Air is a poor conductor, and conditions are, therefore, approximately adiabatic. From

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FIGURE 2.10 Stability of the atmosphere

equation (2.9), substituting γ = 1.414 and R = 287 J kg−1 K−1, the adiabatic temperature lapse rate in the element is δT = 0.01 K m−1. The natural temperature lapse rate δT′ occurring in the atmosphere is found to be of the order of 0.0065 K m−1. Since the lapse rates differ for the ascending element of air and the surrounding atmosphere the changes of density with altitude will also differ and can be calculated from equation (2.15). For example, if ρ1 = density of air at sea level, ρ2 = density of air at an elevation of 1000 m, R = 287 J kg−1 K−1, then: 1.

Assuming δT = 0.01 K m−1, for the air in the displaced element, g 9.81 −−−−−− – 1 = −−−−−−−−−−−−−−− – 1 = 2.418, RδT 287 × 0.01 and from equation (2.15), 0.01 × 1000 2.48 ρ 2 = ρ 1 ⎛⎝ 1 – −−−−−−−−−−−−−−−− ⎞⎠ , 288

2.

ρ2 −− = 0.9181 ρ1

for air in the element. Assuming δT = 0.0065 K m−1 for the surrounding atmosphere, g 9.81 −−−−−− – 1 = −−−−−−−−−−−−−−−−−− – 1 = 4.258, RδT 287 × 0.0065

ρ 2 ⎛ 0.0065 × 1000⎞ 4.258 −− = 1 – −−−−−−−−−−−−−−−−−−−− = 0.9018 ⎠ ρ1 ⎝ 288 for the surrounding air. Thus, the density of the ascending element expanding adiabatically decreases less rapidly than that of the surrounding air; the element eventually becomes denser than the surroundings and tends to fall back to its original level. The atmosphere, therefore, tends to be stable under normal conditions. If, however, the natural temperature lapse rate were to exceed the adiabatic lapse rate, equilibrium would be unstable and an element displaced upwards would continue to rise. Such conditions can arise in thundery weather.

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2.13

Pressure and head

43

2.13 PRESSURE AND HEAD In a fluid of constant density, dpdz = −ρg can be integrated immediately to give p = −ρgz + constant. In a liquid, the pressure p at any depth z, measured downwards from the free surface so that z = −h (Fig. 2.11), will be p = ρgh + constant and, since the pressure at the free surface will normally be atmospheric pressure patm , p = ρgh + patm.

(2.16)

It is often convenient to take atmospheric pressure as a datum. Pressures measured above atmospheric pressure are known as gauge pressures.

FIGURE 2.11 Pressure and head

Since atmospheric pressure varies with atmospheric conditions, a perfect vacuum is taken as the absolute standard of pressure. Pressures measured above perfect vacuum are called absolute pressures: Absolute pressure = Gauge pressure + Atmospheric pressure. Taking patm as zero, equation (2.16) becomes p = ρgh,

(2.17)

which indicates that, if g is assumed constant, the gauge pressure at a point X (Fig. 2.11) can be defined by stating the vertical height h, called the head, of a column of a given fluid of mass density ρ which would be necessary to produce this pressure. Note that when pressures are expressed as head, it is essential that the mass density ρ is given or the fluid named. For example, since from equation (2.17) h = pρg, a pressure of 100 kN m−2 can be expressed in terms of water ( ρH O = 103 kg m−3) as a head of (100 × 103)(103 × 9.81) = 10.19 m of water. Alternatively, in terms of mercury (relative density 13.6) a pressure of 100 kN m−2 will correspond to a head of (100 × 103)(13.6 × 103 × 9.81) = 0.75 m of mercury. 2

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EXAMPLE 2.5

A cylinder contains a fluid at a gauge pressure of 350 kN m−2. Express this pressure in terms of a head of (a) water ( ρH O = 1000 kg m−3), (b) mercury (relative density 13.6). What would be the absolute pressure in the cylinder if the atmospheric pressure is 101.3 kN m−2? 2

Solution From equation (2.17), head, h = pρg. (a) Putting p = 350 × 103 N m−2, ρ = ρH O = 1000 kg m−3, 2

350 × 10 3 Equivalent head of water = −−−−3−−−−−−−−−− = 35.68 m. 10 × 9.81 (b) For mercury ρHg = σρH O = 13.6 × 1000 kg m−3, 2

350 × 10 3 Equivalent head of water = −−−−−−−−−−−−−−3−−−−−−−−−− = 2.62 m, 1.36 × 10 × 9.81 Absolute pressure = Gauge pressure + Atmospheric pressure = 350 + 101.3 = 451.3 kN m−2.

2.14 THE HYDROSTATIC PARADOX From equation (2.17) it can be seen that the pressure exerted by a fluid is dependent only on the vertical head of fluid and its mass density ρ; it is not affected by the weight of the fluid present. Thus, in Fig. 2.12 the four vessels all have the same base area A and are filled to the same height h with the same liquid of density ρ. Pressure on bottom in each case, p = ρgh, Force on bottom = Pressure × Area = pA = ρghA. Thus, although the weight of fluid is obviously different in the four cases, the force on the bases of the vessels is the same, depending on the depth h and the base area A. FIGURE 2.12 The hydrostatic paradox

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2.15

Pressure measurement by manometer

45

2.15 PRESSURE MEASUREMENT BY MANOMETER The relationship between pressure and head is utilized for pressure measurement in the manometer or liquid gauge. The simplest form is the pressure tube or piezometer shown in Fig. 2.13, consisting of a single vertical tube, open at the top, inserted into a pipe or vessel containing liquid under pressure which rises in the tube to a height depending on the pressure. If the top of the tube is open to the atmosphere, the pressure measured is ‘gauge’ pressure: Pressure at A = Pressure due to column of liquid of height h1, pA = ρgh1. Similarly, FIGURE 2.13 Pressure tube or piezometer

EXAMPLE 2.6

Pressure at B = pB = ρgh2. If the liquid is moving in the pipe or vessel, the bottom of the tube must be flush with the inside of the vessel, otherwise the reading will be affected by the velocity of the fluid. This instrument can only be used with liquids, and the height of the tube which can conveniently be employed limits the maximum pressure that can be measured.

What is the maximum gauge pressure of water that can be measured by means of a piezometer tube 2 m high? (Mass density of water ρH O = 103 kg m−3.) 2

Solution Since p = ρgh for maximum pressure, put ρ = ρH O = 103 and h = 2 m, giving 2

Maximum pressure, p = 103 × 9.81 × 2 = 19.62 × 103 N m−2.

The U-tube gauge, shown in Fig. 2.14, can be used to measure the pressure of either liquids or gases. The bottom of the U-tube is filled with a manometric liquid Q which is of greater density ρman and is immiscible with the fluid P, liquid or gas, of

FIGURE 2.14 U-tube manometer

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density ρ, whose pressure is to be measured. If B is the level of the interface in the lefthand limb and C is a point at the same level in the right-hand limb, Pressure pB at B = Pressure pC at C. For the left-hand limb, pB = Pressure pA at A + Pressure due to depth h1 of fluid P = pA + ρgh1. For the right-hand limb, pC = Pressure pD at D + Pressure due to depth h 2 of liquid Q. But

pD = Atmospheric pressure = Zero gauge pressure,

and so pC = 0 + ρman gh 2. Since pB = pC , pA + ρgh1 = ρman gh 2, pA = ρman gh 2 − ρgh1.

EXAMPLE 2.7

(2.18)

A U-tube manometer similar to that shown in Fig. 2.14 is used to measure the gauge pressure of a fluid P of density ρ = 800 kg m−3. If the density of the liquid Q is 13.6 × 103 kg m−3, what will be the gauge pressure at A if (a) h1 = 0.5 m and D is 0.9 m above BC, (b) h1 = 0.1 m and D is 0.2 m below BC?

Solution (a) In equation (2.18), ρman = 13.6 × 103 kg m−3, ρ = 0.8 × 103 kg m−3, h1 = 0.5 m, h 2 = 0.9 m; therefore: pA = 13.6 × 103 × 9.81 × 0.9 − 0.8 × 103 × 9.81 × 0.5 = 116.15 × 103 N m−2. (b) Putting h1 = 0.1 m and h 2 = −0.2 m, since D is below BC: pA = 13.6 × 103 × 9.81 × (−0.2) − 0.8 × 103 × 9.81 × 0.1 = −27.45 × 103 N m−2, the negative sign indicating that pA is below atmospheric pressure.

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2.15

Pressure measurement by manometer

47

FIGURE 2.15 Measurement of pressure difference

In Fig. 2.15, a U-tube gauge is arranged to measure the pressure difference between two points in a pipeline. As in the previous case, the principle involved in calculating the pressure difference is that the pressure at the same level CD in the two limbs must be the same, since the fluid in the bottom of the U-tube is at rest. For the left-hand limb, pC = pA + ρga. For the right-hand limb pD = pB + ρg(b − h) + ρman gh. Since pC = pD, pA + ρga = pB + ρg(b − h) + ρman gh, Pressure difference = pA − pB = ρg(b − a) + hg(ρman − ρ).

EXAMPLE 2.8

(2.19)

A U-tube manometer is arranged, as shown in Fig. 2.15, to measure the pressure difference between two points A and B in a pipeline conveying water of density ρ = ρH O = 103 kg m−3. The density of the manometric liquid Q is 13.6 × 103 kg m−3, and point B is 0.3 m higher than point A. Calculate the pressure difference when h = 0.7 m. 2

Solution In equation (2.19), ρ = 103 kg m−3, ρman = 13.6 × 103 kg m−3, (b − a) = 0.3 m and h = 0.7 m. Pressure difference = pA − pB = 103 × 9.81 × 0.3 + 0.7 × 9.81(13.6 − 1) × 103 N m−2 = 89.467 × 103 N m−2.

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FIGURE 2.16 U-tube with one leg enlarged

In both the above cases, if the fluid P is a gas its density ρ can usually be treated as negligible compared with ρman and the equations (2.18) and (2.19) can be simplified. In forming the connection from a manometer to a pipe or vessel in which a fluid is flowing, care must be taken to ensure that the connection is perpendicular to the wall and flush internally. Any burr or protrusion on the inside of the wall will disturb the flow and cause a local change in pressure so that the manometer reading will not be correct. Industrially, the simple U-tube manometer has the disadvantage that the movement of the liquid in both limbs must be read. By making the diameter of one leg very large as compared with the other (Fig. 2.16), it is possible to make the movement in the large leg very small, so that it is only necessary to read the movement of the liquid in the narrow leg. Assuming that the manometer in Fig. 2.16 is used to measure the pressure difference p1 − p2 in a gas of negligible density and that XX is the level of the liquid surface when the pressure difference is zero, then, when pressure is applied, the level in the right-hand limb will rise a distance z vertically. Volume of liquid transferred from left-hand leg to right-hand leg = z × (π 4)d 2; Fall in level of the left-hand leg Volume transferred z × ( π 4 )d 2 d 2 = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− = −−−−−−−−−−−−−−−2−− = z ⎛ −−⎞ . ⎝ D⎠ ( π 4 )D Area of left-hand leg The pressure difference, p1 − p2, is represented by the height of the manometric liquid corresponding to the new difference of level: p1 − p2 = ρg[z + z(dD)2] = ρgz[1 + (dD)2], or, if D is large compared with d, p1 − p2 = ρgz. If the pressure difference to be measured is small, the leg of the U-tube may be inclined as shown in Fig. 2.17. The movement of the meniscus along the inclined leg, read off on the scale, is considerably greater than the change in level z: Pressure difference, p1 − p2 = ρgz = ρgx sin θ.

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2.15

Pressure measurement by manometer

49

FIGURE 2.17 U-tube with inclined leg

FIGURE 2.18 Inverted U-tube manometer

The manometer can be made as sensitive as may be required by adjusting the angle of inclination of the leg and choosing a liquid with a suitable value of density ρ to give a scale reading x of the desired size for a given pressure difference. The inverted U-tube shown in Fig. 2.18 is used for measuring pressure differences in liquids. The top of the U-tube is filled with a fluid, frequently air, which is less dense than that connected to the instrument. Since the fluid in the top is at rest, pressures at level XX will be the same in both limbs. For the left-hand limb, pXX = pA − ρga − ρman gh. For the right-hand limb, pXX = pB − ρg(b + h). Thus

pB − pA = ρg(b − a) + gh(ρ − ρman),

or, if A and B are at the same level, pB − pA = (ρ − ρman )gh. If the top of the tube is filled with air ρman is negligible compared with ρ and pB − pA = ρgh. On the other hand, if the liquid in the top of the tube is chosen so that ρman is very nearly equal to ρ, and provided that the liquids do not mix, the result will be a very sensitive manometer giving a large value of h for a small pressure difference.

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EXAMPLE 2.9

An inverted U-tube of the form shown in Fig. 2.18 is used to measure the pressure difference between two points A and B in an inclined pipeline through which water is flowing ( ρH O = 103 kg m−3). The difference of level h = 0.3 m, a = 0.25 m and b = 0.15 m. Calculate the pressure difference pB − pA if the top of the manometer is filled with (a) air, (b) oil of relative density 0.8. 2

Solution In either case, the pressure at XX will be the same in both limbs, so that pXX = pA − ρga − ρman gh = pB − ρg(b + h), pB − pA = ρg(b − a) + gh(ρ − ρman). (a) If the top is filled with air ρman is negligible compared with ρ. Therefore, pB − pA = ρg(b − a) + ρgh = ρg(b − a + h). Putting ρ = ρH O = 103 kg m−3, b = 0.15 m, a = 0.25 m, h = 0.3 m: 2

pB − pA = 103 × 9.81(0.15 − 0.25 + 0.3) = 1.962 × 103 N m−2. (b) If the top is filled with oil of relative density 0.8, ρman = 0.8ρH O = 0.8 × 103 kg m−3. 2

pB − pA = ρg(b − a) + gh(ρ − ρman ) = 103 × 9.81(0.15 − 0.25) + 9.81 × 0.3 × 103(1 − 0.8) N m−2 = 103 × 9.81(−0.1 + 0.06) = −392.4 N m−2.

The manometer in its various forms is an extremely useful type of pressure gauge, but suffers from a number of limitations. While it can be adapted to measure very small pressure differences, it cannot be used conveniently for large pressure differences – although it is possible to connect a number of manometers in series and to use mercury as the manometric fluid to improve the range. A manometer does not have to be calibrated against any standard; the pressure difference can be calculated from first principles. However, for accurate work, the temperature should be known, since this will affect the density of the fluids. Some liquids are unsuitable for use because they do not form well-defined menisci. Surface tension can also cause errors due to capillary rise; this can be avoided if the diameters of the tubes are sufficiently large – preferably not less than 15 mm diameter. It is difficult to correct for surface tension, since its effect will depend upon whether the tubes are clean. A major disadvantage of the manometer is its slow response, which makes it unsuitable for measuring fluctuating pressures. Even under comparatively static conditions, slight fluctuations of pressure can make the liquid in the manometer oscillate, so that it is difficult to get a precise reading of the levels of the liquid in the gauge. These oscillations can be reduced by putting restrictions in the manometer connections. It is also essential that the pipes connecting the manometer to the pipe or vessel containing the liquid under pressure should be filled with this liquid and that there should be no air bubbles in the liquid.

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2.17

Pressure distribution in a liquid subject to horizontal acceleration

51

2.16 RELATIVE EQUILIBRIUM If a fluid is contained in a vessel which is at rest, or moving with constant linear velocity, it is not affected by the motion of the vessel; but if the container is given a continuous acceleration, this will be transmitted to the fluid and affect the pressure distribution in it. Since the fluid remains at rest relative to the container, there is no relative motion of the particles of the fluid and, therefore, no shear stresses, fluid pressure being everywhere normal to the surface on which it acts. Under these conditions the fluid is said to be in relative equilibrium.

2.17 PRESSURE DISTRIBUTION IN A LIQUID SUBJECT TO HORIZONTAL ACCELERATION Figure 2.19 shows a liquid contained in a tank which has an acceleration a. A particle of mass m on the free surface at O will have the same acceleration a as the tank and so will be subject to an accelerating force F. From Newton’s law, F = ma.

(2.20)

FIGURE 2.19 Effect of horizontal acceleration

Also, F is the resultant of the fluid pressure force R, acting normally to the free surface at O, and the weight of the particle mg, acting vertically. Therefore, F = mg tan θ.

(2.21)

Comparing equations (2.20) and (2.21), tan θ = ag

(2.22)

and is constant for all points on the free surface. Thus, the free surface is a plane inclined at a constant angle θ to the horizontal. Since the acceleration is horizontal, vertical forces are not changed and the pressure at any depth h below the surface will be ρgh. Planes of equal pressure lie parallel to the free surface.

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2.18 EFFECT OF VERTICAL ACCELERATION If the acceleration is vertical, the free surface will remain horizontal. Considering a vertical prism of cross-sectional area A (Fig. 2.20), subject to an upward acceleration a, then at depth h below the surface, where the pressure is p, Upward accelerating force, F = Force due to p – Weight of prism = pA − ρghA. By Newton’s second law, F = Mass of prism × Acceleration = ρhA × a. Therefore, pA − ρghA = ρhAa, p = ρgh(1 + ag).

(2.23)

FIGURE 2.20 Effect of vertical acceleration

2.19 GENERAL EXPRESSION FOR THE PRESSURE IN A FLUID IN RELATIVE EQUILIBRIUM If ∂p∂x, ∂p∂y and ∂p∂z are the rates of change of pressure p in the x, y and z directions (Fig. 2.21) and ax , ay and az the accelerations,

∂p Force in x direction, Fx = p δ y δ z – ⎛ p + −−− δ x⎞ δ y δ z ⎝ ∂x ⎠ ∂p = – −−− δ x δ y δ z. ∂x By Newton’s second law, Fx = ρδxδyδz × ax ; therefore,

∂p – −−− = ρ a x . ∂x

(2.24)

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2.19

General expression for the pressure in a fluid in relative equilibrium

53

FIGURE 2.21 Relative equilibrium: the general case

Similarly, in the y direction,

∂p – −−− = ρ a y . ∂y

(2.25)

In the vertical z direction, the weight of the element ρgδxδyδz must be considered:

∂p F z = p δ x δ y – ⎛ p + −−− δ z⎞ δ x δ y – ρ g δ x δ y δ z ⎝ ∂z ⎠ ∂p = – −−− δ x δ y δ z – ρ g δ x δ y δ z. ∂z By Newton’s second law, Fz = ρδxδyδz × az ; therefore,

∂p – −−− = ρ ( g + a z ). ∂z

(2.26)

For an acceleration as in any direction in the x − z plane making an angle φ with the horizontal, the components of the acceleration are ax = as cos φ and az = as sin φ. Now

dp ∂ p dx ∂ p dz −−− = −−−−−−−− + −−− −−−. ds ∂ x ds ∂ z ds

(2.27)

For the free surface and all other planes of constant pressure, dpds = 0. If θ is the inclination of the planes of constant pressure to the horizontal, tan θ = dzdx. Putting dpds = 0 in equation (2.27),

∂ p dx ∂ p dz −−− −−− + −−− −−− = 0 ∂ x ds ∂ z ds dz ∂p ∂p −−− = tan θ = – −−− −−− . dx ∂x ∂z

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Substituting from equations (2.24) and (2.26), tan θ = −ax(g + az),

(2.28)

or, in terms of as , a cos φ tan θ = – −−−−−−s−−−−−−−−−−−−. ( g + a s sin φ )

(2.29)

For the case of horizontal acceleration, φ = 0 and equation (2.29) gives tan θ = −asg, which agrees with equation (2.22). (Note effect of sign convention.) For vertical acceleration, φ = 90° giving tan θ = 0, indicating that the free surface remains horizontal. Since, for the two-dimensional case,

∂p ∂p dp = −−− dx + −−− dz, ∂x ∂z the pressure at a particular point in the fluid can be found by integration: p=

dp = −∂∂−x−p dx + −∂∂−−pz dz.

Substituting from equations (2.24) and (2.26) and assuming that ρ is constant: p=

(–ρa ) dx + [–ρ(g + a )] dz + constant x

z

= −ρ (xas cos φ − gz − zas sin φ) + constant, or, since xz = tan θ, p = −zρ (as tan θ cos φ − g − as sin φ) + constant,

(2.30)

where z is positive measured upwards from a horizontal datum fixed relative to the fluid.

EXAMPLE 2.10

A rectangular tank 1.2 m deep and 2 m long is used to convey water up a ramp inclined at an angle φ of 30° to the horizontal (Fig. 2.22). Calculate the inclination of the water surface to the horizontal when (a) the acceleration parallel to the slope on starting from the bottom is 4 m s−2, (b) the deceleration parallel to the slope on reaching the top is 4.5 m s−2. If no water is to be spilt during the journey what is the greatest depth of water permissible in the tank when it is at rest?

Solution The slope of the water surface is given by equation (2.29). (a) During acceleration, as = +4 m s−2.

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2.19

General expression for the pressure in a fluid in relative equilibrium

55

FIGURE 2.22 Acceleration up an inclined plane

– a cos φ 4 cos 30° tan θ A = −−−−−−s−−−−−−−−− = – −−−−−−−−−−−−−−−−−−−−−− g + a s sin φ 9.81 + 4 sin 30° = −0.2933

θA = 163° 39′′. (b) During retardation, as = −4.5 m s−2. ( – 4.5 ) cos 30° tan θ R = – −−−−−−−−−−−−−−−−−−−−−−−− = 0.5154 9.81 – 4.5 sin 30°

θR = 27° 16′′. Since 180° − θR θA, the worst case for spilling will be during retardation. When the water surface is inclined, the maximum depth at the tank wall will be Depth + −12 Length × tan θ, which must not exceed 1.2 m if the water is not to be spilt. Putting length = 2 m, tan θ = tan θR = 0.5154, Depth + (2.02) × 0.5154 = 1.2, Depth = 1.2 − 0.5154 = 0.6846 m.

The equations derived in this section indicate: 1. 2. 3.

if there is no horizontal acceleration, ax = 0 and then θ = 0 so that surfaces of constant pressure are horizontal; in free space, g will be zero so that tan θ = −axaz (surfaces of constant pressure will therefore be perpendicular to the resultant acceleration); since free surfaces of liquids are surfaces of constant pressure, their inclination will be determined by equation (2.29); thus, if ax and ay are zero, the free surface will be horizontal.

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2.20 FORCED VORTEX A body of fluid, contained in a vessel which is rotating about a vertical axis with uniform angular velocity, will eventually reach relative equilibrium and rotate with the same angular velocity ω as the vessel, forming a forced vortex. The acceleration of any particle of fluid at radius r due to rotation will be −ω 2r perpendicular to the axis of rotation, taking the direction of r as positive outward from the axis. Thus, from equation (2.24), dp −−− = – ρω 2 r. dr Figure 2.23 shows a cylindrical vessel containing liquid rotating about its axis, which is vertical. At any point P on the free surface, the inclination θ of the free surface is given by equation (2.28): FIGURE 2.23 Forced vortex

a ω 2 r dz tan θ = – −−−−−x−−− = −−−−− = −−−. g dr g + az

(2.31)

The inclination of the free surface varies with r and, if z is the height of P above O, the surface profile is given by integrating equation (2.31):

−ω−g−−−r dr = −ω−2g−−−r− + constant. r

z=

2

2 2

(2.32)

Thus, the profile of the water surface is a paraboloid. Similarly other surfaces of equal pressure will also be paraboloids. If the container is closed and the fluid has no free surface, the paraboloid drawn to represent the imaginary free surface represents the variation of pressure head with radius. Thus, the pressure p at radius r is given by equation (2.32) as z = pρg = ω 2r 22g + constant, p = ρω 2r 22 + constant.

(2.33)

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Problems

57

Concluding remarks The definitions of pressure and head presented in this chapter are fundamental to the study of fluid mechanics. In particular the development of the hydrostatic equation and the proof that pressure at a depth in a continuous fluid is constant was central to the development of both flow pressure and velocity measuring techniques due to the equation’s application to the design and operation of pressure measuring manometers. The basic expressions derived for manometer use will be utilized later in the text in the treatment of flow measurement, Chapter 6. The treatment of the atmosphere stresses the obvious point that all our activities and structures must be seen as submerged within a fluid.

Summary of important equations and concepts 1.

2.

3.

4.

Pascal’s law, equation (2.3), stating that pressures at a depth are equal in all coordinate directions, coupled with the hydrostatic equation (2.4) that links depth, gravitational acceleration, density and pressure are the fundamental underpinning of this chapter’s treatment of hydrostatic forces. Equation (2.8) underlines the importance of the density–pressure relationship identified in Chapter 1. This becomes essential to the understanding of the atmospheric variations of pressure and density with altitude, Sections 2.7 to 2.12. Despite the predominance of electronic measurement of fluid flow conditions the application of hydrostatics to manometer measurement of pressure, and hence flow conditions, remains important. Sections 2.13 to 2.15 investigate these applications fully. Relative equilibrium is introduced in Section 2.16 and is used to discuss the effects of acceleration and the generation of a forced vortex, concepts returned to in more detail in Chapter 6.

Problems 2.1 Calculate the pressure in the ocean at a depth of 2000 m assuming that salt water is (a) incompressible with a constant density of 1002 kg m−3, (b) compressible with a bulk modulus of 2.05 GN m−2 and a density at the surface of 1002 kg m−3. [(a) 19.66 MN m−2, (b) 19.75 MN m−2] 2.2 What will be (a) the gauge pressure, (b) the absolute pressure of water at a depth of 12 m below the free surface? Assume the density of water to be 1000 kg m−3 and the atmospheric pressure 101 kN m−2. [(a) 117.72 kN m−2, (b) 218.72 kN m−2] 2.3 What depth of oil, specific gravity 0.8, will produce a pressure of 120 kN m−2? What would be the corresponding depth of water? [15.3 m, 12.2 m]

2.4 At what depth below the free surface of oil having a density of 600 kg m−3 will the pressure be equal to 1 bar? [17 m] 2.5 What would be the pressure in kilonewtons per square metre if the equivalent head is measured as 400 mm of (a) mercury of specific gravity 13.6, (b) water, (c) oil of specific weight 7.9 kN m −3, (d ) a liquid of density 520 kg m−3? [(a) 53.4 kN m−2, (b) 3.92 kN m−2, (c) 3.16 kN m−2, (d ) 2.04 kN m−2] 2.6 A mass of 50 kg acts on a piston of area 100 cm2. What is the intensity of pressure on the water in contact with the underside of the piston, if the piston is in equilibrium? [4.905 × 104 N m−2]

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Chapter 2

Pressure and Head

2.7 The pressure head in a gas main at a point 120 m above sea level is equivalent to 180 mm of water. Assuming that the densities of air and gas remain constant and equal to 1.202 kg m−3 and 0.561 kg m−3, respectively, what will be the pressure head in millimetres of water at sea level? [103 mm] 2.8 A manometer connected to a pipe in which a fluid is flowing indicates a negative gauge pressure head of 50 mm of mercury. What is the absolute pressure in the pipe in newtons per square metre if the atmospheric pressure is 1 bar. [93.3 kN m−2] 2.9 An open tank contains oil of specific gravity 0.75 on top of water. If the depth of oil is 2 m and the depth of water 3 m, calculate the gauge and absolute pressures at the bottom of the tank when the atmospheric pressure is 1 bar. [44.15 kN m−2, 144.15 kN m−2] 2.10 A closed tank contains 0.5 m of mercury, 2 m of water, 3 m of oil of density 600 kg m−3 and there is an air space above the oil. If the gauge pressure at the bottom of the tank is 200 kN m−2, what is the pressure of the air at the top of the tank? [96 kN m−2] 2.11 An inverted cone 1 m high and open at the top contains water to half its height, the remainder being filled with oil of specific gravity 0.9. If half the volume of water is drained away find the pressure at the bottom (apex) of the inverted cone. [9033 N m−2]

values are 15 °C and 101.5 kN m−2. Assuming that the temperature decreases uniformly with increasing altitude, estimate the temperature lapse rate and the pressure and density of the air at an altitude of 3000 m. [6.37 °C per 1000 m, 70.22 kN m−2, 0.91 kg m−3] 2.15 Show that the ratio of the atmospheric pressure at an altitude h1 to that at sea level may be expressed as ( pp0) = (TT0) n, a uniform temperature lapse rate being assumed. Find the ratio of the pressures and the densities at 10 700 m and at sea level taking the standard atmosphere as having a sea level temperature of 15 °C and a lapse rate of 6.5 °C per 1000 m to a minimum of −56.5 °C. [0.2337, 0.3082] 2.16 The barometric pressure of the atmosphere at sea level is equivalent to 760 mm of mercury and its temperature is 288 K. The temperature decreases with increasing altitude at the rate of 6.5 K per 1000 m until the stratosphere is reached in which the temperature remains constant at 216.5 K. Calculate the pressure in millimetres of mercury and the density in kilograms per cubic metre at an altitude of 14 500 m. Assume R = 287 J kg−1 K−1. [97.52 mm, 0.209 kg m−3] 2.17 In Fig. 2.24 fluid P is water and fluid Q is mercury. If the specific weight of mercury is 13.6 times that of water and the atmospheric pressure is 101.3 kN m−2, what is the absolute pressure at A when h1 = 15 cm and h2 = 30 cm? [59.8 kN m−2]

2.12 A hydraulic press has a diameter ratio between the two pistons of 8:1. The diameter of the larger piston is 600 mm and it is required to support a mass of 3500 kg. The press is filled with a hydraulic fluid of specific gravity 0.8. Calculate the force required on the smaller piston to provide the required force (a) when the two pistons are level, (b) when the smaller piston is 2.6 m below the larger piston. [(a) 536 N, (b) 626.2 N] 2.13 Show that the ratio of the pressures ( p2p1) and densities ( ρ2 ρ 1) for altitudes h2 and h1 in an isothermal atmosphere is given by

p2 ρ2 −− = −− = e –g ( h p1 ρ1

2

– h 1 )RT

. FIGURE 2.24

What increase in altitude is necessary in the stratosphere to halve the pressure? Assume a constant temperature of −56.5 °C and the gas constant R = 287 J kg−1 K−1. [4390 m] 2.14 From observation it is found that at a certain altitude in the atmosphere the temperature is −25 °C and the pressure is 45.5 kN m−2, while at sea level the corresponding

2.18 A U-tube manometer (Fig. 2.25) measures the pressure difference between two points A and B in a liquid of density ρ1. The U-tube contains mercury of density ρ2. Calculate the difference of pressure if a = 1.5 m, b = 0.75 m and h = 0.5 m if the liquid at A and B is water and ρ2 = 13.6ρ1. [54.4 kN m−2]

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Problems

59

end of 44 mm diameter. The left-hand limb and the bottom of the tube are filled with water and the top of the righthand limb is filled with oil of specific gravity 0.83. The free surfaces of the liquids are in the enlarged ends and the interface between the oil and water is in the tube below the enlarged end. What would be the difference in pressures applied to the free surfaces which would cause the oilwater interface to move 1 cm? [21 N m−2]

FIGURE 2.25 2.19 The top of an inverted U-tube manometer is filled with oil of specific gravity 0.98 and the remainder of the tube with water of specific gravity 1.01. Find the pressure difference in newtons per square metre between two points at the same level at the base of the legs when the difference of water level is 75 mm. [22 N m−2] 2.20 An inclined manometer is required to measure an air pressure difference of about 3 mm of water with an accuracy of ±3 per cent. The inclined arm is 8 mm diameter and the enlarged end is 24 mm diameter. The density of the manometer fluid is 740 kg m−3. Find the angle which the inclined arm must make with the horizontal to achieve the required accuracy assuming an acceptable readability of 0.5 mm. [12° 39′] 2.21 An inclined tube manometer consists of a vertical cylinder of 35 mm diameter to the bottom of which is connected a tube of 5 mm diameter inclined upwards at 15° to the horizontal. The manometer contains oil of relative density 0.785. The open end of the inclined tube is connected to an air duct while the top of the cylinder is open to the atmosphere. Determine the pressure in the air duct if the manometer fluid moves 50 mm along the inclined tube. What is the error if the movement of the fluid in the cylinder is ignored? [107.5 N m−2 7.85 N m−2] 2.22 A manometer consists of a U-tube, 7 mm internal diameter, with vertical limbs each with an enlarged upper

2.23 A vessel 1.4 m wide and 2.0 m long is filled to a depth of 0.8 m with a liquid of mass density 840 kg m−3. What will be the force in N on the bottom of the vessel (a) when being accelerated vertically upwards at 4 m s−1, (b) when the acceleration ceases and the vessel continues to move at a constant velocity of 7 m s−1 vertically upwards? [(a) 25 985 N, (b) 18 458 N] 2.24 A pipe 25 mm in diameter is connected to the centre of the top of a drum 0.5 m in diameter, the cylindrical axis of the pipe and the drum being vertical. Water is poured into the drum through the pipe until the water level stands in the pipe 0.6 m above the top of the drum. If the drum and pipe are now rotated about their vertical axis at 600 rev min−1 what will be the upward force exerted on the top of the drum? [13.26 kN] 2.25 A tube ABCD has the end A open to the atmosphere and the end D closed. The portion ABC is vertical while the portion CD is a quadrant of radius 250 mm with its centre at B, the whole being arranged to rotate about its vertical axis ABC. If the tube is completely filled with water to a height in the vertical limb of 300 mm above C find (a) the speed of rotation which will make the pressure head at D equal to the pressure head at C, (b) the value and position of the maximum pressure head in the curved portion CD when running at this speed. [(a) 84.6 rev min−1, (b) 0.362 m of water and 0.12 m below point D] 2.26 A closed airtight tank 4 m high and 1 m in diameter contains water to a depth of 3.3 m. The air in the tank is at a pressure of 40 kN m−2 gauge. What are the absolute pressures at the centre and circumference of the base of the tank when it is rotating about its vertical axis at a speed of 180 rev min−1? At this speed the water wets the top surface of the tank. [17.01 m absolute, 21.53 m absolute]

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Chapter 3

Static Forces on Surfaces. Buoyancy 3.1 3.2

3.3

3.4 3.5 3.6

Action of fluid pressure on a surface Resultant force and centre of pressure on a plane surface under uniform pressure Resultant force and centre of pressure on a plane surface immersed in a liquid Pressure diagrams Force on a curved surface due to hydrostatic pressure Buoyancy

Equilibrium of floating bodies Stability of a submerged body Stability of floating bodies Determination of the metacentric height 3.11 Determination of the position of the metacentre relative to the centre of buoyancy 3.12 Periodic time of oscillation 3.13 Stability of a vessel carrying liquid in tanks with a free surface 3.7 3.8 3.9 3.10

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The implication of the hydrostatic equation, together with the realization that pressures at any equal depth in a continuous fluid are both equal and act equally in all directions, leads to the treatment of static forces on submerged surfaces. This also defines buoyancy and the stability of floating bodies. This chapter will introduce the techniques available to

determine the forces acting on surfaces as a result of the applied fluid pressure, and will stress the difference between pressure, which is a scalar quantity acting equally in all directions at a particular depth, and the associated force, which is a vector quantity possessing both magnitude and direction. l l l

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3.1

FIGURE 3.1 Forces on a plane surface

ACTION OF FLUID PRESSURE ON A SURFACE

Since pressure is defined as force per unit area, when fluid pressure p acts on a solid boundary – or across any plane in the fluid – the force exerted on each small element of area δA will be pδA, and, since the fluid is at rest, this force will act at right angles to the boundary or plane at the point under consideration. In a body of fluid, the pressure p may vary from point to point, and the forces on each element of area will also vary. If the fluid pressure acts on or across a plane surface, all the forces on the small elements will be parallel (Fig. 3.1) and can be represented by a single force, called the resultant force, acting at right angles to the plane through a point called the centre of pressure. Resultant force, R = Sum of forces on all elements of area R = p1 δA1 + p2 δA2 + · · · + pn δAn = ∑ p δA,

FIGURE 3.2 Forces on a curved surface

where ∑ means ‘the sum of’. If the boundary is a curved surface, the elementary forces will act perpendicular to the surface at each point and will, therefore, not be parallel (Fig. 3.2). The resultant force can be found by resolution or by a polygon of forces, but will be less than ∑ pδA. For example, in the extreme case of the curved surface of a bucket filled with water (Fig. 3.3), the elementary forces acting radially on the vertical wall will balance and the resultant force will be zero. If this were not so, there would be an unbalanced horizontal force in some direction and the bucket would move of its own accord.

3.2 FIGURE 3.3 Forces on a cylindrical surface

RESULTANT FORCE AND CENTRE OF PRESSURE ON A PLANE SURFACE UNDER UNIFORM PRESSURE

The pressure p on a plane horizontal surface in a fluid at rest will be the same at all points, and will act vertically downwards at right angles to the surface. If the area of the plane surface is A, Resultant force = pA. It will act vertically downwards and the centre of pressure will be the centroid of the surface. For gases, the variation of pressure with elevation is small and so it is usually possible to assume that gas pressure on a surface is uniform, even though the surface may not be horizontal. The resultant force is then pA acting through the centroid of the plane surface.

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3.3

Resultant force and centre of pressure on a plane surface immersed in a liquid

63

FIGURE 3.4 Resultant force on a plane surface immersed in a fluid

3.3

RESULTANT FORCE AND CENTRE OF PRESSURE ON A PLANE SURFACE IMMERSED IN A LIQUID

Figure 3.4 shows a plane surface PQ of any area A totally immersed in a liquid of density ρ and inclined at an angle φ to the free surface. Considering one side only, there will be a force due to fluid pressure p acting on each element of area δA. The magnitude of p will depend on the vertical depth y of the element below the free surface. Taking the pressure at the free surface as zero, from equation (2.4), and measuring y downwards, p = ρgy; therefore, Force on element of area, δA = pδA = ρgyδA. Summing the forces on all such elements over the whole surface, since these forces are all perpendicular to the plane PQ, Resultant force, R = ∑ ρ g y δA. If we assume that ρ and g are constant, R = ρg ∑ yδA.

(3.1)

The quantity ∑ yδA is the first moment of area under the surface PQ about the free surface of the liquid and is equal to AD, where A = the area of the whole immersed surface PQ and D = the vertical depth to the centroid G of the immersed surface. Substituting in equation (3.1), Resultant force, R = ρgAD.

(3.2)

This resultant force R will act perpendicular to the immersed surface at the centre of pressure C at some vertical depth D below the free surface, such that the moment of R about any point will be equal to the sum of the moments of the forces on all the elements δA about the same point. Thus, if the plane of the immersed surface cuts the free surface at O, Moment of R about O = Sum of moments of forces on all elements of area δA about O, Force on any small element = ρgyδA = ρgs sin φ × δA, since y = s sin φ.

(3.3)

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Moment of force on element about O = pgs sin φ × δA × s = ρg sin φ × δA × s 2. Since ρ, g and φ are the same for all elements, Sum of the moments of all such forces about O = ρg sin φ ∑ s 2δA. Also R = ρgAD; therefore, Moment of R about O = ρgAD × OC = ρgAD(Dsin φ). Substituting in equation (3.3),

ρgAD(Dsin φ) = ρg sin φ ∑ s 2δA, D = sin2 φ (∑ s 2δA)AD, ∑ s 2δA = Second moment of area of the immersed surface about an axis in the free surface through O = IO = AkO2 , where kO = the radius of gyration of the immersed surface about O. Therefore, D = sin2 φ (IOAD) = sin2 φ ( kO2 /D ).

(3.4)

The values of IO and kO2 can be found if the second moment of area of the immersed surface IG about an axis through its centroid G parallel to the free surface is known by using the parallel axis rule, or,

AkO2 = AkG2 + A(Dsin φ)2.

Thus

D = sin2 φ [ kG2 + (Dsin φ)2D] = sin2 φ ( kG2 D) + D.

(3.5)

The geometrical properties of some common figures are given in Table 3.1. From equation (3.5) it can be seen that the centre of pressure will always be below the centroid G except when the surface is horizontal (φ = 0°). As the depth of immersion increases, the centre of pressure will move nearer to the centroid, since for the given surface the change of pressure between the upper and lower edge becomes proportionately smaller in comparison with the mean pressure, making the pressure distribution more uniform. The lateral position of the centre of pressure can be found by taking moments about the line OG, which is the line of intersection of the immersed surface with a vertical plane through G: R × d = Sum of moments of forces on small elements about OG = ∑ ρgδAyx. Putting

R = ρgAD, d = ( ∑δAπx)Ay.

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3.3

Resultant force and centre of pressure on a plane surface immersed in a liquid

TABLE 3.1 Geometrical properties of some common figures

Area A

65

Second moment of area IGG about axis GG through the centroid

If the area is symmetrical about a vertical plane through the centroid G, the moment of each small element on one side is balanced by that due to a similar element on the other side so that ∑ δAy = 0. Therefore, d = 0 and the centre of pressure will be on the axis of symmetry.

EXAMPLE 3.1

A trapezoidal opening in the vertical wall of a tank is closed by a flat plate which is hinged at its upper edge (Fig. 3.5). The plate is symmetrical about its centreline and is 1.5 m deep. Its upper edge is 2.7 m long and its lower edge is 1.2 m long. The free surface of the water in the tank stands 1.1 m above the upper edge of the plate. Calculate the moment about the hinge line required to keep the plate closed.

Solution The moment required to keep the plate closed will be equal and opposite to the moment of the resultant force R due to the water acting at the centre of pressure C, i.e. R × CB. From equation (3.2), R = ρgAD. Area of plate, A = −12 (2.7 + 1.2) × 1.5 = 2.925 m2.

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FIGURE 3.5 Trapezoidal sluice gate

To find the position of the centroid G, take moments of area about BB′, putting the vertical distance GB = y: A × y = Moment of areas BHE and FJB′ + Moment of EFJH = 2 × ( −12 × 1.5 × 0.75) × 0.5 + (1.2 × 1.5) × 0.75. 2.925y = 0.5625 + 1.35 = 1.9125, y = 0.654 m. Depth to the centre of pressure, D = y + OB = 0.654 + 1.1 = 1.754 m. Substituting in equation (3.2), Resultant force, R = 103 × 9.81 × 2.925 × 1.754 = 50.33 kN. From equation (3.4), Depth to centre of pressure C, D = sin2 φ (IOAD). Using the parallel axis rule for second moments of area, IO = Second moment of EFJH about O + Second moment of BEH and B′FJ about O 1.2 × 1.5 3 1.5 × 1.5 3 = ⎛ −−−−−−−−−−−−− + 1.2 × 1.5 × 1.85 2⎞ + ⎛ −−−−−−−−−−−−− + 1.5 × 0.75 × 1.6 2⎞ m 4 ⎝ ⎠ ⎝ ⎠ 12 36 = 9.5186 m4. As the wall is vertical, sin φ = 1; therefore, 9.5186 Depth to centre of pressure, D = −−−−−−−−−−−−−−−−−−− = 1.8553 m. 2.925 × 1.754 Moment about hinge = R × BC = 50.33(1.8553 − 1.1) = 38.01 kN m.

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3.3

EXAMPLE 3.2

Resultant force and centre of pressure on a plane surface immersed in a liquid

67

The angle between a pair of lock gates (Fig. 3.6) is 140° and each gate is 6 m high and 1.8 m wide, supported on hinges 0.6 m from the top and bottom of the gate. If the depths of water on the upstream and downstream sides are 5 m and 1.5 m, respectively, estimate the reactions at the top and bottom hinges.

FIGURE 3.6 Lock gate

Solution Figure 3.6(a) shows the plan view of the gates. F is the force exerted by one gate on the other and is assumed to act perpendicular to the axis of the lock if friction between the gates is neglected. P is the resultant of the water forces P1 and P2 (Fig. 3.6(b)) acting on the upstream and downstream faces of the gate, and R is the resultant of the forces R T and RB on the top and bottom hinges. Using equation (3.2) Upstream water force, P1 = ρgA1D1 = 103 × 9.81 × (5 × 1.8) × 2.5 = 220.725 × 103 N, Downstream water force, P2 = ρgA2z2 = 103 × 9.81 × (1.5 × 1.8) × 0.75 = 19.865 × 103 N, Resultant water force on one gate, P = P1 − P2 = (220.73 − 19.86) × 103 N = 200.87 × 103 N. The gates are rectangular, and so P1 and P2 will act at one-third of the depth of water (as shown in Fig. 3.6(b)), since, in equation (3.5), φ = 90°, D = d2, kG2 = d 212, where d is the depth of the gate immersed (see also Section 3.4). The height above the base at which the resultant force P acts can be found by taking moments. If P acts at a distance x from the bottom of the gate, then by taking moments about O,

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Px = P1 × (53) − P2 × (1.53) = (220.73 × 53 − 19.86 × 0.5) × 103 = 357.95 × 103 N m. x = (357.95 × 103 )(200.87 × 103 ) = 1.782 m. Assuming that F, R and P are coplanar, they will meet at a point, and, since F is assumed to be perpendicular to the axis of the lock on plan, both F and R are inclined to the gate as shown at an angle of 20° so that F = R and P = F sin 20° + R sin 20° = 2R sin 20°, P 200.87 × 10 3 R = −−−−−−−−−−−− = −−−−−−−−−−−−−−−−−− = 293.65 × 10 3 N. 2 sin 20° 2 × 0.342 If R is coplanar with P it acts at 1.78 m from the bottom of the gate. Taking moments about the bottom hinge, 4.8R T = 1.18R R T = 1.184.8 × 293.65 × 103 = 72.2 × 103 N = 72.2 kN, RB = R − R T = 293.65 − 72.2 = 221.45 kN.

3.4

PRESSURE DIAGRAMS

The resultant force and centre of pressure can be found graphically for walls and other surfaces of constant vertical height for which it is convenient to calculate the horizontal force exerted per unit width. In Fig. 3.7, ABC is the pressure diagram for the vertical wall of the tank containing a liquid, pressure being plotted horizontally against depth vertically. At the free surface A, the (gauge) pressure is zero. At depth y, p = ρgy. The relationship between p and y is linear and can be represented by the triangle ABC. The area of this triangle will be the product of depth (in metres) and pressure (in newtons per square metre), and will represent, to scale, the resultant force R on unit width of the immersed surface perpendicular to the plane of the diagram (in newtons per metre). Area of pressure diagram = −12 AB × BC = −12 H × ρgH.

FIGURE 3.7 Pressure diagram for a vertical wall

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3.4

Pressure diagrams

69

Therefore, Resultant force, R = ρgH 22 for unit width, and R will act through the centroid P of the pressure diagram, which is at a depth of −2 H from A. 3 This result could also have been obtained from equations (3.2) and (3.5), since, for unit width, R = ρgAD = ρ g(H × 1) × −12 H = ρgH 22, and, in equation (3.5), φ = 90°, sin φ = 1, D = H2, kG2 = H 212; therefore, H 212 H 2 D = −−−−−−−−− + −− = −H, H2 2 3 as before. If the plane surface is inclined and submerged below the surface, the pressure diagram is drawn perpendicular to the immersed surface (Fig. 3.8) and will be a straight line extending from p = 0 at the free surface to p = ρgH at depth H. As the immersed surface does not extend to the free surface, the resultant force R is represented by the shaded area, instead of the whole triangle, and acts through the centroid P of this area.

FIGURE 3.8 Pressure diagram for an inclined submerged surface

It is also possible to draw pressure diagrams in three dimensions for immersed areas of various shapes as, for example, the triangular sluice gate in Fig. 3.9. However, such diagrams do little more than provide assistance in visualizing the situation.

FIGURE 3.9 Pressure diagram for a triangular sluice gate

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EXAMPLE 3.3

A closed tank (Fig. 3.10), rectangular in plan with vertical sides, is 1.8 m deep and contains water to a depth of 1.2 m. Air is pumped into the space above the water until the air pressure is 35 kN m−2. If the length of one wall of the tank is 3 m, determine the resultant force on this wall and the height of the centre of pressure above the base.

FIGURE 3.10 Pressure diagram

Solution The air pressure will be transported uniformly over the whole of the vertical wall, and can be represented by the pressure diagram ABCD (Fig. 3.10(b)), the area of which represents the force exerted by the air per unit width of wall. Force due to air, RAir = (p × AB) × Width = 35 × 103 × 1.8 × 3 = 189 × 103 N, and, since the wall is rectangular and the pressure uniform, RAir will act at mid-height, which is 0.9 m above the base. The pressure due to the water will start from zero at the free surface, corresponding to the point E, and reach a value DF equal to ρgh at the bottom. The area of the triangular pressure diagram EFD represents the force exerted by the water per unit width: Force due to water, RH O =

−1 2

× ( ρgh × DE) × Width

=

−1 2

× 103 × 9.81 × 1.2 × 1.2 × 3

2

= 21.19 × 103 N, and, since the wall is rectangular, RH O will act at 1−3 h = 0.4 m from the base. 2

Total force due to both air and water, R = RAir + RH O 2

= (189 + 21.19) × 103 = 210.19 × 103 N.

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3.5

Force on a curved surface due to hydrostatic pressure

71

If x is the height above the base of the centre of pressure through which R acts, R × x = RAir × 0.9 + RH O × 0.4, 2

x = (189 × 0.9 + 21 × 0.4)210.19 = 0.85 m.

3.5

FORCE ON A CURVED SURFACE DUE TO HYDROSTATIC PRESSURE

If a surface is curved, the forces produced by fluid pressure on the small elements making up the area will not be parallel and, therefore, must be combined vectorially. It is convenient to calculate the horizontal and vertical components of the resultant force. This can be done in three dimensions, but the following analysis is for a surface curved in one plane only. In Fig. 3.11(a) and (b), AB is the immersed surface and R h and R v are the horizontal and vertical components of the resultant force R of the liquid on one side of the surface. In Fig. 3.11(a) the liquid lies above the immersed surface, while in Fig. 3.11(b) it acts below the surface. In Fig. 3.11(a), if ACE is a vertical plane through A, and BC is a horizontal plane, then, since element ACB is in equilibrium, the resultant force P on AC must equal the horizontal component Rh of the force exerted by the fluid on AB because there are no other horizontal forces acting. But AC is the projection of AB on a vertical plane; therefore,

FIGURE 3.11 Hydrostatic force on a curved surface

Horizontal component Rh = Resultant force on the projection of AB on a vertical plane. Also, for equilibrium, P and R h must act in the same straight line; therefore, the horizontal component R h acts through the centre of pressure of the projection of AB on a vertical plane.

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Similarly, in Fig. 3.11(b), element ABF is in equilibrium, and so the horizontal component R h is equal to the resultant force on the projection BF of the curved surface AB on a vertical plane, and acts through the centre of pressure of this projection. In Fig. 3.11(a), the vertical component R v will be entirely due to the weight of the fluid in the area ABDE lying vertically above AB. There are no other vertical forces, since there can be no shear forces on AE and BD because the fluid is at rest. Thus, Vertical component, R v = Weight of fluid vertically above AB, and will act vertically downwards through the centre of gravity G of ABDE. In Fig. 3.11(b), if the surface AB were removed and the space ABDE filled with the liquid, this liquid would be in equilibrium under its own weight and the vertical force on the boundary AB. Therefore, Vertical component, R v = Weight of the volume of the same fluid which would lie vertically above AB, and will act vertically upwards through the centre of gravity G of this imaginary volume of fluid. In the case of closed vessels under pressure, a free surface does not exist, but an imaginary free surface can be substituted at a level pρg above a point at which the pressure p is known, ρ being the mass density of the actual fluid. The resultant force R is found by combining the components vectorially. In the general case, the components in three directions may not meet at a point and, therefore, cannot be represented by a single force. However, in Fig. 3.11, if the surface is of uniform width perpendicular to the diagram, R h and R v will intersect at O. Thus, Resultant force, R = ( R 2h + R 2v ),

FIGURE 3.12 Resultant force on a cylindrical surface

EXAMPLE 3.4

and acts through O at an angle θ given by tan θ = R vR h. In the special case of a cylindrical surface, all the forces on each small element of area acting normal to the surface will be radial and will pass through the centre of curvature O (Fig. 3.12). The resultant force R must, therefore, also pass through the centre of curvature O.

A sluice gate is in the form of a circular arc of radius 6 m as shown in Fig. 3.13. Calculate the magnitude and direction of the resultant force on the gate, and the location with respect to O of a point on its line of action.

Solution Since the water reaches the top of the gate,

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3.6

Buoyancy

73

FIGURE 3.13 Sector gate

Depth of water, h = 2 × 6 sin 30° = 6 m, Horizontal component of force on gate = R h per unit length = Resultant force on PQ per unit length = ρg × h × h2 = ρgh 22 = (103 × 9.81 × 36)2 N m−1 = 176.58 kN m−1, Vertical component of force on gate = R v per unit length = Weight of water displaced by segment PSQ = (Sector OPSQ − ∆OPQ)ρ g = [ (60360) × π × 62 − 6 sin 30° × 6 cos 30°] × 103 × 9.81 N m−1 = 32.00 kN m−1, Resultant force on gate, R = ( R 2h + R 2v ) = (176.582 + 32.002) = 179.46 kN m−1. If R is inclined at an angle θ to the horizontal, tan θ = R vRh = 32.00176.58 = 0.181 22

θ = 10.27° to the horizontal. Since the surface of the gate is cylindrical, the resultant force R must pass through O.

3.6

BUOYANCY

The method of calculating the forces on a curved surface applies to all shapes of surface and, therefore, to the surface of a totally submerged object (Fig. 3.14). Considering any vertical plane VV through the body, the projected area of each of the

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FIGURE 3.14 Buoyancy

two sides on this plane will be equal and, as a result, the horizontal forces F will be equal and opposite. There is, therefore, no resultant horizontal force on the body due to the pressure of the surrounding fluid. The only force exerted by the fluid on an immersed body is vertical and is called the buoyancy or upthrust. It will be equal to the difference between the resultant forces on the upper and lower parts of the surface of the body. If ABCD is a horizontal plane, Upthrust on body = Upward force on lower surface ADEC − Downward force on upper surface ABCD = Weight of volume of fluid AECDGFH − Weight of volume of fluid ABCDGFH = Weight of volume of fluid ABCDE, Upthrust on body = Weight of fluid displaced by the body, and will act through the centroid of the volume of fluid displaced, which is known as the centre of buoyancy. This result is known as Archimedes’ principle. As an alternative to the proof given above, it can be seen that, if the body were completely replaced by the fluid in which it is immersed, the forces exerted on the boundaries corresponding to the original body would exactly maintain the substituted fluid in equilibrium. Thus, the upward force on the boundary must be equal to the downward force corresponding to the weight of the fluid displaced by the body. If a body is immersed so that part of its volume V1 is immersed in a fluid of density ρ1 and the rest of its volume V2 in another immiscible fluid of mass density ρ2 (Fig. 3.15),

FIGURE 3.15 Body immersed in two fluids

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Upthrust on upper part, R1 = ρ1 gV1 acting through G1, the centroid of V1, Upthrust on lower part, R2 = ρ2 gV2 acting through G2, the centroid of V2, Total upthrust = ρ1 gV1 + ρ2 gV2. The positions of G1 and G2 are not necessarily on the same vertical line, and the centre of buoyancy of the whole body is, therefore, not bound to pass through the centroid of the whole body.

EXAMPLE 3.5

A rectangular pontoon has a width B of 6 m, a length l of 12 m, and a draught D of 1.5 m in fresh water (density 1000 kg m−3). Calculate (a) the weight of the pontoon, (b) its draught in sea water (density 1025 kg m−3) and (c) the load (in kilonewtons) that can be supported by the pontoon in fresh water if the maximum draught permissible is 2 m.

Solution When the pontoon is floating in an unloaded condition, Upthrust on immersed volume = Weight of pontoon. Since the upthrust is equal to the weight of the fluid displaced, Weight of pontoon = Weight of fluid displaced, W = ρgBlD. (a) In fresh water, ρ = 1000 kg m−3 and D = 1.5 m; therefore, Weight of pontoon, W = 1000 × 9.81 × 6 × 12 × 1.5 N = 1059.5 kN. (b) In sea water, ρ = 1025 kg m−3; therefore, Draught in sea water, D = WρgBl 1059.5 × 10 3 = −−−−−−−−−−−−−−−−−−−−−−−−−−−−− = 1.46 m. 1025 × 9.81 × 6 × 12 (c) For the maximum draught of 2 m in fresh water, Total upthrust = Weight of water displaced = ρgBlD = 1000 × 9.81 × 6 × 12 × 2 N = 1412.6 kN, Load that can be supported

= Upthrust − Weight of pontoon = 1412.6 − 1059.5 = 353.1 kN.

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3.7

EQUILIBRIUM OF FLOATING BODIES

When a body floats in vertical equilibrium in a liquid, the forces present are the upthrust R acting through the centre of buoyancy B (Fig. 3.16) and the weight of the body W = mg acting through its centre of gravity. For equilibrium, R and W must be equal and act in the same straight line. Now, R will be equal to the weight of fluid displaced, ρgV, where V is the volume of fluid displaced; therefore, V = mgρg = mρ. As explained in Section 2.1, the equilibrium of a body may be stable, unstable or neutral, depending upon whether, when given a small displacement, it tends to return to the equilibrium position, move further from it or remain in the displaced position. For a floating body, such as a ship, stability is of major importance. FIGURE 3.16 Body floating in equilibrium

3.8

STABILITY OF A SUBMERGED BODY

For a body totally immersed in a fluid, the weight W = mg acts through the centre of gravity of the body, while the upthrust R acts through the centroid of the body B, which is the centre of buoyancy. Whatever the orientation of the body, these two points will remain in the same positions relative to the body. It can be seen from Fig. 3.17 that a small angular displacement θ from the equilibrium position will generate a moment W × BG × θ. If the centre of gravity G is below the centre of buoyancy B (Fig. 3.17(a)), this will be a righting moment and the body will tend to return to its equilibrium position. However, if (as in Fig. 3.17(b)) the centre of gravity is above the centre of buoyancy, an overturning moment is produced and the body is unstable. Note that, as FIGURE 3.17 Stability of submerged bodies

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the body is totally immersed, the shape of the displaced fluid is not altered when the body is tilted and so the centre of buoyancy remains unchanged relative to the body.

3.9

STABILITY OF FLOATING BODIES

Figure 3.18(a) shows a body floating in equilibrium. The weight W = mg acts through the centre of gravity G and the upthrust R acts through the centre of buoyancy B of the displaced fluid in the same straight line as W. When the body is displaced through an angle θ (Fig. 3.18(b)), W continues to act through G; the volume of liquid remains unchanged since R = W, but the shape of this volume changes and its centre of gravity, which is the centre of buoyancy, moves relative to the body from B to B1. Since R and W are no longer in the same straight line, a turning moment proportional to W × θ is produced, which in Fig. 3.18(b) is a righting moment and in Fig. 3.18(d) is an overturning moment. If M is the point at which the line of action of the upthrust R cuts the original vertical through the centre of gravity of the body G, x = GM × θ, provided that the angle of tilt θ is small, so that sin θ = tan θ = θ in radians. The point M is called the metacentre and the distance GM is the metacentric height. Comparing Fig. 3.18(b) and (d) it can be seen that: 1. 2. 3.

FIGURE 3.18 Stable and unstable equilibrium

If M lies above G, a righting moment W × GM × θ is produced, equilibrium is stable and GM is regarded as positive. If M lies below G, an overturning moment W × GM × θ is produced, equilibrium is unstable and GM is regarded as negative. If M coincides with G, the body is in neutral equilibrium.

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Since a floating body can tilt in any direction, it is usual, for a ship, to consider displacement about the longitudinal (rolling) and transverse (pitching) axes. The position of the metacentre and the value of the metacentric height will normally be different for rolling and pitching.

3.10 DETERMINATION OF THE METACENTRIC HEIGHT The metacentric height of a vessel can be determined if the angle of tilt θ caused by moving a load P (Fig. 3.19) a known distance x across the deck is measured. Overturning moment due to movement of load P = Px.

(3.6)

If GM is the metacentric height and W = mg is the total weight of the vessel including P, FIGURE 3.19 Determination of metacentric height

Righting moment = W × GM × θ.

(3.7)

For equilibrium in the tilted position, the righting moment must equal the overturning moment so that, from equations (3.6) and (3.7), W × GM × θ = Px, Metacentric height, GM = PxWθ.

(3.8)

The true metacentric height is the value of GM as θ → 0.

3.11 DETERMINATION OF THE POSITION OF THE METACENTRE RELATIVE TO THE CENTRE OF BUOYANCY For a vessel of known shape and displacement, the position of the centre of buoyancy B is comparatively easily found and the position of the metacentre M relative to B can be calculated as follows. In Fig. 3.20, AC is the original waterline plane and B the

FIGURE 3.20 Height of metacentre above centre of buoyancy

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centre of buoyancy in the equilibrium position. When the vessel is tilted through a small angle θ, the centre of buoyancy will move to B′ as a result of the alteration in the shape of the displaced fluid. A′C′ is the waterline plane in the displaced position. For small angles of tilt, BM = BB′θ. The movement of the centre of buoyancy, which is the centre of gravity of the displaced fluid, from B to B′ is the result of the removal of a volume of fluid corresponding to the wedge AOA′ and the addition of a wedge COC′. The total weight of fluid displaced remains unchanged, since it is equal to the weight of the vessel; therefore, Weight of wedge AOA′ = Weight of wedge COC′. If a is a small area in the waterline plane at a distance x from the axis of rotation OO, it will generate a small volume, shown shaded, when the vessel is tilted. Volume swept out by a = DD′ × a = axθ. Summing all such volumes and multiplying by the specific weight ρg of the liquid, Weight of wedge AOA′ =

x =AO

∑ ρ gax θ .

(3.9)

x =0

Similarly, Weight of wedge COC′ =

x = CO

∑ ρ gax θ .

(3.10)

x =0

Since there is no change in displacement, we have, from equations (3.9) and (3.10), x = AO

x = CO

x =0

x =0

ρ g θ ∑ ax = ρ g θ ∑ ax, ∑ax = 0. But ∑ax is the first moment of area of the waterline plane about OO; therefore the axis OO must pass through the centroid of the waterline plane. The distance BB′ can now be calculated, since the couple produced by the movement of the wedge AOA′ to COC′ must be equal to the couple due to the movement of R from B to B′. Moment about OO of the weight of fluid swept out by area a = ρgaxθ × x. Total moment due to altered displacement = ρgθ ∑ax2. Putting ∑ax2 = I = Second moment of area of waterline plane about OO, Total moment due to altered displacement = ρgθI,

(3.11)

Moment due to movement of R = R × BB′ = ρgV × BB′,

(3.12)

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where V = volume of liquid displaced. Equating equations (3.11) and (3.12),

ρgV × BB′ = ρgθI, giving

BB′ = θIV,

(3.13)

BM = BB′θ = IV.

(3.14)

The distance BM is known as the metacentric radius.

EXAMPLE 3.6

A cylindrical buoy (Fig. 3.21) 1.8 m in diameter, 1.2 m high and weighing 10 kN floats in salt water of density 1025 kg m−3. Its centre of gravity is 0.45 m from the bottom. If a load of 2 kN is placed on the top, find the maximum height of the centre of gravity of this load above the bottom if the buoy is to remain in stable equilibrium.

FIGURE 3.21 Stability of a cylindrical buoy

Solution In Fig. 3.21, let G be the centre of gravity of the buoy, G1 the centre of gravity of the load at a height Z1 above the bottom, and G′ the combined centre of gravity of the load and the buoy at a height Z′ above the bottom. When the load is in position, let V be the volume of salt water displaced and Z the depth of immersion of the buoy. Buoyancy force = Weight of salt water displaced = ρgV = ρg(π4)d 2Z. For equilibrium, the buoyancy force must equal the combined weight of the buoy and the load (W + W1); therefore, W + W1 = ρg(π 4)d 2Z, Depth of immersion, Z = 4(W + W1)ρgπ d 2 = 4(10 + 2) × 103(1025 × 9.81 × 1.82 × π ) = 0.47 m.

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The centre of buoyancy B will be at the centre of gravity of the displaced water, so that OB = −12 Z = 0.235 m. If the buoy and the load are just in stable equilibrium, the metacentre M must coincide with the centre of gravity G′ of the buoy and load combined. The metacentric height G′M will then be zero and BG′ = BM. From equation (3.14), 1.8 2 I π d 464 BG′ = BM = −− = −−−−−2−−−−− = −−−−−−−−−−−−− = 0.431 m. π d z4 16 × 0.47 V Thus, the position of G′ is given by Z′ = −12 Z + BG′ = 0.235 + 0.431 = 0.666 m. The value of Z1 corresponding to this value of Z′ is found by taking moments about O: W1Z1 + 0.45W = (W + W1)Z′. Maximum height of load above bottom, ( W + W )Z′ – 0.45W Z 1 = −−−−−−−−−−−−−1−−−−−−−−−−−−−−−−−− W1 12 × 10 3 × 0.666 – 0.45 × 10 × 10 3 = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−3−−−−−−−−−−−−−−−−−−− m = 1.746 m. 2 × 10

3.12 PERIODIC TIME OF OSCILLATION The displacement of a stable vessel through an angle θ from its equilibrium position produces a righting moment T which, from equation (3.7), is given by T = W × GM × θ, where W = mg is the weight of the vessel and GM is the metacentric height. This will produce an angular acceleration d2θdt2, and, if I is the mass moment of inertia of the vessel about its axis of rotation, d2 θ T W × GM × θ GM × θ g −−−−2 = −− = – −−−−−−−−−−−−−−−− −− = – −−−−−−−−2−−−−− , dt ( Wg )k 2 k I where k is the radius of gyration from its axis of rotation. The negative sign indicates that the acceleration is in the opposite direction to the displacement. Since this corresponds to simple harmonic motion, Displacement Periodic time, t = 2 π ⎛ −−−−−−−−−−−−−−−−−−−− ⎞ = 2 π ⎝ Acceleration ⎠

= 2π [k 2(GM × g)],

θ −−−−−−−−−−−−−−−−−−−−−−−2− GM × θ × ( gk ) (3.15)

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from which it can be seen that, although a large metacentric height will improve stability, it produces a short periodic time of oscillation, which results in discomfort and excessive stress on the structure of the vessel.

3.13 STABILITY OF A VESSEL CARRYING LIQUID IN TANKS WITH A FREE SURFACE The stability of a vessel carrying liquid in tanks with a free surface (Fig. 3.22) is affected adversely by the movement of the centre of gravity of the liquid in the tanks as the vessel heels. Thus, G1 will move to G′1 and G2 to G′2 . The distance moved is calculated in the same way as the movement BB′ of the centre of buoyancy, given by equation (3.13): G 1 G′1 = θ I1V1 and

G 2 G′2 = θI2 V2 ,

FIGURE 3.22 Vessel carrying liquid in tanks

where I1 and I2 are the second moments of area of the free surfaces, and V1 and V2 the volumes, of the liquid in the tanks. As a result of the movement of G1 and G2, the centre of gravity G of the whole vessel and contents will move to G′. If V is the volume of water displaced by the vessel and ρ is the mass density of water, Weight of vessel and contents = Weight of water displaced = ρgV. If the volume of liquid of density ρ1 in the tanks is V1 and V2, Weight of contents of the first tank

= ρ1 gV1,

Weight of contents of the second tank = ρ1 gV2.

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Taking moments to find the change in the centre of gravity of the vessel and contents,

ρgV × GG′ = ρ1 gV1 × G 1 G′1 + ρ1 gV2 × G 2 G′2 = ρ1 gV1 × θI1V1 + ρ1 gV2 × θI2 V2 , 1 GG′ = −− ( ρ1ρ)θ (I1 + I2 ). V In the tilted position, the new vertical through B′ intersects the original vertical through G at the metacentre M, but the weight W acts through G′ instead of G and its line of action cuts the original vertical at N, reducing the metacentric height from GM to NM. Effective metacentric height, NM = ZB + BM − (ZG + GN), 1 and, since BM = IV and GN = GG′θ = −− ( ρ1ρ)(I1 + I2 ), V 1 NM = ZB − ZG + −− [1 − (ρ1ρ)(I1 + I2)]. V

(3.16)

Thus, the effect of the liquid in the tank is to reduce the effective metacentric height and impair stability, provided that the liquid in the tanks has a free surface so that its centre of gravity moves as the vessel tilts. Lateral subdivision of the tanks improves stability by reducing the sum of the second moments of area I1, I2, etc.

EXAMPLE 3.7

FIGURE 3.23 Barge containing liquids

A barge (Fig. 3.23) has vertical sides and ends and a flat bottom. In plan view it is rectangular, 20 m long by 6 m wide, but with an additional semicircular portion of 3 m radius at one end. The empty barge weighs 200 kN and floats upright in fresh water. The part of the vessel which is rectangular in plan is divided by a wall into two

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compartments 3 m wide by 20 m long. These compartments form open-top tanks which are partly filled with liquid of relative density 0.8 to a depth of 0.8 m in one tank and 1.0 m in the other. The vessel rolls about a horizontal axis, but the flat end remains in a vertical plane. Ignoring the thickness of the material of the barge structure and assuming that the centre of gravity of the barge and contents is 0.45 m above the bottom, find the angle of roll.

Solution In order to be able to determine the angle of roll, we must first find the effective metacentric height from equation (3.16). For the whole vessel lb 3 π b 4 I OO = −−− + −−−−− = 20 × 6312 + π × 64128 m4 = 391.9 m4. 12 128 For each tank ICC = l × ( −12 b)312 = 20 × 3312 = 45 m4. Weight of barge = 200 kN. Weight of liquid load = 0.8 × 103 × 9.81(20 × 3 × 1 + 20 × 3 × 0.8) N = 846 kN. Total weight of barge and contents = 1046 kN. Area of waterline plane of vessel = 20 × 6 + −12 π × 32 = 134.1 m2. Volume of vessel submerged = Weight(Density × g) = 1046 × 103(103 × 9.81) = 106.8 m3. Depth submerged = 106.8134.1 = 0.80 m. Height of centre of buoyancy B above bottom = −12 Depth submerged = 0.4 m. Putting these values in equation (3.16) with ρ1ρ = 0.8, Effective metacentric height, NM = 0.4 − 0.45 + (391.9 − 0.8 × 2 × 45)106.8 = 2.95 m. The overturning moment is caused by the weight of the excess liquid in one tank, P = 0.8 × 103 × 9.81 × 20 × 3(1.0 − 0.8) = 94 kN. The centre of gravity of this excess liquid is 1.5 m from the centreline. Overturning moment due to excess liquid = P × 1.5, Righting moment = W × NM tan θ. Thus, for equilibrium, P × 1.5 = W × NM tan θ, tan θ = 94 × 1.5(1046 × 2.95) = 0.0457, Angle of roll, θ = 2°37′′.

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Problems

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Concluding remarks Arising directly from the hydrostatic equation developed in Chapter 2, this chapter has demonstrated techniques available for determining the forces acting on submerged or partially submerged surfaces. The chapter has also stressed the relationship between pressure and force, and in particular has highlighted the fact that force is a vector quantity, calculated by reference to the applied pressure and the surface area normal to the force direction. This concept, although apparently obvious, will form the basis of later calculations of lift and drag on aerofoils, and the definition of appropriate lift and drag coefficients. The treatment of buoyancy and floating-body stability is a further demonstration of the application of the techniques appropriate to the analysis of solid-body mechanics.

Summary of important equations and concepts 1.

2. 3.

The integration necessary to determine the resultant force acting on a surface is emphasized, Section 3.2, and most importantly the concept of a centre of pressure through which this force acts is introduced, Section 3.3. It will be shown later that centre of pressure movement as flow becomes supersonic can affect wing stability and introduce the need for remedial action, either by control surface activation or by corresponding movement of the aircraft centre of gravity. The treatment of a range of hydrostatic force and moment examples illustrates the interface between hydrostatics and mechanics, Sections 3.4 and 3.5. The treatment of buoyancy and stability of floating bodies continues this linkage.

Problems 3.1 A circular lamina 125 cm in diameter is immersed in water so that the distance of its edge measured vertically below the free surface varies from 60 cm to 150 cm. Find the total force due to the water acting on one side of the lamina, and the vertical distance of the centre of pressure below the surface. [12 639 N, 1.1 m] 3.2 One end of a rectangular tank is 1.5 m wide by 2 m deep. The tank is completely filled with oil of specific weight 9 kN m−3. Find the resultant pressure on this vertical end and the depth of the centre of pressure from the top. [27 kN, 1.33 m]

3.5 A barge in the form of a closed rectangular tank 20 m long by 4 m wide floats in water. If the bottom is 1.5 m below the surface, what is the water force on one long side and at what level below the surface does it act? If a uniform pressure of 50 kN m−2 gauge is applied inside the barge what will be the new magnitude and point of action of the resultant force on the side? The deck is 0.2 m above water level. [220.73 kN, 1.0 m; 1479.27 kN, 0.6 m below surface] 3.6 A rectangular sluice door (Fig. 3.24) is hinged at the top at A and kept closed by a weight fixed to the door.

3.3 What is the position of the centre of pressure of a vertical semicircular plane submerged in a hom*ogeneous liquid with its diameter d at the free surface? [Depth 3π d32] 3.4 A culvert draws off water from the base of a reservoir. The entrance to the culvert is closed by a circular gate 1.25 m in diameter which can be rotated about its horizontal diameter. Show that the turning moment on the gate is independent of the depth of water if the gate is completely immersed and find the value of this moment. [1177 N m]

FIGURE 3.24

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The door is 120 cm wide and 90 cm long and the centre of gravity of the complete door and weight is at G, the combined weight being 9810 N. Find the height of the water h on the inside of the door which will just cause the door to open. [0.88 m] 3.7 A rectangular gate (Fig. 3.25) of negligible thickness, hinged at its top edge and of width b, separates two tanks in which there is the same liquid of density ρ. It is required that the gate shall open when the level in the left-hand tank falls below a distance H from the hinge. The level in the righthand tank remains constant at a height y above the hinge. Derive an expression for the weight of the gate in terms of H, Y, y, b and g. Assume that the weight of the gate acts at its centre of area.

purpose. If the two halves of the container are not secured together, what must be the mass of the upper hemisphere if it just fails to lift off the lower hemisphere? [12.5 kg] 3.11 A sluice gate (Fig. 3.26) consists of a quadrant of a circle of radius 1.5 m pivoted at its centre O. Its centre of gravity is at G as shown. When the water is level with the pivot O, calculate the magnitude and direction of the resultant force on the gate due to the water and the turning moment required to open the gate. The width of the gate is 3 m and it has a mass of 6000 kg.

FIGURE 3.26 [61.6 kN, 57°31′, 35.3 kN m]

FIGURE 3.25

3Y 2 ( y + H ) – H 3 W = 0.77 ρ gb −−−−−−−−−−−−−−−−−−−−−−− Y 3.8 A masonry dam 6 m high has the water level with the top. Assuming that the dam is rectangular in section and 3 m wide, determine whether the dam is stable against overturning and whether tension will develop in the masonry joints. Density of masonry 1760 kg m−3. [Stable, tension on the water face]

3.12 A sector-shaped sluice gate having a radius of curvature of 5.4 m is as shown in Fig. 3.27. The centre of curvature C is 0.9 m vertically below the lower edge A of the gate and 0.6 m vertically above the horizontal axis passing through O about which the gate is constructed to turn. The mass of the gate is 3000 kg per metre run and its centre of gravity is 3.6 m horizontally from the centre O. If the water level is 2.4 m above the lower edge of the gate, find per metre run (a) the resultant force acting on the axis at O, (b) the resultant moment about O.

3.9 A pair of lock gates, each 3 m wide, have their lower hinges at the bottom of the gates and their upper hinges 5 m from the bottom. The width of the lock is 5.5 m. Find the reaction between the gates when the water level is 4.5m above the bottom of one side and 1.5 m on the other. Assuming that this force acts at the same height as the resultant force due to the water pressure find the reaction forces on the hinges. [331 kN; 107.6 kN, 223.4 kN] 3.10 A spherical container is made up of two hemispheres, the joint between the two halves being horizontal. The sphere is completely filled with water through a small hole in the top. It is found that 50 kg of water are required for this

FIGURE 3.27 [(a) 39.2 kN, (b) 106 kN m]

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Problems 3.13 The face of a dam (Fig. 3.28) is curved according to the relation y = x 2 2.4, where y and x are in metres. The height of the free surface above the horizontal plane through A is 15.25 m. Calculate the resultant force F due to the fresh water acting on unit breadth of the dam, and determine the position of the point B at which the line of action of this force cuts the horizontal plane through A.

87

3.16 The shifting of a portion of cargo of mass 25 000 kg through a distance of 6 m at right angles to the vertical plane containing the longitudinal axis of a vessel causes it to heel through an angle of 5°. The displacement of the vessel is 5000 metric tonnes and the value of I is 5840 m4. The density of sea water is 1025 kg m−3. Find (a) the metacentric height and (b) the height of the centre of gravity of the vessel above the centre of buoyancy. [(a) 0.342 m, (b) 0.849 m] 3.17 A buoy floating in sea water of density 1025 kg m−3 is conical in shape with a diameter across the top of 1.2 m and a vertex angle of 60°. Its mass is 300 kg and its centre of gravity is 750 mm from the vertex. A flashing beacon is to be fitted to the top of the buoy. If this unit has a mass of 55 kg what is the maximum height of its centre of gravity above the top of the buoy if the whole assembly is not to be unstable? (The centre of volume of a cone of height h is at a distance −34 h from the vertex.) [1.25 m]

FIGURE 3.28 [1290 kN m−1, 14.15 m] 3.14 A steel pipeline conveying gas has an internal diamter of 120 cm and an external diameter of 125 cm. It is laid across the bed of a river, completely immersed in water and is anchored at intervals of 3 m along its length. Calculate the buoyancy force in newtons per metre and the upward force in newtons on each anchorage. Density of steel = 7900 kg m−3, density of water = 1000 kg m−3. [12 037 N m−1, 13 742 N] 3.15 The ball-operated valve shown in Fig. 3.29 controls the flow from a tank through a pipe to a lower tank, in which it is situated. The water level in the upper tank is 7 m above the 10 mm diameter valve opening. Calculate the volume of the ball which must be submerged to keep the valve closed.

3.18 A rectangular pontoon 10 m by 4 m in plan weighs 280 kN. A steel tube weighing 34 kN is placed longitudinally on the deck. When the tube is in a central position, the centre of gravity for the combined weight lies on the vertical axis of symmetry 250 mm above the water surface. Find (a) the metacentric height, (b) the maximum distance the tube may be rolled laterally across the deck if the angle of heel is not to exceed 5°. [(a) 1.02 m, (b) 0.82 m] 3.19 A rectangular tank 90 cm long and 60 cm wide is mounted on bearings so that it is free to turn on a longitudinal axis. The tank has a mass of 68 kg and its centre of gravity is 15 cm above the bottom. When the tank is slowly filled with water it hangs in stable equilibrium until the depth of water is 45 cm after which it becomes unstable. How far is the axis of the bearings above the bottom of the tank? [0.21 m] 3.20 A cylindrical buoy 1.35 m in diameter and 1.8 m high has a mass of 770 kg. Show that it will not float with its axis vertical in sea water of density 1025 kg m−3. If one end of a vertical chain is fastened to the base, find the pull required to keep the buoy vertical. The centre of gravity of the buoy is 0.9 m from its base. [GM = −0.42 m, 4632 N] 3.21 A solid cylinder 1 m in diameter and 0.8 m high is of uniform relative density 0.85. Calculate the periodic time of small oscillations when the cylinder floats with its axis vertical in still water. [2.90 s]

FIGURE 3.29 [110 cm3]

3.22 A ship has displacement of 5000 metric tonnes. The second moment of area of the waterline section about a fore and aft axis is 12 000 m4 and the centre of buoyancy is 2 m below the centre of gravity. The radius of gyration is 3.7 m. Calculate the period of oscillation. Sea water has a density of 1025 kg m−3. [10.94 s]

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Part II

Concepts of Fluid Flow 4 Motion of Fluid Particles and Streams 90

6 The Energy Equation and its Applications 166

5 The Momentum Equation and its Applications 112

7 Two-dimensional Ideal Flow 212

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The study of fluid motion is complicated by the introduction of viscosity-dependent shear forces that were absent in the preceding treatment of stationary fluids. In the majority of flow cases analysis relies upon a body of empirical work, supported by the concepts of dimensional analysis and similarity. In this part of the text we will establish the analytical techniques that will later be combined with the empirical representation of frictional forces to allow the study of ‘real’ fluid behaviour. In order to deal effectively with flowing fluids it is first necessary to identify flow categories, defined in predominantly mathematical terms, that will allow the appropriate analysis to be undertaken by identifying suitable and acceptable simplifications. Examples of the categories to be introduced include variation of the flow parameters with time (steady or unsteady) or variations along the flow path (uniform or nonuniform). Similarly, compressibility effects may be important in high-speed gas flows but may be ignored in many liquid flow situations. In parallel to setting up these flow categories it is also necessary to develop a series of mathematically expressed principles that will allow the variations in flow parameters as a result of the motion of the fluid to be predicted. The principles of continuity, energy and momentum are developed in this part of the text.

The steady flow energy equation is introduced and will be utilized later to describe the behaviour of real fluids by the inclusion of an empirically based friction term. The momentum equation will be introduced and its application illustrated both for fluid-to-solid boundary transfers, such as the calculation of forces acting on moving vanes or pipe nozzles, and for other flow situations, such as the formation of hydraulic jumps in open-channel flows. While the treatment of the behaviour of real fluid motion requires the introduction of viscous and, possibly, compressibility terms, the study of an ideal fluid freed from these constraints is useful and important, particularly in the consideration of flow patterns away from the influence of solid boundaries. Primarily a mathematical modelling tool, the study of ideal fluid flow has its roots in the work of eighteenthcentury hydrodynamicists and has applications now in aerodynamics as it allows the introduction of a further flow classification, namely rotational or irrotational flow. The study of ideal flow allows flow patterns around aerofoil sections to be considered and therefore naturally leads to considerations of lift and vorticity. Taken together with Part I, this portion of the text provides the foundation upon which the study and application of the behaviour of real fluids may be based.

Opposite: Offshore wind turbines, photo courtesy of the British Wind Energy Association, © NEG Micon

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Chapter 4

Motion of Fluid Particles and Streams 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Fluid flow Uniform flow and steady flow Frames of reference Real and ideal fluids Compressible and incompressible flow One-, two- and three-dimensional flow Analyzing fluid flow

Motion of a fluid particle Acceleration of a fluid particle Laminar and turbulent flow Discharge and mean velocity Continuity of flow Continuity equations for threedimensional flow using Cartesian coordinates 4.14 Continuity equation for cylindrical coordinates 4.8 4.9 4.10 4.11 4.12 4.13

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The prediction of the conditions encountered by, and as a result of, fluids in motion presents a range of problems that must be resolved by reference to the fundamental laws of physics, coupled with the particular fluid properties identified in Chapter 1. The treatment presented in this chapter will lay the foundations for later analysis in that the various fluid flow regimes, whether time dependent or determined by the shear forces assumed to act on the boundaries of the fluid flow or the compressibility of the fluid,

will be identified, including the Reynolds numberdependent laminar and turbulent flow regimes. The presence of velocity profiles within any fluid flow will be emphasized, together with the importance of fluid viscosity in determining the detail conditions within the fluid flow. The application of the conservation of mass relationship across a control volume defined within a flowing fluid will be introduced, and the relationship linking mass flow to local or mean flow velocity values will be detailed. l l l

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4.1

FLUID FLOW

The motion of a fluid is usually extremely complex. The study of a fluid at rest, or in relative equilibrium, was simplified by the absence of shear forces, but when a fluid flows over a solid surface or other boundary, whether stationary or moving, the velocity of the fluid in contact with the boundary must be the same as that of the boundary, and a velocity gradient is created at right angles to the boundary (see Section 1.2). The resulting change of velocity from layer to layer of fluid flowing parallel to the boundary gives rise to shear stresses in the fluid. Individual particles of fluid move as a result of the action of forces set up by differences of pressure or elevation. Their motion is controlled by their inertia and the effect of the shear stresses exerted by the surrounding fluid. The resulting motion is not easily analysed mathematically, and it is often necessary to supplement theory by experiment. If an individual particle of fluid is coloured, or otherwise rendered visible, it will describe a pathline, which is the trace showing the position at successive intervals of time of a particle which started from a given point. If, instead of colouring an individual particle, the flow pattern is made visible by injecting a stream of dye into a liquid, or smoke into a gas, the result will be a streakline or filament line, which gives an instantaneous picture of the positions of all the particles which have passed through the particular point at which the dye is being injected. Since the flow pattern may vary from moment to moment, a streakline will not necessarily be the same as a pathline. When using tracers or dyes it is essential to choose a material having a density and other physical properties as similar as possible to those of the fluid being studied. In analysing fluid flow, we also make use of the idea of a streamline, which is an imaginary curve in the fluid across which, at a given instant, there is no flow. Thus, the velocity of every particle of fluid along the streamline is tangential to it at that moment. Since there can be no flow through solid boundaries, these can also be regarded as streamlines. For a continuous stream of fluid, streamlines will be continuous lines extending to infinity upstream and downstream, or will form closed curves as, for example, round the surface of a solid object immersed in the flow. If conditions are steady and the flow pattern does not change from moment to moment, pathlines and streamlines will be identical; if the flow is fluctuating this will not be the case. If a series of streamlines are drawn through every point on the perimeter of a small area of the stream cross-section, they will form a streamtube (Fig. 4.1). Since there is no flow across a streamline, the fluid inside the streamtube cannot escape through its walls, and behaves as if it were contained in an imaginary pipe. This is a useful concept in dealing with the flow of a large body of fluid, since it allows elements of the fluid to be isolated for analysis.

FIGURE 4.1 A streamtube

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4.3

4.2

Frames of reference

93

UNIFORM FLOW AND STEADY FLOW

Conditions in a body of fluid can vary from point to point and, at any given point, can vary from one moment of time to the next. Flow is described as uniform if the velocity at a given instant is the same in magnitude and direction at every point in the fluid. If, at the given instant, the velocity changes from point to point, the flow is described as non-uniform. In practice, when a fluid flows past a solid boundary there will be variations of velocity in the region close to the boundary. However, if the size and shape of the cross-section of the stream of fluid are constant, the flow is considered to be uniform. A steady flow is one in which the velocity, pressure and cross-section of the stream may vary from point to point but do not change with time. If, at a given point, conditions do change with time, the flow is described as unsteady. In practice, there will always be slight variations of velocity and pressure, but, if the average values are constant, the flow is considered to be steady. There are, therefore, four possible types of flow:

1.

2.

3.

4.

Steady uniform flow. Conditions do not change with position or time. The velocity and cross-sectional area of the stream of fluid are the same at each cross-section: for example, flow of a liquid through a pipe of uniform bore running completely full at constant velocity. Steady non-uniform flow. Conditions change from point to point but not with time. The velocity and cross-sectional area of the stream may vary from cross-section to cross-section, but, for each cross-section, they will not vary with time: for example, flow of a liquid at a constant rate through a tapering pipe running completely full. Unsteady uniform flow. At a given instant of time the velocity at every point is the same, but this velocity will change with time: for example, accelerating flow of a liquid through a pipe of uniform bore running full, such as would occur when a pump is started up. Unsteady non-uniform flow. The cross-sectional area and velocity vary from point to point and also change with time: for example, a wave travelling along a channel.

4.3

FRAMES OF REFERENCE

Whether a given flow is described as steady or unsteady will depend upon the situation of the observer, since motion is relative and can only be described in terms of some frame of reference which is determined by the observer. If a wave travels along a channel, then to an observer on the bank the flow in the channel will appear to vary with time, and, therefore, be unsteady. If, however, the observer were travelling on the crest of the wave, conditions would not appear to the observer to change with time, and the flow would be steady according to the observer’s frame of reference.

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The frame of reference adopted for describing the motion of a fluid is usually a set of fixed coordinate axes, but the analysis of steady flow is usually simpler than that of unsteady flow and it is sometimes useful to use moving coordinate axes to convert an unsteady flow problem to a steady flow problem. The normal laws of mechanics will still apply, provided that the movement of the coordinate axes takes place with uniform velocity in a straight line.

4.4

REAL AND IDEAL FLUIDS

When a real fluid flows past a boundary, the fluid immediately in contact with the boundary will have the same velocity as the boundary. As explained in Section 4.1, the velocity of successive layers of fluid will increase as we move away from the boundary. If the stream of fluid is imagined to be of infinite width perpendicular to the boundary, a point will be reached beyond which the velocity will approximate to the free stream velocity, and the drag exerted by the boundary will have no effect. The part of the flow adjoining the boundary in which this change of velocity occurs is known as the boundary layer. In this region, shear stresses are developed between layers of fluid moving with different velocities as a result of viscosity and the interchange of momentum due to turbulence causing particles of fluid to move from one layer to another. The thickness of the boundary layer is defined as the distance from the boundary at which the velocity becomes equal to 99 per cent of the free stream velocity. Outside this boundary layer, in a real fluid, the effect of the shear stresses due to the boundary can be ignored and the fluid can be treated as if it were an ideal fluid, which is assumed to have no viscosity and in which there are no shear stresses. If the fluid velocity is high and its velocity low, the boundary layer is comparatively thin, and the assumption that a real fluid can be treated as an ideal fluid greatly simplifies the analysis of the flow and still leads to useful results. Even in problems in which the effects of viscosity and turbulence cannot be neglected, it is often convenient to carry out the mathematical analysis assuming an ideal fluid. An experimental investigation can then be made to correct the theoretical analysis for the factors omitted and to bring the results obtained into agreement with the behaviour of a real fluid.

4.5

COMPRESSIBLE AND INCOMPRESSIBLE FLOW

All fluids are compressible, so that their density will change with pressure, but, under steady flow conditions and provided that the changes of density are small, it is often possible to simplify the analysis of a problem by assuming that the fluid is incompressible and of constant density. Since liquids are relatively difficult to compress, it is usual to treat them as if they were incompressible for all cases of steady flow. However, in unsteady flow conditions, high pressure differences can develop (see Chapter 20) and the compressibility of liquids must be taken into account. Gases are easily compressed and, except when changes of pressure and, therefore, density are very small, the effects of compressibility and changes of internal energy must be taken into account.

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4.6

4.6

One-, two- and three-dimensional flow

95

ONE-, TWO- AND THREE-DIMENSIONAL FLOW

Although, in general, all fluid flow occurs in three dimensions, so that velocity, pressure and other factors vary with reference to three orthogonal axes, in some problems the major changes occur in two directions or even in only one direction. Changes along the other axis or axes can, in such cases, be ignored without introducing major errors, thus simplifying the analysis. Flow is described as one-dimensional if the factors, or parameters, such as velocity, pressure and elevation, describing the flow at a given instant, vary only along the direction of flow and not across the cross-section at any point. If the flow is unsteady, these parameters may vary with time. The one dimension is taken as the distance along the central streamline of the flow, even though this may be a curve in space, and the values of velocity, pressure and elevation at each point along this streamline will be the average values across a section normal to the streamline. A one-dimensional treatment can be applied, for example, to the flow through a pipe, but, since in a real fluid the velocity at any cross-section will vary from zero at the pipe wall (Fig. 4.2) to a maximum at the centre, some correction will be necessary to compensate for this (see Chapter 10) if a high degree of accuracy is required. In two-dimensional flow it is assumed that the flow parameters may vary in the direction of flow and in one direction at right angles, so that the streamlines are curves lying in a plane and are identical in all planes parallel to this plane. Thus, the flow over a weir of constant cross-section (Fig. 4.3) and infinite width perpendicular to the plane of the diagram can be treated as two-dimensional. A real weir has a limited width, but it can be treated as two-dimensional over its whole width and then an end correction can be introduced to modify the result to allow for the effect of the disturbance produced by the end walls (see Chapter 16).

FIGURE 4.2 Velocity profiles for one-dimensional flow

FIGURE 4.3 Two-dimensional flow

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A special case of two-dimensional flow occurs when the cross-section of the flow is circular and the flow parameters vary symmetrically about the axis. For example, ideally the velocity distribution in a circular pipe will be the same across any diameter, the velocity varying from zero at the wall to a maximum at the centre. Referred to orthogonal coordinate axes (x in the direction of motion, y and z in the plane of the cross-section) the flow is three-dimensional, but, since it is axisymmetric, it can be reduced to two-dimensional flow by using a system of cylindrical coordinates (x in the direction of flow and r the radius defining the position in the cross-section).

4.7

ANALYZING FLUID FLOW

One difficulty encountered in deciding how to investigate the flow of a fluid is that, in the majority of problems, we are dealing with an endless stream of fluid. We have to decide what part of this stream shall constitute the element or system to be studied and what shall be regarded as the surroundings which act upon this system. There are two main alternatives: 1.

We can study the behaviour of a specific element of the fluid of fixed mass. Such an element constitutes a closed system. Its boundaries are a closed surface which may vary with time, but always contain the same mass of fluid. At any instant, a free body diagram can be drawn showing the forces exerted by the surrounding fluid and any solid boundaries on this element. We can define the system to be studied as a fixed region in space, or in relation to some frame of reference, known as a control volume, through which the fluid flows, forming, in effect, an open system. The boundary of this system is its control surface and its shape does not change with time. The control volume for a particular problem is chosen arbitrarily for reasons of convenience of analysis. However, the control surface will usually follow solid boundaries where these are present, and where it cuts the flow direction it will do so at right angles. Where there are no solid boundaries the control volume may form a streamtube.

2.

4.8

MOTION OF A FLUID PARTICLE

Any particle or element of fluid will obey the normal laws of mechanics in the same way as a solid body. When a force is applied, its behaviour can be predicted from Newton’s laws, which state:

1. 2. 3.

A body will remain at rest or in a state of uniform motion in a straight line until acted upon by an external force. The rate of change of momentum of a body is proportional to the force applied and takes place in the direction of action of that force. Action and reaction are equal and opposite.

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4.8

Motion of a fluid particle

97

Since momentum is the product of mass and velocity, for an element of fixed mass Newton’s second law relates the change of velocity occurring in a given time (i.e. the acceleration) to the applied force. Working in a coherent system of units, such as SI, the proportionality becomes an equality and Newton’s second law can be written Change of velocity Force = Mass × −−−−−−−−−−−−−−−−−−−−−−−−−−−− Time = Mass × Acceleration. The relationships between the acceleration a, initial velocity v1, final velocity v2 and the distance moved s in time t are given by the equations of motion: v2 = v1 + at, s = v1t + −12 at2, v 22 = v 21 + 2as. In any body of flowing fluid, the velocity at a given instant will generally vary from point to point over any specified region, and if the flow is unsteady the velocity at each point may vary with time. In this field of flow, at any given time, a particle at point A will have a different velocity from that of a particle at point B. The velocities at A and B may also change with time. Thus the change of velocity δ v, which occurs when a particle moves from A to B through a distance δ s in time δ t, is given by Difference of Change of Total change velocity between velocity at = + of velocity A and B at the B occurring given instant in time δ t.

(4.1)

The velocity v depends on both distance s and time t. The rate of change of velocity with position at a given time is, therefore, expressed by the partial differential ∂ v∂ s, and the rate of change of velocity with time at a given point is expressed by the partial differential ∂ v∂ t. Since A and B are a distance δs apart,

∂v Difference of velocity between A and B at the given instant = −−− (δ s ). ∂s Also,

∂v Change of velocity at B in time t = −−− (δ t ). ∂t Thus, in symbols, equation (4.1) is

∂v ∂v dv = −−− (δ s ) + −−− (δ t ). ∂s ∂t

(4.2)

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4.9

ACCELERATION OF A FLUID PARTICLE

The forces acting on a particle are related to the resultant acceleration δ vδ t of the particle by Newton’s second law. From equation (4.2) in the limit at δ t → 0, dv ∂ v ds ∂ v Acceleration in the direction of flow, a = −−− = −−− −−− + −−−. dt ∂ s dt ∂ t To denote that the derivative dvdt is obtained by following the motion of a single particle, it is written DvDt, and since dsdt = v, Dv ∂v ∂v a = −−−− = v −−− + −−−. Dt ∂s ∂t

(4.3)

The derivative DDt is known as the substantive derivative. The total acceleration, known as the substantive acceleration, is composed of two parts, as shown in equation (4.3): 1.

2.

the convective acceleration v(∂ v∂s) due to the movement of the particle from one point to another point at which the velocity at the given instant is different; the local or temporal acceleration ∂ v∂ t due to the change of velocity at every point with time.

For steady flow, ∂ v∂t = 0, while for uniform flow, ∂ v∂s = 0. We have so far assumed that the particle is accelerating in a straight line, but, if it is moving in a curved path, its velocity will be changing in direction and consequently there will be an acceleration perpendicular to its path, whether the velocity v is changing in magnitude or not. Figure 4.4 shows a particle moving from A to B along a curved path of length δ s subtending a small angle δθ at the centre of curvature. The change of velocity δ vn will be perpendicular to the direction of motion and, from the velocity diagram,

δ vn = vδθ = vδ sR.

FIGURE 4.4 Change of velocity for a circular path

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4.9

Acceleration of a fluid particle

99

Dividing by δ t, the time in which the change occurs, in the limit the acceleration perpendicular to the direction of motion is dv v ds a n = −−−−n = −− −−− dt R dt or, since ds −−− = v, dt an = v 2R. This is the convective term, and, if v has a component vn towards the instantaneous centre of curvature, there will be a temporal term ∂ vn ∂ t so that the substantive derivative is v2 ∂ vn a n = −− + −−−−. R ∂t In general, the motion of a fluid particle is three-dimensional and its velocity and acceleration can be expressed in terms of three mutually perpendicular components. Thus, if vx, vy and vz are the components of the velocity in the x, y and z directions, respectively, and ax, ay and az the corresponding components of acceleration, the velocity field is described by vx = vx(x, y, z, t), vy = vy(x, y, z, t), vz = vz(x, y, z, t), and the velocity v at any point is given by v = vxi + vyj + vzk, where i, j and k are the unit vectors in the x, y and z directions. The change of the component velocities in each direction as a particle moves in a fluid can now be calculated. Thus, in the x direction,

∂v ∂v ∂v ∂v δ v x = −−−−x (δ x ) + −−−−x (δ y ) + −−−−x (δ z ) + −−−−x (δ t ), ∂x ∂y ∂z ∂t and the acceleration in the x direction, in the limit as δ t → 0, will be

∂ v dx ∂ v dy ∂ v dz ∂ v Dv a x = −−−−−x = −−−−x −−− + −−−−x −−− + −−−−x −−− + −−−−x Dt ∂ x dt ∂ y dt ∂ z dt ∂ t or, since dxdt = vx, dydt = vy , dzdt = vz,

∂v ∂v ∂v ∂ vx Dv a x = −−−−−x = v x −−−−x + v y −−−−x + v z −−−−x + −−−−. Dt ∂x ∂y ∂z ∂t

(4.4)

Similarly,

∂v ∂v ∂v ∂v Dv a y = −−−−−y = v x −−−−y + vy −−−−y + vz −−−−y + −−−−y , Dt ∂x ∂y ∂z ∂t

(4.5)

∂v ∂v ∂ v ∂ vz Dv a z = −−−−−z = v x −−−−z + vy −−−−z + vz −−−−z + −−−−. Dt ∂x ∂y ∂z ∂t

(4.6)

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The first three terms in each of equations (4.4) to (4.6) represent the convective acceleration and the final term the local or temporal acceleration.

4.10 LAMINAR AND TURBULENT FLOW Observation shows that two entirely different types of fluid flow exist. This was demonstrated by Osborne Reynolds in 1883 through an experiment in which water was discharged from a tank through a glass tube (Fig. 4.5). The rate of flow could be controlled by a valve at the outlet, and a fine filament of dye injected at the entrance to the tube. At low velocities, it was found that the dye filament remained intact throughout the length of the tube, showing that the particles of water moved in parallel lines. This type of flow is known as laminar, viscous or streamline, the particles of fluid moving in an orderly manner and retaining the same relative positions in successive cross-sections.

FIGURE 4.5 Reynolds’ apparatus

As the velocity in the tube was increased by opening the outlet valve, a point was eventually reached at which the dye filament at first began to oscillate and then broke up so that the colour was diffused over the whole cross-section, showing that the particles of fluid no longer moved in an orderly manner but occupied different relative positions in successive cross-sections. This type of flow is known as turbulent and is characterized by continuous small fluctuations in the magnitude and direction of the velocity of the fluid particles, which are accompanied by corresponding small fluctuations of pressure. When the motion of a fluid particle in a stream is disturbed, its inertia will tend to carry it on in the new direction, but the viscous forces due to the surrounding fluid will tend to make it conform to the motion of the rest of the stream. In viscous flow, the viscous shear stresses are sufficient to eliminate the effects of any deviation, but in turbulent flow they are inadequate. The criterion which determines whether flow will be viscous or turbulent is therefore the ratio of the inertial force to the viscous force acting on the particle. Suppose l is a characteristic length in the system under consideration, e.g. the diameter of a pipe or the chord of an aerofoil, and t is a typical time; then lengths, areas, velocities and accelerations can all be expressed in terms of l and t. For a small element of fluid of mass density ρ,

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Laminar and turbulent flow

101

Volume of element = k1l 3, Mass of element = k1ρl 3, Velocity of element, v = k2lt, Acceleration of element = k3lt2, where k1, k2 and k3 are constants. By Newton’s second law, Inertial force = Mass × Acceleration = k1ρl 3 × k3lt2 = k1k3ρl 2(lt)2 = (k1k3 k 22 )ρl 2v 2. Similarly, Viscous force = Viscous shear stress × Area on which stress acts. From Newton’s law of viscosity (equation (1.2)), Viscous shear stress = µ × Velocity gradient = µ(vk4l), where µ = coefficient of dynamic viscosity. Area on which shear stress acts = k5l 2. Therefore, Viscous force = µ(vk4l) × k5l 2 = (k5 k4)µ vl. The ratio k k k ρ l 2 v2 Inertial force ρ vl −−−−−−−−−−−−−−−−−−−− = −−1−−2−3−−−4 −−−−−−− = constant × −−−− . Viscous force µ k 2 k5 µ vl Thus, the criterion which determines whether flow is viscous or turbulent is the quantity ρ vlµ, known as the Reynolds number. It is a ratio of forces and, therefore, a pure number and may also be written as vlν, where ν is the kinematic viscosity (ν = µρ). Experiments carried out with a number of different fluids in straight pipes of different diameters have established that if the Reynolds number is calculated by making l equal to the pipe diameter and using the mean velocity C (Section 4.11), then, below a critical value of ρ Cdµ = 2000, flow will normally be laminar (viscous), any tendency to turbulence being damped out by viscous friction. This value of the Reynolds number applies only to flow in pipes, but critical values of the Reynolds number can be established for other types of flow, choosing a suitable characteristic length such as the chord of an aerofoil in place of the pipe diameter. For a given fluid flowing in a pipe of a given diameter, there will be a critical velocity of flow Cc corresponding to the critical value of the Reynolds number, below which flow will be viscous.

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In pipes, at values of the Reynolds number above 2000, flow will not necessarily be turbulent. Laminar flow has been maintained up to Re = 50 000, but conditions are unstable and any disturbance will cause reversion to normal turbulent flow. In straight pipes of constant diameter, flow can be assumed to be turbulent if the Reynolds number exceeds 4000.

4.11 DISCHARGE AND MEAN VELOCITY The total quantity of fluid flowing in unit time past any particular cross-section of a stream is called the discharge or flow at that section. It can be measured either in terms of mass, in which case it is referred to as the mass rate of flow A and measured in units such as kilograms per second, or in terms of volume, when it is known as the volume rate of flow Q, measured in units such as cubic metres per second. In an ideal fluid, in which there is no friction, the velocity u of the fluid would be the same at every point of the cross-section (Fig. 4.2). In unit time, a prism of fluid would pass the given cross-section and, if the cross-sectional area normal to the direction of flow is A, the volume passing would be Au. Thus Q = Au. In a real fluid, the velocity adjacent to a solid boundary will be zero or, more accurately, equal to the wall velocity in the flow direction, a condition known as ‘no slip’, which will be true as long as the flow does not separate from the wall. For a pipe, the velocity profile would be as shown in Fig. 4.6(a) for laminar flow and Fig. 4.6(b) for turbulent flow.

FIGURE 4.6 Calculation of discharge for a circular section, note ‘no slip’ at wall

If u is the velocity at any radius r, the flow δ Q through an annular element of radius r and thickness δ r will be

δQ = Area of element × Velocity = 2π rδ r × u, and, hence,

ur dr. R

Q = 2π

(4.7)

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4.11

Discharge and mean velocity

103

If the relation between u and r can be established, this integral can be evaluated or the integration may be undertaken numerically, see Section 6.8, program VOLFLO. In many problems, the variation of velocity over the cross-section can be ignored, the velocity being assumed to be constant and equal to the mean velocity B, defined as volume rate of discharge Q divided by the area of cross-section A normal to the stream: Mean velocity, B = QA.

EXAMPLE 4.1

Air flows between two parallel plates 80 mm apart. The following velocities were determined by direct measurement. Distance from one plate (mm) Velocity (m s−1)

0 10 20 30 40 50 60 70 80 0 23 28 31 32 29 22 14 0

Plot the velocity distribution curve and calculate the mean velocity.

Solution Figure 4.7 shows the velocity distribution curve. The area enclosed by the curve represents the product of velocity and distance, and since the two plates are parallel Discharge per unit width Mean velocity, B = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−. Distance between plates FIGURE 4.7 Velocity distribution curve

The area under the graph may be determined by the mid-ordinate method, taking values from Fig. 4.7, B = (∑Mid-ordinates8) = (17.5 + 26.0 + 29.6 + 31.9 + 30.7 + 25.4 + 18.1 + 7.7)8 = 23.36 m s−1.

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4.12 CONTINUITY OF FLOW Except in nuclear processes, matter is neither created nor destroyed. This principle of conservation of mass can be applied to a flowing fluid. Considering any fixed region in the flow (Fig. 4.8) constituting a control volume, Increase of mass Mass of fluid Mass of fluid entering of fluid in the = leaving per unit + per unit time control volume time per unit time. FIGURE 4.8 Continuity of flow

For steady flow, the mass of fluid in the control volume remains constant and the relation reduces to Mass of fluid entering Mass of fluid leaving = per unit time per unit time. Applying this principle to steady flow in a streamtube (Fig. 4.9) having a crosssectional area small enough for the velocity to be considered as constant over any given cross-section, for the region between sections 1 and 2, since there can be no flow through the walls of a streamtube, Mass entering per unit time = Mass leaving per unit time at section 1 at section 2. Suppose that at section 1 the area of the streamtube is δA1, the velocity of the fluid u1 and its density ρ1, while at section 2 the corresponding values are δA2, u2 and ρ2; then Mass entering per unit time at 1 = ρ1δA1u1, Mass leaving per unit time at 2 = ρ2δA2u2. FIGURE 4.9 Continuous flow through a streamtube

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4.12

Continuity of flow

105

Then, for steady flow,

ρ1δA1u1 = ρ2δA2u2 = Constant.

(4.8)

This is the equation of continuity for the flow of a compressible fluid through a streamtube, u1 and u2 being the velocities measured at right angles to the crosssectional areas δA1 and δA2. For the flow of a real fluid through a pipe or other conduit, the velocity will vary from wall to wall. However, using the mean velocity B, the equation of continuity for steady flow can be written as

ρ1A1B1 = ρ2A2B2 = A,

(4.9)

where A1 and A2 are the total cross-sectional areas and A is the mass rate of flow. If the fluid can be considered as incompressible, so that ρ1 = ρ2, equation (4.9) reduces to A1B1 = A2B2 = Q.

(4.10)

The continuity of flow equation is one of the major tools of fluid mechanics, providing a means of calculating velocities at different points in a system. The continuity equation can also be applied to determine the relation between the flows into and out of a junction. In Fig. 4.10, for steady conditions, Total inflow to junction = Total outflow from junction,

ρ1Q1 = ρ2Q2 + ρ3Q3. FIGURE 4.10 Applications of the continuity equation

For an incompressible fluid, ρ 1 = ρ 2 = ρ 3 so that Q1 = Q2 + Q3 or

A1C1 = A2C2 + A3C3.

In general, if we consider flow towards the junction as positive and flow away from the junction as negative, then for steady flow at any junction the algebraic sum of all the mass flows must be zero: ∑ ρQ = 0.

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EXAMPLE 4.2

Water flows from A to D and E through the series pipeline shown in Fig. 4.11. Given the pipe diameters, velocities and flow rates below, complete the tabular data for this system.

FIGURE 4.11 Relations between discharge, diameter and velocity

Pipe AB BC CD DE

Diameter (mm) d1 = 50 d2 = 75 d3 = ? d4 = 30

Flow rate (m3 s−1)

Velocity (m s−1)

Q1 = ? Q2 = ? Q3 = 2Q4 Q4 = 0.5Q3

C1 = ? C2 = 2.0 C3 = 1.5 C4 = ?

Solution Adding area A = (227)d 24 to the data table and noting that Q = AC and that Q1 = Q2 = (Q 3 + Q 4) = 1.5Q3 allows the table to be completed as (additions in bold),

Diameter (mm)

Area (m2)

Flow rate (m3 s−1)

Velocity (m s−1)

d 1 = 50

1.9643 × 10−3

Q1 = Q2 = 8.839 × 10−3

C1 = C2 A2A1 = 2.0 × 4.41961.9643 = 4.27

d 2 = 75

4.4196 × 10−3

Q2 = 2.0 × 4.4196 × 10−3 = 8.839 × 10−3

C2 = 2.0

Q3 = Q21.5 = 5.893 × 10−3

C3 = 1.5

Q4 = 0.5Q3 = 0.5 × 5.893 × 10−3 = 2.947 × 10−3

C4 = Q 4A4 = 2.9470.7071 = 4.17

d 3 = [Q3(v3 π4)] 0.5 = (5.893 × 10 −31.5 × 0.786)0.5 = 0.707 d 4 = 30

0.707 × 10 −3

(The calculation route is as follows: calculate areas where possible and then Q2 and hence Q1 and C1. From Q2 calculate Q 3 and Q 4 and hence C4. Calculate d 3 from Q 3 and C3.)

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4.13

Continuity equations for three-dimensional flow using Cartesian coordinates

107

4.13 CONTINUITY EQUATIONS FOR THREE-DIMENSIONAL FLOW USING CARTESIAN COORDINATES The control volume ABCDEFGH in Fig. 4.12 is taken in the form of a small rectangular prism with sides δx, δy and δ z in the x, y and z directions, respectively. The mean values of the component velocities in these directions are vx, vy and vz. Considering flow in the x direction, Mass inflow through ABCD in unit time = ρ vxδyδz.

FIGURE 4.12 Continuity in three dimensions

In the general case, both mass density ρ and velocity vx will change in the x direction and so

∂ Mass outflow through EFGH in unit time = ρ vx + −−− ( ρ v x ) δ x δ y δ z. ∂x Thus,

∂ Net outflow in unit time in x direction = −−− ( ρ vx ) δ x δ y δ z. ∂x Similarly,

∂ Net outflow in unit time in y direction = −−− ( ρ vy ) δ x δ y δ z, ∂y ∂ Net outflow in unit time in z direction = −−− ( ρ vz ) δ x δ y δ z. ∂z Therefore,

∂ ∂ ∂ Total net outflow in unit time = −−− ( ρ vx ) + −−− ( ρ vy ) + −−− ( ρ vz ) δ x δ y δ z. ∂x ∂y ∂z

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Also, since ∂ρ∂t is the change in mass density per unit time,

∂ρ Change of mass in control volume in unit time = − −−− δ x δ y δ z ∂t (the negative sign indicating that a net outflow has been assumed). Then, Total net outflow in unit time = change of mass in control volume in unit time

∂ ∂ ∂ ∂ρ −−− ( ρ vx ) + −−− ( ρ vy ) + −−− ( ρ vz ) δ x δ y δ z = – −−− δ x δ y δ z ∂x ∂y ∂z ∂t or

∂ ∂ ∂ ∂ρ −−− ( ρ v x ) + −−− ( ρ v y ) + −−− ( ρ v z ) = – −−−. ∂x ∂y ∂z ∂t

(4.11)

Equation (4.11) holds for every point in a fluid flow whether steady or unsteady, compressible or incompressible. However, for incompressible flow, the density ρ is constant and the equation simplifies to

∂ vx ∂ vy ∂ vz −−−− + −−−− + −−−− = 0. ∂x ∂y ∂ z

(4.12)

For two-dimensional incompressible flow this will simplify still further to

∂ vx ∂ vy −−−− + −−−− = 0. ∂x ∂y

EXAMPLE 4.3

(4.13)

The velocity distribution for the flow of an incompressible fluid is given by vx = 3 − x, vy = 4 + 2y, vz = 2 − z. Show that this satisfies the requirements of the continuity equation.

Solution For three-dimensional flow of an incompressible fluid, the continuity equation simplifies to equation (4.12):

∂ vx −−−− = –1, ∂x

∂v −−−−y = +2, ∂y

∂ vz −−−− = – 1, ∂z

and, hence,

∂ vx ∂ vy ∂ vz −−−− + −−−− + −−−− = – 1 + 2 – 1 = 0, ∂x ∂ y ∂ z which satisfies the requirement for continuity.

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Concluding remarks

109

4.14 CONTINUITY EQUATION FOR CYLINDRICAL COORDINATES The form of the continuity equation for a system of cylindrical coordinates r, θ and z, in which r and θ are measured in a plane corresponding to the x−y plane for Cartesian coordinates, can be found by using the relations between polar and Cartesian coordinates: x 2 + y 2 = r 2, ( yx) = tan θ, vx = vr cos θ − vθ sin θ, vy = vr sin θ + vθ cos θ,

∂ ∂ ∂ r ∂ ∂θ −−− = −−− −−− + −−− −−−, ∂x ∂ r ∂x ∂θ ∂x

∂ ∂ ∂ r ∂ ∂θ −−− = −−− −−− + −−− −−− . ∂ y ∂ r ∂y ∂θ ∂y

This results in equation (4.12) becoming 1 ∂ v ∂v 1 ∂ − −−− ( rv r ) + − −−−−θ + −−−−z = 0. r ∂θ ∂ z r ∂r

(4.14)

In the case of two-dimensional flow, this can be simplified further. Putting ∂ vz ∂ z = 0 and writing

∂v ∂ −−− ( rv ) = ⎛ r −−−−r + v r⎞ , ⎝ ∂r ⎠ ∂r r equation (4.14) becomes v r ∂ v r 1 ∂ vθ −− + −−−− + − −−−− = 0. r ∂ r r ∂θ

Concluding remarks This chapter has provided the frame of reference for much of the later material in this text. The classification of flows based on time or distance dependence, together with the influence of viscosity via Reynolds number, is fundamental. In defining the flow into laminar and turbulent regimes much of the observed fluid flow behaviour becomes understandable. The presence of velocity profiles across a fluid stream between boundaries is similarly fundamental as it heralds the later work on the development of the boundary layer and the recognition of the condition of ‘no slip’ at fluidsurface interfaces. The introduction of the continuity equation, whether in its volumetric or its more widely applicable mass flow form, provides one of the recurring tools for fluid flow analysis, which, when the storage terms are introduced, will find application

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throughout this text for both compressible and incompressible flows under steady or unsteady conditions.

Summary of important equations and concepts 1.

2.

3.

4.

Chapter 4 introduces definitions of flow conditions, such as steady and unsteady, relating these to changes in flow condition, Section 4.2, together with the concept of the boundary layer, Section 4.4, which will become central to later chapters. Movement in more than one dimension is introduced, with examples, emphasizing the simplifications possible if flow can be considered as one-dimensional, Sections 4.6 to 4.9. The classification of flows into laminar and turbulent regimes, following Reynolds, is an essential concept and one that will be returned to continuously in later chapters. It is important to recognize that the ratio of forces represented by the Reynolds number applies to a whole range of flow geometries, not restricted to pipeflow, Section 4.10. The concept of continuity of mass flow is established and shown in the special case of incompressible flow to reduce to volumetric flow continuity that allows mean velocities to be calculated in series and parallel pipe networks, equations (4.9) and (4.10).

Problems 4.1 The velocity of a fluid varies with time t. Over the period from t = 0 to t = 8 s the velocity components are u = 0 m s−1 and v = 2 m s−1, while from t = 8 s to t = 16 s the components are u = 2 m s−1 and v = −2 m s−1. A dye streak is injected into the flow at a certain point commencing at time t = 0 and the path of a particle of fluid is also traced from that point starting at t = 0. Draw to scale the streakline, the pathline of the particle and the streamlines at time t = 12 s. 4.2 The velocity distribution for a two-dimensional field of flow is given by 2 u = −−−−−− m s –1 3+t

and

t2 v = 2 – −−− m s –1 . 32

For the period of time from t = 0 to t = 12 s draw a streakline for an injection of dye through a certain point A and a pathline for a particle of fluid which was at A when t = 0. Draw also the streamlines for t = 6 s and t = 12 s. 4.3 A nozzle is formed so that its cross-sectional area converges linearly along its length. The inside diameters are 75 mm and 25 mm at inlet and exit and the length of the nozzle is 300 mm. What is the convective acceleration at a section halfway along the length of the nozzle if the discharge is constant at 0.014 m3 s−1? [337.94 m s−2]

4.4 During a wind tunnel test on a sphere of radius r = 150 mm it is found that the velocity of flow u along the longitudinal axis of the tunnel passing through the centre of the sphere at a point upstream which is a distance x from the centre of the sphere is given by r3 u = U 0 ⎛ 1 – −−3⎞ ⎝ x⎠ where U0 is the mean velocity of the undisturbed airstream. If U0 = 60 m s−1 what is the convective acceleration when the distance x is (a) 300 mm, (b) 150 mm? [(a) 3937.5 m s−2, (b) 0] 4.5 The velocity along the centreline of a nozzle of length L is given by x u = 2t ⎛ 1 – 0.5 −−⎞ ⎝ L⎠

2

where u is the velocity in metres per second, t is the time in seconds from the commencement of flow, x is the distance from the inlet to the nozzle. Find the convective acceleration and the local acceleration when t = 3 s, x = --12- L and L = 0.8 m. [18.99 m s−2, 1.125 m s−2] 4.6 Water flows through a pipe 25 mm in diameter at a velocity of 6 m s−1. Determine whether the flow will be laminar or turbulent assuming that the dynamic viscosity

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Problems

111

of water is 1.30 × 10−3 kg m−1 s−1 and its density 1000 kg m−3. If oil of specific gravity 0.9 and dynamic viscosity 9.6 × 10−2 kg m−1 s−1 is pumped through the same pipe, what type of flow will occur? [ Turbulent, Re = 115 385; laminar, Re = 1406]

4.13 During a test on a circular duct 2 m in diameter it was found that the fluid velocity was zero at the duct surface and 6 m s−1 on the axis of the duct when the flow rate was 9 m3 s−1. Assuming the velocity distribution to be given by

4.7 An air duct is of rectangular cross-section 300 mm wide by 450 mm deep. Determine the mean velocity in the duct when the rate of flow is 0.42 m3 s−1. If the duct tapers to a cross-section 150 mm wide by 400 mm deep, what will be the mean velocity in the reduced section assuming that the density remains unchanged? [3.11 m s−1, 7.0 m s−1]

u = c1 − c2 r n,

4.8 A sphere of diameter 300 mm falls axially down a 305 mm diameter vertical cylinder which is closed at its lower end and contains water. If the sphere falls at a speed of 150 mm s−1, what is the mean velocity relative to the cylinder wall of the water in the gap surrounding the midsection of the sphere? [4.46 m s−1] 4.9 The air entering a compressor has a density of 1.2 kg m−3 and a velocity of 5 m s−1, the area of the intake being 20 cm2. Calculate the mass flow rate. If air leaves the compressor through a 25 mm diameter pipe with a velocity of 4 m s−1, what will be its density? [12 × 10−3 kg s−1, 6.11 kg m−3] 4.10 Water flows through a pipe AB 1.2 m in diameter at 3 m s−1 and then passes through a pipe BC which is 1.5 m in diameter. At C the pipe forks. Branch CD is 0.8 m in diameter and carries one-third of the flow in AB. The velocity in branch CE is 2.5 m s−1. Find (a) the volume rate of flow in AB, (b) the velocity in BC, (c) the velocity in CD, (d ) the diameter of CE. [ (a) 3.393 m3 s−1, (b) 1.92 m s−1, (c) 2.25 m s−1, (d ) 1.073 m] 4.11 A closed tank of fixed volume is used for the continuous mixing of two liquids which enter at A and B and are discharged completely mixed at C. The diameter of the inlet pipe at A is 150 mm and the liquid flows in at the rate of 56 dm3 s−1 and has a specific gravity of 0.93. At B the inlet pipe is of 100 mm diameter, the flow rate is 30 dm3 s−1 and the liquid has a specific gravity of 0.87. If the diameter of the outlet pipe at C is 175 mm, what will be the mass flow rate, velocity and specific gravity of the mixture discharged? [78.18 kg s−1, 3.58 m s−1, 0.909] 4.12 In a 0.6 m diameter duct carrying air the velocity profile was found to obey the law u = −5r 2 + 0.45 m s−1 where u is the velocity at radius r. Calculate the volume rate of flow of the air and the mean velocity. [0.0636 m3 s−1, 0.225 m s−1]

where u is the fluid velocity at any radius r, determine the values of the constants c1, c2 and n, specifying the units of c1 and c2. Evaluate the mean velocity and determine the radial position at which a Pitot tube must be placed to measure this mean velocity. [6 m s−1, 6 m −0.825 s−1, 1.825; 2.86 m s−1, 0.701 m] 4.14 Air flows through a rectangular duct which is 30 cm wide by 20 cm deep in cross-section. To determine the volume rate of flow experimentally the cross-section is divided into a number of imaginary rectangular elements of equal area and the velocity measured at the centre of each element with the following results:

Distance from bottom of duct (cm)

Distance from side of duct (cm) 3

9

15

21

27

18 14 10 6 2

1.6 1.9 2.1 2.0 1.8

2.0 3.4 6.8 3.5 2.0

2.2 6.9 10.0 7.0 2.3

2.0 3.7 7.0 3.8 2.1

1.7 2.0 2.3 2.1 1.9

Calculate the volume rate of flow and the mean velocity in the duct. [0.202 m3 s−1, 3.364 m s−1] 4.15 If a two-dimensional flow field were to have velocity components u = U(x 3 + xy 2) and v = U(y 3 + yx 2) would the continuity equation be satisfied?

[Yes]

4.16 Determine whether the following expressions satisfy the continuity equation: (a) u = 10xt, v = −10yt, ρ = constant

[Yes]

(b) u = U ( y δ )17 , v = 0, ρ = constant.

[Yes]

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Chapter 5

The Momentum Equation and its Applications 5.1 5.2

5.3 5.4 5.5 5.6 5.7 5.8 5.9

Momentum and fluid flow Momentum equation for two- and three-dimensional flow along a streamline Momentum correction factor Gradual acceleration of a fluid in a pipeline neglecting elasticity Force exerted by a jet striking a flat plate Force due to the deflection of a jet by a curved vane Force exerted when a jet is deflected by a moving curved vane Force exerted on pipe bends and closed conduits Reaction of a jet

5.10 Drag exerted when a fluid flows over a flat plate 5.11 Angular motion 5.12 Euler’s equation of motion along a streamline 5.13 Pressure waves and the velocity of sound in a fluid 5.14 Velocity of propagation of a small surface wave 5.15 Differential form of the continuity and momentum equations 5.16 Computational treatment of the differential forms of the continuity and momentum equations 5.17 Comparison of CFD methodologies

FM5_C05.fm Page 113 Wednesday, September 21, 2005 9:20 AM

The analysis of fluid flow phenomena fundamentally depends upon the application of Newton’s laws of motion, together with a recognition of the special properties of fluids in motion. The momentum equation relates the sum of the forces acting on a fluid element to its acceleration or rate of change of momentum in the direction of the resultant force. This relationship is, perhaps, when taken with the conservation of mass and the energy equation, the foundation upon which all fluid flow analysis is based. This chapter will introduce the application of the momentum equation to a range of fluid flow conditions, including forces exerted upon and by a fluid as a result of changes in direction and

impact upon both stationary and moving surfaces, as well as introducing the application of the momentum equation to determine engine thrust as a result of changes to fluid momentum. The application of the momentum equation to the prediction of the rate of propagation of pressure or surface wave discontinuities will be presented. By utilizing the momentum equation, together with the conservation of mass, this chapter will also introduce Euler’s equation for motion along a streamline under general conditions. Bernoulli’s equation, the special form of Euler’s equation applicable to incompressible inviscid flows, will be introduced and its application demonstrated. l l l

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5.1

MOMENTUM AND FLUID FLOW

In mechanics, the momentum of a particle or object is defined as the product of its mass m and its velocity v: Momentum = mv. The particles of a fluid stream will possess momentum, and, whenever the velocity of the stream is changed in magnitude or direction, there will be a corresponding change in the momentum of the fluid particles. In accordance with Newton’s second law, a force is required to produce this change, which will be proportional to the rate at which the change of momentum occurs. The force may be provided by contact between the fluid and a solid boundary (e.g. the blade of a propeller or the wall of a bend in a pipe) or by one part of the fluid stream acting on another. By Newton’s third law, the fluid will exert an equal and opposite force on the solid boundary or body of fluid producing the change of velocity. Such forces are known as dynamic forces, since they arise from the motion of the fluid and are additional to the static forces (see Chapter 3) due to pressure in a fluid; they occur even when the fluid is at rest. To determine the rate of change of momentum in a fluid stream consider a control volume ABCD (Fig. 5.1). As the fluid flow is assumed to be steady and nonuniform in nature the continuity of mass flow across the control volume may be expressed as FIGURE 5.1 Momentum in a flowing fluid

ρ2A2v2 = ρ1A1v1 = A,

(5.1)

i.e. there is no storage within the control volume and A is the fluid mass flow. The rate at which momentum exits the control volume across boundary CD may be defined as

ρ2A2v2v2. Similarly the rate at which momentum enters the control volume across AB may be expressed as

ρ1A1v1v1. Thus the rate of change of momentum across the control volume may be seen to be

ρ2 A2 v2 v2 − ρ1A1v1 v1

(5.2)

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5.2

Momentum equation for two- and three-dimensional flow along a streamline

115

or, from the continuity of mass flow equation,

ρ1A1v1(v2 − v1) = A(v2 − v1) = Mass flow per unit time × Change of velocity.

(5.3)

Note that this is the increase of momentum per unit time in the direction of motion, and according to Newton’s second law will be caused by a force F, such that F = A(v2 − v1).

(5.4)

This is the resultant force acting on the fluid element ABCD in the direction of motion. By Newton’s third law, the fluid will exert an equal and opposite reaction on its surroundings.

MOMENTUM EQUATION FOR TWO- AND THREEDIMENSIONAL FLOW ALONG A STREAMLINE

5.2

In Section 5.1, the momentum equation (5.4) was derived for one-dimensional flow in a straight line, assuming that the incoming and outgoing velocities v1 and v2 were in the same direction. Figure 5.2 shows a two-dimensional problem in which v1 makes an angle θ with the x axis, while v2 makes a corresponding angle φ. Since both momentum and force are vector quantities, they can be resolved into components in the x and y directions and equation (5.4) applied. Thus, if Fx and Fy are the components of the resultant force on the element of fluid ABCD, FIGURE 5.2 Momentum equation for two-dimensional flow

Fx = Rate of change of momentum of fluid in x direction = Mass per unit time × Change of velocity in x direction = A(v2 cos φ − v1 cos θ ) = A(vx2 − vx1). Similarly, Fy = A(v2 sin φ − v1 sin θ ) = A(vy2 − vy1).

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These components can be combined to give the resultant force, F = ( F 2x + F 2y ). Again, the force exerted by the fluid on the surroundings will be equal and opposite. For three-dimensional flow, the same method can be used, but the fluid will also have component velocities vz1 and vz2 in the z direction and the corresponding rate of change of momentum in this direction will require a force Fz = A(vz2 − vz1). To summarize the position, we can say, in general, that Total force exerted on Rate of change of momentum the fluid in a control in the given direction of = volume in a given the fluid passing through direction the control volume, F = A(vout − vin). The value of F is positive in the direction in which v is assumed to be positive. For any control volume, the total force F which acts upon it in a given direction will be made up of three component forces: F1 = Force exerted in the given direction on the fluid in the control volume by any solid body within the control volume or coinciding with the boundaries of the control volume. F2 = Force exerted in the given direction on the fluid in the control volume by body forces such as gravity. F3 = Force exerted in the given direction on the fluid in the control volume by the fluid outside the control volume. Thus, F = F1 + F2 + F3 = A(vout − vin).

(5.5)

The force R exerted by the fluid on the solid body inside or coinciding with the control volume in the given direction will be equal and opposite to F1 so that R = −F1.

5.3

MOMENTUM CORRECTION FACTOR

The momentum equation (5.5) is based on the assumption that the velocity is constant across any given cross-section. When a real fluid flows past a solid boundary, shear stresses are developed and the velocity is no longer uniform over the cross-section. In a pipe, for example, the velocity will vary from zero at the wall to a maximum at the centre. The momentum per unit time for the whole flow can be found by summing the momentum per unit time through each element of the cross-section, provided that

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5.3

Momentum correction factor

117

these are sufficiently small for the velocity perpendicular to each element to be taken as uniform. Thus, if the velocity perpendicular to the element is u and the area of the element is δA, Mass passing through element in unit time = ρδA × u, Momentum per unit time passing through element = Mass per unit time × Velocity = ρδAu × u = ρu2δA, Total momentum per unit time passing whole = cross-section

ρu dA. 2

(5.6)

To evaluate this integral, the velocity distribution must be known. If we consider turbulent flow through a pipe of radius R (Fig. 5.3), the velocity u at any distance y from the pipe wall is given approximately by Prandtl’s one-seventh power law: u = umax(yR)17,

FIGURE 5.3 Calculation of momentum correction factor

the maximum velocity, umax, occurring at the centre of the pipe. Since the velocity is constant at any radius r = R − y, it is convenient to take the element of area δA in equation (5.6) as an annulus of radius r and width δr,

δA = 2π rδ r, and, from equation (5.6), for the whole cross-section,

ρu dA R

Total momentum per unit time =

2

=

R

ρ u 2max ( yR )

27

2 πρ 2 π r dr = ⎛ −−−2−−⁄ 7 ⎞ u 2max ⎝R ⎠

y R

27

r dr.

(5.7)

Since r = R − y, dr = −dy, and so, substituting for r and dr in equation (5.7) and changing the limits (because y = 0 when r = R),

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2 πρ u 2max Total momentum per unit time = −−−−−−27 −−−−− R 2 πρ u 2max = −−−−−−27 −−−−− R

y 0

27

( R – y ) ( – dy )

R

2 πρ u 2max ⎛ 7 167 7 97 ⎞ 0 ( y 97 – Ry 27 ) dy = −−−−−−27 −−−−− ⎝ −−−y – −Ry ⎠ 16 9 R R R 0

2 πρ u 2max 167 ⎛ 7 7 ⎞ 49 = −−−−−−27 −−−−− R − – −− = −−− πρ R 2 u 2max . ⎝9 − 16 ⎠ 72 R

(5.8)

In practice, it is usually more convenient to use the mean velocity B instead of the maximum velocity umax: Total volume per unit time passing section Mean velocity, B = -----------------------------------------------------------------------------------------------------------Total area of cross-section 1 = −−−−−2 πR

uδA. R

Putting u = umax(yR)17 and δA = 2π rδ r, 1 B = −−−−−2 πR

R

y u max ⎛ −−⎞ ⎝ R⎠ 0

17

2u max 2 π r dr = −−−− −−− R 157

y R

17

r dr.

Putting r = R − y, dr = −dy, and changing the limits, 2u max B = −−−− −−− R 157

2u max y 17 ( R – y ) ( – dy ) = −−−− −−− R 157 R

(y 0

87

– Ry 17 ) dy

R

2u max ⎛ 7 157 7 87 ⎞ 0 49 −−− −−−y – −Ry ⎠ = −−−u max , = −−−− 60 R 157 ⎝ 15 8 R 60 u max = −−−B. 49

(5.9)

Substituting from equation (5.9) in equation (5.8), 49 60 2 Total momentum per unit time = −−− πρ R 2 ⎛ −−−⎞ B 2 = 1.02ρπR2B2, ⎝ 72 49⎠

(5.10)

or, since ρπR2B = mass per unit time, Momentum per unit time = 1.02 × Mass per unit time × Mean velocity. If the momentum per unit time of the stream had been calculated from the mean velocity without considering the velocity distribution, the value obtained would have been ρπR2B2. To take the velocity distribution into account, a momentum correction factor β must be introduced, so that, for the whole stream, True momentum per unit time = β × Mass per unit time × Mean velocity. The value of β depends upon the shape of the cross-section and the velocity distribution.

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5.4

Gradual acceleration of a fluid in a pipeline neglecting elasticity

119

GRADUAL ACCELERATION OF A FLUID IN A PIPELINE NEGLECTING ELASTICITY

5.4

It is frequently the case that the velocity of the fluid flowing in a pipeline has to be changed, thus causing the momentum of the whole mass of fluid in the pipeline to change. This will require the action of a force, which can be calculated from the rate of change of momentum of the mass of fluid and is produced as a result of a change in the pressure difference between the ends of the pipeline. In the case of liquids flowing in rigid pipes, an approximate value of the change of pressure can be obtained by neglecting the effects of elasticity – provided that the acceleration or deceleration is small.

EXAMPLE 5.1

Water flows through a pipeline 60 m long at a velocity of 1.8 m s−1 when the pressure difference between the inlet and outlet ends is 25 kN m−2. What increase of pressure difference is required to accelerate the water in the pipe at the rate of 0.02 m s−2? Neglect elasticity effects.

Solution Let A = cross-sectional area of the pipe, l = length of pipe, ρ = mass density of water, a = acceleration of water, δ p = increase in pressure at inlet required to produce acceleration a. As this is not a steady flow problem, consider a control mass comprising the whole of the water in the pipe. By Newton’s second law, Force due to δ p in Rate of change of momentum of water in = direction of motion the whole pipe = Mass of water in pipe × Acceleration,

(I)

Force due to δ p = Cross-sectional area × δp = Aδp, Mass of water in pipe = Mass density × Volume = ρAl. Substituting in (I), Aδ p = ρAla,

δ p = ρla = 103 × 60 × 0.02 N m−2 = 1.2 kN m−2. In this example, the change of pressure difference is small because the acceleration is small, but very large pressures can be developed by sudden accelerations or decelerations, such as may occur when valves are shut suddenly. The elasticity of the fluid and of the pipe must then be taken into account, as explained in Chapter 20.

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5.5

FORCE EXERTED BY A JET STRIKING A FLAT PLATE

Consider a jet of fluid striking a flat plate that may be perpendicular or inclined to the direction of the jet, or indeed may be moving in the initial direction of the jet (Fig. 5.4). A control volume encapsulating the approaching jet and the plate may be established, this control volume being fixed relative to the plate and therefore moving with it. It is helpful to consider components of the velocity and force vectors perpendicular and parallel to the surface of the plate. In cases where the plate is itself in motion the most helpful technique is to reduce the plate, and therefore the associated control volume, to rest by the superimposition on the system of an equal but opposite plate velocity, as illustrated in Fig. 5.4. This reduces all the cases illustrated to a simple consideration of a jet striking a stationary plain surface, the initial motion of the surface being reflected in the amendment of the jet velocity relative to the plate. In each of the cases illustrated the impingement of the jet on the plate surface reduces the jet velocity component normal to the plate surface to zero. In general terms the jet velocity thus destroyed may be expressed as vnormal = (v − u) cos θ.

FIGURE 5.4 Force exerted on a flat plate

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5.5

Force exerted by a jet striking a flat plate

121

The mass flow entering the control volume is also affected by the superposition of a velocity equal and opposite to the plate velocity and may be expressed as A = ρA(v − u) cos θ, which reduces to A = ρAv if the plate is stationary. (Note that the sign convention adopted is positive in the direction of the initial jet velocity.) Thus the rate of change of momentum normal to the plate surface is given by dMomentumdt = ρA(v − u)(v − u) cos θ.

Clearly this expression reduces to dMomentumdt = ρAv 2 cos θ if the plate is stationary, and dMomentumdt = ρAv 2 if the plate is both stationary and perpendicular to the initial jet direction. (Note that the plate velocity, represented by the superposition of an equal and opposite velocity on the system as a whole to bring the control volume to rest, appears in both the mass flow and relative jet velocity terms.) There will therefore be a force exerted upon the plate equal to the rate of momentum destroyed normal to the plate, given in the general case by an expression of the form Force normal to plate = ρA(v − u)(v − u) cos θ. There will be an equal and opposite reaction force exerted on the jet by the plate. In a direction parallel to the plate, the force exerted will depend upon the shear stress between the fluid and the surface of the plate. For an ideal fluid there would be no shear stress and hence no force parallel to the plate. The fluid would flow out over the plate so that the total momentum in unit time parallel to the plate remained unchanged.

EXAMPLE 5.2

A jet of water from a fixed nozzle has a diameter of 25 mm and strikes a flat plate inclined to the jet direction. The velocity of the jet is 5 m s−1, and the surface of the plate may be assumed frictionless. (a) Indicate in tabular form the reduction in the force normal to the plate surface as the inclination of the plate to the jet varies from 90° to 0°. (b) Indicate in tabular form the force normal to the plate surface as the plate

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velocity changes from 2 m s−1 to −2 m s−1 in the direction of the jet, given that the plate is itself perpendicular to the approaching jet.

Solution For each case the control volume taken is fixed relative to the plate. If the plate is in motion the control volume is brought to rest by the superposition of an equal and opposite velocity on the system as a whole. As a force applied normal to the plate is required in each case, the components of velocity and force normal and parallel to the plate surface are considered. From equation (5.5) the gravity force is negligible and, if the fluid jet is assumed to be parallel sided and passing through a region at atmospheric pressure, there is no force exerted on the jet by fluid outside the control volume. Thus the force exerted normal to the plate in the general case is given by Force normal to plate = ρA(v − u)(v − u) cos θ, which may be utilized in the following tables drawn up to answer (a) and (b) above. The cross-sectional area of the jet is A = π 0.02524 = 4.9 × 10− 4 and the density of the water jet ρ = 1000.0 kg m−3. The jet velocity v = 5.0 m s−1.

TABLE (A) Variation of force exerted normal to the plate with plate angle

θ (deg) 0 15 30 45 60 75 90

TABLE (B) Variation of force exerted normal to the plate with plate velocity

θ (deg) 0 0 0 0 0

v cos θ (m s−1)

ρAv (kg s−1)

Force = ρAv 2 cos θ (N)

5.00 4.83 4.33 3.54 2.50 1.29 0.00

2.46 2.46 2.46 2.46 2.46 2.46 2.46

12.28 11.86 10.63 8.68 6.14 3.18 0.00

v (m s−1)

u (m s−1)

v−u (m s−1)

ρA(v − u) (kg s−1)

Force = ρA(v − u)2 (N)

5.0 5.0 5.0 5.0 5.0

2.0 1.0 0.0 −1.0 −2.0

3.0 4.0 5.0 6.0 7.0

1.47 1.96 2.46 2.94 3.43

4.41 7.84 12.28 17.64 24.01

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5.6

Force due to the deflection of a jet by a curved vane

123

FORCE DUE TO THE DEFLECTION OF A JET BY A CURVED VANE

5.6

Both velocity and momentum are vector quantities and, therefore, even if the magnitude of the velocity remains unchanged, a change in direction of a stream of fluid will give rise to a change of momentum. If the stream is deflected by a curved vane (Fig. 5.5), entering and leaving tangentially without impact, a force will be exerted between the fluid and the surface of the vane to cause this change of momentum. It is usually convenient to calculate the components of this force parallel and perpendicular to the direction of the incoming stream by calculating the rate of change of momentum in these two directions. The components can then be combined to give the magnitude and direction of the resultant force which the vane exerts on the fluid, and the equal and opposite reaction of the fluid on the vane.

FIGURE 5.5 Force exerted on a curved vane

EXAMPLE 5.3

A jet of water from a nozzle is deflected through an angle θ = 60° from its original direction by a curved vane which it enters tangentially (see Fig. 5.5) without shock with a mean velocity C1 of 30 m s−1 and leaves with a mean velocity C2 of 25 m s−1. If the discharge A from the nozzle is 0.8 kg s−1, calculate the magnitude and direction of the resultant force on the vane if the vane is stationary.

Solution The control volume will be as shown in Fig. 5.5. The resultant force R exerted by the fluid on the vane is found by determining the component forces Rx and Ry in the x and y directions, as shown. Using equation (5.5), Rx = −F1 = F2 + F3 − A(vout − vin)x . Neglecting force F2 due to gravity and assuming that for a free jet the pressure is constant everywhere, so that F3 = 0, Rx = A(vin − vout)x ,

(I)

and, similarly, Ry = A(vin − vout)y .

(II)

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Since the nozzle and vane are fixed relative to each other, Mass per unit Mass per unit time entering = A = time leaving control volume nozzle. In the x direction, vin = Component of C1 in x direction = C1, vout = Component of C2 in x direction = C2 cos θ. Substituting in (I), Rx = A(C1 − C2 cos θ ).

(III)

Putting A = 0.8 kg s−1, C1 = 30 m s−1, C2 = 25 m s−1, θ = 60°, Rx = 0.8(30 − 25 cos 60°) = 14 N. In the y direction, vin = Component of C1 in y direction = 0, vout = Component of C2 in y direction = C2 sin θ. Thus, from (II), Ry = AC2 sin θ.

(IV)

Putting in the numerical values, Ry = 0.8 × 25 sin 60° = 17.32 N. Combining the rectangular components Rx and Ry, Resultant force exerted = ( R 2x – R 2y ) by fluid on vane, R = ( 14 2 + 17.32 2 ) = 22.27 N. This resultant force R will be inclined to the x direction at an angle φ = tan−1(Ry Rx) = tan−1(17.3214) = 51°3′.

5.7

FORCE EXERTED WHEN A JET IS DEFLECTED BY A MOVING CURVED VANE

If a jet of fluid is to be deflected by a moving curved vane without impact at the inlet to the vane, the relation between the direction of the jet and the tangent to the curve of the vane at inlet must be such that the relative velocity of the fluid at inlet is tangential to the vane. The force in the direction of motion of the vane will be equal to the rate of change of momentum of the fluid in the direction of motion, i.e. the mass

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5.7

Force exerted when a jet is deflected by a moving curved vane

125

deflected per second multiplied by the change of velocity in that direction. The force at right angles to the direction of motion will be equal to the mass deflected per second times the change of velocity at right angles to the direction of motion.

EXAMPLE 5.4

A jet of water 100 mm in diameter leaves a nozzle with a mean velocity C1 of 36 m s−1 (Fig. 5.6) and is deflected by a series of vanes moving with a velocity u of 15 m s−1 in a direction at 30° to the direction of the jet, so that it leaves the vane with an absolute mean velocity C2 which is at right angles to the direction of motion of the vane. Owing to friction, the velocity of the fluid relative to the vane at outlet Cr2 is equal to 0.85 of the relative velocity Cr1 at inlet. Calculate (a) the inlet angle α and outlet angle β of the vane which will permit the fluid to enter and leave the moving vane tangentially without shock, and (b) the force exerted on the series of vanes in the direction of motion u.

FIGURE 5.6 Force exerted on a series of moving vanes

Solution If the absolute velocity C2 is to be at right angles to the direction of motion, the vane must turn the fluid so that it leaves with a relative velocity Cr2 , which has a component velocity equal and opposite to u as shown in the outlet velocity triangle (Fig. 5.6). (a) To determine the inlet angle α, consider the inlet velocity triangle. The velocity of the fluid relative to the vane at inlet, Cr1, must be tangential to the vane and make an angle α with the direction of motion, tan α = CDBC = C1 sin 30°(C1 cos 30° − u). Putting C1 = 36 m s−1 and u = 15 m s−1, tan α = 36 × 0.5(36 × 0.866 − 15) = 1.113,

α = 48°3′. To determine the outlet angle β, if C2 has no component in the direction of motion, the outlet velocity triangle is right angled, cos β = uCr2, but Cr2 = 0.85Cr1 and, from the inlet triangle, Cr1 = CDsin α = C1 sin 30°sin α.

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Therefore 15 × 0.744 u sin α cos β = −−−−−−−−−−−−−−−−−−− = −−−−−−−−−−−−−−−−−−−−− = 0.729, 0.85v 1 sin30° 0.85 × 36 × 0.5

β = 43°11′. (b) Since the jet strikes a series of vanes, perhaps mounted on the periphery of a wheel, so that as each vane moves on its place is taken by the next in the series, the average length of the jet does not alter and the whole flow from the nozzle of diameter d is deflected by the vanes. Neglecting the force due to gravity and assuming a free jet that does not fill the space between the vanes completely, so that the pressure is constant everywhere, the component forces in the x and y directions (Fig. 5.6) can be found from equation (5.5) putting R = −F1 and F2 = F3 = 0. In the direction of motion, which is the x direction, Rx = A(vin − vout)x

(I)

Mass per unit time Mass per unit time entering control = A = leaving nozzle volume = ρ(π4)d 2C1, vin = Component of C1 in x direction = C1 cos 30°, vout = Component of C2 in x direction = C2 cos 90° = 0. Substituting in (I), Force on vanes in = Rx = ρ(π4)d 2C1 × C1 cos 30°. direction of motion Putting in the numerical values, Force on vanes in 2 direction of motion = 1000 × (π4)(0.1) × 36 × 36 × 0.866 N = 8816 N.

5.8

FORCE EXERTED ON PIPE BENDS AND CLOSED CONDUITS

Figure 5.7 shows a bend in a pipeline containing fluid. When the fluid is at rest, it will exert a static force on the bend because the lines of action of the forces due to pressures p1 and p2 do not coincide. If the bend tapers, the magnitude of the static forces will also be affected. When the fluid is in motion, its momentum will change as it passes round the bend due to the change in its direction and, if the pipe tapers, any consequent change in magnitude of its velocity. There must, therefore, be an additional force acting between the fluid and the pipe.

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5.8

EXAMPLE 5.5

Force exerted on pipe bends and closed conduits

127

A pipe bend tapers from a diameter of d1 of 500 mm at inlet (see Fig. 5.7) to a diameter of d2 of 250 mm at outlet and turns the flow through an angle θ of 45°. Measurements of pressure at inlet and outlet show that the pressure p1 at inlet is 40 kN m−2 and the pressure p2 at outlet is 23 kN m−2. If the pipe is conveying oil which has a density ρ of 850 kg m−3, calculate the magnitude and direction of the resultant force on the bend when the oil is flowing at the rate of 0.45 m3 s−1. The bend is in a horizontal plane.

FIGURE 5.7 Force on a tapering bend

Solution Referring to Fig. 5.7, take the x direction parallel to the incoming velocity C1 and the y direction as shown. The control volume is bounded by the inside wall of the bend and the inlet and outlet sections 1 and 2. Mass per unit time entering control volume = ρQ. The forces acting on the fluid will be F1 exerted by the walls of the pipe, F2 due to gravity (which will be zero), and F3 due to the pressures p1 and p2 of the fluid outside the control volume acting on areas A1 and A2 at sections 1 and 2. The force exerted by the fluid on the bend will be R = −F1. Using equation (5.5), putting F2 = 0 and resolving in the x direction: (F1 + F3)x = A(vout − vin)x and, since Rx = −(F1)x, Rx = (F3)x − A(vout − vin)x.

(I)

Now (F3)x = p1A1 − p2A2 cos θ, vout = Component of C2 in x direction = C2 cos θ, vin = Component of C1 in x direction = C1. Substituting in (I), Rx = p1A1 − p2A2 cos θ − ρQ(C2 cos θ − C1). Resolving in the y direction, (F1 + F3)y = A(vout − vin)y

(II)

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and, since Ry = −(F1)y , Ry = (F3)y − A(vout − vin)y .

(III)

Now, (F3)y = 0 + p2A2 sin θ, vout = Component of C2 in y direction = −C2 sin θ, vin = Component of C1 in y direction = 0. Substituting in (III), Ry = p2A2 sin θ + ρQC2 sin θ.

(IV)

For the given problem, A1 = (π 4) d 21 = (π 4)(0.5)2 = 0.196 35 m2, A2 = (π 4) d 22 = (π 4)(0.25)2 = 0.049 09 m2, Q = 0.45 m3 s−1, C1 = QA1 = 0.450.196 35 = 2.292 m s−1, C2 = QA2 = 0.450.049 09 = 9.167 m s−1. Putting ρ = 850 kg m−3, θ = 45°, p1 = 40 kN m−2, p2 = 23 kN m−2, and substituting in equation (II), Rx = 40 × 103 × 0.196 35 − 23 × 103 × 0.049 09 cos 45° − 850 × 0.45(9.167 cos 45° − 2.292) N = 103(7.855 − 0.798 − 1.603) N = 5.454 × 103 N. Substituting in equation (IV), Ry = 23 × 103 × 0.049 09 sin 45° + 850 × 0.45 × 9.167 sin 45° N = 10 3(0.798 + 2.479) N = 3.277 × 103 N. Combining the x and y components, Resultant force on bend, R = (Rx2 + Ry2 ) = (5.4542 + 3.2772) kN = 6.362 kN. The inclination of R to the x direction is given by

φ = tan−1(Ry Rx) = tan−1(3.2775.454) = 31°.

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5.9

Reaction of a jet

129

REACTION OF A JET

5.9

Whenever the momentum of a stream of fluid is increased in a given direction in passing from one section to another, there must be a net force acting on the fluid in that direction, and, by Newton’s third law, there will be an equal and opposite force exerted by the fluid on the system which is producing the change of momentum. A typical example is the reaction force exerted when a fluid is discharged in the form of a high-velocity jet, and which is applied to the propulsion of ships and aircraft through the use of propellers, pure jet engines and rocket motors. The propulsive force can be determined from the application of the linear momentum equation (5.5) to flow through a suitable control volume.

EXAMPLE 5.6

A jet of water of diameter d = 50 mm issues with velocity C = 4.9 m s−1 from a hole in the vertical side of an open tank which is kept filled with water to a height of 1.5 m above the centre of the hole (Fig. 5.8). Calculate the reaction of the jet on the tank and its contents (a) when it is stationary, (b) when it is moving with a velocity u = 1.2 m s−1 in the opposite direction to the jet while the velocity of the jet relative to the tank remains unchanged. In the latter case, what would be the work done per second?

FIGURE 5.8 Reaction of a jet

Solution Take the control volume shown in Fig. 5.8. In equation (5.5), the direction under consideration will be that of the issuing jet, which will be considered as positive in the direction of motion of the jet: therefore, F2 = 0, and, if the jet is assumed to be at the same pressure as the outside of the tank, F3 = 0. Force exerted by fluid system in = −F1 = −A(vout − vin), direction of motion, R or in words, Reaction force in direction Mass discharged Increase of velocity × = opposite to that per unit time in direction of jet. of the jet

(I)

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In the present problem, Mass discharged = ρ(π 4)d 2C = 1000 × (π4)(0.05)2 × 4.9 kg s−1 per unit time, A = 9.62 kg s−1. (a) If the tank is stationary, vout = C = 4.9 m s−1, vin = Component of velocity of the free surface in the direction of the jet = 0. Substituting in equation (I), Reaction of jet on tank = 9.62 × (4.9 − 0) N = 47.14 N in the direction opposite to that of the jet. (b) If the tank is moving with a velocity u in the opposite direction to that of the jet, the effect is to superimpose a velocity of −u on the whole system: vout = C − u, vin = −u, vout − vin = C. Thus, the reaction of the jet R remains unaltered at 47.14 N. Work done per second = Reaction × Velocity of tank = R × u = 47.14 × 1.2 = 56.57 W.

A rocket motor is, in principle, a simple form of engine in which the thrust is developed as the result of the discharge of a high-velocity jet of gas produced by the combustion of the fuel and oxidizing agent. Both the fuel and the oxidant are carried in the rocket and so it can operate even in outer space. It does not require atmospheric air, either for combustion or for the jet to push against; the thrust is entirely due to the reaction developed from the momentum per second discharged in the jet.

EXAMPLE 5.7

The mass of a rocket mr is 150 000 kg and, when ready to launch, it carries a mass of fuel mf0 of 300 000 kg. The initial thrust of the rocket motor is 5 MN and fuel is consumed at a constant rate A. The velocity Cr of the jet relative to the rocket is 3000 m s−1. Assuming that the flight is vertical, and neglecting air resistance, find (a) the burning time, (b) the speed of the rocket and the height above ground at the moment when all the fuel is burned, and (c) the maximum height that the rocket will reach. Assume that g is constant and equal to 9.81 m s−2.

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131

Solution (a) From equation (I), Example 5.6, Initial thrust, T = ACr, Rate of fuel consumption, A = TCr = 5 × 1063000 = 1667 kg s−1, Initial mass of fuel, mf0 = 300 000 kg, Burning time = mf0A = 300 0001667 = 180 s. (b) If there is no air resistance, the forces acting on the rocket and the fuel which it contains during vertical flight are the thrust T acting upwards and the weight (mr + mft)g acting downwards, where mft is the mass of the fuel in the rocket at time t. From Newton’s second law, dv T − (mr + mft)g = (mr + mft) −−−−t , dt where vt is the velocity of the rocket at time t. dv t T – ( m r + m ft )g −−− = −−−−−−−−−−−−−−−−−−−−−−. dt m r + m ft Since the fuel is being consumed at a rate A, Mass of fuel at time t, mft = mf0 − At. Also, T = ACr and so dv t m˙ C r – ( m r + m f0 – m˙t )g −−− = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−. m r + m f0 – m˙t dt Substituting numerical values, dv t 1667 × 3000 – ( 150 000 + 300 000 – 1667t ) × 9.81 −−− = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− dt 150 000 + 300 000 – 1667t 3000 = –9.81 + −−−−−−−−−−−−−− m s –2 . 269.95 – t Integrating, vt = −9.81t − 3000 log e (269.95 − t) + constant. Putting vt = 0 when t = 0, the value of the constant is 3000 log e 269.95, giving vt = −9.81t − 3000 log e (1 − t269.95). From (a), all the fuel will be burnt out when t = 180 s. Substituting in equation (I), vt = −9.81 × 180 − 3000 log e (1 − 180269.95) m s−1 = −1765.8 + 3296.9 = 1531.2 m s−1.

(I)

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The height at time t = 180 s is given by Z1 =

180

vt dt = – 9.81 0

180

t dt – 3000 0

180

log e ( 1 – t269.95 ) dt 0

= – ( 4.9t 2 ) 180 + 3000{269.95(1 − t269.95)[log e(1 − t269.95) − 1]}180 0 0 = −158 760 + 243 451.9 = 84 691.9 m = 84.692 km. (c) When the fuel is exhausted, the rocket will have reached an altitude of 84 692 m and will have kinetic energy m r v2t 2g. It will, therefore, continue to rise a further distance Z2 until this kinetic energy has been converted into an increase of potential energy. Z2 = v2t 2g = 1531.22(2 × 9.81) = 119 499 m, Maximum height reached = Z1 + Z2 = 84 692 + 119 499 m = 204.2 km.

For aircraft or missiles propelled in the atmosphere it is not necessary to employ a self-contained system, the propulsive force being obtained from the reaction of a jet of atmospheric air which is taken in and accelerated by means of a propeller, turboprop or jet engine and expelled at the rear of the craft. In the case of the jet engine, air is taken in at the front of the engine and mixed with a small amount of fuel which, on burning, produces a stream of hot gas to be discharged at a much higher velocity at the rear. Figure 5.9(a) shows a jet engine moving through still air. It is convenient to take a control volume which is fixed relative to the engine and to reduce the system to a steady state by imposing a rearward velocity v upon it (Fig. 5.9(b)). Relative to the control volume,

FIGURE 5.9 Jet engine

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Reaction of a jet

133

Intake velocity, C1 = v, Jet velocity = Cr, Total force exerted on fluid in the control = Increase of momentum in volume in the direction direction of the jet, of the jet F = A2C2 − A1C1. Since the mass per unit time of the hot gases discharged will be greater than that of the air entering the control volume, owing to the addition of fuel, A2 = A1 + Af . Putting C2 = Cr and C1 = v, F = (A1 + Af)Cr − A1v = A1(Cr − v) + A f Cr. If r is the ratio of the mass of fuel burned to the mass of air taken in, F = A1[(1 + r)Cr − v]. If T is the thrust exerted on the engine by the fluid, taken as positive in the direction of the jet, the force F1 exerted by the engine on the fluid is equal to −T. There will be no gravity forces acting on the fluid in horizontal flight, but there will be a force ( p1A1 − p2A2) exerted on the fluid due to the fluid outside the control volume, so that F = −T + ( p1A1 − p2A2). Substituting for F in the previous equation, T = ( p1A1 − p2A2) − A1[(1 + r)Cr − v], Force on engine in forward = −T = A1[(1 + r)Cr − v] − ( p1A1 − p2A2). direction

EXAMPLE 5.8

(5.11)

A jet engine consumes 1 kg of fuel for each 40 kg of air passing through the engine. The fuel consumption is 1.1 kg s−1 when the aircraft is travelling in still air at a speed of 200 m s−1. The velocity of the gases which are discharged at atmospheric pressure from the tailpipe is 700 m s−1 relative to the engine. Calculate (a) the thrust of the engine, (b) the work done per second, and (c) the efficiency.

Solution (a) From equation (5.11), putting r = 140, A1 = Afr = 40 × 1.1 = 44 kg s−1; v = 200 m s−1, Cr = 700 m s−1, p1 = p2 = 0;

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therefore, 1⎞ ⎛ −− 700 – 200 = 22 770 N Thrust = 44 ⎝ 1 + − 40⎠ = 22.77 kN. (b) Work done per second = Thrust × Forward velocity = T × v = 22.77 × 200 = 4554 kW. (c) In addition to the useful work done on the aircraft, work is also done in giving the exhaust gases discharge from the tailpipe kinetic energy. Relative to the ground, the velocity of the air at outlet is (Cr − v), while at intake it is zero for still air. Since the mass discharge is A1(1 + r), Loss of kinetic energy 1 = −2 A1(1 + r)(Cr − v)2 per second 1 = −12 × 44 ⎛ 1 + −−−⎞ ( 700 – 200 ) 2 W ⎝ 40⎠ = 5638 kW. Efficiency

Work done per second = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Work done per second + Loss 4554 = −−−−−−−−−−−−−−−−− = 0.447 = 44.7 per cent. 4554 + 5638

FIGURE 5.10 Jet propulsion of vessels. (a) Intake in direction of motion. (b) Intake in side of vessel

Jet propulsion can also be applied to boats. Water is taken in through an opening either in the bows of the vessel (Fig. 5.10(a)) or on either side (Fig. 5.10(b)) and pumped out of a jet pipe at the stern at high velocity. In both cases, the control volume taken for analysis is fixed relative to the vessel. The two cases differ in that the water entering at the bows has a velocity relative to the vessel in the direction of the jet equal to the absolute velocity of the vessel u, while in Fig. 5.10(b), for side intake, the water entering has no component velocity in the direction of the jet.

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EXAMPLE 5.9

Reaction of a jet

135

Derive a formula for the propulsion efficiency of a jet-propelled vessel in still water if u is the absolute velocity of the vessel, vr the velocity of the jet relative to the vessel, when the intake is (a) at the bows facing the direction of motion, (b) amidships at right angles to the direction of motion.

Solution (a) For intake in the direction of motion, Mass of fluid entering control volume = ρQ, in unit time Mean velocity of water at inlet in vin = direction of motion relative = u, to control volume Mean velocity of water at outlet in = vr. vout = direction of motion relative to control volume From equation (5.11), assuming that the pressure in the water is the same at outlet and inlet, Propelling force = ρQ(vout − vin ) = ρQ(vr − u), Work done per unit time = Propelling force × Speed of vessel = ρQ(vr − u)u. In unit time, a mass of water ρQ enters the pump intake with a velocity u and leaves with a velocity vr. Kinetic energy per unit 1 = −2 ρQu 2, time at inlet Kinetic energy per unit 1 = −2 ρQv 2r , time at outlet Kinetic energy per unit 1 = − ρQ(v 2r − u 2), time supplied by pump 2 Work done per unit time Hydraulic efficiency = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Energy supplied per unit time = ρQ(vr − u)u −12 ρ Q ( v 2r – u 2 ) , = 2u(Vr + u). (b) For intake at right angles to the direction of motion (Fig. 5.10(b)), the control volume used will be the same as in (a), as will the rate of change of momentum through the control volume, and therefore the propelling force. Hence, Work done per unit time = ρQ(vr − u)u.

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As, however, the intake to the pumps is at right angles to the direction of motion, the forward velocity of the vessel will not assist the intake of water to the pumps and, therefore, the whole of the energy of the outgoing jet must be provided by the pumps. Energy supplied per unit time = −12 ρ Qv 2r , Work done per unit time Hydraulic efficiency = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Energy supplied per unit time = ρQ(vr − u)u −12 ρ Qv 2r = 2(Vr − u)uV r2.

5.10 DRAG EXERTED WHEN A FLUID FLOWS OVER A FLAT PLATE When a fluid flows over a stationary flat surface, such as the upper surface of the smooth flat plate shown in Fig. 5.11, there will be a shear stress τ0 between the surface of the plate and the fluid, acting to retard the fluid. At a section AB of the flow well upstream of the tip of the plate O, the velocity will be undisturbed and equal to U. The fluid in contact with the surface of the plate will be at rest, and, at a cross-section such as CD, the velocity u of the adjacent fluid will increase gradually with the distance y away from the plate until it approximates to the free stream velocity at the outside of the boundary layer when y = δ. The limit of this boundary layer, in which the drag of the stationary boundary affects the velocity of the fluid, is defined as the distance δ at which uU = 0.99. The value of δ will increase from zero at the leading edge O, since the drag force D exerted on the fluid due to the shear stress τ0 will increase as x increases. The value of D can be found by applying the momentum equation.

FIGURE 5.11 Drag on a flat plate

Consider a control volume PQSR (Fig. 5.12) consisting of a section of the boundary layer of length ∆x at a distance x from the upstream edge of the plate. Fluid enters the control volume through section PQ and through the upper edge of the boundary layer QS, leaving through section RS. Applying the momentum equation, Force acting on fluid Rate of increase of momentum in in control volume in = x direction of fluid passing x direction through control volume.

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Drag exerted when a fluid flows over a flat plate

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FIGURE 5.12 Momentum equation applied to a boundary layer

Since the velocity u in the boundary layer varies with the distance y from the surface of the plate, the momentum efflux through RS must be determined by integration. Consider an element of thickness δy, through which the velocity in the x direction is u2. For a width B perpendicular to the diagram, Momentum per second = Mass per second × Velocity passing through element = ρBδyu2 × u2, Total momentum per second = ρB passing through RS

δ2

u dy. 2 2

(5.12)

Similarly, for the control surface PQ, Total momentum per second = ρB passing through PQ

δ1

u dy. 2 1

(5.13)

where u1 is the velocity through PQ at a distance y from the surface. For the control surface QS, for continuity of flow, Rate of flow into Rate of flow Rate of flow = − the control volume, Q through RS through PQ Q=B

δ2

δ1

u 2 dy – B

u dy, 1

Momentum in x direction = ρQU entering through QS ⎛ = ρB ⎝

δ2

δ1

u 2 dy –

Force exerted on the fluid by = −τ0B∆x. the boundary in the x direction

u dy⎞⎠ U 1

(5.14)

(5.15)

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Equating the force given by equation (5.15) with the sum of the x momenta from equations (5.12), (5.13) and (5.14),

– τ 0 B∆x = ρ B

δ2

u 22 dy –

δ1

⎛ u 21 dy – U ⎝

δ2

δ2

δ1

u 2 dy –

1

δ1

(u – Uu ) dy – (u – Uu ) dy

= ρB

u dy⎞⎠

2 2

2 1

2

1

.

The term in the square brackets is the difference between 1 δ0 (u2 − Uu) dy at sections RS and PQ, which can be written as ∆[1 δ0 u(u − U ) dy] so that δ

– τ 0 B∆x = ρ B∆

u(u – U ) dy . 0

In the limit, as ∆x tends to zero,

d τ 0 = ρ U 2 −−− dx

δ

−Uu− ⎛⎝1 – −Uu−⎞⎠ dy.

(5.16)

The drag D on one surface of the plate will be given by

τ dx. x

D=B

If the fluid acts on both the upper and the lower surface of the plate, this force will of course be doubled.

5.11 ANGULAR MOTION In Section 4.8, we set out the equations of motion for a particle or element of fluid moving in a straight line. If the particle or element is rotating about a fixed point, similar equations can be written to describe its angular motion. Angular displacement will be measured as the angle θ in radians through which the particle or element has moved about the centre measured from a reference direction. Angular velocity ω will be the rate of change of displacement θ with time, i.e. dθ ω = θ˙ = −−−, dt and the angular acceleration α will be the rate of change of ω with time, so that

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d2θ α = θ¨ = −−−−. dt 2

The laws of angular motion will be similar to those for linear motion (see Section 4.8):

ω 2 = ω1 + αt, θ = ω1t + −12 α t 2, ω 22 = ω 21 + 2 αθ . For a particle (Fig. 5.13) which at a given instant is rotating about a fixed point with angular velocity ω at a radius r, Tangential linear velocity, vθ = ω r, Momentum of particle, mvθ = mω r.

FIGURE 5.13 Angular motion

If the angular velocity changes from ω to zero in time t under the influence of a force F acting at radius r, Rate of change of momentum of particle = mrω t. By Newton’s second law, F = mrω t. This force produces a turning moment or torque T about the centre of rotation, T = Fr = mr2ω t = Angular momentumTime. Now consider a particle moving in a curved path, so that in time t it moves from a position at which it has an angular velocity ω1 at radius r1 to a position in which the corresponding values are ω2 and r2. The effect will be equivalent to first applying a torque to reduce the particle’s original angular momentum to zero, and then applying

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a torque in the opposite direction to produce the angular momentum required in the second position: Torque required to eliminate = mr12ω1t, original angular momentum Torque required to produce = mr22ω 2 t, new angular momentum Torque required to produce = (mt)(ω 2r 22 − ω1r 12 ) change of angular momentum = (mt)(vθ 2r2 − vθ 1r1), where vθ = tangential velocity = ω r. This analysis applies equally to a stream of fluid moving in a curved path, since mt is the mass flowing per unit time, A = ρQ. The torque which must be acting on the fluid will be T = ρQ(vθ 2r2 − vθ 1r1),

(5.17)

and, of course, the fluid will exert an equal and opposite reaction.

EXAMPLE 5.10

A water turbine rotates at 240 rev min−1. The water enters the rotating impeller at a radius of 1.2 m with an absolute mean velocity which has a tangential component of 2.3 m s−1 in the direction of motion and leaves with a tangential component of 0.2 m s−1 at a radius of 1.6 m. If the volume rate of flow through the turbine is 10 m3 s−1, calculate the torque exerted and the theoretical power output.

Solution In equation (5.17), ρ = 1000 kg m−3, Q = 10 m3 s−1, Cθ 2 = 0.2 m s−1, Cθ 1 = 2.3 m s−1, r2 = 1.6 m, r1 = 1.2 m. Hence, Torque acting on fluid = 1000 × 10(0.2 × 1.6 − 2.3 × 1.2) N m = 10 000(0.32 − 2.76) = −24 400 N m. The torque exerted by the fluid on the rotor will be equal and opposite: Torque exerted by fluid = 24 400 N m. If n is the rotational speed in revolutions per second, n = 24060 = 4, Power output = 2π nT = 2π × 4 × 24 400 W = 613 318 W = 613.32 kW.

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Euler’s equation of motion along a streamline

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5.12 EULER’S EQUATION OF MOTION ALONG A STREAMLINE From consideration of the rate of change of momentum from point to point along a streamline and the forces acting due to the effects of the surrounding pressures and changes of elevation, it is possible to derive a relationship between velocity, pressure, elevation and density along a streamline. Figure 5.14 shows a short section of a streamtube surrounding the streamline and having a cross-sectional area small enough for the velocity to be considered constant over the cross-section. AB and CD are two cross-sections separated by a short distance δs. At AB the area is A, velocity v, pressure p and elevation z, while at CD the corresponding values are A + δA, v + δ v, p + δ p and z + δ z. The surrounding fluid will exert a pressure pside on the sides of the element and, if the fluid is assumed to be inviscid, there will be no shear stresses on the sides of the streamtube and pside will act normally. The weight of the element mg will act vertically downward at an angle θ to the centreline. FIGURE 5.14 Euler’s equation

Mass per unit time flowing = ρAv, Rate of increase of momentum = ρAv[(v + δ v) − v] = ρAvδ v. from AB to CD

(5.18)

The forces acting to produce this increase of momentum in the direction of motion are Force due to p in direction of motion

= pA,

Force due to p + δp opposing motion

= ( p + δ p)(A + δA),

Force due to pside producing a component = psideδA, in the direction of motion Force due to mg producing a component = mg cos θ, opposing motion Resultant force in the direction of motion =

pA − (p + δ p)(A + δA) + psideδA − mg cos θ.

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The value of pside will vary from p at AB to p + δp at CD and can be taken as p + kδp, where k is a fraction, Weight of element, mg = ρg × Volume = ρg(A + −12 δA)δs, cos θ = δ zδ s, Resultant force in the −pδA − Aδ p − δ pδA + pδA + kδ p · δA = direction of motion − ρg(A + −12 δA)δ s · (δ zδ s). Neglecting products of small quantities, Resultant force in the direction of motion = −Aδ p − ρgAδ z.

(5.19)

Applying Newton’s second law from equations (5.18) and (5.19),

ρAvδ v = −Aδ p − ρgAδ z. Dividing by ρAδs, 1 δp δv δz − −−− + v −−− + g −−− = 0, ρ δs δs δs

(5.20)

or, in the limit as δs → 0, dv dz 1 dp − −−− + v −−− + g −−− = 0. ds ds ρ ds

(5.21)

This is known as Euler’s equation, giving, in differential form, the relationship between pressure p, velocity v, density ρ and elevation z along a streamline for steady flow. It cannot be integrated until the relationship between density ρ and pressure p is known. For an incompressible fluid, for which ρ is constant, integration of equation (5.21) along the streamline, with respect to s, gives pρ + v 22 + gz = constant.

(5.22)

The terms represent energy per unit mass. Dividing by g, pρg + v 22g + z = constant = H,

(5.23)

in which the terms represent the energy per unit weight. Equation (5.23) is known as Bernoulli’s equation and states the relationship between pressure, velocity and elevation for steady flow of a frictionless fluid of constant density. An alternative form is p + −12 ρv 2 + ρgz = constant,

(5.24)

in which the terms represent the energy per unit volume. These equations apply to a single streamline. The sum of the three terms is constant along any streamline, but the value of the constant may be different for different streamlines in a given stream.

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Pressure waves and the velocity of sound in a fluid

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If equation (5.21) is integrated along the streamline between any two points indicated by suffixes 1 and 2, p1ρg + v 21 2g + z1 = p2 ρg + v 22 2g + z2.

(5.25)

For a compressible fluid, the integration of equation (5.21) can only be partially completed, to give v −−− + −−− + z = H. dp ρ g 2g 2

The relationship between ρ and p must then be inserted for the given case. For gases, this will be of the form pρ n = constant, varying from adiabatic to isothermal conditions, while, for a liquid, ρ(dpdρ) = K, the bulk modulus.

5.13 PRESSURE WAVES AND THE VELOCITY OF SOUND IN A FLUID In a real fluid, any change of pressure at a point or any cross-section will be associated with a change in density of the fluid, so that the particles of fluid will change their positions, moving closer together or further apart. Adjacent particles will, in turn, change their positions, and so the change of pressure and density will spread very rapidly through the fluid. Clearly, if the fluid were incompressible, every particle would have to change its position simultaneously and the speed of propagation of the disturbance or pressure wave would, theoretically, be infinite. However, the elasticity of a compressible fluid allows the particles to adjust their positions one after the other, so that the disturbance spreads with a finite velocity. The speed of propagation of a pressure change is very rapid and, in some problems, it is sufficient to assume that pressure changes are propagated instantaneously throughout the fluid. However, when studying abrupt changes of pressure, such as those occurring when a valve on a pipeline is closed suddenly, or when fluid velocities are high relative to a solid body (as in the case of aircraft in flight), the speed of propagation of pressure changes in the fluid can be a factor of major importance from the practical point of view. In Fig. 5.15, a pressure wave is moving through a fluid from left to right with a velocity c relative to a stationary observer. The fluid to the right of the wavefront will not have been affected by the pressure wave and will have its original pressure p, velocity u relative to the observer and density ρ as indicated. To the left, the fluid behind

FIGURE 5.15 Pressure wave. Unsteady flow relative to a stationary observer

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FIGURE 5.16 Pressure wave. Steady state relative to a moving observer

the wavefront will be at the new pressure p + δp, velocity u + δu and density ρ + δρ. From a terrestrial frame of reference, conditions in the fluid are not steady, since at a point fixed with reference to a stationary observer conditions will change with time. The usual equations for steady flow cannot, therefore, be applied. However, to an observer moving with the wave at velocity c, the wave will appear stationary; conditions will not change with time and the flow is steady and can be analysed as such. As shown in Fig. 5.16, the effect is equivalent to imposing a backward velocity c on the system from right to left. Considering an element of cross-sectional area δA perpendicular to the direction of flow, Mass per unit time flowing Mass per unit time flowing = on the left of wavefront on the right of wavefront ( ρ + δρ)(u + δu − c)δA = ρ(u − c)δA,

ρδu + uδρ + δuδρ − cδρ = 0, (c − u)δρ = (ρ + δρ)δu.

(5.26)

Owing to the pressure difference δp across the wavefront, there will be a force acting to the right, in the direction of flow, which will cause an increase in momentum per unit time in this direction. Force due to δp = Increase of momentum per unit time to the right = Mass per unit time × Increase of velocity,

δp × δA = ρ(u − c)δA × [u − (u + δ u)], δp = ρ(u − c)(−δ u), δ u = δpρ(c − u).

(5.27)

Substituting from equation (5.27) for δ u in equation (5.26), (c − u)δρ = ( ρ + δρ)δpρ(c − u),

δρ δ p 2 ( c – u ) = ⎛ 1 + −−−⎞ −−−. ⎝ ρ ⎠ δρ

(5.28)

If the change of pressure and density across the wavefront is small, the pressure wave is said to be weak and, in the limit as δp and δρ tend to zero, equation (5.28) gives (c – u) =

⎛⎝ d−−−ρ⎞⎠ . dp

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Pressure waves and the velocity of sound in a fluid

145

Now (c − u) is the velocity of the wavefront relative to the fluid, so that Velocity of propagation of a weak pressure wave = c – u =

⎛⎝ d−−−ρ⎞⎠ . dp

(5.29)

For a mass m of fluid of volume V and density ρ,

ρV = m. Differentiating,

ρ dV + Vdρ = 0, dρ = −( ρV ) dV. If K is the bulk modulus, then from equation (1.9), dp dp K = – V −−−− = ρ −−− , dV dρ dp K −−− = −−. dρ ρ Therefore, Velocity of propagation of ˙ ρ). = c – u = ( K a weak pressure wave

(5.30)

This equation applies to solids, liquids and gases. Note, however, that when c represents the velocity of sound in still air, u = 0. Since sound is propagated in the form of very weak pressure waves, equation (5.30) gives the velocity of sound or sonic velocity, with u = 0. In a gas, the pressure and temperature changes occurring due to the passage of a sound wave are so small and so rapid that the process can be considered as reversible and adiabatic, so that pργ = constant. Differentiating, dp γ p −−− = −−− ρ dρ or, since pρ = RT, dp −−− = γ RT. dρ Substituting in equation (5.29), with u = 0, Sonic velocity, c = (γ pρ) = (γ RT ).

(5.31)

The above equations apply only to weak pressure waves in which the change of pressure is very small compared with the pressure of the fluid. The pressure change

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involved in the passage of a sound wave in atmospheric air, for example, varies from about 3 × 10−5 N m−2 for a barely audible sound to 100 N m−2 for a sound so loud that it verges on the painful. These are small in comparison with atmospheric pressure of 105 N m−2. For relatively large pressure changes, the velocity of propagation of the pressure wave would be greater. The sonic velocity is important in fluid mechanics, because when the velocity of the fluid exceeds the sonic velocity, i.e. becomes supersonic, small pressure waves cannot be propagated upstream. At subsonic velocities, lower than the sonic velocity, small pressure waves can be propagated both upstream and downstream. This results in the flow pattern around an obstacle, for example, differing for supersonic and subsonic flow with consequent differences in the forces exerted. The ratio of the fluid velocity u to the sonic velocity c − u is known as the Mach number Ma = u(c − u). If Ma 1, flow is supersonic; if Ma 1, flow is subsonic.

5.14 VELOCITY OF PROPAGATION OF A SMALL SURFACE WAVE For flow with a free surface, as, for example, in open channels, the pressure cannot vary from point to point along the free surface. A disturbance in the fluid will be propagated as a surface wave rather than as a pressure wave. Using the approach adopted in Section 5.13, assume that a surface wave of height δZ (Fig. 5.17(a)) is being propagated from left to right in the view of a stationary observer. If this wave is brought to rest relative to the observer by imposing a velocity c equal to the wave velocity on the observer, conditions will now appear steady, as shown in Fig. 5.17(b). Considering a width B of the flow, perpendicular to the plane of the diagram,

FIGURE 5.17 Velocity of propagation of a small surface wave. (a) Unsteady flow as seen by a stationary observer (b) Steady flow as seen by a moving observer

Mass per unit time Mass per unit time flowing on the left = flowing on the right of the wavefront of the wavefront,

ρ × B(Z + δZ) × (u + δu − c) = ρ + B × Z × (u − c), the mass density ρ being the same on both sides of the wavefront since the pressure is unchanged. Simplifying, Zδ u + uδZ + δZδ u − cδZ = 0, (c − u)δZ = (Z + δZ)δ u.

(5.32)

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Velocity of propagation of a small surface wave

147

The change of momentum occurring as a result of the change of velocity across the wavefront is produced as a result of the hydrostatic force due to the difference of level δZ acting on the cross-sectional area BZ. By Newton’s second law, Hydrostatic force Mass per unit Change of = × due to δZ time velocity,

ρgδZBZ = BZ(u − c) × (− δ u), δ u = gδZ(c − u). Substituting from equation (5.32) for δu, (c − u)δZ(Z + δZ) = gδZ(c − u), (c − u)2 = (Z + δZ)g = gZ if the wave height δZ is small. Velocity of propagation of the = c – u = ( gZ ). wave relative to the fluid

(5.33)

Taking velocities in a downstream direction as positive, the wave velocity c relative to the bed of the channel is given by (gZ) + u if the wave is travelling downstream, and ( gZ) − u if it is travelling upstream. Thus, if the stream velocity u ( gZ), the wave cannot travel upstream relative to the bed, while if u ( gZ) a surface wave will be propagated in both directions. The ratio of the stream velocity u to the velocity of propagation c − u of the wave in the fluid is known as the Froude number Fr: Fr = u ( c – u ) = u ( gZ ).

(5.34)

Thus the condition for a wave to be stationary is that the Froude number is unity, i.e. Fr = 1. The Froude number is also a criterion of the type of flow present in an open channel. If Fr 1 the flow is defined as supercritical, rapid or shooting, and is characterized by shallow and fast fluid motion. If Fr 1 the flow is defined as subcritical, tranquil or streaming, and is characterized, relative to supercritical flow, as slow and deep fluid motion. Analogies can be drawn between compressible flow and flow with a free surface, which can be utilized in experimental investigations of the former. While it is convenient to develop free surface relationships with reference to rectangular channels, many free surface flow conditions occur in uniform but not rectangular channels, for example, the whole range of free surface flow conditions in partially filled pipes to be found in sewer and urban drainage and storm drainage applications. The fundamental concepts remain the same and it is possible to develop analogous expressions for wave speed. It may be shown that for a uniform channel of non-rectangular cross-section the wave speed as defined above is given by c = ( gAT )

(5.35)

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where A is the flow cross-sectional area of surface width T at any particular depth. (It may be noted that AT is a form of ‘average’ depth relative to a rectangular channel, but this is a misleading interpretation.) The same criterion based on Froude number applies and the same flow-type nomenclature is utilized.

5.15 DIFFERENTIAL FORM OF THE CONTINUITY AND MOMENTUM EQUATIONS The differential form of the continuity equation was developed in Section 4.13 as

∂ρ ∂ ∂ ∂ ------ + ------ ( ρ v x ) + ------ ( ρ v y ) + ----- ( ρ v z ) = 0. ∂t ∂x ∂y ∂z

(5.36)

This expression is applicable to every point in a fluid flow, whether steady or unsteady, compressible or incompressible. The derivation as set out in Section 4.13 considered the continuity of flow and mass storage across an infinitesimal cuboid control volume within the flow. A similar approach may be taken to the determination of a differential form of the momentum equation that would be equally valid across the flow conditions listed above. Figure 5.18 illustrates the flow in three dimensions through an element of the fluid, together with the forces acting on each surface of the element. Considering the x direction as an exemplar from which the other directional equations will be derived, the total acceleration in the x direction may be written as

δ vx ∂v ∂v ∂v ∂v -------- = v x --------x + v x --------x + v x -------z + --------x , δt ∂x ∂y ∂z ∂t

(5.37)

from Section 4.9. The rate of change of momentum in the x direction may then be written as

∂ Mx ∂v ∂v ∂v ∂v ----------- = ρδ x δ y δ z ⎛ v x --------x + v y --------x + v z --------x + --------x⎞ . ⎝ ∂x ∂t ∂y ∂z ∂t ⎠ FIGURE 5.18 Definition of coordinate axes and normal and shear stress notation. Stresses identified for the x axis derivation only

(5.38)

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Differential form of the continuity and momentum equations

149

The force acting in the x direction may be determined from a summation of the normal stress, σx, on the element surfaces perpendicular to the x direction, having area δyδz, and the shear stresses, τyx and τzx, acting on the element surfaces parallel to the x direction, having areas δzδx and δxδy, where the shear stress suffixes represent the flow direction considered and the separation of the two faces of the element. In the x direction, therefore, the net force due to the normal stress on the perpendicular faces and the shear stress on the tangential faces of the element is

∂σ ∂τ yx ∂τ zx⎞ - + --------- δ x δ y δ z, F x = ⎛ ρ X – ---------x + --------⎝ ∂x ∂y ∂z ⎠

(5.39)

where X is the body force in the x direction, comprising, for example, gravitational or Coriolis forces as appropriate, and the element is sufficiently small for the change of stress or mass flow with distance to be assumed linear. Therefore from equations (5.38) and (5.39) the general form of the momentum equation in each of the three dimensions may be written as

∂σ ∂τ yx ∂τ zx ∂v ∂v ∂v ∂v - + --------- = ρ ⎛ --------x + v x --------x + v y -------x- + v z --------x⎞ , ρ X – ---------x + --------⎝ ∂t ∂x ∂y ∂z ∂x ∂y ∂z ⎠

(5.40)

∂τ xy ∂σ y ∂τ zy ∂v ∂v ∂v ∂v - + -------- + --------- = ρ ⎛ --------y + v x --------y + v y --------y + v z --------y⎞ , ρ Y + --------⎝ ∂t ∂x ∂y ∂z ∂x ∂y ∂z ⎠

(5.41)

∂τ xz ∂τ yz ∂σ z ∂v ∂v ∂v ∂v + --------- – -------- = ρ ⎛ -------z + v x -------z + v y -------z + v z -------z⎞ . ρ Z + --------⎝ ∂x ∂y ∂z ∂t ∂x ∂y ∂z ⎠

(5.42)

While these momentum equations are entirely general they cannot be integrated without reference to expressions defining the stresses assumed to act normal and tangential to the element surfaces. In inviscid flow the shear stress terms disappear and the normal stress, σ, terms may be replaced by pressure, p, terms, becoming the general form of the onedimensional steady flow Euler equation presented in Section 5.12, equation (5.20), where the body force is gravitational. Newtonian fluids, as previously defined in Section 1.4, display properties that allow stress to be related to velocity gradients, for both normal and shear components so that the viscous stresses are proportional to the rate of deformation, defined in terms of a linear deformation by the coefficient of dynamic viscosity, µ, and a second viscosity coefficient, λ, to cover volumetric deformation, defined as the sum of the velocity gradients along each of the three coordinate axes. The stress velocity gradient expressions, known as the constitutive equations, may be defined as

∂v ∂v ∂v ∂v σ x = p – 2 µ --------x – λ ⎛⎝ --------x + --------y + -------z⎞⎠ , ∂x ∂x ∂y ∂z

∂v ∂v τ xy = µ ⎛⎝ --------x + --------y⎞⎠ , ∂y ∂x

(5.43)

∂v ∂v ∂v ∂v σ y = p – 2 µ --------y – λ ⎛⎝ --------x + --------y + -------z⎞⎠ , ∂y ∂x ∂y ∂z

∂v ∂v τ xz = µ ⎛⎝ --------x + -------z⎞⎠ , ∂z ∂x

(5.44)

∂v ∂v ∂v ∂v σ z = p – 2 µ -------z – λ ⎛⎝ --------x + --------y + -------z⎞⎠ , ∂z ∂x ∂y ∂z

∂v ∂v τ yz = µ ⎛⎝ --------y + -------z⎞⎠ . ∂z ∂y

(5.45)

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The effect of the second viscosity coefficient, λ, is small in practice. A good approximation is to set λ = − 23 µ, i.e. the Stokes hypothesis, and pressure may be seen to be the average of the three normal stresses from equations (5.43) to (5.45). Using equation (5.40) as an exemplar it may be seen that for a hom*ogeneous fluid, i.e. one where the properties are not affected by position, substitution for the normal and shear stress terms from equation (5.43) and using the Stokes hypothesis allows the left-hand side (LHS) of equation (5.40) to be recast as

∂ 2v 2 ∂ ∂ v ∂ v ∂ v ∂p LHS = ρ X – ------ + 2 µ ---------2-x – --- µ ------ ⎛ --------x + --------y + --------z⎞ ∂x ∂x 3 ∂x ⎝ ∂x ∂y ∂z ⎠ ∂ ⎛ ∂ v x ∂ v y⎞ ∂ ⎛ ∂ v x ∂ v z⎞ - + -------- + ----- -------- + ------+ µ ∂-----y- ⎝ ------∂y ∂x ⎠ ∂z ⎝ ∂z ∂x ⎠ , ∂ 2v ∂ 2v ∂ 2v 2 ∂ ∂v ∂v ∂v ∂p LHS = ρ X – ------ + µ ⎛ ---------2-x + ---------2-x + ---------2-x⎞ – --- µ ------ ⎛ --------x + --------y + -------z⎞ ⎝ ∂x ∂x ∂y ∂z ⎠ 3 ∂x ⎝ ∂x ∂y ∂z ⎠ ∂ ∂v ∂v ∂v + µ ------ ⎛ --------x + --------y + -------z⎞ , ∂x ⎝ ∂x ∂y ∂z ⎠ so that if the right-hand side (RHS) of equation (5.40) is set to Dv x RHS = ρ ---------, Dt the expression in the x direction becomes Dv x ∂ 2v ∂ 2v ∂ 2v 1 ∂ ∂v ∂v ∂v ∂p ρ X – ------ + µ ⎛⎝ ---------2-x + ---------2-x + ---------2-x⎞⎠ + --- µ ------ ⎛⎝ -------x- + --------y + -------z⎞⎠ = ρ ---------, 3 ∂x ∂x ∂y ∂z ∂x Dt ∂x ∂y ∂z

(5.46)

with equivalent expressions for the y and z coordinate axes. If the flow is steady and incompressible then, by reference to the continuity equation, equation (5.46) may be reproduced in each of the three coordinate directions as 2 2 v x ∂ 2 v x⎞ Dv x ∂p v-x ∂--------- + ---------- = ρ ---------, + ρ X – ------ + µ ⎛⎝ ∂--------2 ∂x Dt ∂x ∂ y2 ∂ z2 ⎠

(5.47)

Dv y ∂ 2v ∂ 2v ∂ 2v ∂p ρ Y – ------ + µ ⎛⎝ ---------2-y + ---------2-y + ---------2-y⎞⎠ = ρ ---------, ∂y Dt ∂x ∂y ∂z

(5.48)

Dv ∂ 2v ∂ 2v ∂ 2v ∂p ρ Z – ------ + µ ⎛⎝ ---------2-z + ---------2-z + ---------2-z⎞⎠ = ρ ---------z . ∂z Dt ∂x ∂y ∂z

(5.49)

Equations (5.47) to (5.49) are known as the Navier–Stokes equations following their independent derivation by these two nineteenth-century researchers. While in laminar flow the shear stress is proportional to the viscosity and the rate of shear strain, equation (1.3), turbulent stresses are complex and no wholly satisfactory model exists

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to be used in developing analogous forms of the Navier–Stokes equations. The introduction of an eddy viscosity, Section 11.4, which includes the turbulence effects and is based on the turbulence models discussed in Chapter 11, allows the Navier–Stokes equations to become central to the developing field of computational fluid dynamics.

5.16 COMPUTATIONAL TREATMENT OF THE DIFFERENTIAL FORMS OF THE CONTINUITY AND MOMENTUM EQUATIONS The use of computer-based models and simulations to describe fluid flow conditions has numerous advantages for the designer and researcher. The development of computing capacity over the past decade has been exponential and has made possible the implementation of long-recognized numerical solutions through the sledgehammer of fast computing. It is now possible to assess the likely effects of design changes without recourse to costly, both in time and resource, physical testing. However, care must be exercised, and computational fluid dynamics (CFD) must be recognized as being itself still in the developmental stage. In particular the problem of turbulent flow description has not been wholly solved and care must be taken with the resulting simulation predictions. While recognizing that caution is necessary, the benefits of the use of computational methods to deal with flow conditions, both steady and transient, previously thought too complex, or at least too time consuming, are clear. Examples of unsteady and transient simulations will be developed later in Parts VI and VII of this text, while routine application of a computational approach will be found throughout the text. The literature on CFD is now extensive and it would be inappropriate for this text to provide more than an introduction. The availability of high-speed computing allows the time or distance grids used to become very small and this in turn leads to the application of relatively straightforward numerical methods to solve the governing equations for each case studied. Within the rapidly growing application of computational methods three main approaches to CFD may be usefully identified, namely finite difference methods, the finite element method and the finite volume method, each with its own exponents and literature. The finite difference method utilizes a time–distance grid of nodes and a truncated Taylor series approach to determine the conditions at any particular node one time step in the future based on the conditions at adjacent nodes at the current time. A brief coverage of the application of the Taylor series and the nodal grid will illustrate several points fundamental to flow simulation, points considered again later in dealing with unsteady flow simulation. Figure 5.19 illustrates a nodal grid superimposed on a duct or pipe, where the termination of the duct may be connection to a further duct, connection to some fitting, such as a damper, or energy source, such as a fan or pump, or connection to atmospheric or room conditions. The flow conditions along the duct are known at time zero. For full bore flows these conditions might be zero flow and atmospheric pressure, while for partially filled pipe or channel flow the initial conditions could be a set uniform flow and depth.

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FIGURE 5.19 Nodal grid basis for a finite difference method representation of flow conditions Note: Nodes A, B, C and P are linked by explicit formulation, nodes A, O, B, P, C and Q by an implicit approach.

At time, t, the profile of variable y with x may be described by a truncated Taylor series as y(x0 ± ∆x) = y(x0) ± y′(x0) ∆x + y ″(x0) ∆x2/2! ± y′″(x0) ∆x 3/3! + . . . , (5.50) where the value of x at node i is x0 and the inclusion of the ± notation allows equation (5.50) to be used in either a forward or backward difference approach. From equation (5.50) it may be seen that the forward or backward first differential based on the first two terms only may be defined as Forward difference

y ( x 0 + ∆x ) – y ( x 0 ) - + Neglected terms, y′(x0) = -------------------------------------------∆x (5.51)

y ( x 0 – ∆x ) – y ( x 0 ) - + Neglected terms. Backward difference y′(x0) = − -------------------------------------------∆x (5.52) Note that the neglected terms are of the first order of ∆x and hence these expressions are referred to as first order accurate. Summation of the forward/backward application of equation (5.50) also yields a central difference expression for the first derivative of variable y at x0: Central difference

y ( x 0 + ∆x ) – y ( x 0 – ∆x ) -, y′(x0) = − --------------------------------------------------------2∆x

(5.53)

which may be seen to be second order accurate as the neglected terms are of the order ∆x 2. This approach is the equivalent of fitting a three-point parabolic curve through values of variable y at the nodes considered and has second order accuracy. A more useful result is the summation of the second order forward/backward forms of the truncated Taylor series to determine the second differential of variable y at x0: y ( x 0 + ∆x ) – 2y ( x 0 ) + y ( x 0 – ∆x ) . y″(x0) = − -------------------------------------------------------------------------------2 ∆x

(5.54)

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Figure 5.19 illustrates a grid in terms of ∆x and ∆t. The rate of change of the variable with time may be written as dy y ( x 0,t+∆t ) – y ( x 0, t ) ------ = -------------------------------------------. ∆t dt

(5.55)

If the flow relationship is known then these expressions allow conditions at nodes i = 1 to i = n to be calculated one time step in the future based on known conditions along the duct at time zero. Clearly on the next time step this reduces to i = 2 to i = n − 1 unless some information is available as to the way in which conditions change at the boundary nodes, i = 1 and i = n + 1. Figure 5.19 illustrates this limitation and thereby introduces boundary equation considerations that will be returned to in more detail in the discussion of unsteady flow simulations. It will be appreciated therefore that a differential equation may be replaced by finite difference approximations that allow a numerical solution to proceed. In cases where the formulation is dependent upon base conditions known at a particular time the resulting solution is referred to as explicit. If the approximations involve unknown conditions at a time step into the future then the solution is referred to as implicit. Figure 5.19 illustrates this as conditions at A, B and C are known at time t, thus allowing an explicit solution for conditions at P. However, if the approximations involve time-averaged values based on conditions at AO, BP and CQ then direct solution becomes impossible without recourse to matrix methods to solve the complete set of approximations for all later time nodes simultaneously. Many flow phenomena are commonly described by differential equations of the form

∂ f ∂ f ∂ f a --------2 + b ------------- + c --------2 = 0, ∂ y ∂ x ∂x ∂y 2

2

2

where f may represent flow velocity, temperature or contaminant concentration. The definition of the form of the equation depends on the relative values of the coefficients a, b, c. For example when b2 − 4ac > 0 the relationship becomes 2 ∂ f 2∂ f --------2 – c --------2 = 0, ∂y ∂x 2

which is hyperbolic and defines unsteady flow conditions. If b2 − 4ac = 0 then the equation becomes parabolic,

∂f ∂ f ------ – c --------2 = 0, ∂x ∂y 2

and defines unsteady heat transfer and contamination decay. When b2 − 4ac < 0 the equation becomes the Laplace relationship defining equilibrium flows, including seepage flows,

∂ f ∂ f -------- + --------2 = 0. ∂x ∂y 2

2

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The terms explicit and implicit may be understood in terms of a solution for a typical parabolic differential equation: for example, the Fourier unsteady heat transfer equation,

∂ f ∂f ----- = c --------2 . ∂t ∂x 2

(5.56)

The RHS and LHS of this expression may be written in finite difference format if the node considered is ‘i’ (Fig. 5.19) and the time step is from time t to time t + ∆t, t

t

t

t t t t 2 1 f i +1 – f i f i – f i −1⎞ ⎛ f i +1 – 2f i + f i −1⎞ ∂ f ∂ (∂ f ) - , – ------------------ = ------------------------------------------2 = --------------- = ------- ⎛ -----------------2 ⎠ ∆x ⎝ ∆x ∆x ⎠ ⎝ ∂x ∆x ∂x

and similarly t+∆t

t

∂f f i – f i ----- = --------------------- , ∆t ∂t so that t

t

t

t+∆t t f i +1 – 2f i + f i −1⎞ fi –fi - , --------------------- = c ⎛ ----------------------------------2 ⎝ ⎠ ∆t ∆x

which yields an ‘explicit’ solution for the value of the variable f at one time step into the future at the node under consideration. However, this approach does not include any allowance for the effect of the future value of the variable at that node. This may be improved by taking a time average as opposed to averages based wholly on known conditions at time t. This introduces unknowns into both sides of the equation as it will be necessary to write for each node a time-averaged value of the variable: for example, at node i, t+∆t

t

fi –fi -. f it+∆t/2 = -------------------2 Thus while the values of f are known at all nodes at time t, this is not the case for time t + ∆t, and the solution then becomes ‘implicit’ and will require more complex modes of solution, commonly involving matrix methods. Equations (5.54) and (5.55) may be reordered to demonstrate the explicit formulation as f(x0,t + ∆t) = f(x0 t)(1 − 2F ) + F [ f(x0 + ∆x,t) + f(x0 − ∆x,t)], c∆t F = ---------2 , ∆x

(5.57)

where the value of the variable f at one time step into the future is directly calculated from known current values.

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Alternatively, the use of a time-averaged value of the f variable at each of the three nodes would yield a form of equation (5.57):

t+∆t

fx

F t+∆t t t t+∆t t ( 1 + F ) = f x ( 1 – F ) + --- ( f x +∆x + f x +∆x + f x −∆x + f x −∆x ), 2 0

(5.58)

where direct solution for the value of f is not possible – an example of an implicit solution requiring simultaneous solution of a complete set of equations, including suitable boundary expressions. Stability is a concern in finite difference solutions. Referring to the value of F in equation (5.57) common practice is to set this to 0.25 to prevent divergence of the simulation. This is based on an inspection of the equation, which would suggest values below 0.5 to prevent a change of sign for one of the y terms, and experience: for example, in the solution of the Fourier unsteady heat conduction equation. In addition while computing power has increased and accuracy improved there are still ‘rounding errors’ associated with each value used in the solution and truncation errors arising from the order of the Taylor series used to develop the simulation. Reducing the time step or increasing the number of nodes will generally improve the situation but may also lead to other potential hazards, such as numerical dispersion or attenuation of a wave front. Reducing the time step or increasing the number of nodes is not a panacea as any advantage can be lost to increasingly important rounding errors. In the unsteady simulations discussed in Parts VI and VII stability is determined by conformance to the Courant criterion, which links time step and internodal distance to the local fluid velocity and wave propagation speed.

5.17 COMPARISON OF CFD METHODOLOGIES The finite element method (FEM) was initially developed for structural analysis but has been utilized for fluid flow predictions as it offers the advantage of a non-regular grid. This allows FEM simulations to address complex boundary geometries. FEM also has an advantage in that the base equations describing flow conditions within each ‘cell’ have a higher degree of accuracy than those used in FDM; however, the methodologies used are more complex than FDM, where, as shown above, relatively easily understood techniques are applied. The finite volume method draws together the best attributes of FDM and FEM in that it is capable of simulating complex boundary geometries and accurately modelling conservation for each cell while at the same time utilizing relatively straightforward finite difference relationships to represent the governing differential equations. The physics of almost every fluid flow and heat transfer phenomenon is governed by three fundamental principles, namely mass conservation, momentum conservation (or Newton’s second law) and energy conservation taken together with appropriate initial or boundary conditions. These three principles and conditions may be expressed mathematically, in most cases through integral or partial differential equations, whose close-form analytical solutions rarely exist. The ability to seek the numerical solutions of these governing equations under a given set of boundary and initial conditions has led to the development of computational

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FIGURE 5.20 The ability to model rotating components makes it possible to represent flow machines such as fans, pumps and compressors (CFD result courtesy of FLUENT Inc)

FIGURE 5.21 Modelling flow pathlines illustrates the vulnerability of a building ventilation intake to pollutants from a stack exhaust. Air flow around buildings is used to study pollutant transport from stacks or vents for a range of assumed wind conditions (CFD result courtesy of FLUENT Inc)

fluid dynamics (CFD), a new discipline in fluid dynamics that had to await the availability of computing power. Along with the traditional approaches of experimental and analytical fluid science, CFD is now widely used within a wide range of engineering applications, from fan designers concerned with detailed internal flow predictions to the description of air flow patterns around proposed building complexes. CFD is applied by all the fluid mechanics related disciplines from aeronauticalaerospace engineering where fluid dynamics is crucial to the industrial applications such as HVAC (heating, ventilation and air conditioning) design for buildings. Examples of the wide diversity of successful applications of CFD codes and packages are illustrated in Figs 5.20–5.24, ranging from the flow internal to a centrifugal fan, through the pollution spread from a process chimney to air flows within a building envelope and around a jet fighter.

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FIGURE 5.22 Internal flow and temperature modelling used to investigate natural ventilation in the Building Research Establishment’s Environmental Building (CFD application, reproduced by permission of BRE Ltd)

FIGURE 5.23 Air flow around a jet fighter design illustrating the application of CFD to areas previously the province of wind tunnel testing (STAR-CD from Computational Dynamics Ltd)

FIGURE 5.24 3-D Euler solution on an unstructured mesh for an Airbus research configuration (courtesy of Airbus UK Ltd, Filton, Bristol)

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The numerical solutions obtained through CFD for a flow problem represent the values of the physical variables of the fluid field. To achieve this, various techniques are applied, including manipulating the defining equations, dividing the fluid domain into a large number of small cells or control volumes (also called a mesh or grid), transforming the integrals or partial derivatives into discretized algebraic forms (so that the governing equations become linear algebraic equations in a discretized flow field) and, finally, solving the algebraic equations at the grid points. All these numerical algorithms were developed in the early 1970s, the first commercial CFD packages emerging some ten years later as computing power escalated. The exciting CFD history is briefly, yet colourfully, reviewed by Anderson, see the Further Reading section. Since then new techniques have been added and new models have been developed to enhance the capability of CFD codes and packages to simulate more accurately ‘real’ problems. Modern CFD can handle flow around a geometry of great complexity in which all details of flow significance have been faithfully represented. Fluid flow associated with other phenomena, such as chemical reactions, turbulence, multi-phase or free surface problems, and radiative heat transfer, can all be simulated by the commercial CFD packages now available, with a suite of built-in models describing these processes. Because of the rapid increase in computer power, memory and affordability, CFD is no longer confined to advanced research projects of great commercial or defence significance. It is an integral part of the engineering design and analysis environment in an increasing number of companies due to its ability to predict the performance of a novel design or simulate an industrial process before manufacture or implementation. Compared with experimental testing, computer modelling offers potential savings, removing the necessity for a sophisticated physical model and offering the possibility of fewer iterations to the final design with fewer expensive prototypes to produce. Naturally these advantages rely entirely on the validation of the CFD models and the level of confidence in any particular representation chosen – an area that still requires attention, particularly in the treatment of turbulence.

5.17.1 Structure of a CFD code A CFD code has three basic components: pre-processor, solver and post-processor. The solver is the heart of the code, carrying out the major computations and providing the numerical solutions. The pre-processor and post-processor are the front and end of the code, providing the usermachine interface that allows a CFD operator to communicate with the solver: inputting data to define the problem to be simulated and commanding the solver to use certain models and schemes to carry out the simulation and, finally, presenting for study the computed results. Apart from these key elements, a commercial package aimed at multi-purpose modelling will have a suite of models for various flow problems, such as various turbulence models to cover a range of turbulence conditions and assumptions. The packages will also have a library of material properties for defining the fluid media and solid boundaries in the computational domain. Experience will guide the user in the choice of appropriate model and boundary condition. The pre-processor has a number of functions that allow users to define a fluid domain, known as the computational domain, and build up the physical geometry of the zone considered, creating a meshgrid system throughout the domain and tuning this mesh to improve computation quality, including accuracy and speed. The user may also specify the properties of the fluid and other materials in contact with the fluid at this stage.

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The pre-processor allows the user to define the fluid flow phenomena to be modelled, to choose from the appropriate physical or chemical models provided by the package, and to select the numerical parameters and initial and boundary conditions for the computation. It is clearly apparent that experience will dictate success, as an inappropriate choice of boundary conditions or numerical parameters may lead to wholly erroneous results or computational instability. All of these activities are crucial in ensuring a quality CFD modelling task. For example, the meshgrid refinement in this stage affects directly the accuracy of the solution and the cost of computation, in terms of computation time and hardware requirement. An optimized meshgrid system uses less computer memory space, requires less computing, and yet gives satisfactory accuracy. The same applies to defining the computational domain – these two activities constitute more than half of all time spent in a CFD modelling exercise. The solver is the heart of a CFD code, although it is very often treated as a ‘black box’ by many CFD operators. It is a collection of various algorithms and numerical techniques that perform the major computation tasks described above. It consists essentially of two components that provide a discretization for the defining equations and subsequent solution. The first component uses a discretizing scheme to express the governing equations in a discretized form for all the meshgrid elements over the whole computational domain and discretizes the boundary equations appropriate at the boundary elements. In summary, this section converts the partial differential equations and boundary condition formulae into a group of algebraic equations. The second part uses an iterative procedure to find solutions that satisfy these boundary conditions for the algebraic equations defining the flow domain. A solver in any CFD code is based on one of the available discretization methods. Currently there are three major methods, namely finite difference (FD), finite volume (FV) and finite element (FE). The finite element method was developed originally for structural stress analysis and is much more widely used in that area than in fluid dynamics. Over 90 per cent of CFD codes are based on either the finite difference or finite volume methods. The latter was developed as a special formulation of the former. As the finite volume method has been very well established and thoroughly validated, it is applied in most commercial CFD packages used worldwide, such as FLUENT, CFX, START-CD and FLOW-3D. The distinguishing feature of the FV method is that it integrates the governing equations in the finite volumes (known as control volumes) over the whole computational domain. Hence there is one generic form of equation for one flow variable, φ, which could be a velocity component, enthalpy or species concentration:

∂ρφ ---------- = – ∇ ( ρ V φ ) + ∇ ( Γ φ ∇ φ ) + S φ , ∂t

(5.64)

where: φ = dependent variable Γφ = exchange coefficient (laminar + turbulent) V = flow velocity vector ρ = flow density Sφ = source of the variable. This integrated expression has a clear physical meaning, namely that the rate of change of φ in the control volume with respect of time is equal to the sum of the net flux of φ due to convection into the control volume, the net flux of φ due to

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diffusion into the control volume and the net generation rate of φ inside the control volume. Following integration the solver uses various approximations, which are based on an application of the finite difference method, to replace all the terms in the integrated equation, namely the time–change rate, the convective and diffusive fluxes and the source term. The process converts the equation into a set of algebraic equations, which are ready to be solved by an iterative method. The iteration is actually the third and last action the solver does in the whole computational task. The output of the solver is a set of flow variable values at the meshgrid nodes. The post-processor allows the CFD operator to construct a picture of the simulated flow problem by displaying the geometry of the problem, the computational domain and the meshgrid system. The post-processor may then display contour or isosurface plots for the flow variables, including contours plotted over specified surfaces, such as on a solidfluid interface or a iso-surface of a second flow variable. In many cases velocity vector plotting is important, as is streamline presentation or particle tracking. In some cases animation of the fluid flow or a flow process may be appropriate and, finally, it may be desirable to provide hardcopy printouts. A complete CFD simulation often requires repeating the procedure a number of times. It is not rare to run a large number of trials before reaching a set of reasonable solutions. The procedure includes tuning the mesh system, adjusting boundary conditions, selecting numerical parameters, finding the right physical model, monitoring iteration and, finally, viewing the results. This again highlights the need for experience within the application of CFD code. To support CFD predictions, model validation is the key issue in carrying out a CFD exercise. Considerable effort is required to ensure that the computer model developed is robust and that modelling quality is ensured. Validation normally has two parts: first, a mathematicalnumerical verification, including convergence, grid-independence and stability test, and, second, physical validation through comparing predicted results with experimental data; this is the ultimate measure of model validation. However, very often it is hard to provide a physical validation for some obvious reasons: time, cost, or safety prohibitions to carrying out the experiment. This is often true in industrial applications, where the validation is limited to qualitative comparison with existing flow cases. The CFD operator’s experience of various flow problems becomes important in ensuring a quick computer prediction with reasonable accuracy.

5.17.2 CFD model considerations In addition to model validation, there are some other issues that need to be considered during a CFD exercise, some of these having already been mentioned. For example: Explicit vs. implicit. These are two schemes of using finite difference type approximations to convert the governing partial differential equations into an algebraic format. Implicit methods allow arbitrarily large time-step sizes to be used in calculations so that the CPU time required can be reduced. Hence these methods are rather popular in many CFD codes. However, they require iterative solution methods that depend on the character of an under-relaxation in each iteration. This feature may introduce significant errors or very slow convergence in some circ*mstances, such as control volumes with large aspect ratios. In addition, the implicit methods are not accurate for convective processes. Explicit numerical methods, on the other hand, require less computational effort although there is a restriction on selection of time-step.

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Their numerical stability requirements are equivalent to accuracy requirements. More detailed discussion on this account can be found in Anderson (1995). Implicit methods gain their time-step independence by introducing diffusive effects into the approximating equations. The addition of numerical diffusion to physical diffusion, e.g. to heat conduction, may not cause a serious problem as it only modifies the diffusion rate. However, adding numerical diffusion to convective processes completely changes the character of the physical phenomena being modelled. In FLOW-3DX time-steps are automatically controlled by the program to ensure time-accurate approximations. Body-fitted coordinate vs. cartesian coordinate. In a cartesian coordinate system complex fluidsolid interfaces require very fine gridmesh definition to reduce errors introduced by stepping the interface surface to approximate the actual surface profile. This also results in the necessity for non-uniform grid sizes. This results in extra computational time, storage and difficulties if it is necessary to transform the predictions for this grid into a regular or uniform one. In cases where the stepped surface approximation is not considered satisfactory or acceptable, a new mesh definition approach is required, such as body-fitted coordinates and unstructured coordinates as used in finite element analysis, allowing fluid phenomena such as heat transfer and shear stress to be satisfactorily modelled. These approaches do, however, require more computation than the regular grid technique, and while this may not be a long-term problem in view of advances in computational power, elegant solutions require a minimization of such costs and effort. This issue is addressed in the literature, for example, Anderson (1995). Relaxation and convergence criteria. Implicit schemes also need to select one or more numerical parameters to control convergence and relaxation. It is crucial to make a wise choice, as poor ones often lead to either divergence or slow convergence. To reduce the chances of making a poor selection some commercial CFD codes have reduced the range of choice or even developed devices to pre-select the parameters automatically. Fluidsolid interface. Flow becomes more complex at the fluidsolid interface, as the variables are more likely to experience radical changes due to the presence of heat transfer, turbulence and wall shear stress at the surface and within the boundary layer. The size of the control volume can significantly affect the accuracy of the calculation, particularly in some CFD packages where a heat exchange coefficient is calculated locally. In addition, the mesh size also determines the estimation interface area so that total heat transfer prediction through the surface is influenced. Hence finer meshgrid choices at the interface are reflected in the cost of increasing computation time. Selecting a ‘right physical model’ and the ‘right boundary conditions’. Commercial CFD codes always carry a collection of physical models and a suite of boundary conditions to make the codes user-friendly. It is important to select the right one that requires less computation and yet reveals enough details of any real, complex, flow phenomenon, e.g. there are many approximation equations developed to represent complex flow turbulence, and many new models are being formulated. A model that best suits one problem may be totally inadequate for another. Very often new equations are formulated to represent a complex flow phenomenon: the k–ε turbulence model is a typical example. Boundary conditions can be implemented differently from one code to another. Test runs and user manuals are essential to help find out the most suitable boundary condition for a particular problem. The description above is clearly only the tip of the CFD iceberg. However, the issues raised may be seen to be echoed in all the computational examples included in this text, and particularly in the unsteady flow simulations included in Chapter 21

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where the importance of transforming the governing equations into an algebraic format is emphasized, as is the stability conditions determining time-step size and the central importance of boundary condition selection. While commercial packages often major in the output data presentation phase, these constraints are common and have to be addressed if a realistic prediction of the flow condition is to be realized. The issue of validation is still not wholly resolved in many applications, and hence user experience, grounded in a full understanding of the basics of fluid mechanics, becomes an imperative. In many cases, CFD appears more like art than science, as formulating mathematical equations, selecting numerical parameters and many other decisions made in an exercise are more likely to be based on experience or intuition, rather than scientific deduction. More practice makes a better CFD operator.

Concluding remarks The application of the momentum equation to fluid flow situations provides the second of the fundamental tools to be used in understanding fluid flow. Its application to the derivation of Euler’s equation has been demonstrated and the simplification of this equation for steady uniform inviscid flow into Bernoulli’s equation linking flow pressure, velocity and elevation will be utilized continuously. While the use of the momentum equation to determine the forces acting between a flow and its boundaries is important, including the application to turbines and the calculation of engine thrust, the momentum equation has a much wider range of application. Its use in determining the rate of propagation of pressure and surface wave discontinuities has been shown, and later in the text it will be used to calculate drag forces and to analyse unsteady flow situations that can lead to destructive situations. The development of the Navier–Stokes equations has been included in this chapter and has led to the introduction of finite difference numerical solution techniques that will be of application later in the text. A brief description of the available CFD methodologies has also been included, although it is stressed that this subject area now has an extensive literature and application outwith the scope of this text, as represented by the Further Reading list presented here. However, the importance of a fundamental understanding of fluid flow mechanisms prior to embarking upon the use of CFD packages is reinforced by the discussion provided, as is the underlying commonality to the flow simulations presented later in the current text. Chapter 5 should be seen as a resource upon which later analysis and discussion will be based.

Summary of important equations and concepts 1.

2.

3.

The statements of the momentum equation found in Section 5.2 and its application to a range of flow to structure interface applications in Sections 5.5 to 5.8 demonstrate the central importance of this chapter. The use of the momentum equation to determine thrust, drag and torque, equations (5.11), (5.16) and (5.17), are all applications deriving directly from the momentum equation. Euler’s equation is developed in Section 5.12 as a form of the momentum equation and may be seen under certain flow conditions to be analogous to

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Problems

4.

5.

163

Bernoulli’s, which may also be derived by reference to the conservation of energy laws, Chapter 6. Equation (5.21) illustrates the importance of defining the flow as compressible or incompressible as in the latter case integration is possible, leading to a form of Bernoulli’s equation. The propagation of information through a fluid system at the appropriate wave or acoustic velocity depends upon the use of the momentum equation to define wave speed. Equations (5.31) and (5.35) demonstrate these relationships for compressible and free surface flows. The Navier–Stokes equations derive from the momentum considerations introduced in this chapter. Modern computing, allied to finite difference schemes, allows numerical simulation and solution of a wide range of flow conditions. Sections 5.15 to 5.17 introduce these methodologies and reinforce the importance of user experience in defining the appropriate CFD model for any fluid flow phenomenon. The finite difference methods will be returned to in Chapters 18 to 21.

Further reading Abbott, M. B. and Basco, D. R. (1989). Computational Fluid Dynamics – An Introduction for Engineers. Longman, Harlow. Anderson, J. D. Jr (1995). Computational Fluid Dynamics. McGraw-Hill, New York. Smith, G. D. (1985). Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd edition. Clarendon Press, Oxford. Versteeg, H. K. and Malalasekera, W. (1995). An Introduction to Computational Fluid Dynamics, the Finite Volume Method. Longman, Harlow. Zienkiewicz, O. C. and Taylor, R. L. (1991). The Finite Element Method. Volume 2: Solid and Fluid Mechanics. McGraw-Hill, New York.

Problems 5.1 Oil flows through a pipeline 0.4 m in diameter. The flow is laminar and the velocity at any radius r is given by u = (0.6 − 15r2) m s−1. Calculate (a) the volume rate of flow, (b) the mean velocity, (c) the momentum correction factor. [(a) 0.0377 m3 s−1, (b) 0.30 m s−1, (c) 1.333] 5.2 A liquid flows through a circular pipe 0.6 m in diameter. Measurements of velocity taken at intervals along a diameter are: Distance from wall m Velocity m s−1

0 0

0.05 0.1 2.0 3.8

Distance from wall m Velocity m s−1

0.4 0.5 4.5 3.7

0.2 0.3 4.6 5.0

0.55 0.6 1.6 0

(a) Draw the velocity profile, (b) calculate the mean velocity, (c) calculate the momentum correction factor. [(b) 2.82 m s−1, (c) 1.33]

5.3 Calculate the mean velocity and the momentum correction factor for a velocity distribution in a circular pipe given by (vv0) = (yR)1n, where v is the velocity at a distance y from the wall of the pipe, v0 is the centreline velocity, R the radius of the pipe and n an unspecified power. 2 2v n 2 ( n + 1 ) ( 2n + 1 ) −−−−−−−−−−−0−−−−−−−−−− , −−−−−−−−−−−−−−−−−−−−−−− ( n + 1 ) ( 2n + 1 ) 4n 2 ( n + 2 )

5.4 A pipeline is 120 m long and 250 mm in diameter. At the outlet there is a nozzle 25 mm in diameter controlled by a shut-off valve. When the valve is fully open water issues as a jet with a velocity of 30 m s−1. Calculate the reaction of the jet. If the valve can be closed in 0.2 s what will be the resulting rise in pressure at the valve required to bring the water in the pipe to rest in this time? Assume no change in density of the water and no expansion of the pipe. [437.5 N, 180 kN m−2]

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5.5 A uniform pipe 75 m long containing water is fitted with a plunger. The water is initially at rest. If the plunger is forced into the pipe in such a way that the water is accelerated uniformly to a velocity of 1.7 m s−1 in 1.4 s what will be the increase of pressure on the face of the plunger assuming that the water and the pipe are not elastic? If instead of being uniformly accelerated the plunger is driven by a crank 0.25 m long and making 120 rev min−1 so that the plunger moves with simple harmonic motion, what would be the maximum pressure on the face of the piston? [91 kN m−2, 2962.5 kN m−2] 5.6 A flat plate is struck normally by a jet of water 50 mm in diameter with a velocity of 18 m s−1. Calculate (a) the force on the plate when it is stationary, (b) the force on the plate when it moves in the same direction as the jet with a velocity of 6 m s−1, (c) the work done per second and the efficiency in case (b). [(a) 636.2 N, (b) 282.7 N, (c) 1696.2 W, 29.6 per cent] 5.7 A jet of water 50 mm in diameter with a velocity of 18 m s−1 strikes a flat plate inclined at an angle of 25° to the axis of the jet. Determine the normal force exerted on the plate (a) when the plate is stationary, (b) when the plate is moving at 4.5 m s−1 in the direction of the jet, and (c) determine the work done and the efficiency for case (b). [(a) 269 N, (b) 151.2 N, (c) 287.55 W, 5 per cent]

FIGURE 5.25 5.11 Water flows through the pipe bend and nozzle arrangement shown in Fig. 5.26 which lies with its axis in the horizontal plane. The water issues from the nozzle into the atmosphere as a parallel jet with a velocity of 16 m s−1 and the pressure at A is 128 kN m−2 gauge. Friction may be neglected. Find the moment of the resultant force due to the water on this arrangement about a vertical axis through the point X. [65.4 N m counterclockwise]

5.8 A jet of water delivers 85 dm3 s−1 at 36 m s−1 onto a series of vanes moving in the same direction as the jet at 18 m s−1. If stationary, the water which enters tangentially would be diverted through an angle of 135°. Friction reduces the relative velocity at exit from the vanes to 0.80 of that at entrance. Determine the magnitude of the resultant force on the vanes and the efficiency of the arrangement. Assume no shock at entry. [2546 N, 0.783] 5.9 A 5 cm diameter jet delivering 56 litres of water per second impinges without shock on a series of vanes moving at 12 m s−1 in the same direction as the jet. The vanes are curved so that they would, if stationary, deflect the jet through an angle of 135°. Fluid resistance has the effect of reducing the relative velocity by 10 per cent as the water traverses the vanes. Determine (a) the magnitude and direction of the resultant force on the vanes, (b) the work done per second by the vanes and (c) the efficiency of the arrangement. [(a) 1632 N at 21°15′, (b) 18.25 kW, (c) 79.7 per cent] 5.10 Figure 5.25 shows a cross-section of the end of a circular duct through which air (density 1.2 kg m−3) is discharged to atmosphere through a circumferential slot, the exit velocity being 30 m s−1. Find the force exerted on the duct by the air if the gauge pressure at A is 2065 N m−2 below the pressure at outlet. [720.7 N]

FIGURE 5.26 5.12 A ram-jet engine consumes 20 kg of air per second and 0.6 kg of fuel per second. The exit velocity of the gases is 520 m s−1 relative to the engine and the flight velocity is 200 m s−1 absolute. What is the power developed? [1340 kW] 5.13 The resistance of a ship is given by 5.55u6 + 978u1.9 N at a speed of u m s−1. It is driven by a jet propulsion system with intakes facing forward, the efficiency of the jet drive being 0.8 and the efficiency of the pumps 0.72. The vessel is to be driven at 3.4 m s−1. Find (a) the mass of water to be pumped astern per second, (b) the power required to drive the pump. [(a) 10 928 kg s−1, (b) 109.66 kW]

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Problems 5.14 A rocket is fired vertically starting from rest. Neglecting air resistance, what velocity will it attain in 68 s if its initial mass is 13 000 kg and fuel is burnt at the rate of 124 kg s−1, the gases being ejected at a velocity of 1950 m s−1 relative to the rocket? If the fuel is exhausted after 68 s what is the maximum height that the rocket will reach? Take g = 9.8 m s−2. [1372 m s−1, 130.8 km] 5.15 A submarine cruising well below the surface of the sea leaves a wake in the form of a cylinder which is symmetrical about the longitudinal axis of the submarine. The wake velocity on the longitudinal axis is equal to the speed of the submarine through the water, which is 5 m s−1, and decreases in direct proportion to the radius to zero at a radius of 6 m. Calculate the force acting on the submarine and the minimum power required to keep the submarine moving at this speed. Density of sea water = 1025 kg m−3. [483 kN, 2415 kW] 5.16 If u = ay + by represents the velocity of air in the boundary layer of a surface, a and b being constants and y the perpendicular distance from the surface, calculate the shear stress acting on the surface when the speed of the air relative to the surface is 75 m s−1 at a distance of 1.5 mm from the surface and 105 m s−1 when 3 mm from the surface. The viscosity of the air is 18 × 10 −6 kg m−1 s−1. [1.17 N m−2] 2

165

5.17 A lawn sprinkler consists of a horizontal tube with nozzles at each end normal to the tube but inclined upward at 40° to the horizontal. A central bearing incorporates the inlet for the water supply. The nozzles are of 3 mm diameter and are at a distance of 12 cm from the central bearing. If the speed of rotation of the tube is 120 rev min−1 when the velocity of the jets relative to the nozzles is 17 m s−1, calculate (a) the absolute velocity of the jets, (b) the torque required to overcome the frictional resistance of the tube and bearing. [(a) 18.2 m s−1, (b) 0.374 N m s−1] 5.18 Derive an expression for the velocity of transmission of a pressure wave through a fluid of bulk modulus K and mass density ρ. What will be the velocity of sound through water if K = 2.05 × 109 N m−2 and ρ = 1000 kg m−3? [1432 m s−1] 5.19 Calculate the velocity of sound in air assuming an adiabatic process if the temperature is 20 °C, γ = 1.41 and R = 287 J kg−1 K−1. [344.34 m s−1] 5.20 Calculate the velocity of propagation relative to the fluid of a small surface wave along a very wide channel in which the water is 1.6 m deep. If the velocity of the stream is 2 m s−1 what will be the Froude number? [3.96 m s−1, 0.505]

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Chapter 6

The Energy Equation and its Applications Mechanical energy of a flowing fluid 6.2 Steady flow energy equation 6.3 Kinetic energy correction factor 6.4 Applications of the steady flow energy equation 6.5 Representation of energy changes in a fluid system 6.6 The Pitot tube 6.7 Determination of volumetric flow rate via Pitot tube 6.8 Computer program VOLFLO 6.9 Changes of pressure in a tapering pipe 6.10 Principle of the venturi meter 6.1

6.11 Pipe orifices 6.12 Limitation on the velocity of flow in a pipeline 6.13 Theory of small orifices discharging to atmosphere 6.14 Theory of large orifices 6.15 Elementary theory of notches and weirs 6.16 The power of a stream of fluid 6.17 Radial flow 6.18 Flow in a curved path. Pressure gradient and change of total energy across the streamlines 6.19 Vortex motion

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While chapter 5 introduced the momentum equation, the consideration of energy transfers within a flowing fluid is also fundamental to the study and prediction of fluid flow phenomena. This chapter will revisit the development of Bernoulli’s equation and demonstrate that it is merely one special form of a more general energy equation that can accommodate apparent energy losses, due to frictional and separation effects, by application of the conservation of energy principle and the concept of changes in the internal energy of the flowing fluid. The transfer of energy into, or out of, a fluid flow system, by the introduction of mechanical devices such as fans,

pumps or turbines, will be accommodated within the principle of conservation of energy across a predetermined control volume, leading to the introduction of the general steady flow energy equation. The representation of apparent energy losses due to friction and separation losses will be defined and the application of the energy equation to the measurement of fluid flow rate and fluid flow velocity demonstrated for a range of pipe flow and free surface flow conditions. A computer program designed to provide mass flow at a duct crosssection based on velocity traverse data is included. Finally, vortex flow will be introduced. l l l

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MECHANICAL ENERGY OF A FLOWING FLUID

6.1

An element of fluid, as shown in Fig. 6.1, will possess potential energy due to its height z above datum and kinetic energy due to its velocity v, in the same way as any other object. For an element of weight mg, Potential energy = mgz, Potential energy per unit weight = z,

(6.1)

Kinetic energy = mv , −1 2

2

Kinetic energy per unit weight = v 22g.

(6.2)

FIGURE 6.1 Energy of a flowing fluid

A steadily flowing stream of fluid can also do work because of its pressure. At any given cross-section, the pressure generates a force and, as the fluid flows, this crosssection will move forward and so work will be done. If the pressure at a section AB is p and the area of the cross-section is A, Force exerted on AB = pA. After a weight mg of fluid has flowed along the streamtube, section AB will have moved to A′B′: Volume passing AB = mgρg = mρ. Therefore, Distance AA′ = mρA, Work done = Force × Distance AA′ = pA × mρA, Work done per unit weight = pρg.

(6.3)

The term pρg is known as the flow work or pressure energy. Note that the term pressure energy refers to the energy of a fluid when flowing under pressure as part of a continuously maintained stream. It must not be confused with the energy stored in

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Mechanical energy of a flowing fluid

169

a fluid due to its elasticity when it is compressed. The concept of pressure energy is sometimes found difficult to understand. In solid-body mechanics, a body is free to change its velocity without restriction and potential energy can be freely converted to kinetic energy as its level falls. The velocity of a stream of fluid which has a steady volume rate of flow depends on the cross-sectional area of the stream. Thus, if the fluid flows, for example, in a uniform pipe and is incompressible, its velocity cannot change and so the conversion of potential energy to kinetic energy cannot take place as the fluid loses elevation. The surplus energy appears in the form of an increase in pressure. As a result, pressure energy can, in a sense, be regarded as potential energy in transit. Comparing the results obtained in equations (6.1), (6.2) and (6.3) with equation (5.23) it can be seen that the three terms of Bernoulli’s equation are the pressure energy per unit weight, the kinetic energy per unit weight and the potential energy per unit weight; the constant H is the total energy per unit weight. Thus, Bernoulli’s equation states that, for steady flow of a frictionless fluid along a streamline, the total energy per unit weight remains constant from point to point although its division between the three forms of energy may vary: Pressure Kinetic Potential Total energy per energy per + energy per + energy per = = constant, unit weight unit weight unit weight unit weight pρg × v 22g + z = H.

(6.4)

Each of these terms has the dimension of a length, or head, and they are often referred to as the pressure head pρg, the velocity head v 22g, the potential head z and the total head H. Between any two points, suffixes 1 and 2, on a streamline, equation (6.4) gives p 1 v 21 p 2 v 22 + −−− + z 1 = −−−− + −−− + z −−−− ρ 1 g 2g ρ 2 g 2g 2

(6.5)

or Total energy per unit weight at 1 = Total energy per unit weight at 2, which corresponds with equation (5.25). In formulating equation (6.5), it has been assumed that no energy has been supplied to or taken from the fluid between points 1 and 2. Energy could have been supplied by introducing a pump; equally, energy could have been lost by doing work against friction or in a machine such as a turbine. Bernoulli’s equation can be expanded to include these conditions, giving Energy Total energy Total energy Loss per Work done supplied per unit = per unit + unit + per unit − per unit weight at 1 weight at 2 weight weight weight p 1 v 21 p 2 v 22 −−−− + −−− + z 1 = −−−− + −−− + z + h + w – q . ρ 1 g 2g ρ 2 g 2g 2

(6.6)

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EXAMPLE 6.1

The Energy Equation and its Applications

A fire engine pump develops a head of 50 m, i.e. it increases the energy per unit weight of the water passing through it by 50 N m N−1. The pump draws water from a sump at A (Fig. 6.2) through a 150 mm diameter pipe in which there is a loss of energy per unit weight due to friction h1 = 5u 21 2g varying with the mean velocity u1 in the pipe, and discharges it through a 75 mm nozzle at C, 30 m above the pump, at the end of a 100 mm diameter delivery pipe in which there is a loss of energy per unit weight h2 = 12u 22 2g. Calculate (a) the velocity of the jet issuing from the nozzle at C and (b) the pressure in the suction pipe at the inlet to the pump at B.

FIGURE 6.2

Solution (a) We can apply Bernouilli’s equation in the form of equation (6.6) between two points, one of which will be C, since we wish to determine the jet velocity u3, and the other a point at which the conditions are known, such as a point A on the free surface of the sump where the pressure will be atmospheric, so that pA = 0, the velocity vA will be zero if the sump is large, and A can be taken as the datum level so that zA = 0. Then, Energy per Total energy Total energy Loss in Loss in unit weight per unit = per unit + inlet − + discharge supplied by weight at A weight at C pipe pipe, pump Total energy p A v A2 per unit = −−− + −−− + z = 0, ρ g 2g A weight at A Total energy p u2 per unit = −−−C + −−−3 + z 3 , ρ g 2g weight at C pC = Atmospheric pressure = 0, z3 = 30 + 2 = 32 m.

(I)

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171

Therefore, Total energy per unit = 0 + u 23 2g + 32 = u 23 2g + 32 m. weight at C Loss in inlet pipe, h1 = 5 u 21 2g, Energy per unit weight supplied by pump = 50 m, Loss in delivery pipe, h2 = 12 u 22 2g. Substituting in (I), 0 = ( u 23 2g + 32) + 5 u 21 2g − 50 + 12 u 22 2g, u 23 + 5 u 21 + 12 u 22 = 2g × 18.

(II)

From the continuity of flow equation, (π4) d 21 u1 = (π4) d 22 u2 = (π4) d 23 u3; therefore, d 2 75 2 1 u 1 = ⎛ −−3⎞ u 3 = ⎛ −−−−−⎞ u 3 = −u 3 , ⎝ d 1⎠ ⎝ 150⎠ 4 d 2 75 2 9 u 2 = ⎛ −−3⎞ u 3 = ⎛ −−−−−⎞ u 3 = −−−u 3 . ⎝ d 2⎠ ⎝ 100⎠ 16 Substituting in equation (II), u 23 [1 + 5 × ( −14 )2 + 12 × ( −169−− )2] = 2g × 18, 5.109 u 23 = 2g × 18 u3 = 8.314 m s−1. (b) If pB is the pressure in the suction pipe at the pump inlet, applying Bernoulli’s equation to A and B, Total energy Total energy Loss in per unit = per unit + inlet weight at A weight at B pipe, 0 = (pB ρg + u 21 2g + z2) + 5 u 21 2g, pB ρg = −z2 − 6 u 21 2g, z2 = 2 m, u1 = −14 u3 = 8.3144 = 2.079 m s−1, pB ρg = − (2 + 6 × 2.07922g) = −(2 + 1.32) = −3.32 m, pB = −1000 × 9.81 × 3.32 = 32.569 kN m−2 below atmospheric pressure.

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6.2

STEADY FLOW ENERGY EQUATION

Bernoulli’s equation and its expanded form, as given in equation (6.6), were developed from Euler’s equation (5.21) which, in turn, was derived from the momentum equation. It is possible to develop an energy equation for the steady flow of a fluid from the principle of conservation of energy, which states: For any mass system, the net energy supplied to the system equals the increase of energy of the system plus the energy leaving the system. Thus, if ∆E is the increase of energy of the system, ∆Q is the energy supplied to the system and ∆W the energy leaving the system, then, considering the energy balance for the system, ∆E = ∆Q − ∆W. The energy of a mass of fluid will have the following forms: 1. 2. 3.

internal energy due to the activity of the molecules of the fluid forming the mass; kinetic energy due to the velocity of the mass of fluid itself; potential energy due to the mass of fluid being at a height above the datum level and acted upon by gravity.

Suppose that at section AA (Fig. 6.3) through a streamtube the cross-sectional area is A1, the pressure p1, velocity v1, density ρ1, internal energy per unit mass e1 and height above datum z1, while the corresponding values at BB are A2, p2, v2, ρ2, e2 and z2. The fluid flows steadily with a mass flow rate A and between sections AA and BB the fluid receives energy at the rate of q per unit mass and loses energy at the rate of w per unit mass. For example, q may be in the form of heat energy, while w might take the form of mechanical work. Energy entering Kinetic Potential Internal at AA in unit = + + = A( −12 v 21 + gz1 + e1), energy energy energy time, E1 Energy leaving at BB in unit = A( −12 v 22 + gz2 + e2). time, E2 FIGURE 6.3 Steady flow energy equation

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6.2

Steady flow energy equation

173

Therefore ∆E = E2 − E1 = A [ −12 ( v 22 − v 21 ) + g(z2 − z1) + (e2 − e1)].

(6.7)

This change of energy has occurred because energy has entered and left the fluid between AA and BB. Also, work is done on the fluid in the control volume between the two sections AA and BB by the fluid entering at AA and by the fluid in the control volume as it leaves at BB. Energy entering per unit time between AA and BB = Aq, Energy leaving per unit time between AA and BB = Aw. As the fluid flows, work will be done by the fluid entering at AA since a force p1A1 is exerted on the cross-section by the pressure p1 and, in unit time t, the fluid which was at AA will move a distance s1 to A′A′: Work done in unit time on the fluid at AA = p1A1s1t. But, Mass passing per unit time, A = ρ1A1s1t; therefore, A1s1 = Aρ1, Work done per unit time on the fluid at AA = p1Aρ1. Similarly, Change of energy = Work done on fluid at AA of the system, ∆E − Work done by fluid at BB + Energy entering between AA and BB − Energy leaving between AA and BB = Ap1ρ1 − Ap2ρ2 + Aq − Aw = A(q − w + p1ρ1 − p2ρ2).

(6.8)

Comparing equations (6.7) and (6.8), 1 − 2

( v 22 − v 21 ) + g(z2 − z1) + (e2 − e1) = p1ρ1 − p2ρ2 + q − w.

Thus, gz1 + −12 v 21 + ( p1ρ1 + e1) + q − w = gz2 + −12 v 22 + ( p2ρ2 + e2).

(6.9)

The terms ( p1ρ1 + e1) and ( p2ρ2 + e2) can be replaced by the enthalpies H1 and H2, giving gz1 + −12 v 21 + H1 + q − w = gz2 + −12 v 22 + H2.

(6.10)

This steady flow energy equation can be applied to all fluids, real or ideal, whether liquids, vapours or gases, provided that flow is continuous and energy is transferred

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steadily to or from the fluid at constant rates q and w, conditions remaining constant with time and all quantities being constant across the inlet and outlet sections. In thermodynamics, it is usual to distinguish between heat and work and to treat q as the net inflow of heat and w as the net outflow of mechanical work per unit mass.

6.3

KINETIC ENERGY CORRECTION FACTOR

The derivation of Bernoulli’s equation and the steady flow energy equation has been carried out for a streamtube assuming a uniform velocity across the inlet and outlet sections. In a real fluid flowing in a pipe or over a solid surface, the velocity will be zero at the solid boundary and will increase as the distance from the boundary increases. The kinetic energy per unit weight of the fluid will increase in a similar manner. If the cross-section of the flow is assumed to be composed of a series of small elements of area δA and the velocity normal to each element is u, the total kinetic energy passing through the whole cross-section can be found by determining the kinetic energy passing through an element in unit time and then summing by integrating over the whole area of the section. Mass passing through element = ρδA × u, in unit time Kinetic energy per unit time = passing through element

−1 2

× Mass per unit time × (Velocity)2

= −12 ρδAu3, Total kinetic energy passing in = unit time Total weight passing in unit time =

ρu δA, −1 2

3

ρguδA.

Thus, taking into account the variation of velocity across the stream, True kinetic energy 1 1ρ u 3 δ A = −−− −−−−−−−−−−−, per unit weight 2g 1ρ u δA

(6.11)

which is not the same as B 22g, where B is the mean velocity: B=

(uA) dA.

Thus, True kinetic energy per unit weight = α B 22g,

(6.12)

where α is the kinetic energy correction factor, which has a value dependent on the shape of the cross-section and the velocity distribution. For a circular pipe, assuming Prandtl’s one-seventh power law, u = u max( yR)17, for the velocity at a distance y from the wall of a pipe of radius R, the value of α = 1.058.

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6.4

6.4

Applications of the steady flow energy equation

175

APPLICATIONS OF THE STEADY FLOW ENERGY EQUATION

A comparison of equations (6.6) and (6.9) is helpful in applying the steady flow energy equation to a wide range of fluid flow conditions. Reference to the control volume AA′BB′ in Fig. 6.3 allows the steady flow energy equation to be recast, from equation (6.9) for a constant density, i.e. incompressible flow, as p1 + −12 ρv 2 + ρgz1 + ρq − ρw = p2 + −12 ρv 2 + ρgz2 + ρ∆e

(6.13)

where, as in equation (5.24), the terms represent energy per unit volume. However, it is also clear that, in order to maintain dimensional hom*ogeneity, each term in this representation of the steady flow energy equation has the dimensions of pressure. It will be shown how it is possible to utilize this particular form of the steady flow energy equation with remarkable ease in the definition of a wide range of fluid flow conditions. The terms ρq and ρw may be identified, for example in Fig. 6.4, as the pressure rise across a pump or fan maintaining flow through a pipe or duct, or the pressure drop across a turbine. Clearly each of these terms has values dependent upon the particular flow rate passing through the system, identified in this form of the steady flow energy equation by the mean flow velocity at the control volume boundaries AA′ or BB′. The term ρ∆e represents an energy ‘loss’ due to frictional or separation losses between the boundaries of the control volume. As energy ‘loss’ cannot occur within the control volume it follows that this term represents a transfer of energy from one category to another, in this case into the fluid internal energy as identified earlier. Again the value of ρ∆e will depend upon the flow rate in the system and on the fluid and conduit parameters; appropriate expressions defining these energy transfers in

FIGURE 6.4 Energy addition or extraction at rotodynamic machines

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terms of the flow and pipe parameters will be developed later. It is sufficient at this stage to state that the pressure changes associated with these transfers, i.e. the ρ∆e term, are dependent upon the square of flow velocity. Thus the steady flow energy equation may be seen as an energy audit across a user-defined control volume. The appropriate choice of control volume makes the steady flow energy equation an immensely powerful tool in defining a wide range of flow conditions.

6.4.1 Choice of control volume boundary conditions for the steady flow energy equation Referring to the general definition of the steady flow energy equation in Fig. 6.3, equation (6.13) may be written as p1 + −12 ρv 2 + ρgz1 + ∆pinput − ∆pout = p2 + −12 ρv 2 + ρgz2 + ∆pF+S , where ∆pinput and ∆pout refer to the pressure rise experienced across a fan, or pump, and the pressure drop across a turbine, respectively. Suffixes 1 and 2 refer to the entry and exit boundary conditions of the control volume. The pressure loss experienced as a result of friction and separation of the flow from the walls of the conduit is encapsulated in the ∆pF+S term and will be shown to be defined by a term of the form −12 ρKu2, where u is the local flow velocity and K is a constant dependent upon the conduit parameters, i.e. length, diameter, roughness or fitting type. The steady flow energy equation may thus be written as p1 + −12 ρv 2 + ρgz1 + ∆pinput − ∆pout = p2 + −12 ρv 2 + ρgz2 + −12 ρKu 2, where all terms are defined in the dimensions of pressure and hence are amenable to direct experimental measurement for any particular flow condition. The steady flow energy equation in this form may be easily applied to a range of flow conditions by ‘dropping’ terms that are irrelevant to the particular case to be studied. A range of common examples are presented below, many of which will be returned to later in the text in more detail. The steady flow energy equation may be applied across a control volume whose boundaries may be taken as the water surfaces in each reservoir (Fig. 6.5). FIGURE 6.5 Flow between two reservoirs open to atmosphere

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Applications of the steady flow energy equation

177

As there are no subtractions or additions of energy due to the presence of turbines or pumps in this system the steady flow energy equation reduces to p1 + −12 ρv 2 + ρgz1 = p2 + −12 ρv 2 + ρgz2 + −12 ρKu 2. Further simplifications may be made by a careful study of the conditions at the system boundaries. In a gauge pressure frame of reference, the values of p1 and p2, the atmospheric pressure at the reservoir open surface, may be taken as zero. Further, as the surface areas of the two reservoirs may be assumed to be very large compared with the cross-sectional area of the connecting pipeline, it follows from an application of the continuity of flow equation between the two reservoirs that the values of the reservoir surface velocity, either v1 vertically down at 1 or v2 vertically up at 2, may be disregarded when compared with the flow velocity, u, in the actual pipeline. Therefore it is acceptable to neglect the surface kinetic energy terms in comparison with the combined friction and separation loss term, reducing the steady flow energy equation to

ρg(z1 − z2) = −12 ρKu 2, i.e. the expected result that the difference in reservoir surface level, or potential energy, is solely responsible for overcoming the frictional and separation losses incurred in a flow between the two reservoirs. Therefore the choice of pipeline, in terms of its diameter, roughness or length, or the setting of any valves along the pipe, determines the throughflow – an expected result that conforms to our knowledge of the physical world. If the upstream reservoir were to be replaced by a large pressurized tank at a pressure pt above atmosphere, so that the continuity equation continued to support the dropping of the surface velocity terms, then the form of the steady flow energy equation would become pt + ρg(z1 − z2 ) = −12 ρKu 2, and for any pipeline condition the flow delivered would rise compared with the open surface reservoir case – again a result that could be predicted. An identical process may be seen to apply in the consideration of a simple ventilation system extracting air from a room at atmospheric pressure and discharging it to atmosphere at approximately the same elevation (Fig. 6.6). If the boundaries of the control volume are positioned sufficiently far from the ductwork entry and exit grilles, then the local air velocity, and hence the associated kinetic energy, may be ignored in comparison with the ductwork air flow velocity and the associated friction and separation loss term. In this special case the pressure terms at the boundaries (points 1 and 2 in Fig. 6.6) may also be ignored as both are atmospheric, and as the fluid is air and the elevation difference across the control volume is stated to be small, the potential energy terms may also be dropped. Hence the steady flow energy equation reduces to the almost trivial ∆pinput = −12 ρKu 2. FIGURE 6.6 Room ventilation

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FIGURE 6.7 Pressurized room air supply and extract ventilation

It will be appreciated that this form of the steady flow energy equation may also be utilized to represent the case where it is necessary either to supply or to extract air from a space held above atmospheric pressure. Examples of this would be clean rooms in electronics facilities or hospital operating theatres. In both cases it is required that any air leakage be out of the space (Fig. 6.7). In the case where the fan is expected to supply air from atmosphere to a room held above atmospheric pressure, the steady flow energy equation becomes ∆pinput = proom + −12 ρKu 2, and the extract fan receives support from the pressure gradient existing between the room and the external atmosphere, as illustrated by the appropriate form of the steady flow energy equation, ∆pout = −proom + −12 ρKu 2.

6.5

REPRESENTATION OF ENERGY CHANGES IN A FLUID SYSTEM

The changes of energy, and its transformation from one form to another which occurs in a fluid system, can be represented graphically. In a real fluid system, the total energy per unit weight will not remain constant. Unless energy is supplied to the system at some point by means of a pump, it will gradually decrease in the direction of motion due to losses resulting from friction and from the disturbance of flow at changes of pipe section or as a result of changes of direction. In Fig. 6.8, for example, the flow of FIGURE 6.8 Energy changes in a fluid system

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Representation of energy changes in a fluid system

179

water from the reservoir at A to the reservoir at D is assisted by a pump which develops a head h p, thus providing an addition to the energy per unit weight of h p. At the surface of reservoir A, the fluid has no velocity and is at atmospheric pressure (which is taken as zero gauge pressure), so that the total energy per unit weight is represented by the height HA of the surface above datum. As the fluid enters the pipe with velocity u1, there will be a loss of energy due to disturbance of the flow at the pipe entrance and a continuous loss of energy due to friction as the fluid flows along the pipe, so that the total energy line will slope downwards. At B there is a change of section, with an accompanying loss of energy, resulting in a change of velocity to u2. The total energy line will continue to slope downwards, but at a greater slope since u2 is greater than u1 and friction losses are related to velocity. At C, the pump will put energy into the system and the total energy line will rise by an amount h p. The total energy line falls again due to friction losses and the loss due to disturbance at the entry to the reservoir, where the total energy per unit weight is represented by the height of the reservoir surface above datum (the velocity of the fluid being zero and the pressure atmospheric). If a piezometer tube were to be inserted at point 1, the water would not rise to the level of the total energy line, but to a level u 21 2g below it, since some of the total energy is in the form of kinetic energy. Thus, at point 1, the potential energy is represented by z1, the pressure energy by p1ρg and the kinetic energy by u 21 2g, the three terms together adding up to the total energy per unit weight at that point. Similarly, at points 2 and 3, the water would rise to levels p2ρg and p3ρg above the pipe, which are u 22 2g and u 23 2g, respectively, below the total energy line. The line joining all the points to which the water would rise, if an open stand pipe was inserted, is known as the hydraulic gradient, and runs parallel to the total energy line at a distance below it equal to the velocity head. If, as in Fig. 6.9, a pipeline rises above the hydraulic gradient, the pressure in the portion PQ will be below atmospheric pressure and will form a siphon. Under reduced pressure, air or other gases may be released from solution or a vapour pocket may form and interrupt the flow. While earlier examples have concentrated on the application of the steady flow energy equation between the extremities of a system, in order to benefit from the resulting simplifications it is clear that the boundaries of the chosen control volume may be placed at any two points of interest along the system conduits. Figure 6.9 illustrates one example where this may be helpful and where a concentration on the extremities of the system may give quite misleading results. Application of the steady state flow energy equation between the open supply reservoir and the apex of the siphon allows the practicality of the siphon to be assessed. FIGURE 6.9 Pipeline rising above hydraulic gradient

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Between the system extremities, i.e. the open reservoir surfaces at 1 and 2, the steady flow energy equation implies that the flow is governed by the expression

ρg(z1 − z2) = −12 ρKu 2, regardless of the intermediate elevation of the pipeline and its possible failure to operate under subatmospheric conditions. However, it is possible to apply the steady flow energy equation between the reservoir surface at 1 and the apex of the pipeline at A in order to assess the practicality of the siphon. Thus between 1 and A in Fig. 6.9, the steady flow energy equation becomes p1 + −12 ρv 2 + ρgz1 = pA + −12 ρ vA + ρgzA + −12 ρKu 2, where the friction and separation loss term, −12 ρKu 2, refers to the losses from 1 to A. The surface velocity v1 may be neglected relative to the flow velocity in the pipeline, vA. In this case the pipeline flow velocity u used in the loss calculation is identical to the local velocity at A, vA, as the pipe up to A has been assumed to be of constant diameter. Thus the steady flow energy equation applied from the entry reservoir surface to the apex of the siphon becomes pA = ρg(z1 − zA ) − −12 ρu 2(1 + K ). Setting pA to gas release pressure, fluid vapour pressure or indeed absolute zero, yields information as to the acceptability of any zA or pipe length to the apex of the siphon value, the latter being contained in the loss coefficient K, which also depends on the other pipeline parameters mentioned previously, i.e. diameter and roughness.

6.6

THE PITOT TUBE

The Pitot tube is used to measure the velocity of a stream and consists of a simple Lshaped tube facing into the oncoming flow (Fig. 6.10(a)). If the velocity of the stream at A is u, a particle moving from A to the mouth of the tube B will be brought to rest so that u0 at B is zero. By Bernoulli’s equation,

FIGURE 6.10 Pitot tube

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Determination of volumetric flow rate via Pitot tube

181

Total energy per unit Total energy per unit = weight at A weight at B, u 22g + pρg = u 20 2g + p0ρg, p0ρg = u 22g + pρg, since u0 = 0. Thus, p0 will be greater than p. Now pρg = z and p0ρg = h + z. Therefore, u22g = ( p0 − p)ρg = h, Velocity at A = u = (2gh). When the Pitot tube is used in a channel, the value of h can be determined directly (as in Fig. 6.10(a)), but, if it is to be used in a pipe, the difference between the static pressure and the pressure at the impact hole must be measured with a differential pressure gauge, using a static pressure tapping in the pipe wall (as in Fig. 6.10(b)) or a combined Pitot–static tube (as in Fig. 6.10(c)). In the Pitot–static tube, the inner tube is used to measure the impact pressure while the outer sheath has holes in its surface to measure the static pressure. While, theoretically, the measured velocity u = (2gh), Pitot tubes may require calibration. The true velocity is given by u = C(2gh), where C is the coefficient of the instrument and h is the difference of head measured in terms of the fluid flowing. For the Pitot–static tube shown in Fig. 6.10(c), the value of C is unity for values of Reynolds number ρuDµ 3000, where D is the diameter of the tip of the tube.

6.7

DETERMINATION OF VOLUMETRIC FLOW RATE VIA PITOT TUBE

It will be shown in later chapters that relationships for the velocity distribution across fully developed pipe and duct flow may be utilized to determine the relationship between the velocity at the pipe centreline, or any other identified location, and the theoretical mean velocity in the conduit. Thus volumetric flow rate may be determined by a single Pitot tube or hot-wirefilm anemometer. However, in practice this approach is flawed as it depends upon the flow conforming to a particular theoretical velocity distribution in circular cross-section flows and is particularly dubious for noncircular ducts. A more common device utilized particularly in the study of fan characteristics is to mount a grid of Pitot tubes across the flow and to determine the volume flow by recording local velocities within preset areas; a subsequent summation yields the overall duct flow rate. Figure 6.11 illustrates the guidance offered by a leading fan manufacturer for the flow integration in circular and rectangular ducts. A suitable method for circular ducts is to divide the duct cross-section into three or four concentric equal areas and to determine the velocity in each by averaging six velocity readings taken at 60° intervals round this annulus. Rectangular ducts should be divided into at least 25 equal rectangular areas by subdividing each side into five equal length increments. For ‘long,

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FIGURE 6.11

Circular ducts: three zone, 18 point, traverse

Location of velocity measurements in ducts (Woods Air Movement Ltd)

d d d -----1 -----2 -----3 D D D 0.032 0.135 0.321 four zone, 24 point, traverse d1 d d d4 ---------2 -----3 ----D D D D 0.021 0.117 0.184 0.345

Rectangular ducts: five zones per side, 25 points a1 a2 a3 ---------A A A 0.074 0.288 0.500 6 × 5 zones, 30 points b b b -----1 -----2 -----3 B B B 0.061 0.235 0.437 7 × 5 zones, 35 points b1 b b b ---------2 -----3 -----4 B B B B 0.053 0.203 0.366 0.500

thin’ cross-sections better accuracy may be obtained by increasing the subdivision of the ‘long’ side to six or seven increments, yielding 30 or 35 areas across the flow. Flow velocity is then measured within each area. The duct volumetric flow may then be calculated from the relationship ∑ ( V A local ) Vmean = −−−−−−−local −−−−−−−−−−−− . A duct If the output of the Pitot tubes and the duct static pressure are recorded, either as pressures or as heights of manometer fluid, then this expression may be modified to yield volumetric flow directly from these readings by substituting for the local velocity. See Section 4.11 and Example 4.1.

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6.9

6.8

COMPUTER PROGRAM

Changes of pressure in a tapering pipe

183

VOLFLO

Program VOLFLO allows the determination of volumetric flow at a duct section from either Pitot–static pressure readings or velocity values following a traverse of the section. The program may use the duct area subdivision suggested in Fig. 6.11 or any other user-specified configuration of traverse measurement locations. The program handles both circular and rectangular section ducts and accepts velocity data in m s−1 or pressure data in mm of manometer fluid or N m−2. The data required for either circular or rectangular ducts are velocity or pressure, in the latter case either in mm or N m−2. Constant static pressure at the traverse location is required if the traverse only records Pitot pressure at each location. The density of the flow and the manometer fluid may be required together with the dimensions of the duct and the number of sampling points and zones across the section, see Fig. 6.11.

6.8.1 Application example For a rectangular section 0.3 m wide, 0.2 m deep, having five width and five depth increments and hence 25 traverse locations, velocity data are available as follows: 1.6 2.0 2.2 2.0 1.7

1.9 3.4 6.9 3.7 2.0

2.1 2.0 6.8 3.5 10.0 7.0 7.0 3.8 2.3 2.1

1.8 2.0 2.3 2.1 1.9

The VOLFLO determined flow rate is 0.202 m3 s−1 with an average velocity of 3.36 m s−1.

6.8.2 Additional investigations using VOLFLO The computer program calculation may also be used to investigate the divergence in predicted volumetric flow rate when coarser grid settings are used for both rectangular or circular section ducts.

6.9

CHANGES OF PRESSURE IN A TAPERING PIPE

Changes of velocity in a tapering pipe were determined by using the continuity of flow equation (Section 4.12). Change of velocity will be accompanied by a change in the kinetic energy per unit weight and, consequently, by a change in pressure, modified by any change of elevation or energy loss, which can be determined by the use of Bernoulli’s equation.

EXAMPLE 6.2

A pipe inclined at 45° to the horizontal (Fig. 6.12) converges over a length l of 2 m from a diameter d1 of 200 mm to a diameter d2 of 100 mm at the upper end. Oil of relative density 0.9 flows through the pipe at a mean velocity C1 at the lower end of 2 m s−1. Find the pressure difference across the 2 m length ignoring any loss of energy, and the difference in level that would be shown on a mercury manometer connected

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FIGURE 6.12 Pressure change in a tapering pipe

across this length. The relative density of mercury is 13.6 and the leads to the manometer are filled with the oil.

Solution Let A1, C1, p1, d1, z1 and A2, C2, p2, d2, z2 be the area, mean velocity, pressure, diameter and elevation at the lower and upper sections, respectively. For continuity of flow, assuming the density of the oil to be constant, A1C1 = A2C2 , so that

C2 = (A1A2)C1 ,

where

A1 = (π4) d 12 and A2 = (π4) d 22 .

Thus,

C2 = (d1d2 )2C1 = (0.20.1)2 × 2 = 8 m s−1.

Applying Bernoulli’s equation to the lower and upper sections, assuming no energy losses, Total energy per unit Total energy per unit = weight at section 1 weight at section 2, p1ρ0 g + C 21 2g + z1 = p2ρ0 g + C 22 2g + z2 , p1 − p2 = −12 ρ0( C 22 − C 21 ) + ρ0 g(z2 − z1).

(I)

z2 − z1 = l sin 45° = 2 × 0.707 = 1.414 m

Now,

and, since the relative density of the oil is 0.9, if ρH O = density of water, then ρoil = 0.9ρH O = 0.9 × 1000 = 900 kg m−3. Substituting in equation (I), 2

2

p1 − p2 =

−1 2

× 900(82 − 22 ) + 900 × 9.81 × 1.414 N m−2

= 8829(3.058 + 1.414) = 39 484 N m−2.

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6.10

Principle of the venturi meter

185

For the manometer, the pressure in each limb will be the same at level XX; therefore, p1 + ρoil gz1 = p2 + ρoil g(z2 − h) + ρman gh, ( p1 − p2)ρoil g + z1 − z2 = h( ρman ρoil − 1), p –p ρ −−−−−− ⎞ ⎛ −−1−−−−−−−2 + z 1 – z 2⎞ . h = ⎛ −−−−−−−oil ⎝ ρ man – ρ oil ⎠ ⎝ ρ oil g ⎠ Putting ρoil = 0.9 ρ H O = 900 kg m−3 and ρman = 13.6 ρ H O , 2

2

h = [0.9(13.6 − 0.9)][39 484(900 × 9.81) − 1.414] = 0.217 m.

6.10 PRINCIPLE OF THE VENTURI METER As shown by equation (I) in Example 6.2, the pressure difference between any two points on a tapering pipe through which a fluid is flowing depends on the difference of level z2 − z1, the velocities C2 and C1, and, therefore, on the volume rate of flow Q through the pipe. Hence, the pressure difference can be used to determine the volume rate of flow for any particular configuration. The venturi meter uses this effect for the measurement of flow in pipelines. As shown in Fig. 6.13, it consists of a short converging conical tube leading to a cylindrical portion, called the throat, of smaller

FIGURE 6.13 Inclined venturi meter and U-tube

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diameter than that of the pipeline, which is followed by a diverging section in which the diameter increases again to that of the main pipeline. The pressure difference from which the volume rate of flow can be determined is measured between the entry section 1 and the throat section 2, often by means of a U-tube manometer (as shown). The axis of the meter may be inclined at any angle. Assuming that there is no loss of energy, and applying Bernoulli’s equation to sections 1 and 2, z1 + p1ρg + v 21 2g = z2 + p2 ρg + v 22 2g, v 22 − v 21 = 2g[( p1 − p2)ρg + (z1 − z2)].

(6.14)

For continuous flow, A1v1 = A2v2 or v2 = (A1A2)v1. Substituting in equation (6.14), v 21 [(A1A2)2 − 1] = 2g[( p1 − p2)ρg + (z1 − z2)], A v 1 = −−−−− −−−−−2−−−−−−− ( A 21 – A 22 ) 12

p –p 2g ⎛ −−1−−−−−−−2 + z 1 – z 2⎞ . ⎝ ρg ⎠

Volume rate of flow, Q = A1v1 = [A1A2 ( A 21 − A 22 )12 ](2gH), where H = ( p1 − p2)ρg + (z1 − z2) or, if m = Area ratio = A1A2, Q = [A1(m2 − 1)12 ](2gH ).

(6.15)

In practice, some loss of energy will occur between sections 1 and 2. The value of Q given by equation (6.15) is a theoretical value which will be slightly greater than the actual value. A coefficient of discharge Cd is, therefore, introduced: Actual discharge, Qactual = Cd × Qtheoretical. The value of H in equation (6.15) can be found from the reading of the U-tube gauge (Fig. 6.13). Assuming that the connections to the gauge are filled with the fluid flowing in the pipeline, which has a density ρ, and that the density of the manometric liquid in the bottom of the U-tube is ρman, then, since pressures at level XX must be the same in both limbs, pX = p1 + ρg(z1 − z) = p2 + ρg(z2 − z − h) + ρmanhg. Expanding and rearranging, H = ( p1 − p2)ρg + (z1 − z2 ) = h ( ρman ρ − 1). Equation (6.15) can now be written

Q = [ A 1 ( m 2 – 1 ) 12 ]

ρ man ⎞ 2gh ⎛ −−− −− – 1 . ⎝ ρ ⎠

(6.16)

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Pipe orifices

187

Note that equation (6.16) is independent of z1 and z2, so that the manometer reading h for a given rate of flow Q is not affected by the inclination of the meter. If, however, the actual pressure difference ( p1 − p2) is measured and equation (6.14) or (6.15) used, the values of z1 and z2, and, therefore, the slope of the meter, must be taken into account.

EXAMPLE 6.3

A venturi meter having a throat diameter d2 of 100 mm is fitted into a pipeline which has a diameter d1 of 250 mm through which oil of specific gravity 0.9 is flowing. The pressure difference between the entry and throat tappings is measured by a U-tube manometer, containing mercury of specific gravity 13.6, and the connections are filled with the oil flowing in the pipeline. If the difference of level indicated by the mercury in the U-tube is 0.63 m, calculate the theoretical volume rate of flow through the meter.

Solution Using equation (6.16), Area at entry, A1 = (π4) d 21 = (π4)(0.25)2 = 0.0491 m2, Area ratio, m = A1A2 = (d1d2)2 = (0.250.10)2 = 6.25, h = 0.63 m, ρHg = ρman = 13.6 × ρ H O , 2

ρoil = 0.9 ρ H O , 2

where ρHg = density of mercury, ρH O = density of water and ρoil = density of oil. Substituting in equation (6.16), 2

Q = [0.0491(6.252 − 1)12][2 × 9.81 × 0.63(13.60.9 − 1)] = 0.105 m3 s−1.

6.11 PIPE ORIFICES The venturi meter described in Section 6.10 operates by changing the cross-section of the flow, so that the cross-sectional area is less at the downstream pressure tapping than at the upstream tapping. A similar effect can be achieved by inserting an orifice plate which has an opening in it smaller than the internal diameter of the pipeline (as shown in Fig. 6.14). The orifice plate produces a constriction of the flow as shown, the cross-sectional area A2 of the flow immediately downstream of the plate being FIGURE 6.14 Pipe orifice meter

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approximately the same as that of the orifice. The arrangement is cheap compared with the cost of a venturi meter, but there are substantial energy losses. The theoretical discharge can be calculated from equation (6.14) or (6.15), but the actual discharge may be as little as two-thirds of this value. A coefficient of discharge must, therefore, be introduced in the same way as for the venturi meter, a typical value for a sharpedged orifice being 0.65.

6.12 LIMITATION ON THE VELOCITY OF FLOW IN A PIPELINE Since Bernoulli’s equation requires that the total energy per unit weight of a flowing fluid shall, if there are no losses, remain constant, any increase in velocity or elevation must be accompanied by a reduction in pressure. Furthermore, since the pressure can never fall below absolute zero, there will be a maximum velocity for a given configuration of a pipeline which cannot be exceeded. For a flowing liquid, the pressure will never fall to absolute zero since air or vapour will be released and form pockets in the flow well before this can occur.

6.13 THEORY OF SMALL ORIFICES DISCHARGING TO ATMOSPHERE An orifice is an opening, usually circular, in the side or base of a tank or reservoir, through which fluid is discharged in the form of a jet, usually into the atmosphere. The volume rate of flow discharged through an orifice will depend upon the head of the fluid above the level of the orifice and it can, therefore, be used as a means of flow measurement. The term ‘small orifice’ is applied to an orifice which has a diameter, or vertical dimension, which is small compared with the head producing flow, so that it can be assumed that this head does not vary appreciably from point to point across the orifice. Figure 6.15 shows a small orifice in the side of a large tank containing liquid with a free surface open to the atmosphere. At a point A on the free surface, the pressure pA is atmospheric and, if the tank is large, the velocity vA will be negligible. In the

FIGURE 6.15 Flow through a small orifice

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Theory of small orifices discharging to atmosphere

189

region of the orifice, conditions are rather uncertain, but at some point B in the jet, just outside the orifice, the pressure pB will again be atmospheric and the velocity vB will be that of the jet v. Taking the datum for potential energy at the centre of the orifice and applying Bernoulli’s equation to A and B, assuming that there is no loss of energy, Total energy per Total energy per = unit weight at A unit weight at B, zA + v A2 2g + pA ρg = zB + v B2 2g + pB ρg. Putting zA − zB = H, vA = 0, vB = v and pA = pB , Velocity of jet, v = (2gH).

(6.17)

This is a statement of Torricelli’s theorem, that the velocity of the issuing jet is proportional to the square root of the head producing flow. Equation (6.17) applies to any fluid, H being expressed as a head of the fluid flowing through the orifice. For example, if an orifice is formed in the side of a vessel containing gas of density ρ at a uniform pressure p, the value of H would be pρg. Theoretically, if A is the crosssectional area of the orifice, Discharge, Q = Area × Velocity = A(2gH).

(6.18)

In practice, the actual discharge is considerably less than the theoretical discharge given by equation (6.18), which must, therefore, be modified by introducing a coefficient of discharge Cd, so that Actual discharge, Qactual = CdQtheoretical = CdA(2gH).

(6.19)

There are two reasons for the difference between the theoretical and actual discharges. First, the velocity of the jet is less than that given by equation (6.17) because there is a loss of energy between A and B: Actual velocity at B = Cv × v = Cv(2gH),

(6.20)

where Cv is a coefficient of velocity, which has to be determined experimentally and is of the order of 0.97. Second, as shown in Fig. 6.16, the paths of the particles of the fluid converge on the orifice, and the area of the issuing jet at B is less than the area of the orifice A at C. FIGURE 6.16 Contraction of issuing jet

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In the plane of the orifice, the particles have a component of velocity towards the centre and the pressure at C is greater than atmospheric pressure. It is only at B, a small distance outside the orifice, that the paths of the particles have become parallel. The section through B is called the vena contracta. Actual area of jet at B = Cc A,

(6.21)

where Cc is the coefficient of contraction, which can be determined experimentally and will depend on the profile of the orifice. For a sharp-edged orifice of the form shown in Fig. 6.16, it is of the order of 0.64. We can now determine the actual discharge from equations (6.20) and (6.21): Actual discharge = Actual area at B × Actual velocity at B = Cc A × Cv(2gH ) = Cc × Cv A(2gH ).

(6.22)

Comparing equation (6.22) with equation (6.19), we see that the relation between the coefficients is Cd = Cc × Cv . The values of the coefficient of discharge, the coefficient of velocity and the coefficient of contraction are determined experimentally and values are available for standard configurations in British Standard specifications. To determine the coefficient of discharge, it is only necessary to collect, or otherwise measure, the actual volume discharged from the orifice in a given time and compare this with the theoretical discharge given by equation (6.18). measured discharge Coefficient of discharge, C d = Actual −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− . Theoretical discharge Similarly, the actual area of the jet at the vena contracta can be measured, of jet at vena contracta Coefficient of contraction, C c = Area −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− . Area of orifice In the same way, if the actual velocity of the jet at the vena contracta can be found, at vena contracta Coefficient of velocity, C v = Velocity −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− . Theoretical velocity If the orifice is not in the bottom of the tank, one method of measuring the actual velocity of the jet is to measure its profile.

EXAMPLE 6.4

A jet of water discharges horizontally into the atmosphere from an orifice in the vertical side of a large open-topped tank (Fig. 6.17). Derive an expression for the actual velocity v of a jet at the vena contracta if the jet falls a distance y vertically in a horizontal distance x, measured from the vena contracta. If the head of water above the orifice is H, determine the coefficient of velocity.

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191

FIGURE 6.17 Determination of the coefficient of velocity

If the orifice has an area of 650 mm2 and the jet falls a distance y of 0.5 m in a horizontal distance x of 1.5 m from the vena contracta, calculate the values of the coefficients of velocity, discharge and contraction, given that the volume rate of flow is 0.117 m3 and the head H above the orifice is 1.2 m.

Solution Let t be the time taken for a particle of fluid to travel from the vena contracta A (Fig. 6.17) to the point B. Then

or

x = vt

and y = −12 gt 2,

v = xt

and t = (2yg).

Eliminating t, Velocity at the vena contracta, v = (gx22y). This is the actual velocity of the jet at the vena contracta. From equation (6.17), Theoretical velocity = (2gH), Actual velocity Coefficient of velocity = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−− = v ( 2gH ) = (x 24yH ). Theoretical velocity Putting x = 1.5 m, y = 0.5 m, H = 1.2 m and area, A = 650 × 10−6 m2, Coefficient of velocity, Cv = ( x 24yH ) = [ 1.5 ( 4 × 0.5 × 1.2 ) ] 2

= 0.968, Coefficient of discharge, Cd = Qactual A(2gH ) = (0.11760)[650 × 10−6(2 × 9.81 × 1.2)] = 0.618, Coefficient of contraction, Cc = Cd Cv = 0.6180.968 = 0.639.

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6.14 THEORY OF LARGE ORIFICES If the vertical height of an orifice is large, so that the head producing flow is substantially less at the top of the opening than at the bottom, the discharge calculated from the formula for a small orifice, using the head h measured to the centre of the orifice, will not be the true value, since the velocity will vary very substantially from top to bottom of the opening. The method adopted is to calculate the flow through a thin horizontal strip across the orifice (Fig. 6.18), and integrate from top to bottom of the opening to obtain the theoretical discharge, from which the actual discharge can be determined if the coefficient of discharge is known. FIGURE 6.18 Flow through a large orifice

EXAMPLE 6.5

A reservoir discharges through a rectangular sluice gate of width B and height D (Fig. 6.18). The top and bottom of the opening are at depths H1 and H2 below the free surface. Derive a formula for the theoretical discharge through the opening. If the top of the opening is 0.4 m below the water level and the opening is 0.7 m wide and 1.5 m in height, calculate the theoretical discharge (in cubic metres per second), assuming that the bottom of the opening is above the downstream water level. What would be the percentage error if the opening were to be treated as a small orifice?

Solution Since the velocity of flow will be much greater at the bottom than at the top of the opening, consider a horizontal strip across the opening of height δ h at a depth h below the free surface: Area of strip = Bδ h, Velocity of flow through strip = (2gh), Discharge through strip, δQ = Area × Velocity = B(2g)h12δ h. For the whole opening, integrating from h = H1 to h = H2, Discharge, Q = B ( 2g )

H2

h 12 dh

H1

= −23 B(2g)( H 32 − H 32 2 1 ).

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Elementary theory of notches and weirs

193

Putting B = 0.7 m, H1 = 0.4 m, H2 = 1.9 m, Theoretical discharge, Q =

−2 3

× 0.7 × (2 × 9.81)(1.932 − 0.432)

= 2.067(2.619 − 0.253) = 4.891 m3 s−1. For a small orifice, Q = A(2gh), where A is the area of the orifice and h is the head above the centreline. Putting A = BD = 0.7 × 1.5 m2, h = −12 (H1 + H2) = −12 (0.4 + 1.9) = 1.15 m, Q = 0.7 × 1.5(2 × 9.81 × 1.15) = 4.988 m3 s−1. This result is greater than that obtained by the large-orifice analysis. Error = (4.988 − 4.891)4.891 = 0.0198 = 1.98 per cent.

6.15 ELEMENTARY THEORY OF NOTCHES AND WEIRS A notch is an opening in the side of a measuring tank or reservoir extending above the free surface. It is, in effect, a large orifice which has no upper edge, so that it has a variable area depending upon the level of the free surface. A weir is a notch on a large scale, used, for example, to measure the flow of a river, and may be sharp edged or have a substantial breadth in the direction of flow. The method of determining the theoretical flow through a notch is the same as that adopted for the large orifice. For a notch of any shape (Fig. 6.19), consider a horizontal strip of width b at a depth h below the free surface and height δ h. FIGURE 6.19 Discharge from a notch

Area of strip = bδ h, Velocity through strip = (2gh), Discharge through strip, δQ = Area × Velocity = bδ h(2gh).

(6.23)

Integrating from h = 0 at the free surface to h = H at the bottom of the notch,

bh H

Total theoretical discharge, Q = ( 2g )

12

dh.

(6.24)

Before the integration of equation (6.24) can be carried out, b must be expressed in terms of h.

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FIGURE 6.20 Rectangular and vee notches

For a rectangular notch (Fig. 6.20(a)), put b = constant = B in equation (6.24), giving

h H

Q = B ( 2g )

12

2

dh = −3 B ( 2g )H 32 .

(6.25)

For a vee notch with an included angle θ (Fig. 6.20(b)), put b = 2(H − h) tan(θ2) in equation (6.24), giving

(H – h)h H

Q = ( 2g ) tan ( θ2 )

12

dh

= 2 ( 2g ) tan ( θ2 ) ( −3 Hh 32 – −5 h 52 ) h0 2

Q=

8 −−− 15

2

( 2g ) tan ( θ2 ) H 52 .

(6.26)

Inspection of equations (6.25) and (6.26) suggests that, by choosing a suitable shape for the sides of the notch, any desired relationship between Q and H could be achieved, but certain laws do lead to shapes which are not feasible in practice. As in the case of orifices, the actual discharge through a notch or weir can be found by multiplying the theoretical discharge by a coefficient of discharge to allow for energy losses and the contraction of the cross-section of the stream at the bottom and sides.

EXAMPLE 6.6

It is proposed to use a notch to measure the flow of water from a reservoir and it is estimated that the error in measuring the head above the bottom of the notch could be 1.5 mm. For a discharge of 0.28 m3 s−1, determine the percentage error which may occur, using a right-angled triangular notch with a coefficient of discharge of 0.6.

Solution For a vee notch, from equation (6.26), Q = C d −15−− ( 2g ) tan ( θ2 ) H 52 . 8

Putting Cd = 0.6 and θ = 90°, 8

Q = 0.6 × −15−− × ( 19.62 ) × 1 × H 52 = 1.417H 52.

(I)

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195

When Q = 0.28 m3 s−1, H = (0.281.417)25 = 0.5228 m. The error δQ in the discharge, corresponding to an error δH in the measurement of H, can be found by differentiating equation (I):

δQ = 2.5 × 1.417H 32δH = 2.5QδHH, δQQ = 2.5δHH. Putting δH = 1.5 mm and H = 0.5228 m, Percentage error = (δQQ) × 100 = (2.5 × 0.00150.5228) × 100 = 0.72 per cent.

In the foregoing theory, it has been assumed that the velocity of the liquid approaching the notch is very small so that its kinetic energy can be neglected; it can also be assumed that the velocity through any horizontal element across the notch will depend only on its depth below the free surface. This is a satisfactory assumption for flow over a notch or weir in the side of a large reservoir, but, if the notch or weir is placed at the end of a narrow channel, the velocity of approach to the weir will be substantial and the head h producing flow will be increased by the kinetic energy of the approaching liquid to a value x = h + α C 22g,

(6.27)

where C is the mean velocity of the liquid in the approach channel and α is the kinetic energy correction factor to allow for the non-uniformity of velocity over the crosssection of the channel. Note that the value of C is obtained by dividing the discharge by the full cross-sectional area of the channel itself, not that of the notch. As a result, the discharge through the strip (shown in Fig. 6.19) will be

δQ = bδ h(2gx), and, from equation (6.27), δ h = δx, so that

δQ = b(2g)x12 dx.

(6.28)

At the free surface, h = 0 and x = α C 22g, while, at the sill, h = H and x = H + aC 22g. Integrating equation (6.28) between these limits, Q = ( 2g )

( H +α C 2 2g )

bx 12 dx. α C 2 2g

For a rectangular notch, putting b = B = constant,

α C 2 32 α C 2 32 2 . Q = −3 B ( 2g )H 32 ⎛⎝ 1 + −−−−−−⎞⎠ – ⎛⎝ −−−−−−⎞⎠ 2gH 2gH

(6.29)

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The Energy Equation and its Applications

A long rectangular channel 1.2 m wide leads from a reservoir to a rectangular notch 0.9 m wide with its sill 0.2 m above the bottom of the channel. Assuming that, if the velocity of approach is neglected, the discharge over the notch, in SI units, is given by Q = 1.84BH 32, calculate the discharge (in cubic metres per second) when the head over the bottom of the notch H is 0.25 m (a) neglecting the velocity of approach, (b) correcting for the velocity of approach assuming that the kinetic energy correction factor α is 1.1.

Solution (a) Neglecting the velocity of approach, Q1 = 1.84BH 32. Putting B = 0.9 m and H = 0.25 m, Q1 = 1.84 × 0.9 × 0.2532 = 0.207 m3 s−1. (b) Taking the velocity of approach into account, from equation (6.29) the correction factor k will be k = [(1 + α C 22gH)32 − (α C 22gH)32 ] , and the corrected value of Q will be Q2 = Q1 × k, so that

α C 2 32 α C 2 32 . Q 2 = 1.84BH 32 ⎛⎝ 1 + −−−−−−⎞⎠ – ⎛⎝ −−−−−−⎞⎠ 2gH 2gH Putting B = 0.9 m, H = 0.25 m and α = 1.1, 32 32 1.1C 2 1.1C 2 Q 2 = 1.84 × 0.9 × 0.25 32 ⎛⎝ 1 + −−−−−−−−−−−−−−−−− ⎞⎠ – ⎛⎝ −−−−−−−−−−−−−−−−− ⎞⎠ 19.62 × 0.25 19.62 × 0.25

= 0.207[(1 + 0.224 C 2 )32 − (0.224 C 2 )32 ].

(I)

Now, Discharge V = Velocity in approach channel = −−−−−−−−−−−−−−−−−−−−−−−− Area of channel = Q21.2(H + 0.2).

(II)

Using (II), the solution to (I) can be found by successive approximation, taking C = 0 for the first approximation − which gives Q = 0.207 m3 s−1. Inserting this value of Q in (II), with H = 0.25 m, C = 0.2071.2 × 0.45 = 0.3833 m s−1. Putting C = 0.3833 m s−1 in (I), Q = 0.207[(1.0329)32 − (0.0329)32] = 0.2161 m3 s−1.

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The power of a stream of fluid

197

For the next approximation, C = 0.21611.2 × 0.45 = 0.4002 m s−1, giving Q = 0.207[(1.0359)32 − (0.0359)32] = 0.2168 m3 s−1. A further approximation gives C = 0.21681.2 × 0.45 = 0.4015 m s−1 and Q = 0.207[(1.0360)32 − (0.0360)32] = 0.2169 m3 s−1.

6.16 THE POWER OF A STREAM OF FLUID In Section 6.1, it was shown that a stream of fluid could do work as a result of its pressure p, velocity v and elevation z and that the total energy per unit weight H of the fluid is given by H = pρg + v 22g + z. If the weight per unit time of fluid flowing is known, the power of the stream can be calculated, since Weight Energy Power = Energy per unit time = −−−−−−−−−−−−−− × −−−−−−−−−−−−−−−−−− . Unit time Unit weight If Q is the volume rate of flow, weight per unit time = ρgQ, Power = ρgQH = ρgQ( pρg + v 22g + z) = pQ + −12 ρ v 2Q + ρgQz.

EXAMPLE 6.8

(6.30)

Water is drawn from a reservoir, in which the water level is 240 m above datum, at the rate of 0.13 m3 s−1. The outlet of the pipeline is at the datum level and is fitted with a nozzle to produce a high speed jet to drive a turbine of the Pelton wheel type. If the velocity of the jet is 66 m s−1, calculate (a) the power of the jet, (b) the power supplied from the reservoir, (c) the head used to overcome losses and (d) the efficiency of the pipeline and nozzle in transmitting power.

Solution (a) The jet issuing from the nozzle will be at atmospheric pressure and at the datum level so that, in equation (6.30), p = 0 and z = 0. Therefore, Power of jet = −12 ρ v 2Q.

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Putting ρ = 1000 kg m−3, v = 66 m s−1, Q = 0.13 m3 s−1, Power of jet =

−1 2

× 1000 × 662 × 0.13 = 283 140 W = 283.14 kW.

(b) At the reservoir, the pressure is atmospheric and the velocity of the free surface is zero so that, in equation (6.30), p = 0, v = 0. Therefore, Power supplied from reservoir = ρgQz. Putting ρ = 1000 kg m−3, Q = 0.13 m3 s−1, z = 240 m, Power supplied from reservoir = 1000 × 9.81 × 0.13 × 240 W = 306.07 kW. (c) If H1 = total head at the reservoir, H2 = total head at the jet, and h = head lost in transmission, Power supplied from reservoir = ρgQH1 = 306.07 kW, Power of issuing jet = ρgQH2 = 283.14 kW, Power lost in transmission = ρgQh = 22.93 kW, Power lost Head lost in pipeline = h = −−−−−−−−−−−−−−− ρ gQ 22.93 × 10 3 = −−−−−−−−−−−−−−−−−−−−−−−−−− = 17.98 m. 1000 × 9.81 × 0.13 Power of jet (d) Efficiency of transmission = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Power supplied by reservoir = 283.14306.07 = 92.5 per cent.

6.17 RADIAL FLOW When a fluid flows radially inwards, or outwards from a centre, between two parallel planes as in Fig. 6.21, the streamlines will be radial straight lines and the streamtubes will be in the form of sectors. The area of flow will therefore increase as the radius increases, causing the velocity to decrease. Since the flow pattern is symmetrical, the total energy per unit weight H will be the same for all streamlines and for all points along each streamline if we assume that there is no loss of energy. If v is the radial velocity and p the pressure at any radius r, H = pρg + v 22g = constant.

(6.31)

Applying the continuity of flow equation and assuming that the density of the fluid remains constant, as would be the case for a liquid, Volume rate of flow, Q = Area × Velocity = 2π rb × v, where b is the distance between the planes. Thus, v = Q2π rb

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Flow in a curved path. Pressure gradient and change of total energy across the streamlines

199

FIGURE 6.21 Radial flow

and, substituting in equation (6.31), pρg + Q 28π 2r 2b2g = H, p = ρg[H − (Q 28π 2b2g) × (1r 2)].

(6.32)

If the pressure p at any radius r is plotted as in Fig. 6.21(c), the curve will be parabolic and is sometimes referred to as Barlow’s curve. If the flow discharges to atmosphere at the periphery, the pressure at any point between the plates will be below atmospheric; there will be a force tending to bring the two plates together and so shut off flow. This phenomenon can be observed in the case of a disc valve. Radial flow under the disc will cause the disc to be drawn down onto the valve seating. This will cause the flow to stop, the pressure between the plates will return to atmospheric and the static pressure of the fluid on the upstream side of the disc will push it off its seating again. The disc will tend to vibrate on the seating and the flow will be intermittent.

6.18 FLOW IN A CURVED PATH. PRESSURE GRADIENT AND CHANGE OF TOTAL ENERGY ACROSS THE STREAMLINES Velocity is a vector quantity with both magnitude and direction. When a fluid flows in a curved path, the velocity of the fluid along any streamline will undergo a change due to its change of direction, irrespective of any alteration in magnitude which may

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FIGURE 6.22 Change of pressure with radius

also occur. Considering the streamtube (shown in Fig. 6.22), as the fluid flows round the curve there will be a rate of change of velocity, that is to say an acceleration, towards the centre of curvature of the streamtube. The consequent rate of change of momentum of the fluid must be due, in accordance with Newton’s second law, to a force acting radially across the streamlines resulting from the difference of pressure between the sides BC and AD of the streamtube element. In Fig. 6.22, suppose that the control volume ABCD subtends an angle δθ at the centre of curvature O, has length δs in the direction of flow and thickness b perpendicular to the diagram. For the streamline AD, let r be the radius of curvature, p the pressure and v the velocity of the fluid. For the streamline BC, the radius will be r + δ r, the pressure p + δ p and the velocity v + δ v, where δ p is the change of pressure in a radial direction. From the velocity diagram, Change of velocity in radial direction, δ v = vδθ, or, since δθ = δ sr, Radial change of velocity δs = v −−−, between AB and CD r Mass per unit time flowing = Mass density × Area × Velocity through streamtube = ρ × (b × δ r) × v, Change of momentum per unit Mass per unit time × Radial = time in radial direction change of velocity = ρbδ rv 2δ sr.

(6.33)

This rate of change of momentum is produced by the force due to the pressure difference between faces BC and AD of the control volume: Force = [( p + δp) − p]bδ s.

(6.34)

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Flow in a curved path. Pressure gradient and change of total energy across the streamlines

201

Equating equations (6.33) and (6.34), according to Newton’s second law,

δ pbδ s = ρ bδ rv2δ sr, δ pδ r = ρv 2r.

(6.35)

For an incompressible fluid, ρ will be constant and equation (6.35) can be expressed in terms of the pressure head h. Since p = ρgh, we have δ p = ρgδ h. Substituting in equation (6.35),

ρgδ hδ r = ρ v 2r, δ hδ r = v 2gr, or, in the limit as δ r tends to zero, Rate of change of pressure head in radial direction dh v 2 = −−− = −−− . dr gr

(6.36)

To produce the curved flow shown in Fig. 6.22, we have seen that there must be a change of pressure head in a radial direction. However, since the velocity v along streamline AD is different from the velocity v + δ v along BC, there will also be a change in the velocity head from one streamline to another: Rate of change of velocity head radially = [(v + δ v)2 − v 2 ]2gδ r, v δv = − −−−, neglecting products of small quantities, g δr v dv = − −−−, as δ r tends to zero. g dr

(6.37)

If the streamlines are in a horizontal plane, so that changes in potential head do not occur, the change of total head H – i.e. the total energy per unit weight – in a radial direction, δHδ r, is given by

δHδ r = Change of pressure head + Change of velocity head. Substituting from equations (6.36) and (6.37), in the limit, dH v 2 v dv Change of total energy with radius, −−−− = −−− + − −−− dr gr g dr dH v ⎛ v dv⎞ −−−− = − − + −−− . dr g ⎝ r dr⎠

(6.38)

The term (vr + dvdr) is also known as the vorticity of the fluid (see Section 7.2).

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In obtaining equation (6.38), it has been assumed that the streamlines are horizontal, but this equation also applies to cases where the streamlines are inclined to the horizontal, since the fluid in the control volume is in effect weightless, being supported vertically by the surrounding fluid. If the streamlines are straight lines, r = ∞ and dvdr = 0. From equation (6.38) for a stream of fluid in which the velocity is uniform across the cross-section, and neglecting friction, we have dHdr = 0 and the total energy per unit weight H is constant for all points on all streamlines. This applies whether the streamlines are parallel or inclined, as in the case of radial flow (Section 6.17).

6.19 VORTEX MOTION In vortex motion, the streamlines form a set of concentric circles and the changes of total energy per unit weight will be governed by equation (6.38). The following types of vortex are recognized.

6.19.1 Forced vortex or flywheel vortex The fluid rotates as a solid body with constant angular velocity ω, i.e. at any radius r, v = ω r so that

dv −−− = ω dr

v and − = ω . r

From equation (6.38), dH ω r 2ω 2r −−−− = −−−− ( ω + ω ) = −−−−−−−. dr g g Integrating, H = ω 2r 2g + C,

(6.39)

where C is a constant. But, for any point in the fluid, H = pρg + v 2 2g + z = pρg + ω 2r 22g + z. Substituting in equation (6.39), pρg + ω 2r 22g + z = ω 2r 2g + C, pρg + z = ω 2r 22g + C.

(6.40)

If the rotating fluid has a free surface, the pressure at this surface will be atmospheric and therefore zero (gauge).

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6.19

Vortex motion

203

FIGURE 6.23 Forced vortex

Putting pρg = 0 in equation (6.40), the profile of the free surface will be given by z = ω 2r 22g + C.

(6.41)

Therefore, the free surface will be in the form of a paraboloid (Fig. 6.23). Similarly, for any horizontal plane, for which z will be constant, the pressure distribution will be given by pρg = ω 2r 22g + (C − z).

EXAMPLE 6.9

(6.42)

A closed vertical cylinder 400 mm in diameter and 500 mm high is filled with oil of relative density 0.9 to a depth of 340 mm, the remaining volume containing air at atmospheric pressure. The cylinder revolves about its vertical axis at such a speed that the oil just begins to uncover the base. Calculate (a) the speed of rotation for this condition and (b) the upward force on the cover.

Solution (a) When stationary, the free surface will be at AB (Fig. 6.24), a height Z2 above the base. Volume of oil = π r 21 Z 2 . When rotating at the required speed ω, a forced vortex is formed and the free surface will be the paraboloid CDE. Volume of oil = Volume of cylinder PQRS − Volume of paraboloid CDE = π r 21 Z 1 − −12 π r 20 Z 1 , since the volume of a paraboloid is equal to half the volume of the circ*mscribing cylinder.

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FIGURE 6.24 Forced vortex example

No oil is lost from the container; therefore,

π r 21 Z 2 = π r 21 Z 1 − −12 π r 20 Z 1 , r 20 = 2r 21 (1 − Z2Z1), r0 = r1[2(1 − Z2Z1)] = r1[2(1 − 340500)] = 0.8r1 = 0.8 × 200 = 160 mm. Also, for the free surface of the vortex from equation (6.41), z = ω 2r 22g + constant, or, between points C and D, taking D as the datum level, ZD = 0 when r = 0 and ZC = Z1 when r = r0 , giving Z1 − 0 = ω 2 r 20 2g,

ω = (2gZ1 r 20 ) = (2 × 9.81 × 0.50.162 ) = 19.6 rad s−1. (b) The oil will be in contact with the top cover from radius r = r0 to r = r1. If p is the pressure at any radius r, the force on an annulus of radius r and width δ r is given by

δF = p × 2π rδ r. Integrating from r = r0 to r = r1, Force on top cover, F = 2 π

r1

r0

From equation (6.42), pρg = ω 2r 22g + C.

pr δ r.

(I)

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6.19

Vortex motion

205

Since the pressure at r0 is atmospheric, p = 0 when r = r0, so that C = −ω 2 r 20 2g, and

ω 2r 2 ω 2r 2 ρω 2 p = ρ g ⎛ −−−−−− – −−−−−−0 ⎞ = −−−−− ( r 2 – r 20 ). ⎝ 2g 2g ⎠ 2 Substituting in (I),

ρω 2 F = 2 π −−−−− 2

(r – r )r dr r1

2

2 0

r0

(r – r r) dr r1

= ρω 2 π

3

2 0

r0

r1

= ρω 2 π ( −14 r 4 – −12 r 20 r 2 ) r

= ρω π ( r – r – r r + −12 r 40 ) 2

1 − 41 4

1 − 40 4

1 − 20 21 2

ρω 2 π π = −−−−−−−− ( r 41 + r 40 – 2r 20 r 21 ) = − ρω 2 ( r 21 – r 20 ) 2 4 4 π = − × ( 0.9 × 1000 ) × 19.6 2 × ( 0.2 2 – 0.16 2 ) 2 N = 56.3 N. 4

6.19.2 Free vortex or potential vortex In this case, the streamlines are concentric circles, but the variation of velocity with radius is such that there is no change of total energy per unit weight with radius, so that dHdr = 0. Substituting in equation (6.38), v v dv 0 = − ⎛− + −−−⎞ , g ⎝ r dr⎠ dv dr −−− + −−− = 0. v r Integrating, loge v + loge r = constant, or

vr = C,

where C is a constant known as the strength of the vortex at any radius r; v = Cr.

(6.43)

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FIGURE 6.25 Free vortex

Since, at any point, z + pρg + v 22g = H = constant, substituting for v from equation (6.43) z + pρg + C 22gr 2 = H. If the fluid has a free surface, pρg = 0 and the profile of the free surface is given by H − z = C 22gr 2,

(6.44)

which is a hyperbola asymptotic to the axis of rotation and to the horizontal plane through z = H, as shown in Fig. 6.25. For any horizontal plane, z is constant and the pressure variation is given by pρg = (H − z) − C 22gr 2.

(6.45)

Thus, in the free vortex, pressure decreases and circumferential velocity increases as we move towards the centre.

EXAMPLE 6.10

A point A on the free surface of a free vortex is at a radius rA = 200 mm and a height zA = 125 mm above datum. If the free surface at a distance from the axis of the vortex, which is sufficient for its effect to be negligible, is 180 mm above datum, what will be the height above datum of a point B on the free surface at a radius of 100 mm?

Solution For point A, from equation (6.44), H − zA = C 2 2gr 2A ; therefore,

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207

C 22g = r A2 (H − zA). Now H is the head above datum at an infinite distance from the axis of rotation, where the effect of the vortex is negligible, so that H = 180 mm = 0.18 m. Also zA = 0.125 m and rA = 0.2 m. Substituting, C2 −−− = 0.2 2 ( 0.18 – 0.125 ) = 2.2 × 10 –3 m 3 . 2g For point B, H − zB = C 2 2gr B2 zB = H − C 2 2gr B2 = 0.18 − (2.2 × 10−3)0.12 = −0.04 = 40 mm below datum.

6.19.3 Compound vortex In the free vortex, v = Cr and thus, theoretically, the velocity becomes infinite at the centre. The velocities near the axis would be very high and, since friction losses vary as the square of the velocity, they will cease to be negligible, and the assumption that the total head H remains constant will cease to be true. The central part of the vortex tends to rotate as a solid body, thus forming a forced vortex surrounded by a free vortex. Figure 6.26 shows the free surface profile of such a compound vortex, and also represents the variation of pressure with radius on any horizontal plane in the vortex. The velocity at the common radius R must be the same for the two vortices.

FIGURE 6.26 Compound vortex

For the free vortex, if y1 = depression of the surface at radius R below the level of the surface at infinity, y1 = C 22gR2 = v 22g = ω 2R22g.

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For the forced vortex, if y2 = height of the surface at radius R above the centre of the depression, y2 = v 22g = C 22gR2. Thus, Total depression = y2 + y1 = C 2gR2 = ω 2R2g.

(6.46)

For the forced vortex, the velocity at radius R is ωR, while for the free vortex, from equation (6.43), the velocity at radius R is CR. Therefore, the common radius, at which these two velocities will be the same, is given by

ωR = CR or R = (Cω). In Section 7.9 it will be shown that C = Γ2π, where Γ is the circulation, so that Common radius, R = (Cω) = ( Γ2πω).

Concluding remarks Chapters 4 and 5 introduced two of the fundamental relationships of fluid mechanics, namely the conservation of mass flow and the momentum equation. This chapter has introduced the third fundamental relationship, the steady flow energy equation, based on the conservation of energy principle. It is interesting to note the relationship between this expression and the simplified form of Euler’s equation, Bernoulli’s equation, generated from the application of the equation of momentum to steady, uniform, inviscid flow. The steady flow energy equation will be used throughout this text in the analysis of flow in pipes, ducts and open channels, as well as providing the basis for the study of the matching of machines with their fluid networks. Similarly the energy equation is essential in the measurement of flow velocity, this application being historically linked to the application of manometer techniques based on the hydrostatic equation. An understanding of the manner in which the available energy in a fluid network changes in response to frictional or separation losses or to the presence of fans, pumps or turbines is essential. This chapter has laid the foundations for that understanding.

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Problems

209

Summary of important equations and concepts 1.

2.

3.

4.

5.

This chapter introduces the steady flow energy equation (SFEE), equation (6.13) and shows how it applies to a wide range of flow conditions, Section 6.4. This is probably the most important single equation to be met in the application of fluid theory to a wide range of actual applications, particularly when the frictional and separation loss terms are fully understood. The SFEE is an essential tool. In addition to the application of the SFEE to conduit flow conditions, and its use in explaining the energy transfers within a fluid system, it also forms the basis for the measurement of flow velocity and flow rate, by Pitot–static tube or by venturi meter or orifice plate, Sections 6.5 through to 6.10. Velocity profiles in real fluid flow situations make volumetric flow measurement difficult in some cases and the use of Pitot–static tube data in conjunction with an array and summing program are demonstrated, program VOLFLO. The energy concepts demonstrated allow orifice flow to be summed and the various coefficients of velocity and discharge used to align ‘real’ flow with the theoretical to be introduced and defined. This application of the energy equation to vortex flow allows the definition of both forced and free vortices, equations (6.42) and (6.45).

Problems 6.1 The suction pipe of a pump rises at a slope of 1 vertical in 5 along the pipe and water passes through it at 1.8 m s−1. If dissolved air is released when the pressure falls to more than 70 kN m−2 below atmospheric pressure, find the greatest practicable length of pipe neglecting friction. Assume that the water in the sump is at rest. [34.9 m]

of 4.7 m s−1. The internal energy of the air at exit is greater than that at entry by 85 kJ kg−1. The compressor is fitted with a cooling system which removes heat at the rate of 60 kJ s−1. Calculate the power required to drive the compressor and the cross-sectional areas of the inlet and outlet pipes. [115.2 kW, 0.0664 m2, 0.0170 m2]

6.2 A jet of water is initially 12 cm in diameter and when directed vertically upwards reaches a maximum height of 20 m. Assuming that the jet remains circular determine the rate of water flowing and the diameter of the jet at a height of 10 m. [0.224 m3 s−1, 14.27 cm]

6.5 A Pitot–static tube is used to measure air velocity. If a manometer connected to the instrument indicates a difference in pressure head between the tappings of 4 mm of water, calculate the air velocity assuming the coefficient of the Pitot tube to be unity. Density of air = 1.2 kg m−3. [8.08 m s−1]

6.3 A pipe AB carries water and tapers uniformly from a diameter of 0.1 m at A to 0.2 m at B over a length of 2 m. Pressure gauges are installed at A, B and also at C, the midpoint of AB. If the pipe centreline slopes upwards from A to B at an angle of 30° and the pressures recorded at A and B are 2.0 and 2.3 bar, respectively, determine the flow through the pipe and pressure recorded at C neglecting all losses. [0.0723 m3 s−1, 2.29 bar] 6.4 Air enters a compressor at the rate of 0.5 kg s−1 with a velocity of 6.4 m s−1, specific volume 0.85 m3 kg−1 and a pressure of 1 bar. It leaves the compressor at a pressure of 6.9 bar with a specific volume of 0.16 m3 kg−1 and a velocity

6.6 A liquid flows through a circular pipe 0.6 m in diameter. Measurements of velocity taken at intervals along a diameter are: Distance from the wall m Velocity m s−1

0 0

0.05 0.1 2.0 3.8

Distance from the wall m Velocity m s−1

0.4 0.5 4.5 3.7

0.2 0.3 4.6 5.0

0.55 0.6 1.6 0

Draw the velocity profile, calculate the mean velocity and determine the kinetic energy correction factor. [2.82 m s−1, 1.899]

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The Energy Equation and its Applications

6.7 A venturi meter with a throat diameter of 100 mm is fitted in a vertical pipeline of 200 mm diameter with oil of specific gravity 0.88 flowing upwards at a rate of 0.06 m3 s−1. The venturi meter coefficient is 0.96. Two pressure gauges calibrated in kilonewtons per square metre are fitted at tapping points, one at the throat and the other in the inlet pipe 320 mm below the throat. The difference between the two gauge pressure readings is 28 kN m−2. Working from Bernoulli’s equation determine the difference in level in the two limbs of a mercury manometer if it is connected to the tapping points and the connecting pipes are filled with the same oil. [202 mm] 6.8 An orifice plate is to be used to measure the rate of air flow through a 2 m diameter duct. The mean velocity in the duct will not exceed 15 m s−1 and a water tube manometer, having a maximum difference between water levels of 150 mm, is to be used. Assuming the coefficient of discharge to be 0.64, determine a suitable orifice diameter to make full use of the manometer range. Take the density of air as 1.2 kg m−3. [1.31 m] 6.9 A sharp-edged orifice, 5 cm in diameter, in the vertical side of a large tank discharges under a head of 5 m. If Cc = 0.62 and Cv = 0.98, determine (a) the diameter of the jet at the vena contracta, (b) the velocity of the jet at the vena contracta and (c) the discharge in cubic metres per second. [(a) 3.94 cm, (b) 9.71 m s−1, (c) 0.0118 m3 s−1] 6.10 Find the diameter of a circular orifice to discharge 0.015 m3 s−1 under a head of 2.4 m using a coefficient of discharge of 0.6. If the orifice is in a vertical plane and the jet falls 0.25 m in a horizontal distance of 1.3 m from the vena contracta, find the value of the coefficient of contraction. [6.82 cm, 0.715] 6.11 A tank has a circular orifice 20 mm diameter in the vertical side near the bottom. The tank contains water to a depth of 1 m above the orifice with oil of relative density 0.8 for a depth of 1 m above the water. Acting on the upper surface of the oil is an air pressure of 20 kN m−2 gauge. The jet of water issuing from the orifice travels a horizontal distance of 1.5 m from the orifice while falling a vertical distance of 0.156 m. If the coefficient of contraction of the orifice is 0.65, estimate the value of the coefficient of velocity and the actual discharge through the orifice. [0.97, 1.72 dm3 s−1]

6.12 Water flows from a reservoir through a rectangular opening 2 m high and 1.2 m wide in the vertical face of a dam. Calculate the discharge in cubic metres per second when the free surface in the reservoir is 0.5 m above the top of the opening assuming a coefficient of discharge of 0.64. [8.16 m3 s−1] 6.13 A vertical triangular orifice in the wall of a reservoir has a base 0.9 m long, 0.6 m below its vertex and 1.2 m below the water surface. Determine the theoretical discharge. [1.19 m3 s−1] 6.14 A rectangular channel 1.2 m wide has at its end a rectangular sharp-edged notch with an effective width after allowing for side contractions of 0.85 m and with its sill 0.2 m from the bottom of the channel. Assuming that the velocity head averaged over the channel is αV 22g where V is the mean velocity and α = 1.1, calculate the discharge in cubic metres per second when the head is 250 mm above the sill allowing for the velocity of approach. [0.204 m3 s−1] 6.15 In an experiment on a 90° vee notch the flow is collected in a 0.9 m diameter vertical cylindrical tank. It is found that the depth of water increases by 0.685 m in 16.8 s when the head over the notch is 0.2 m. Determine the coefficient of discharge of the notch. [0.613] 6.16 A pump discharges 2 m3 s−1 of water through a pipeline. If the pressure difference between the inlet and the outlet of the pump is equivalent to 10 m of water, what power is being transmitted to the water from the pump? [196.2 kW] 6.17 Inward radial flow occurs between two horizontal discs 0.6 m in diameter and 75 mm apart, the water leaving through a central pipe 150 mm in diameter in the lower disc at the rate of 0.17 m3 s−1. If the absolute pressure at the outer edge of the disc is 101 kN m−2 calculate the pressure at the outlet. Find also the resultant force on the upper disc. [90 kN m−2 abs, 373 N] 6.18 Two horizontal discs are 12.5 mm apart and 300 mm in diameter. Water flows radially outwards between the discs from a 50 mm diameter pipe at the centre of the lower disc. If the pressure at the outer edge of the disc is atmospheric, calculate the pressure in the supply pipe when the velocity of the water in the pipe is 6 m s−1. Find also the resultant force on the upper disc, neglecting impact force. [−17.5 kN m−2, −92.4 N]

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Problems 6.19 A hollow cylindrical drum with its axis vertical has an internal diameter of 600 mm and is full of water. A set of paddles 200 mm in diameter rotates concentrically with the axis of the drum at 120 rev min−1 and produces a compound vortex in the water. Assuming that all the water in the 200 cm core rotates as a forced vortex with the paddles and

211

that the water outside this core moves as a free vortex, determine (a) the velocity of the water at 75 mm and 225 mm from the centre, and (b) the pressure head at these radii above the pressure head at the centre. [(a) 0.943 m s−1, 0.56 m s−1, (b) 45.3 mm, 145 mm]

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Chapter 7

Two-dimensional Ideal Flow 7.1 7.2 7.3 7.4 7.5

7.6

Rotational and irrotational flow Circulation and vorticity Streamlines and the stream function Velocity potential and potential flow Relationship between stream function and velocity potential. Flow nets Straight line flows and their combinations

Combined source and sink flows. Doublet 7.8 Flow past a cylinder 7.9 Curved flows and their combinations 7.10 Flow past a cylinder with circulation. Kutta–Joukowsky’s law 7.11 Computer program ROTCYL 7.7

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An ideal flow is a purely theoretical concept as such flows possess no viscosity, compressibility, surface tension or vaporization pressure limit. However, the mathematical treatment of such flows was fundamental in the development of modern fluid mechanics and finds application in the development of aerofoil lift, fanpump blade design and groundwater flow predictions. This chapter will introduce the fundamental definitions of idealized flow, including circulation and vorticity, stream

function, velocity potential and the techniques necessary for the generation of flow nets. The representation of flow conditions by a combination of rectilinear flows, sources and sinks will be demonstrated, leading to both the representation of flow over a cylinder, and, with the inclusion of circulation, the prediction of lift forces. A computer program to predict the lift coefficient and location of the stagnation point on a rotating cylinder is introduced. l l l

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Two-dimensional Ideal Flow

In Chapter 4, a distinction was made between real and ideal fluids. The former exhibits the effects of viscosity and will be dealt with in the next part of the book, whereas the latter will be considered in this chapter. An ideal fluid is a purely hypothetical fluid which is assumed to have no viscosity and no compressibility, and, in the case of liquids, no surface tension and no vaporization. The study of flow of such a fluid stems from the eighteenth-century hydrodynamics developed by mathematicians, who, by making the above assumptions regarding the fluid, aimed at establishing mathematical models for fluid flow. Although the assumptions of ideal flow appear to be very far fetched, the introduction of the boundary layer concept by Prandtl in 1904 enabled the distinction to be made between two regimes of flow: that adjacent to the solid boundary, in which viscosity effects are predominant and, therefore, the ideal flow treatment would be erroneous, and that outside the boundary layer, in which viscosity has negligible effect so that the idealized flow conditions may be applied. This argument is developed further in Chapter 12 when dealing with external flow. The ideal flow theory may also be extended to situations in which fluid viscosity is very small and velocities are high, since they correspond to very high values of Reynolds number, at which flows are independent of viscosity. Thus, it is possible to see ideal flow as that corresponding to an infinitely large Reynolds number and to zero viscosity. The applications of ideal flow theory are found in aerodynamics, in accelerating flow, tides and waves. The study of ideal flow provides mathematical expressions for streamlines in elementary or basic flow patterns. By combining these basic flow patterns in various ways, it is possible to obtain complex flow patterns which, in many cases, resemble remarkably closely the real situations outside the boundary layer and any associated wakes.

7.1

ROTATIONAL AND IRROTATIONAL FLOW

Considerations of ideal flow lead to yet another flow classification, namely the distinction between rotational and irrotational flow. Basically, there are two types of motion: translation and rotation. The two may exist independently or simultaneously, in which case they may be considered as one superimposed on the other. If a solid body is represented by a square, then pure translation or pure rotation may be represented as shown in Fig. 7.1(a) and (b), respectively.

FIGURE 7.1 Translation and rotation

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7.1

Rotational and irrotational flow

215

FIGURE 7.2 Linear and angular deformation

FIGURE 7.3 Rotation, translation and deformation

If we now consider the square to represent a fluid element, it may be subjected to deformation. This can be either linear or angular, as shown in Fig. 7.2(a) and (b), respectively. Now, consider a motion of a fluid in which rotation of fluid elements is superimposed on their translation. In time dt, then, point A on the fluid element aAb moves to A′ and the element assumes position a′A′b′, as shown in Fig. 7.3. The two angles of rotation α and β will not be the same if deformation takes place and, therefore, the average rate of rotation in time dt will be

α + β 1 1 (α + β) ω = −−−−−−− × −−− = − −−−−−−−−−−, 2 dt 2 dt but, for small values and taking anticlockwise rotation as positive,

∂v ∂v Arc 1 α = −−−−−−−−−− = −−−−y dx dt −−− = −−−−y dt, Radius ∂ x dx ∂ x and

∂v –∂ v 1 β = – −−−−x dy dt −−− = −−−−−−x dt. ∂y dy ∂y

The rate of rotation about the z axis is, therefore,

∂v 1 ∂v 1 1 ∂v ∂v ω z = − ⎛⎝ −−−−y dt – −−−−x dt⎞⎠ −−− = − ⎛⎝ −−−−y – −−−−x⎞⎠ . 2 ∂x ∂y dt 2 ∂ x ∂ y The expression in brackets,

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Two-dimensional Ideal Flow

∂ vy ∂ vx −−−− – −−−− = ζ , ∂x ∂y

(7.1)

is called the vorticity and is denoted by ζ. Thus,

ζ = 2ωz ,

(7.2)

where ωz is the angular velocity of the fluid elements about their mass centre in the x–y plane. In three-dimensional flow, ωz would represent only one of three components of the angular velocity ω and vorticity would be equal to 2ω. The expression (7.1) was obtained by stipulating rotation of fluid elements to exist and to be superimposed on their translation. Such a flow is known as rotational. It follows, therefore, that if there is no rotation, the expression (7.1) and, hence, the vorticity must be equal to zero. Thus, if the motion of particles is purely translational and the distortion is symmetrical, the flow is irrotational and the condition which it must satisfy is

∂ vy ∂ vx −−−− – −−−− = 2 ω z = 0. ∂x ∂y

(7.3)

The distinction between rotational and irrotational flow is important because, for example, it will be shown later that Bernoulli’s equation derived for a streamline applies to all streamlines in the flow field only if the flow is irrotational. It will also be shown that the generation of lift by such surfaces as aerofoils is associated with irrotational flow. Also, a useful and practical procedure of determining ‘flow nets’ can only be applied to irrotational flow.

7.2

CIRCULATION AND VORTICITY

Consider a fluid element ABCD in rotational motion. Let the velocity components along the sides of the element be as shown in Fig. 7.4. Since the element is rotating,

FIGURE 7.4 Circulation

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7.2

Circulation and vorticity

217

being part of rotational flow, there must be a ‘resultant’ peripheral velocity. However, since the centre of rotation is not known, it is more convenient to relate rotation to the sum of products of velocity and distance around the contour of the element. Such a sum is, of course, the line integral of velocity around the element and it is called circulation, denoted by Γ. Thus,

Circulation, Γ =

v ds. s

(7.4)

Circulation is, by convention, regarded as positive for anticlockwise direction of integration. Thus, for the element ABCD, starting from side AD,

∂v ∂v Γ ABCD = v x dx + ⎛ v y + −−−−y dx⎞ dy – ⎛ v x + −−−−x dy⎞ dx – v y dy ⎝ ⎠ ⎝ ∂x ∂y ⎠ ∂v ∂v = −−−−y dx dy – −−−−x dy dx ∂x ∂y ∂v ∂v = ⎛ −−−−y – −−−−x⎞ dx dy, ⎝ ∂x ∂y ⎠ but

∂ v y ∂ v x⎞ ⎛− – =ζ ⎝ ∂−−x− −∂−−y−⎠

for two-dimensional flow in the x–y plane and, therefore, is the vorticity of the element about the z axis, ζz. The product dx dy is the area of the element dA. Thus,

∂v ∂v ΓABCD = ⎛ −−−−y – −−−−x⎞ dx dy = ζ z dA. ⎝ ∂x ∂y ⎠ It is seen, therefore, that the circulation around a contour is equal to the sum of the vorticities within the area of the contour. This is known as Stokes’ theorem and may be stated mathematically, for a general case of any contour C (Fig. 7.5), as ΓC =

v cos θ ds = ζ dA.

(7.5)

A

The concept of circulation is very important in the theory of lifting surfaces such as aerofoils, hydrofoils and blades of rotodynamic machines.

FIGURE 7.5 Circulation and vorticity

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The above considerations indicate that, for irrotational flow, since vorticity is equal to zero, the circulation around a closed contour through which fluid is flowing must be equal to zero.

7.3

STREAMLINES AND THE STREAM FUNCTION

In Section 4.1, a distinction was made between streaklines, pathlines and streamlines. Of the three, the streamline is the one which is a purely theoretical line in space, defined as being tangential to instantaneous velocity vectors. From this definition of a streamline, it follows that there can be no flow across it, simply because a line cannot be tangential to a velocity vector which at the same time crosses it. The concept of the streamline is very useful, especially in ideal flow, because it enables the fluid flow to be conceived as occurring in patterns of streamlines. These patterns may be described mathematically so that the whole system of analysis may be based on it. It requires, however, a mathematical definition of a streamline. Consider, in a two-dimensional case, the velocity and displacement vectors of a fluid at a point, together with their orthogonal components, as shown in Fig. 7.6.

FIGURE 7.6 Velocity and displacement components

Since, by definition of a streamline, ds || v, it follows that dy || vy and dx || vx. Thus, the velocity triangle and the displacement triangle are similar and, therefore, dx dy −−− = −−− . vx vy

(7.6)

This constitutes the equation of a streamline. Since the streamlines in a flow pattern describe it, it is useful to label them by some numerical system. Furthermore, it is possible to relate the numerical labels to the flow rate of the pattern which is being described. Thus, let aa and bb be two streamlines in a flow bounded by solid boundaries AA and BB in Fig. 7.7 If the streamline aa is denoted by Ψa, which will be labelled by a numerical value representing the flow rate per unit depth between AA and the streamline aa, then, Ψa = QOc and, similarly, if Ψb = QOe,

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7.3

Streamlines and the stream function

219

FIGURE 7.7 The stream function

it follows that dΨ = Ψb − Ψa = Qce, so that dΨ = vx dy − vy dx

(7.7)

and Ψ, which is called the stream function, is given by

Ψ=

v dy –v dx. x

y

(7.8)

Thus, the stream function depends upon position coordinates, Ψ = f(x, y) and, hence, the total derivative:

∂Ψ ∂Ψ dΨ = −−−− dx + −−−− dy. ∂x ∂y

(7.9)

Comparing equations (7.9) and (7.7), the relationships between the stream function and the velocity components are obtained:

∂Ψ v x = −−−− ∂y

and

∂Ψ v y = – −−−− . ∂x

(7.10)

(Note: the sign convention adopted here is for the flow to be positive from left to right.) Since the value of a stream function represents the flow rate between a given streamline described by the stream function and a reference boundary, it follows that it must be constant for the given streamline in order to satisfy the continuity equation combined with the requirement of no flow across a streamline.

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FIGURE 7.8 Stream function in polar coordinates

In some curved flows, it is more convenient to use the polar coordinates in mathematical analysis. In these, since Ψ = f (r, θ ), by differentiation,

∂Ψ ∂Ψ dΨ = −−−− dr + −−−− dθ . ∂r ∂θ

(7.11)

The sign convention in polar coordinates is that the tangential velocity is positive in the direction of positive θ, i.e. anticlockwise; the radial velocity is positive in the outward direction. Consider, now, two curved streamlines Ψa and Ψb, as shown in Fig. 7.8. Assuming that AB = dr when dθ = 0 and applying the continuity equation, the following relationship is obtained: vr(r dθ ) − vθ dr = dΨ.

(7.12)

Comparing equations (7.11) and (7.12), the following relationships are deduced:

∂Ψ vθ = – −−−− ∂r

7.4

and

1 ∂Ψ v r = − −−−−. r ∂θ

(7.13)

VELOCITY POTENTIAL AND POTENTIAL FLOW

In connection with flow nets mentioned earlier, an important concept is that of the velocity potential. It is defined as

v ds, B

Φ=

s

A

(7.14)

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7.4

Velocity potential and potential flow

221

FIGURE 7.9 Different paths in the potential field

where A and B are two points in a potential field and vs is the velocity tangential to the elementary path s. To understand the above concept, it is necessary to appreciate the meaning of the term ‘potential’, so often used in mechanics as well as in other situations which satisfy the same specific conditions. Consider, for example, points P and P′ in a gravitational field. If P′ has a greater potential than P, the difference in their potential δW is defined as the work required to move a particle from P to P′ against the gravitational force. If the distance between the points is δ s and the force required is F then

δW = Fδ s,

F ds. P′

or

W=

(7.15)

P

Clearly, the work done in such a case is independent of the path taken in doing the work, and this property is a characteristic of potential fields only. In any other case, say against friction, the work done would depend upon the path taken. Therefore, not all fields are potential, but only those in which the path taken is immaterial. Returning to our velocity field, if points A and B in Fig. 7.9 belong to some potential field, then the integral

v ds B

A

is independent of the path taken. Therefore,

v

v ds.

B

B

m

ds =

(7.16)

n

A

A

From the analogy with the gravitational field, it is apparent that the condition of equation (7.16) will only be satisfied if the field is potential. Fluid flow in such a field is known as potential flow. Consider, now, the circulation around AnBm: Γ AnBm =

B

A

=

m

ds

B

B

A

v A

v n ds +

v B

v n ds –

m

ds .

(7.17)

A

But, for the flow to be potential, the two integrals in equation (7.17) must be equal and, therefore, ΓAnBm = 0. If the circulation is equal to zero, it follows that vorticity must also be equal to zero and, therefore, the condition for potential flow is

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∂ vy ∂ vx −−−− – −−−− = 0, ∂x ∂y

(7.18)

which is identical with the condition for irrotational flow. Thus, potential flow is irrotational and vice versa. In irrotational (potential) flow, therefore, the function (7.14)

v ds B

Φ=

s

A

exists, from which it follows that, if vx and vy are the orthogonal components of vs , then,

∂Φ v x = −−−− ∂x

∂Φ v y = −−−−, ∂y

and

(7.19)

so that Φ=

v dx + v dy. x

y

(7.20)

Similarly, in polar coordinates, if vθ and vr are tangential and radial components of vs, then Φ=

v dr + v r dθ, θ

r

(7.21)

from which

∂Φ v r = −−−− ∂r

and

∂Φ vθ = −−−−−. r ∂θ

(7.22)

It is now appropriate to consider the implications of potential flow to the applicability of Bernoulli’s equation. It was originally derived in Section 5.12 to apply along a streamline, but not necessarily across streamlines, i.e. from one streamline to a neighbouring one. Furthermore, it was shown in Section 6.18 that in some curved flows the Bernoulli constant (or total head), defined as H = pρg + v 22g + z, varies across streamlines, the variation in general being governed by equation (6.38), namely dH v ⎛ v dv⎞ −−−− = − − + −−− . dr g ⎝ r dr⎠

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223

FIGURE 7.10 Circulation in polar coordinates

However, in some flows the Bernoulli constant is the same for all streamlines, so that Bernoulli’s equation may be applied to any points in the flow field. Clearly, for this to happen dHdr must be equal to zero. One such obvious case is when dr → ∞, i.e. in the case of straight line flows. The other possibility is for the expression in brackets to be equal to zero, i.e. v dv − + −−− = 0. r dr

(7.23)

Let us examine such a case by considering an element of fluid in a curved flow, as shown in Fig. 7.10. The circulation around the element ABCD is ΓADCB = vrδθ − (v + dv)(r + dr) δθ = −v drδθ − dv drδθ − r dvδθ. Neglecting infinitesimals of the third order, this reduces to ΓADCB = −v drδθ − r dvδθ. But the area of the element is rδθ dr, so that the vorticity is given by Γ – v drδθ – r dvδθ ζ = −−−−−−− = −−−−−−−−−−−−−−−−−−−−−−−− Area r δθ dr v dv = – ⎛− + −−−⎞ , ⎝ r dr⎠

(7.24)

which is the same as the left-hand side of equation (7.23). Thus, if the vorticity is zero, there is no variation of the Bernoulli constant. This condition applies to irrotational (potential) flow. In potential flow, then, Bernoulli’s equation applies to the whole flow field and is not limited to individual streamlines.

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7.5

RELATIONSHIP BETWEEN STREAM FUNCTION AND VELOCITY POTENTIAL. FLOW NETS

Comparing equations (7.10) and (7.19), from (7.10)

∂Ψ v x = −−−− ∂y

and

∂Ψ v y = – −−−−; ∂x

and

∂Φ v y = −−−−. ∂y

from (7.19),

∂Φ v x = −−−− ∂x

Thus, equating for vx and vy, we obtain

∂Ψ ∂Φ −−−− = −−−− ∂y ∂x

and

∂Φ ∂Ψ −−−− = – −−−− . ∂y ∂x

(7.25)

These equations are known as Cauchy–Riemann equations and they enable the stream function to be calculated if the velocity potential is known and vice versa in a potential flow. It is now possible to return to the condition for potential flow and to restate it in terms of the stream function. The condition is

∂ vy ∂ vx −−−− – −−−− = 0, ∂x ∂y but

∂Ψ v y = – −−−− ∂x

and

∂Ψ v x = −−−−, ∂y

so that, by substitution,

∂ ∂ ∂Ψ ∂Ψ −−− ⎛ – −−−−⎞ – −−− ⎛ −−−−⎞ = 0 ∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠ and

∂ 2Ψ ∂ 2Ψ −−−−−−2 + −−−−−−2 = 0. ∂x ∂y

(7.26)

This is the Laplace equation for the stream function, which must be satisfied for the flow to be potential. It is also interesting to note that the Laplace equation for the velocity potential must also be satisfied. This follows by substitution of equations (7.19) into the continuity equation for steady, incompressible, two-dimensional flow (equation (4.13)):

∂ vx ∂ vy −−−− + −−−− = 0. ∂x ∂y

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Substituting, now, for vx and vy from equations (7.19),

∂ ∂Φ ∂ ∂Φ −−− ⎛ −−−−⎞ + −−− ⎛ −−−−⎞ = 0, ⎝ ⎠ ∂x ∂x ∂y ⎝ ∂y ⎠ so that

∂ 2Φ ∂ 2Φ −−−−−2 + −−−−−2 = 0. ∂x ∂y

(7.27)

Thus, the Laplace equation for the velocity potential must also be satisfied. The fact that, for potential flow, both the stream function and the velocity potential satisfy the Laplace equation indicates that Ψ and Φ are interchangeable (Cauchy–Riemann equations) and that the lines of constant Ψ, i.e. streamlines, and the lines of constant Φ, called equipotential lines, are mutually perpendicular. This means that, if streamlines are plotted, points can be marked on them which have the same value of Φ and can be joined to form equipotential lines. Thus, a flow net of streamlines and equipotential lines is formed. When the streamlines converge, the velocity increases and, therefore, for a given increment of δ Ψ, the distance between the equipotential lines will also decrease. The method of drawing a flow net consists of drawing by eye streamlines equispaced at δ Ψ at some section where the flow is rectilinear, such as AG or DE in Fig. 7.11, which shows an example of a network drawn for a rather unusual converging section, of which the upper half constitutes a sudden contraction but the lower half provides a smooth transition. The number of streamlines drawn, or rather, the size of intervals between them, depends upon the accuracy required. The more streamlines one uses, the more accurate will be the result, but, equally, the time spent in drawing the net will be greater. The set of equipotential lines is drawn next at intervals δ Φ = δ Ψ and in such a way that they cross each streamline at right angles. Thus a set of ‘squares’ is obtained. The process is done by eye and requires a series of successive adjustments to both streamlines and equipotential lines until a satisfactory network of ‘squares’ is achieved. As a final check, diagonals through the ‘squares’ may be drawn. They, too, should be smooth lines and should form a net of squares. A pair of such diagonals from A′ to F′ is shown in Fig. 7.11.

FIGURE 7.11 Example of a flow net

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Where abrupt changes of the outer boundary occur, such as at points B and C, it can be seen that the streamline AA′D cannot follow the contour and separates from the boundary. At B, where the streamline turns towards the fluid, the velocity at the separation area will be zero and the fluid trapped there will be stagnant. At point C, the streamline turns away from the fluid, indicating high velocity in the separation bubble. This velocity is spent in rotation of considerable vigour. Certainly, therefore, the assumption of irrotational flow is not valid there. In general, then, wherever the streamlines diverge or converge abruptly, separation may occur. Because GFE is smooth and converging, no separation will occur there. Should, however, the flow direction be reversed, although the flow net would remain the same, separation might be expected downstream of F due to the divergence of flow. Separation phenomena are discussed fully in Chapter 11 in connection with boundary layer. Constructing flow nets is a useful exercise which requires a lot of patience and experience. The alternative is to use precise mathematical expressions for stream function and velocity potential describing the flow from which a flow net can be plotted exactly. The following sections of this chapter deal with such mathematical expressions for some basic flows which may then be combined to represent more complex flow patterns.

EXAMPLE 7.1

In a two-dimensional incompressible flow the fluid velocity components are given by vx = x − 4y and vy = −y − 4x. Show that the flow satisfies the continuity equation and obtain the expression for the stream function. If the flow is potential, obtain also the expression for the velocity potential.

Solution For two-dimensional incompressible flow, the continuity equation is

∂ vx ∂ vy −−−− + −−−− = 0, ∂x ∂y but

vx = x − 4y and vy = −y − 4x,

and

∂ vx −−−− = 1, ∂x

∂v −−−−y = – 1 ; ∂y

therefore, 1 − 1 = 0 and the flow satisfies the continuity equation. To obtain the stream function, using equations (7.10),

∂Ψ v x = −−−− = x – 4 y, ∂y ∂Ψ v y = – −−−− = – ( y + 4x ) . ∂x Therefore, from (I), Ψ=

(x – 4y) dy + f (x) + C

= xy − 2y 2 + f (x) + C.

(I) (II)

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But, if Ψ0 = 0 at x = 0 and y = 0, which means that the reference streamline passes through the origin, then C = 0 and Ψ = xy − 2y 2 + f (x).

(III)

To determine f (x), differentiate partially the above expression with respect to x and equate to −vy, equation (II):

∂Ψ ∂ −−−− = y + −−− f ( x ) = y + 4 x, ∂x ∂x f(x) =

4x dx = 2x . 2

Substitute into (III), Ψ = 2x 2 + xy − 2y2. To check whether the flow is potential, there are two possible approaches: (a) Since

∂v ∂v −−−−y – −−−−x = 0, ∂x ∂y but vy = −(4x + y) and vx = (x − 4y), therefore,

∂v −−−−y = – 4 ∂x

and

∂ vx −−−− = – 4 , ∂y

so that

∂v ∂v −−−−y – −−−−x = – 4 + 4 = 0 ∂x ∂y and the flow is potential. (b) Laplace’s equation must be satisfied:

∂ 2Ψ ∂ 2Ψ −−−−−2 + −−−−−2 = 0, ∂x ∂y Ψ = 2x 2 + xy − 2y 2. Therefore,

∂Ψ −−−− = 4x + y ∂x

and

∂Ψ −−−− = x – 4 y, ∂y

∂ 2Ψ −−−−−2 = 4 ∂x

and

∂ 2Ψ −−−−−2 = – 4 . ∂y

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Therefore 4 − 4 = 0 and so the flow is potential. Now, to obtain the velocity potential,

∂Φ −−−− = v x = x – 4 y; ∂x therefore, Φ=

(x – 4y) dx + f(y) + G.

But Φ0 = 0 at x = 0 and y = 0, so that G = 0. Therefore Φ = x22 − 4yx + f( y). Differentiating with respect to y and equating to vy, d ∂Φ −−−− = – 4x + −−− f ( y ) = – ( 4x + y ) ∂y dy d −−− f ( y ) = – y dy

and

y2 f ( y ) = – −− , 2

so that Φ = x 22 − 4yx − y 22.

7.6

STRAIGHT LINE FLOWS AND THEIR COMBINATIONS

The simplest flow patterns are those in which the streamlines are all straight lines parallel to each other (Fig. 7.12).

FIGURE 7.12 Rectilinear flow

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229

The convention for numbering the streamlines is that the stream function is considered to increase to the left of an observer looking downstream, i.e. in the direction of flow along the streamlines, as indicated in Fig. 7.12. If the velocity of the rectilinear flow v is inclined to the x axis at an angle α, then its components are vx = v cos α and vy = v sin α. The stream function is obtained simply by substitution of the above expressions into dΨ = vx dy − vy dx, whereupon Ψ=

v cos α dy – v sin α dx + constant.

Since in a uniform flow v = constant and in a straight line flow α is also constant, the expression for the stream function becomes Ψ = vy cos α − vx sin α + constant. The constant of integration may be made zero by choosing the reference streamline Ψ0 = 0 to pass through the origin, so that when x = 0 and y = 0 the stream function Ψ = Ψ0 = 0. Thus Ψ = v( y cos α − x sin α).

(7.28)

Since vx and vy are constant, then ∂ vx∂y and ∂ vy ∂x are both zero and, therefore, the flow is potential. The velocity potential is obtained from

∂Φ ∂Φ dΦ = −−−− dx + −−−− dy = v x dx + v y dy. ∂x ∂y Therefore, by substitution and integration, Φ=

v cos α dx + v sin α dy + constant,

but, if Φ = Φ0 = 0 at x = 0 and y = 0, then Φ = v(x cos α + y sin α).

(7.29)

Some simple straight line flows may be illustrated as follows. 1.

Uniform, straight line flow in the direction Ox, velocity u, shown in Fig. 7.13. Let streamline Ψ0 = 0 be along the x axis. Now,

∂Ψ v x = −−−− = u = constant. ∂y

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FIGURE 7.13 Straight line flow: Ψ = uy

Therefore,

∂ Ψ = u∂ y. Integrating, Ψ = uy + constant, but Ψ0 = 0 at x = 0 and y = 0 so that constant = 0, and the equation of the stream function becomes Ψ = uy. Alternatively, the volume flowing between the x axis and any streamline, per unit depth, is q = uy and, therefore, Ψ = uy. 2.

Uniform, straight line flow in the direction Oy, velocity v, shown in Fig. 7.14. Let streamline Ψ0 = 0 be along the y axis. Now,

∂Ψ v y = – −−−− = v = constant. ∂y Therefore,

∂ Ψ = −v dx.

FIGURE 7.14 Straight line flow: Ψ = −vx

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231

FIGURE 7.15 Combination of straight line flows

Integrating, Ψ = −vx + constant, but Ψ0 = 0 at x = 0 and y = 0 so that constant = 0, and the equation of the stream function is Ψ = −vx. 3.

Combined flow consisting of a uniform flow u = 10 m s−1 along Ox and uniform flow v = 20 m s−1 along Oy, shown in Fig. 7.15. Choose a suitable scale for x and y, say 10 mm = 20 m. Draw horizontal streamlines Ψ0 = uy = 10y and label them. Draw vertical streamlines Ψb = −vx = −20x and label them. At point A the steam function due to v is Ψb = 20 and the stream function due to u is Ψa = −20. Therefore, the combined stream function (scalar quantity) is Ψ = 20 − 20 = 0. Similarly, it is zero at point B and the origin. Hence the stream function for the streamline passing through AOB is Ψ = 0. By the same method, at point A′ the stream function due to v is Ψb = 0 and the stream function for the streamline due to u is Ψa = −20. Therefore, the combined stream function is Ψ = 0 − 20 = −20. Similarly, the combined stream function at B′ is also equal to −20. Thus, the straight line passing through A′B′ represents a streamline of the combined flow whose stream function is −20. By repeating the process of drawing lines through points at which the combined value of the stream function is the same, a new set of streamlines is obtained and it represents the combined flow pattern.

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FIGURE 7.16 Radial flow: a source

The same results may be obtained by the algebraic method. Since Ψ = Ψa + Ψb, it follows that Ψ = uy − vx = 10y − 20x. This equation represents a family of straight lines, each line being defined by the particular value of Ψ assigned to it. The other basic flow patterns in which the streamlines are straight lines are those in which the fluid flows radially either outwards from a point, in which case it is known as a source, or inwards into a point, in which case it is known as a sink. A sink flow is simply treated as a negative source flow and, thus, the mathematics of both may be explained by considering only the source flow, which is shown in Fig. 7.16. Radial flows and their applications were discussed in Section 6.17, but here we are concerned with the mathematical expressions for their stream function and velocity potential which lead to more complex and useful flow combinations. In radial flows, it is seen that, since the velocity passes through the origin and is a function of θ only, the tangential component of velocity does not exist and v = vr. Consider now a source of unit depth and let the steady rate of flow be q, known as the strength of the source. Then, at any radius r, the radial velocity is given by vr = q2π r.

(7.30)

The stream function and the velocity potential are obtained in a similar manner as for the rectilinear flow, but polar coordinates are used. Since, from equation (7.12), dΨ = rvr dθ − vθ dr, for radial flow vθ = 0, and for a source vr = q2π r, it follows that dΨ = r(q2π r) dθ = (q2π) dθ.

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Integrating, Ψ = qθ2π + constant. If, however, Ψ = Ψ0 = 0 when θ = 0, the constant of integration becomes zero and Ψ = qθ2π for a source,

(7.31)

Ψ = −qθ2π for a sink.

(7.32)

Similarly, it may be shown that Φ = (q2π) log e r for a source,

(7.33)

Φ = −(q2π) log e r for a sink.

(7.34)

The simplest case of combining flow patterns is that in which a source is added to a uniform rectilinear flow. This is accomplished by the additions of the stream functions of the two types of flow. The stream function for a uniform rectilinear flow parallel to the x axis is ΨR = v0 y = v0 r sin θ, and that for a source is ΨS = qθ 2π. Thus, the stream function for the combined flow is Ψ = ΨR + ΨS = v0 r sin θ + qθ 2π.

(7.35)

Figure 7.17 shows graphically that this is the superposition of a system of radial streamlines onto a system of straight streamlines parallel to the x axis. By definition, a given streamline is associated with one particular value of the stream function and, therefore, if we join the points of intersection of the radial streamlines with the rectilinear streamlines where the sum of the stream functions is a given constant value, the resulting line will be one streamline of the combined flow pattern. If this procedure

FIGURE 7.17 Combination of rectilinear flow and a source

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is repeated for a number of values of the combined stream function, the result will be a picture of the combined flow pattern. This is shown in Fig. 7.17, where numerical values were assigned to stream functions in order to illustrate the point. Observe, for example, the streamline of the combined flow Ψ = 6. It passes through points A, B, C and D such that: at A, ΨU = 2.5 and ΨS = 3.5; therefore Ψ = 2.5 + 3.5 = 6 at B, ΨU = 3.0 and ΨS = 3.0; therefore Ψ = 3.0 + 3.0 = 6 at C, ΨU = 3.5 and ΨS = 2.5; therefore Ψ = 3.5 + 2.5 = 6 at D, ΨU = 5.0 and ΨS = 1; therefore Ψ = 5.0 + 1.0 = 6. All streamlines of the combined flow are obtained in this manner. It is interesting to note, in this particular flow pattern, that the resulting streamlines are grouped into two distinct sets. In one set all the streamlines emerge from the origin (Ψ = 3, 3.5, 3.75) and in the other they approach the rectilinear flow asymptotically at some distance upstream (Ψ = 4.5, 5, 6). The two sets are separated by the streamline Ψ = 4, which passes through point S. This point is a stagnation point, where the velocity from the source is equal to the uniform velocity of the parallel flow, so that the resultant velocity at S is zero. The distance OS = a may, therefore, be determined by equating the uniform velocity to that from the source at radius a. Thus, v0 = q2π a, so that a = q2π v0.

(7.36)

The value of the stream function for the streamline passing through point S is obtained by substituting θ = π and r = a = q2π v0 into (7.35). Thus, ΨS = v0(q2π v0) sin π + qπ 2π, which simplifies to ΨS = −12 q.

(7.37)

Since there can be no flow across a streamline, then the streamline ΨS passing through S may be replaced by a solid boundary of an object under investigation, such as a hill or the nose of an aerofoil. In the latter case, the flow pattern below the x axis must also be used as shown in Fig. 7.18. It is then known as a half-body or Rankine body. The general equation for the streamline through point S is Ψ = −12 q = v0 r sin θ + qθ 2π,

(7.38)

from which the radial distance rS to any point on this streamline is rS = q(π − θ )2π v0 sin θ,

(7.39)

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235

FIGURE 7.18 Rankine body

and it describes the contour of the Rankine body. It can be appreciated that as x → ∞ this streamline becomes parallel to the x axis and, there, the perpendicular distance from the x axis to the streamline represents the maximum half-width of the Rankine body. The perpendicular distance is given by y = r sin θ = q(π − θ )2π v0. But, as x → ∞, the radius r → ∞ and θ → 0, so that ymax = q2v0.

EXAMPLE 7.2

(7.40)

In the ideal flow around a half-body, the free stream velocity is 0.5 m s−1 and the strength of the source is 2.0 m2 s−1. Determine the fluid velocity and its direction at a point, r = 1.0 m and θ = 120°.

Solution The stream function for the flow around a half-body is given by Ψ = v0 r sin θ + qθ 2π. In this case, v0 = 0.5 m s−1, q = 2.0 m2 s−1. To determine the fluid velocity and its velocity vector at a point, it is first necessary to determine its tangential and radial components. These are 1 ∂Ψ v r = − −−−− r ∂θ

and

∂Ψ vθ = – −−−− . ∂r

Therefore, q 1 1 2 v r = − ⎛ v 0 r cos θ + −−−⎞ = − ⎛ 0.5 × 1 × cos 120° + −−−⎞ 2 π⎠ 1 ⎝ r⎝ 2 π⎠ = −0.25 + 0.318 = 0.068 m s−1, vθ = −v0 sin θ = −0.5 sin 120° = −0.433 m s−1,

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FIGURE 7.19

which is in the clockwise direction. Therefore, v = ( v 2r + vθ2 ) = (0.0047 + 0.188) = 0.438 m s−1. If β is the angle the velocity vector makes with the horizontal, as shown in Fig. 7.19, then

β=θ−α and

tan α = vθ vr = 0.4380.068 = 6.44.

Therefore

α = 81.2°, and β = 120 − 81.2 = 38.8°.

7.7

COMBINED SOURCE AND SINK FLOWS. DOUBLET

Let us consider the flow pattern resulting from the combination of a source and a sink of equal strength, which means that the flow rate from the source is equal to the flow rate into the sink. Also, let them be placed symmetrically about the origin and on the x axis, as shown in Fig. 7.20. Let the stream function for the source be Ψ1, for the sink be Ψ2 and let the flow rate be q. Since the convention for stream functions is that they increase to the left while looking downstream, it follows that the stream functions for the source increase as the angle θ1 increases and those for the sink decrease as angle θ2 increases. As discussed earlier, the value of a combined stream function is obtained by the addition of the values of stream functions at their intersection. For example, if the combined stream function is Ψ = 5, it will pass through points of intersection of Ψ1 and Ψ2 such that their values add up to 5 (e.g. Ψ1 = 1 and Ψ2 = 4 or Ψ1 = 2 and Ψ2 = 3 and so on, as shown in Fig. 7.20). Figure 7.20 also shows that the combined streamlines of a source and a sink of equal strength are circles passing through the point source and the point sink. Mathematically, Ψ = Ψ1 + Ψ2 = qθ1 2π − qθ2 2π, = (q2π)(θ1 − θ2),

(7.41)

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Combined source and sink flows. Doublet

237

FIGURE 7.20 Source and sink

which, since Ψ and q are constant for any given streamline, is a condition satisfied by a circle. Figure 7.20 shows only half of the flow pattern, the other half, below the x axis, being the mirror image of that above it. The velocity potential for such a combined flow is also obtained by the addition of the velocity potentials for the source and the sink. Thus Φ = Φsource + Φsink = (q2π) log e r1 − (q2π) log e r2, Φ = (q2π)(log e r1 − log e r2).

(7.42)

7.7.1 Doublet If a sink and a source of equal strength are brought together in such a way that the product of their strength and the distance between them remain constant, the resulting flow pattern is known as a doublet. Consider point P (Fig. 7.21) on the velocity potential of a doublet. Let the velocity potential for the doublet be ΦD; then ΦD = (q2π) loge (r + dr) − (q2π) log e r FIGURE 7.21 Source and sink

q dr r + dr q = −−− log e ⎛ −−−−−−−−⎞ = −−− log e ⎛ 1 + −−−⎞ . ⎝ ⎝ r ⎠ 2π 2π r⎠

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Expanding, dr dr 1 dr 2 log e ⎛ 1 + −−−⎞ = −−− – − ⎛ −−−⎞ + · · · . ⎝ ⎠ r r 2⎝ r ⎠ Neglecting terms of second order and higher, q dr Φ D = −−− −−−, 2π r but, since by definition of a doublet ds → 0, it follows that dr d s cos θ and ΦD = (q2πr) ds cos θ. Also for a doublet, by definition q ds = constant. Let this constant, known as the strength of the doublet, be denoted by m; then m = q ds and ΦD = (m2π r) cos θ

(7.43)

or, in rectangular coordinates, m x −−−−−−⎞ . Φ D = −−− ⎛ −−− 2 π ⎝ x 2 + y 2⎠

(7.44)

From the above equations, the expressions for the stream function may be obtained, namely ΨD = −(m2π r) sin θ = −(m2π)[ y(x 2 + y 2 )].

(7.45)

Note that the above equations were derived for a doublet in which the source and the sink were placed on the x axis. Such a doublet is shown in Fig. 7.22(a). If, however, the FIGURE 7.22 Doublets

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Combined source and sink flows. Doublet

239

source and the sink are placed on the y axis, the resulting doublet is oriented as in Fig. 7.22(b) and the expressions for the stream function and the velocity potential become ΦD(yy) = (m 2π r) sin θ = (m 2π)[ y(x 2 + y 2)],

(7.46)

ΨD(yy) = −(m2π r) cos θ = −(m2π)[x(x 2 + y 2)].

(7.47)

The flow is always from the source to the sink, so that, if they are placed as shown in Fig. 7.21, the flow in the doublet is as shown in Fig. 7.22(a). If, however, the positions of the source and the sink are reversed, the flow directions are also reversed, which means that the expressions for the stream function and the velocity potential change signs.

EXAMPLE 7.3

A source of strength 10 m2 s−1 at (1, 0) and a sink of the same strength at (−1, 0) are combined with a uniform flow of 25 m s−1 in the −x direction. Determine the size of Rankine body formed by the flow and the difference in pressure between a point far upstream in the uniform flow and the point (1, 1).

Solution For the source (Fig. 7.23), y q 10 Ψ source = −−− θ = −−− tan –1 ⎛ −−−−−− ⎞ ; ⎝ x – 1⎠ 2π 2π for the sink, y q 10 Ψ sink = – −−− θ = – −−− tan –1 ⎛ −−−−−−−⎞ ; ⎝ x + 1⎠ 2π 2π for the uniform flow, ΨU = −25y. Thus, the combined flow is represented by the stream function y y 10 Ψ = −−− tan –1 ⎛⎝ −−−−−− ⎞⎠ – tan –1 ⎛⎝ −−−−−−−⎞⎠ – 25y . x–1 x+1 2π

FIGURE 7.23

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To obtain stagnation points, vx = 0. Thus, x–1 x+1 ∂ Ψ 10 v x = −−−− = −−− −−−−−−−−−−−2−−−−−−2 – −−−−−−−−−−−2−−−−−−2 – 25 . ∂y 2π (x – 1) + y (x + 1) + y Now, vx = 0 at y = 0, i.e. on the x axis; therefore, 10 1 1 −−− ⎛ −−−−−− – −−−−−−−⎞ = 25, ⎝ 2 π x – 1 x + 1⎠ (x + 1) − (x − 1) = 5π (x − 1)(x + 1), 25π = x 2 − 1, x 2 = 25π + 1 = 1.127, x12 = ±1.062 m, and the length of the Rankine body is l = x1 + x2 = 2.124 m. To obtain the width of the Rankine body, it is necessary to determine the maximum value of y on the contour of the body, i.e. on Ψ0. This will occur because of the symmetry at vy = 0: 10 y ∂Ψ y v y = – −−−− = − −−− −−−−−−−−−−−2−−−−−−2 – −−−−−−−−−−−2−−−−−−2 = 0. ( x + 1 ) +y ( x – 1 ) + y 2π ∂x Therefore, y y −−−−−−−−−−−2−−−−−−2 = −−−−−−−−−−−2−−−−−−2 , (x + 1) + y (x – 1) + y but, since y ≠ 0, (x − 1)2 + y 2 = (x + 1)2 + y 2, x 2 − 2x + 1 = x 2 + 2x + 1. −4x = 0, x = 0, which is expected from the symmetry of the source and sink about the origin. To find the value of ymax which will give the width of the body, substitute the above value of x = 0 into Ψ = 0: 10 0 = −−− [ tan –1 ( – y ) – tan –1 y ] – 25 y max , 2π but, since | y | = | −y |, 25 × 2 π −−−−−−−−−− y max = 2 tan –1 y, 10 ymax = 0.127 tan−1 y,

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from which ymax = 0.047 m and the width of the Rankine body is 2ymax = 0.094 m. At point (1, 1), 10 v x = −−− ( – −25 ) – 25 = – 25.63 m s –1 , 2π 10 v y = – −−− ( 1 – −15 ) = – 1.27 m s –1 . 2π Therefore, v = ( v 2x + v 2y ) 12 = 25.66 m s –1 . Since the flow is potential, Bernoulli’s equation may be applied to any two points, such as one in the free stream (v∞ = 25 m s−1) and point (1, 1): p∞ ρg + v 2∞ 2g = p(1, 1) ρg + v 22g, and, hence, p(1, 1) − p∞ = ( ρ2)(v 2 − v 2∞ ) = ( ρ2)33.43 N m−2.

7.8

FLOW PAST A CYLINDER

A flow pattern equivalent to that of an ideal fluid passing a stationary cylinder, with its axis perpendicular to the direction of flow, is obtained by combining a doublet with rectilinear flow. Figure 7.24 shows the resulting streamlines and the stagnation points S which are formed. The combined stream function and the velocity potential are Ψc = ΨD + ΨR = −(m 2π r) sin θ + v0 r sin θ

and

Ψc = (v0 r − m2π r) sin θ,

(7.48)

Φc = (v0 r + m2π r) cos θ.

(7.49)

Since the flow pattern corresponds to that around a cylinder, it is of interest to obtain an expression for the radius of this cylinder. Since the distance between the two

FIGURE 7.24 Flow around a cylinder

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stagnation points is the diameter of this cylinder, say 2a, and the flow at a stagnation point is zero, it follows that the streamline passing through S is Ψ0 = 0. Thus Ψ0 = (v0 a − m2π a) sin θ = 0, so that v0 a = m2π a and

a = (m2π v0).

(7.50)

For a given velocity of the uniform flow and a given strength of the doublet, the radius a is constant, which proves that the body so derived is circular. It is also possible to plot the flow pattern around a cylinder of radius a with uniform velocity v0. From equation (7.50), the strength of the doublet is m = 2π v0 a 2, and the combined stream function becomes Ψc = (v0r − 2π v0 a22π r) sin θ = v0 (r − a 2r) sin θ.

(7.51)

Similarly, the velocity potential is Φc = v0 (r + a 2r) cos θ.

EXAMPLE 7.4

(7.52)

If a 40 mm diameter cylinder is immersed in a stream having a velocity of 1.0 m s−1, determine the radial and normal components of velocity at a point on a streamline where r = 50 mm and θ = 135°, measured from the positive x axis. Assume flow to be ideal. Also determine the pressure distribution with radial distance along the y axis.

Solution By equations (7.13), the velocity components are given by 1 ∂Ψ v r = − −−−− r ∂θ

and

∂Ψ v θ = – −−−− , ∂r

but, for the ideal flow around a cylinder, the stream function is a2 Ψ c = v 0 ⎛ r – −−⎞ sin θ . ⎝ r⎠ Therefore, a2 v 1 ∂ a2 v r = − −−− v 0 ⎛⎝ r – −−⎞⎠ sin θ = −−0 ⎛ r – −−⎞ cos θ r r ∂θ r⎝ r⎠ a2 = ⎛ 1 – −−2 ⎞ v 0 cos θ , ⎝ r ⎠

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and

Flow past a cylinder

243

a2 a2 1 v θ = – −−− v 0 ⎛⎝ r – −− ⎞⎠ sin θ = – v 0 ⎛ 1 + −−2 ⎞ sin θ . ⎝ r ∂r r ⎠

Substituting the numerical values a = 2 cm, r = 5 cm, v0 = 1.0 m s−1, θ = 135°, the following values for velocity components are obtained: 4 21 1 v r = ⎛ 1 – −−−⎞ cos 135° = – −−− −−− = − 0.594 m s –1 , ⎝ 25⎠ 25 2 4 29 1 v θ = – ⎛ 1 + −−−⎞ sin 135 ° = – −−− −−− = − 0.820 m s –1 . ⎝ 25⎠ 25 2 Remembering the sign convention for cylindrical coordinates, these components are as shown in Fig. 7.25.

FIGURE 7.25

To obtain the pressure distribution along Ay it is first necessary to determine the velocity variation so that it may be used in applying Bernoulli’s equation. Since for Ay, θ = 90°, it follows that vr = 0, and, hence, vθ = −(1 + a 2r 2)v0, which is the required velocity distribution for Ay. Now, applying Bernoulli’s equation to a point far upstream where the velocity is v0, the pressure is p0, and to the section in the equation p0ρ + v 20 2 = pρ + vθ2 2, gives

p − p0 = ( ρ2)( v 20 − vθ2 ) = ( ρ2)[ v 20 − (1 + a2r 2)2 v 20 ] = ( ρ2) v 20 [1 − (1 + 2a2r 2 + a 4r 4)] = −( ρ v 20 2)(2a 2r 2 + a 4r 4).

This equation shows that when r → ∞, the pressure p approaches p0, but at the surface of the cylinder (at point A), where r = a, the pressure is lower than that upstream by an amount equal to --32- ρ v 20 .

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7.9

CURVED FLOWS AND THEIR COMBINATIONS

The previous sections of this chapter dealt with flows whose basic components were straight line flows, either rectilinear or radial. The third basic type of flow is such that the streamlines are concentric circles, as shown in Fig. 7.26. Such flows are known as vortex flows. Their characteristic is that the radial component of velocity vr = 0. This is so because, of course, there cannot be any flow across streamlines and, since in vortex flows they are circular, the flow must be confined to purely circular paths. Thus in any vortex flow vr = 0 and v = vθ.

(7.53)

There are two fundamental types of vortex flow distinguished by the nature of flow, namely rotational and irrotational. From these two basic types, various combinations of flows are possible. FIGURE 7.26 Vortex flow

Vortex flows and their applications were discussed in Section 6.19. They were not, however, defined there with respect to rotational or irrotational flow, nor were the mathematics of their stream function and velocity potential appropriately discussed. Let us first consider the irrotational vortex flow, which is known as the free vortex. Because it is irrotational, the vorticity and circulation across the stream must be equal to zero. Consider, in a free vortex flow, an element of fluid between streamlines Ψ and (Ψ + dΨ), as shown in Fig. 7.27. The circulation round the element, starting from A in the anticlockwise direction, is ΓABCD = 0 − (vθ + dvθ)(r + dr) dθ + 0 + vθ r dθ, and, neglecting infinitesimals of the third order, ΓABCD = −(vθ dr + r dvθ) dθ. This, by the definition of irrotational flow, must be equal to zero. Therefore, −(vθ dr + r dvθ) dθ = 0,

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Curved flows and their combinations

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FIGURE 7.27 An element of vortex flow

so that vθ dr + r dvθ = 0, but this is a differential of a product, d(rvθ) = 0, which, when integrated, gives rvθ = constant.

(7.54)

This equation defines the relationship between the velocity and radius for a free vortex. It shows that the velocity increases towards the centre of the vortex and tends to infinity when the radius tends to zero. The velocity decreases as the radius increases and tends to zero as the radius tends to infinity. One practical example of this type of vortex flow is the emptying of a container through a central hole. The constant in equation (7.54) may be established by making use of the singularity which exists in the free vortex flow, namely the infinite velocity at the centre of the vortex which we mentioned above. At this point, the vorticity, which is given (equation (7.24)) by

∂v v – ⎛ −−−−θ + −−θ⎞ , ⎝ ∂r r ⎠ becomes indeterminate on substitution of r → 0 and vθ → ∞. It can, however, be determined by evaluating the circulation around the centre, i.e. along any of the concentric streamlines. This does not violate the condition for irrotational flow, by which the free vortex is defined, because the condition states that vorticity (and circulation) must be zero for any closed loop across the flow (see Section 7.1). The circulation around a circular streamline, ΓC =

v ds = Circumference × Tangential velocity = 2πrv , θ

but, since vθ r = constant, it follows that this particular circulation is constant for any streamline and, therefore, for the whole vortex field. It may, therefore, be used to

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measure the intensity of the vortex and is known as the vortex strength. Thus, vortex strength ΓC = 2π rvθ for the anticlockwise vortex. The free vortex equation (7.54) may now be rewritten as vθ r = ΓC 2π.

(7.55)

The stream function may be obtained from equation (7.12): dΨ = vr r dθ − vθ dr. Since vr = 0,

Ψ = – v θ dr, and, substituting for vθ from equation (7.55),

Γ Γ Ψ = – −−−−C− dr = – −−−C log e r + constant . 2πr 2π The constant of integration is made zero by taking Ψ = 0 at r = 1, so that, finally, Ψ = −(ΓC 2π) log e r

(7.56)

for anticlockwise rotation. The sign of the above expression becomes positive for clockwise rotation. The velocity potential follows (equation (7.21)) from dΦ = vr dr + rvθ dθ, whereas, upon substitution and making Φ0 = 0 at θ = 0, Φ=

−2Γ−−π dθ = −2Γ−−π θ. C

C

(7.57)

Since the free vortex is irrotational, the Bernoulli constant remains the same for all streamlines.

EXAMPLE 7.5

A two-dimensional fluid motion takes the form of concentric, horizontal, circular streamlines. Show that the radial pressure gradient is given by v2 dp −−− = ρ −−, r dr where ρ = density, v = tangential velocity, r = radius. Hence, evaluate the pressure gradient for such a flow defined by Ψ = 2 log e r, where Ψ = stream function, at a radius of 2 m and fluid density of 103 kg m−3.

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Solution For two concentric streamlines the variation of total head or the Bernoulli constant is, in general, given by dH vθ ⎛ dvθ vθ⎞ −−−− = −− −−−− + −− , g ⎝ dr r ⎠ dr but, for horizontal flow, z = 0 and, for vortex flow, v = vθ , so that H = pρg + vθ2 2g; therefore, differentiating, dH 1 dp v dvθ −−−− = −−− −−− + −−θ −−−−. dr ρ g dr g dr Equating the two equations, v θ ⎛ dvθ v θ⎞ 1 dp v dvθ −− −−−− + −− = −−− −−− + −−θ −−−−, ⎝ ⎠ g dr r ρ g dr g dr from which dv v 2 dv vθ2 1 dp − −−− = v θ −−−−θ + −−θ – v θ −−−−θ = −−. ρ dr dr r dr r Therefore, vθ2 dp −−− = ρ −−. r dr The stream function Ψ = 2 log e r represents a free vortex, for which Ψ = (ΓC 2π) log e r, and, hence, ΓC 2π = 2, but, for a free vortex, ΓC 2π = vθ r, so that vθ r = 2 and

4 vθ2 = −−2 r

and, therefore, 4 4 dp −−− = ρ −−3 = 10 3 × −−3 = 500 N m –3 . r 2 dr

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The most common example of a rotational vortex, which is considered of fundamental importance, is a forced vortex. It will be shown in what follows that in a forced vortex the fluid rotates as a solid body with a constant rotational velocity. Consider circulation around a segmental element (such as in Fig. 7.27) of a forced vortex, remembering that vr = 0: ΓABCD = −(vθ dr + r dvθ) dθ. The area of the element is A = r dθ dr, so that vorticity (as already shown in Section 7.2, equation (7.5)) is given by Γ v dv ζ = −−−ABCD −−−−− = – ⎛ −−θ + −−−−θ⎞ , ⎝ dA r dr ⎠ but if, for a solid body, rotation ω is the angular velocity, which at any radius r is related to the tangential velocity vθ by

ω = vθ r,

(7.58)

it follows, therefore, that for a forced vortex the vorticity

ζ = −2ω

(7.59)

and is constant for a given vortex. The flow is rotational and there is, therefore, variation of the Bernoulli constant with radius. Using equation (6.38), dH vθ ⎛ vθ dvθ⎞ 2ω −−−− = −− −− + −−−− = v θ × −−−− dr g ⎝ r dr ⎠ g and, since vθ = ω r, dH 2 ω 2 r −−−− = −−−−−−− . g dr

(7.60)

In order to determine the pressure distribution or the surface gradient in a forced vortex, the above expression must be used in conjunction with Bernoulli’s equation, as was shown in Section 6.19. The stream function for a forced vortex is obtained in the same manner as for the free vortex, but using the appropriate relationship (equation (7.58)), namely that vθ = ω r, which yields

Ψ = – ω r dr = – −12 ω r 2 + constant . But for Ψ = 0 at r = 0, Ψ = − −12 ω r 2 for anticlockwise rotation.

(7.61)

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FIGURE 7.28 Free spiral vortex

Since the forced vortex is rotational, there is no velocity potential corresponding to it and the Laplace equations are not satisfied. A free spiral vortex, mentioned in Section 6.19, is, in contrast to a forced vortex, irrotational and represents a potential flow. The free spiral vortex is the combination of a free vortex and radial flow. It is, therefore, obtained by superposition of the stream functions of a free vortex with either a sink or a source flow depending upon the direction of the radial flow. For outward flow using a source and for a clockwise vortex, Ψsv = Ψsource + Ψfree vortex = qθ2π + (ΓC2π) log e r = (12π)(qθ + ΓC log e r) and

(7.62)

Φsv = Φsource + Φfree vortex = (q2π) log e r + (ΓC2π)θ = (12π)(q log e r + ΓCθ ).

(7.63)

The resulting flow is shown in Fig. 7.28.

7.10 FLOW PAST A CYLINDER WITH CIRCULATION. KUTTA–JOUKOWSKY’S LAW Flow past a stationary cylinder may be obtained by superposition of a parallel flow and a doublet. This was discussed in Section 7.8. However, in the first half of the nineteenth century the German physicist H. G. Magnus observed experimentally that

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FIGURE 7.29 if the cylinder in a parallel flow stream is rotated about its axis, a transverse force, which tends to move the cylinder across the parallel flow stream, is generated. This is known as the Magnus effect or aerodynamic lift. The hydrodynamic equivalent of rotating a cylinder in a flow stream is to add circulation by means of a free vortex to the doublet in a parallel flow. Such a flow pattern can be obtained directly from the results of Sections 7.8 and 7.9 by adding the stream function of a free vortex (equation (7.56)) for clockwise rotation to the stream function of flow past a cylinder, given by equation (7.51). Thus, the combined stream function is Ψ = v0 (r − a2r) sin θ + (ΓC2π) log e r.

(7.64)

The addition of circulation to the ideal flow past a cylinder gives rise to an asymmetric flow pattern, as shown in Fig. 7.29. There is an increase of velocity on one side of the cylinder and a decrease on the other. In consequence of this, the stagnation points move from the axis of the parallel flow. Their positions depend upon the magnitude of the circulation and can be determined using equation (7.64), remembering that at stagnation points, vθ = 0. Thus, the tangential velocity is given by Γ a2 ∂Ψ v θ = – −−−− = – v 0 ⎛ 1 + −−2 ⎞ sin θ – −−−−C− = 0. ⎝ ∂r 2πr r⎠ Also, on the contour of the cylinder, r = a; therefore, −2v0 sin θ − ΓC2π a = 0 or sin θ = −ΓC 4π av0.

(7.65)

The negative sign indicates that for the positive parallel flow (from left to right) and clockwise circulation, the stagnation points lie below the x axis. Furthermore, if the value of circulation ΓC 4π av0, then 0 sin θ −1 and the stagnation points will lie in positions such as those shown in Fig. 7.29(b). For ΓC = 4π av0, the stagnation points merge on the negative y axis, as shown in Fig. 7.29(c), and, finally, for ΓC 4π av0, the stagnation points will be as shown in Fig. 7.29(d). Since sin θ cannot be greater than 1, this is only possible if a is increased, which means that the stagnation point moves away from the surface of the cylinder. In order to establish the magnitude of the transverse force acting on the cylinder and mentioned earlier, it is necessary to obtain first the pressure distribution around

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251

the cylinder and then the forces arising from it. Since the flow is irrotational, Bernoulli’s equation may be applied to a point some distance upstream in the parallel flow and to a point on the surface of the cylinder. Thus, pρ + vθ2 2 = p0 ρ + v 20 2, where p is the pressure on the cylinder and it varies with θ, and p0 is the pressure in the parallel flow some distance upstream, where the velocity is v0. Rearranging and solving for pressure difference: p − p0 = ( ρ2)( v 20 − vθ2 ) = ( ρ v 20 2)(1 − vθ2 v 20 ), but

vθ = −2v0 sin θ − ΓC 2π a,

so that p − p0 = ( ρ v 20 2)[1 − (−2 sin θ − ΓC 2π av0) 2] = ( ρ v 20 2)(1 − 4 sin2 θ − 2ΓC sin θ π av0 − ΓC2 4π 2a 2 v 20 ).

(7.66)

Consider, now, an element of the cylinder’s surface. The force due to pressure acting on it is (p − p0)a dθ, and it may be resolved into vertical and horizontal components. The transverse force will, in our case, be the sum of the vertical components. Thus, the upward force L=–

2π

( p – p 0 ) a sin θ d θ .

In equation (7.66), let 1 − ΓC2 4π 2a 2 v 20 = A; then

p − p0 = ( ρ v 20 2)(A − 4 sin2 θ − 2ΓC sin θ π av0) L=–

and

2π

2π

( ρ v 20 2 ) ( A – 4 sin 2 θ – 2 Γ C sin θ π av 0 ) a sin θ d θ

=–

( ρ v 20 a2 ) ( A sin θ – 4 sin 3 θ – 2 Γ C sin 2 θ π av 0 ) d θ .

But

2π

sin θ d θ = 0

and

2π

sin 3 θ d θ = 0,

so that

ρ av 2 L = −−−−−−0 2

L = ρ v0ΓC.

2π

2Γ C sin 2 θ ρ v0 ΓC ⎛ 1 sin 2 θ ⎞ 2 π −−−−−−−−−−−−−− d θ = −−−−− −−− ⎝ − θ – −−−−−−−− ⎠ 4 π av 0 π 2 0

(7.67)

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Thus, the force perpendicular to the direction of the parallel flow, or a main free stream, which in general is known as the lift, is, for a rotating cylinder of infinite length, independent of the diameter of the cylinder and equal to the product of fluid density, free stream velocity and circulation. This statement is known as Kutta– Joukowsky’s law and gives theoretical justification for the experimentally observed Magnus effect. It is interesting to note that the horizontal component of the force on the cylinder due to pressure, which in general is called the drag, and for our case is given by

D=

2π

( p – p 0 )a cos θ d θ ,

is equal to zero. This result, obtained on the assumption of the ideal flow, is not supported by experiments. This is so because in real fluids viscous friction provides resistance to flow and separation may occur.

7.11 COMPUTER PROGRAM

ROTCYL

Program ROTCYL determines the angular positions of the stagnation points on the surface of a cylinder rotating in a uniform fluid stream and calculates the lift coefficient and the values of the pressure coefficient along the flow axis upstream of the cylinder, Fig. 7.29. If there is only one stagnation point then the program calculates its distance from the cylinder surface. Calculations are based upon potential flow theory developed in this chapter, in particular equations (7.65), (7.66) and (7.67), together with the definitions of lift coefficient, equation (12.5) and pressure coefficient (9.1). The required data are cylinder diameter, D (mm), cylinder rotational speed, N (rev min−1) and free stream velocity, U (m s−1).

7.11.1 Application example For a 60 mm diameter cylinder rotating at 1245 rev min−1 in a free stream of upstream velocity 20 m s−1, the stagnation points on the cylinder surface are located at −5.61° and 185.61° and the lift coefficient C L is 1.23. The values of pressure coefficient along the negative x axis are: X (mm) 30 −0.962 CP

150 −7.7E-2

270 −2.4E-2

390 −1.2E-2

510 630 −7E-3 −4E-3

7.11.2 Additional investigations using ROTCYL ROTCYL may also be used to investigate the effect of cylinder rotational speed, diameter and free stream velocity on stagnation point location and both the lift and pressure coefficient values.

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Concluding remarks While the concept of an ideal fluid is purely theoretical it will be demonstrated that the techniques introduced in this chapter do contribute to the understanding of fluid flow, particularly under flow conditions where the viscosity effects are minimal, namely at high Reynolds numbers and well away from any boundary layer or wake effect. The approach developed may therefore be of use in the study of external flows, in the study of forces acting on aerofoil shapes (Chapter 12), or in the study of rotodynamic machinery (Chapter 22). The mathematical techniques introduced to allow the combination of sources, sinks and rectilinear flow are often helpful in determining the actual flow under complex conditions, provided that the restrictions mentioned above are recognized.

Summary of important equations and concepts 1.

2.

3.

4.

5.

6.

Chapter 7 defines an ideal fluid flow as having no viscosity, compressibility, surface tension or vaporization limits and shows that the theoretical study of such flows was fundamental to the development of modern fluid mechanics. Definitions of vorticity, equation (7.2), and irrotational flow, equation (7.3), are presented, together with definitions of circulation, equation (7.4), and Stokes’ theorem, equation (7.5). The equation of the streamline and the definition of stream function are presented in Section 7.3, while Section 7.4 addresses velocity potential and potential flow and its implications for the application of Bernoulli’s equation to straight line and curved flows, Fig. 7.10. Flow nets are introduced in Section 7.5, together with the Cauchy–Riemann and Laplace equations (7.25) and (7.26), and the requirement that the Laplace equation be satisfied for velocity potential. Combinations of straight line flows are introduced in Section 7.6, and this concept is then expanded to include sources and sinks, Fig. 7.23; doublets are introduced to illustrate the generation of stagnation points for a body in a uniform flow, Fig. 7.23. Flow past a cylinder is introduced in Section 7.8. Combinations of curved flows are introduced in Section 7.9 with reference to forced vortices. Lift on a cylinder with rotation and the Kutta–Joukowsky law is introduced in Section 7.10, with the lift force defined by equation (7.67). A computer program, ROTCYL, is included to determine lift and stagnation point positions for a rotating cylinder in a uniform flow.

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Two-dimensional Ideal Flow

Problems 7.1 The x and y components of fluid velocity in a twodimensional flow field are u = x and v = −y respectively. (a) Determine the stream function and plot the streamlines Ψ = 1, 2, 3. (b) If a uniform flow defined by Ψ = y is superimposed on the above flow, plot the resulting streamlines and label them with Ψ values. (c) Determine the stream function and the velocity potential for the above combined flow. [(a), Ψ = xy, (c) Ψ = y + xy, Φ = x 22 + x − y 22] 7.2 The stream function for the two-dimensional flow of a liquid is given by Ψ = 2xy. In the range of values of x and y between 0 and 5 plot the streamlines and equipotential lines passing through coordinates (1, 1), (1, 2), (2, 2). Also determine the velocity in magnitude and direction at the point (1, 2). [4.47, 63.4°] 7.3 A flow has a potential function Φ given by Φ = V (x3 − 3xy 2). Derive the corresponding stream function Ψ and show that some of the streamlines are straight lines passing through the origin of coordinates. Find the inclinations of these lines. Evaluate also the magnitude and direction of the velocity at an arbitrary point x, y. [±60°, u = V(3x2 − 3y2), v = 6xyV ] 7.4 A source of strength 30 m2 s−1 is located at the origin, and another source of strength 20 m2 s−1 is located at (1, 0). Find the velocity components u and v at (−1, 0) and (1, 1). Also, if the dynamic pressure at infinity is zero for ρ = 2.0 kg m−3 calculate the dynamic pressure at the above points. [u = −6.37, v = 0, 40.58 N m−2, 36.74 N m−2] 7.5 A source of strength m at the origin and a uniform flow of 15 m s−1 are combined in two-dimensional flow so that a stagnation point occurs at (1, 0). Obtain the velocity potential and stream function for this case. [ Ψ = +15 tan−1(yx) − 15y, Φ = +7.5 log e(x 2 + y 2) − 15x] 7.6 A source discharging 20 m3 s−1 is located at (−1, 0) and a sink of twice the strength is located at (2, 0). For the pressure at the origin of 100 N m−2 and density of 1.8 kg m−3, find the velocity and pressure at points (0, 1) and (1, 1). [4.15 m s−1, 84.5 N m−2, 5.14 m s−1, 76.2 N m−2]

7.7 Show that the potential function for the flow generated by a source in a two-dimensional system is a ln (x 2 + y 2) where a is a constant. Hence derive an expression for the potential function for a doublet and show that the streamlines in a flow generated by a doublet are circular. Sketch these streamlines. 7.8 Show that the potential function Φ = ax(x2 + y2) represents the flow generated by a doublet. In which direction is the doublet oriented? A cylinder of radius 4 cm is held with its centre at the point (0, 0) in a fluid stream. At large distances from the cylinder the fluid velocity is constant at 30 m s−1 parallel to the x axis and in the direction of x increasing. Calculate the components of the fluid velocity at the point x = −4 cm, y = 1 cm. [−5.08 m s−1, −17.85 m s−1] 7.9 Under what circ*mstances does potential flow analysis give an accurate prediction of the flow of real fluids? Show that the potential function a2 x ⎞ −−−−−− Φ = U ⎛ x + −−− ⎝ x 2 + y 2⎠ gives the potential flow around a cylinder of radius a. A small particle whose velocity is at all times equal to that of the fluid immediately surrounding it passes through the point (−3a, 0) at time t = 0. At what time will it pass through the point (−2a, 0)? [1.2203aU0] 7.10 Show that a free vortex is an example of irrotational motion. A hollow cylinder 1 m diameter, open at the top, spins about its axis which is vertical, thus producing a forced vortex motion of the liquid contained in it. Calculate the height of the vessel so that the liquid just reaches the top when the minimum depth is 15 cm at 150 rev min−1. [3.29 m] 7.11 Prove that, in the forced vortex motion of a liquid, the rate of increase of pressure p with respect to the radius r at a point in the liquid is given by dpdr = ρω 2r in which ω is the angular velocity of liquid and ρ its density. What will be the thrust on the top of a closed vertical cylinder of 15 cm diameter, if it rotates about its axis at 400 rev min−1 and is completely filled with water? [43.5 N]

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Problems 7.12 A compound vortex in a large tank of water comprises a forced vortex core surrounded by a free vortex. Determine the depth of water at the centre of the core below the free vortex level if the velocity is 2.5 m s−1 at the common radius of 18 cm. [0.636 m]

255

7.13 Define vorticity and discuss the significance of irrotational motion. Give the vorticity at 1 m and 3 m radius in a vortex whose speed is 1 m s−1 throughout, and calculate the difference in pressure between these two places, if the axis of rotation is (a) vertical, and (b) horizontal. [20.72 kN m−2]

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Part III

Dimensional Analysis and Similarity 8 Dimensional Analysis 258 9 Similarity 282

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The application of fluid mechanics in design, perhaps more than most engineering subjects, relies on the use of empirical results built up from an extensive body of experimental research. In many areas empirical data are supplied in the form of tables and charts that the designer may apply directly, an example being the values of friction factor for pipe flow and separation loss coefficients for duct and pipe fittings. However, even here, the tables and the underlying experimental work become too unwieldy and time consuming if no way can be found to replace the relationship between any two variables by generalized groupings. It is, therefore, in the organization of experimental work and the presentation of its results that dimensional analysis plays such an important role. This technique, which is dealt with first in this part of the text, commences with a survey of all the likely variables affecting any phenomenon, and, to the experienced researcher, then suggests the formation of groupings of more than one variable. Experimental work may then be based on these groups rather than on individual variables, considerably reducing the testing programme and leading to simplified design guides, such as the Moody charts mentioned in Chapter 10. The application of results from one test series, involving say a particular pipe flow situation, to another

case, depends on the full understanding of the principles of geometric and dynamic similarity which are covered in the second part of this section. Although similarity is inherent in the formation of relationships such as the Moody chart, it is more commonly associated with the use of models and model testing techniques. Examples of such applications as wind tunnel tests and river and harbour models are mentioned; however, the basic principles depend upon the equivalence of variable groupings formed initially by the use of dimensional analysis. Again, it will be appreciated that mathematics alone is not sufficient in the application of the similarity laws; in many cases it will be found that total equivalence of all the dimensionless groupings will be mutually impossible and here the experience of the researcher will be called upon, examples being found in the cases of ship model tests and pump or turbine modelling techniques utilizing gas in place of water. Together, dimensional analysis, similarity and model testing techniques allow the design engineer to predict accurately and economically the performance of the prototype system, whether it is an aircraft wing, ship hull, dam spillway or harbour construction. The basis of these interactive techniques is presented in this part.

Opposite: Wind tunnel evaluation of aircraft model performance, photo courtesy of the Low Speed Wind Tunnel Facility, British Aerospace, Filton, Bristol

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Chapter 8

Dimensional Analysis 8.1 8.2 8.3

8.4 8.5 8.6

Dimensional analysis Dimensions and units Dimensional reasoning, hom*ogeneity and dimensionless groups Fundamental and derived units and dimensions Additional fundamental dimensions Dimensions of derivatives and integrals

Units of derived quantities Conversion between systems of units, including the treatment of dimensional constants 8.9 Dimensional analysis by the indicial method 8.10 Dimensional analysis by the group method 8.11 The significance of dimensionless groups 8.7 8.8

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Fluid mechanics is essentially an empirically based discipline that has relied upon the utilization of model flow representation to determine the likely performance of prototypes, whether aerofoils, ships or flow over and through buildings. In order to allow the results of model testing to be meaningful it is essential that the parameters determining the flow are identified and organized so that testing may be both directed and of value. This chapter will introduce the mathematical techniques of dimensional analysis whereby the parameters considered to be likely to

affect the flow can be combined into a number of dimensionless groupings, thereby facilitating testing and reducing the overall test programme. The chapter will concentrate upon the introduction of parameter dimensions and the techniques available for parameter combination. Buckingham’s π theorem will be introduced and shown to have general application. However, for the majority of fluid mechanics applications not involving heat transfer or temperature changes, a system based on mass, length and time will be shown to be appropriate. l l l

A floating production storage and off-loading (FPSO) vessel model being tested in ‘design storm’ conditions in the Marintek wave basin, Trondheim, Norway, image courtesy of Marintek.

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8.1

DIMENSIONAL ANALYSIS

The roots of fluid mechanics lie in the experimental investigation of the mechanisms of fluid flow. In order to determine the form of the dependence of one variable upon a range of other controlling parameters in the absence of an analytical solution it is necessary to undertake an experimental investigation; however, simply recording the effect of one variable on another with all others held constant and repeating until all the possible combinations are exhausted is not an option, in terms neither of time nor the utility of the outcome. Dimensional analysis offers a route out of this dilemma by allowing the identification of groups of variables whose interrelationships may be determined experimentally. Dimensional analysis therefore offers a qualitative route to the understanding of fluid flow mechanisms; the quantitative understanding is provided experimentally.

8.2

DIMENSIONS AND UNITS

Any physical situation involving an object or a system may be described in terms of its fundamental properties, which must include its possible mass, length (which obviously also describes its area and volume), velocity or acceleration (combinations of length and time), or density (based on mass and length) or the forces or stresses acting on the system (defined in terms of mass, length and time). Similarly, thermodynamic and electrical properties may also be included. These properties of the system are fundamental and universal and are known as its dimensions. While dimensions are universal, units are chosen as convenient and therefore have a long history, from the ancient cubit (allegedly based on the length of the forearm), or rural distances measured by the number of cigarettes smoked on the journey between villages, to the modern standard kilometre or light year. Modern units therefore provide a convenient and standardized measure of the dimension under consideration so that conversion between different measures of the same dimension is possible. A physical property, such as density or mass per unit volume, may thus be defined in terms of its dimensions as [ML−3], where the [] brackets indicate that we are only interested in the qualitative dimensions of the property and not its quantitative value. The units for density would be expressed in the system of units currently in place, which in SI terms are kgm3.

8.3

DIMENSIONAL REASONING, hom*oGENEITY AND DIMENSIONLESS GROUPS

Dimensional reasoning is predicated on the proposition that, for an equation to be true, then both sides of the equation must be numerically and dimensionally identical. To take a simple example, the expression x + y = z when x = 1, y = 2 and z = 3 is clearly numerically true but only if the dimensions of x, y and z are identical. Thus

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8.4

Fundamental and derived units and dimensions

261

1 elephant + 2 aeroplanes = 3 days is clearly nonsense but 1 metre + 2 metre = 3 metre is wholly accurate. An equation is only dimensionally hom*ogeneous if all the terms have the same dimensions. In general any equation of the form a 1m b 1n c 1p + a 1m b 2n c 2p + . . . + = X 1

1

1

2

2

2

will be physically true if, in addition to being numerically correct, the terms are dimensionally the same so that [a 1m b 1n c 1p ] = [a 1m b 2n c 2p ] = [X ], 1

1

1

2

2

2

where [a 1m b 1n c 1p ] means the dimensions of the group a 1m b 1n c 1p . Note that this introduces the concept of the dimensions of a group of variables rather than that of a single parameter. The dimensions of the group are determined by the normal rules of algebra: for example, as Newton’s second law states that Force = Mass × Acceleration it follows that the dimensions of force are given by [MLT−2]. It follows from the identification of the dimensions of a group that the ratio of two parameters or two dimensionally identical groups of variables will yield a dimensionless group. For example, the ratio of a vehicle’s speed to that of sound yields Vc; dimensionally [V ][c] = [L0][T0] = [1]. Similarly, strain is defined as Extension Original length or [L][L] = [1], a non-dimensional ratio. Dimensional hom*ogeneity may be used to check the accuracy of any equation, remembered or derived. This is an important tool and should be used as a matter of course. 1

8.4

1

1

1

1

1

FUNDAMENTAL AND DERIVED UNITS AND DIMENSIONS

Dimensional analysis requires the definition of fundamental dimensions that will allow all the parameters involved in a particular flow system to be described. It is not practical to assign a fundamental dimension to every physical property. In this book we will usually choose mass, length and time as the fundamental dimensions used to derive the dimensions of all the other parameters normally encountered in fluid systems. It may also be necessary to include temperature and heat flow. Application of the [MLT] system leads to the following derived dimensions. Area and volume may be defined as having derived dimensions of [L2] and [L3]. In kinematics, time becomes a necessary dimension so that Velocity = Distance Time becomes [LT−1] and acceleration, defined as VelocityTime, becomes [LT−2]. Referring to angular motion, an angle may be measured in radians, where Angle = Length of arcLength of radius, so that [Angle] = {L}[L] = [L0] = 1, indicating that

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Chapter 8

Dimensional Analysis

angle is a dimensionless quantity. It follows that angular velocity and acceleration may be defined as [Angular velocity] = [Angle][Time] = 1[T] = [T−1] and [Angular acceleration] = [Angular velocity[Time] = [T−1][T] = [T−2]. In dynamics, Newton’s second law provides the basis for the definition of derived dimensions, e.g. Force = Mass × Acceleration = Mass × VelocityTime = [M][LT−1][T] so that [Force] = [MLT −2]. Because Newton’s second law links force to mass and acceleration and hence to distance and time, any variable such as work, power, viscosity, etc. can be expressed in terms that eventually reach an expression of Newton’s second law. For example, Pressure or stress = ForceArea = Mass × AccelerationArea, so that

[Pressure or stress] = [M] [LT−2][L−2] = [ML−1T−1]

Density = MassVolume so that

[Density] = [M][L3] = [ML−3]

Power = Rate of doing work = Force × DistanceTime = Mass × Acceleration × DistanceTime so that

[Power] = [M] [LT−2 ] [L][T] = [ML2T−3]

Similarly, viscosity, µ, may be defined in terms of shear stress, τ, and velocity gradient ∂u as τ = µ ------ , so that ∂y [Viscosity] = [Stress][Velocity gradient] = [ML−1T−1][LT−1L] = [ML−1T−2] Table 8.1 presents the derived dimensions for a wide range of variables commonly occurring in fluid flow, all derived using the techniques demonstrated above and based on Newton’s second law and our choice of the fundamental dimensions mass, length and time.

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8.5

TABLE 8.1 Dimensions of quantities in mechanics (based on Newton’s second law)

Additional fundamental dimensions

263

Defining equation

Dimensions, MLT system

[M0 L0 T 0 ] [L]

Area Volume First moment of area Second moment of area Strain

ArcRadius (a ratio) (Including all linear measurement) Length × Length Area × Length Area × Length Area × Length2 ExtensionLength

Kinematic Time Velocity, linear Acceleration, linear Velocity, angular Acceleration, angular Volume rate of discharge

– DistanceTime Linear velocityTime AngleTime Angular velocityTime VolumeTime

[T] [LT −1] [LT −2] [T −1] [T −2] [L3 T −1]

ForceAcceleration Mass × Acceleration Force MassVolume WeightVolume DensityDensity of water ForceArea ForceArea StressStrain Force × Time Mass × Length2 Mass × Linear velocity Moment of inertia × Angular velocity Force × Distance WorkTime Force × Distance Shear stressVelocity gradient Dynamic viscosity Mass density EnergyArea

[M] [MLT −2 ] [MLT −2 ] [ML−3 ] [ML−2T −2 ] [M0 L0 T 0 ] [ML−1T −2 ] [ML−1T −2 ] [ML−1T −2 ] [MLT −1] [ML2 ] [MLT −1] [ML2 T −1]

Quantity Geometrical Angle Length

Dynamic Mass Force Weight Mass density Specific weight Specific gravity Pressure intensity Stress Elastic modulus Impulse Mass moment of inertia Momentum, linear Momentum, angular Work, energy Power Moment of a force Viscosity, dynamic Viscosity, kinematic Surface tension

8.5

[L2] [L3] [L3] [L4] [L0]

[ML2 T −2] [ML2 T −3] [ML2 T −2 ] [ML−1 T −1] [L2 T −1] [MT −2 ]

ADDITIONAL FUNDAMENTAL DIMENSIONS

Flow conditions that include thermal considerations introduce the additional dimensions of temperature and heat – seen here as an energy term expressed as [ML2T−2]. Temperature, θ , may be determined from the relationship linking heat input, H, to the temperature rise of a body of mass m and specific heat c, namely H = cmθ , where if c is taken as a non-dimensional ratio, it follows that

Gas constant, R Coeff. of heat transfer Mechanical equivalent of heat

Specific heat ratio Thermal conductivity

Specific heat

Thermal capacity

Temperature, θ Heat quantity, H Enthalpy Entropy, S Coeff. of thermal expansion

Quantity

Time rate of heat transmission per unit area and temp. gradient EnergyMass × Temp.

dS = dHθ Change of lengthunit lengthdegree Heat required per degree temp. rise Thermal capacity per unit mass

Defining equation

[L2 T −2 Θ−1] [M0 L0 T 0 Θ0 ] [MLT −3Θ −1] [L2 T −2 Θ −1] [MT −3Θ −1] [M0 L0 T0 ]

[M0 L0 T 0 Θ0 ] [M0 L0 T 0 Θ0 ] [ML−1 T −1]

[M0 L0 T 0 Θ0 ] [ML−2 T −1] [L2 T −2 Θ−1]

[M0 L0 T 0 ] [M0 L0 T 0 ] [ML−1 T −1]

[M0 L0 T 0 ] [ML−2 T −1] [M0 L0 T 0 ]

– [HL−2 T −1Θ −1] –

[H0 L0 T0 Θ0 ] [HL−1T −1Θ −1]

–

[HΘ−1]

[ML2 T −2 Θ−1]

[Θ] [H] [H] [HΘ−1] [Θ−1]

HLTΘ system

[Θ] [ML2 T −2] [ML2 T −2] [ML2 T −2 Θ −1] [Θ −1]

Dynamic

[M]

[Θ] [MΘ] [MΘ] [M] [Θ−1]

Thermal

[M]

[L2 T −2] [ML2 T −2] [ML2 T −2] [M] [L−2 T2 ]

MLT system

MLTΘ systems

[L2T −2 Θ−1] [HL−2 T −1Θ−1] [H−1ML2 T −2]

[H0 M0 L0 T0 Θ0 ] [HL−1T −1Θ −1]

[HM−1Θ−1]

[HΘ−1]

[Θ] [H] [H] [HΘ−1] [Θ−1]

HMLTΘ system

Chapter 8

Dimensions

264

TABLE 8.2 Dimensions of common quantities in thermodynamics

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Dimensional Analysis

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8.6

DIMENSIONS OF DERIVATIVES AND INTEGRALS

265

[θ ] = [Hm] = [ML2T−2][M] = [L2T−2]. The dimensions of other thermal quantities are defined in Table 8.2, column 1. In some cases it is useful to include temperature. In the resulting MLTΘ system, heat energy becomes [H ] = [MΘ], and the derived dimensions for other quantities are shown in Table 8.2, column 2, or heat energy may be expressed as [ML2T−2], in which case Table 8.2, column 3 is applicable. Heat H may also be used as a fundamental dimension, and Table 8.2, column 4 and column 5 present derived dimensions for the resulting HLTΘ and HMLTΘ systems. It should be noted that the appropriate units for heat will vary according to the system chosen, calories for the [MΘ] definition and joules for the [ML2T−2] representation.

DIMENSIONS OF DERIVATIVES AND INTEGRALS

8.6

The dimensions of derivatives may be determined easily, if it is remembered that, by definition, dydx is the limiting value of the ratio δ yδ x as δ x tends to zero, where δ y is the finite value of y corresponding to a finite increment δ x of x. The dimensions of δ x and δ y are the same as x and y, and hence dy δy x −−− = −−− = − . dx δx y Similarly, for the second or higher differential it follows that d2 y d dy Increment ( dy/dx ) −−−−−2 = −−− ⎛ −−− ⎞ = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−, dx dx ⎝ dx⎠ Increment ( x ) so that dy dx d2y y x y −−−−−2 = −−−−−−−−− = −−−−− = −−−2 dx x x x or, in general,

dny −−−−−n = dx

y −−n . x

The dimensions of integrals may be determined in the same way. The term a

y dx b

means the limit of the sum of all the products of yδ x between x = a and x = b. Thus, the dimensions of the integral will be the same as those of yδ x, and, since [δ x] = [x], the dimensions for the single integral will be

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a

y dx = [yx]. b

Similarly, a double integral 11a dz1 dz2 means the limit sum of the products aδ z1δ z2. Since [δ z1] = [z1] and [δ z2] = [z2], the dimensions are

a dz dz 1

8.7

2

= [ az 1 z 2 ].

UNITS OF DERIVED QUANTITIES

The units of any derived variable may be determined by simply substituting the standard unit for the corresponding dimension: for example, kilogram for mass, metre for length, second for time. The SI system is a rationalized system of metric units in which all units may be derive from six basic units, namely: Length Mass Time Electric current Absolute temperature Luminous intensity

metre kilogram second ampere kelvin candela

However, some derived units have been given names to commemorate outstanding scientists and engineers. For example, force, whose unit would be kgms2, is known as the newton (N), and pressure, whose unit would be Nm2, is known as the pascal (Pa). Details of basic and derived units are given in Table 8.3.

8.8

CONVERSION BETWEEN SYSTEMS OF UNITS, INCLUDING THE TREATMENT OF DIMENSIONAL CONSTANTS

While the SI system of units is intended to become the accepted standard there remain alternative systems that are used internationally – miles and miles per hour in Englishspeaking countries, various volumetric measures such as the Imperial and US gallon, and flowrate measurements in cubic feet per minute. Conversion factors transform these quantities into SI units: kilometres, kilometres per hour, litres or cubic metres, and cubic metres per second. The physical magnitude of any quantity must be the same regardless of its measurement system, so that if a quantity Q is found to have a value n1 when measured in units u1 and n2 when measured in units u2 it follows that Q = n1 u1 = n2 u2.

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TABLE 8.3 SI units

8.8

Conversion between systems of units, including the treatment of dimensional constants

Quantity Basic units Length Mass Time Electric current Absolute temperature Luminous intensity Geometry Angle, plane solid Area Volume First moment of area Second moment of area

267

Unit

Symbol

metre kilogram second ampere kelvin candela

m kg s A K cd

radian steradian square metre metre cubed metre cubed metre to fourth power

rad sr m2 m3 m3 m4

hertz

Hz

metre per second radian per second

m s−1 rad s−1

metre per second squared radian per second squared newton (= kilogram metre per second squared) kilogram per metre cubed newton per metre cubed

m s−2 rad s−2 N (= kg m s−2 )

kilogram metre per second kilogram metre squared per second kilogram metre squared newton metre pascal (= newton per metre squared)

kg m s−1 kg m2 s−1 kg m2 Nm Pa (= N m−2 )

newton second per metre squared (= 10 poise) metre squared per second newton per metre joule (= newton metre) watt (= joule per second)

N s m−2

degree kelvin expansion per unit length per kelvin joule watt joule per kelvin joule per kelvin watt per metre per kelvin watt per metre squared per kelvin

K K−1 J W J K−1 J K−1 W m−1 K−1 W m−2 K−1

Derived units Mechanics Frequency Velocity, linear angular Acceleration, linear angular Force Density, mass Specific weight Momentum, linear angular Moment of inertia Moment of force Pressure or stress (intensity) Viscosity, dynamic kinematic Surface tension Energy, work Power Heat Temperature interval Linear expansion coefficient Heat quantity Heat flow rate Entropy Thermal capacity Thermal conductivity Coefficient of heat transfer

kg m−3 N m−3

m2 s−1 N m−1 J (= N m) W (= J s−1)

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Thus, the numerical value of a quantity is inversely proportional to the size of the units in which it is measured. If the quantity Q has dimensions [MaLb T c ], the unit of measurement will be a derived unit. Suppose that the units of mass, length and time are m1, l1 and t1 in the first system and m2, l2 and t2 in the second system: then the derived units in these systems are u 1 = m a1 l b1 t c1

and

u 2 = m a2 l b2 t c2 .

The fundamental units in each system will be related: therefore m1 = km m2, l1 = kl l2 and t1 = kt t2, where km,l,t are numerical constants. Therefore, as before, it follows that the quantity Q is given by Q = N 1 u 1 = N 1 m a1 l b1 t c1 = N 1 ( k am m a2 ) ( k bl l b2 ) ( k ct t c2 ) = N 1 k am k bl k ct m a2 l b2 t c2 . But

N1 u 1 = N2u2 and u2 = m a2 l b2 t c2 , so that N 2 u 2 = N 1 k am k bl k ct u 2 and N2 = N 1 k am k bl k ct

where the k terms are the numbers of units in the second system required to make the corresponding fundamental units in the first system. Care must be taken in dealing with dimensional constants used in some equations. If these are dimensionless then there is no problem; however, some ‘constants’ have dimension and therefore must be converted between systems of units in the manner discussed above.

EXAMPLE 8.1

An engine produces 57 horsepower. What is the corresponding value in kilowatts and what is the conversion factor?

Solution Horsepower and kilowatts are multiples of the basic unit of power in the British and SI systems. If n1 is the horsepower and N1 is the corresponding number of British units (ft lbfsecond), N1 = 550n1. Similarly, for the SI system, N2 = 1000n2. The dimensional formula for power is [ML2T−3]. Using suffix 1 for the British system and suffix 2 for SI units,

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8.9

Dimensional analysis by the indicial method

269

N2 = N1km kl2 k t−3 where the k terms are the ratio of mass, length and time in system 1 to system 2. The ratio values are as follows:

Quantity Mass Length Time

System 1 British

System 2 SI

Ratio BritishSI

slug foot second

kilogram metre second

14.6 0.3048 1.0

Therefore, N2 = N1(14.6)(0.3048)2(1)−3 = 1.356N1, 1000n2 = 550n1(1.356) and n2 = 0.746 n1.

so that

Therefore, value in kilowatts = 0.746 × value in horsepower. Output = 0.746 × 57 = 42.5 kW, Conversion factor = n 2 n 1 = 0.746.

8.9

DIMENSIONAL ANALYSIS BY THE INDICIAL METHOD

If the variables involved in any flow situation can be identified, then the form of the relationship determining the dependence of one parameter on the others may be largely determined by dimensional analysis. This is a consequence of the dimensional hom*ogeneity condition that effectively limits the number of possible combinations of variables. Clearly, numerical constants will not be determined; however, the form of the relationship will guide experimental work that will yield the required constants. Dimensional analysis has therefore played a major role in defining the fluid mechanics relationships we now accept. The indicial method presented below is a typical example that demonstrates the principles involved.

EXAMPLE 8.2

The thrust, F, of a propeller depends upon its diameter, d, speed of advance, v, revolutions per second, N, the fluid density, ρ, and viscosity, µ. Find an expression for F in terms of these quantities.

Solution The general relationship must be F = φ (d, v, N, ρ, µ), which may be expressed as

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F = K(d m, v p, Nr, ρ q, µ s), where K is a numerical constant. The dimensions of these variables are listed in Table 8.1 and may be conveniently expressed in the table below: Variable

F

d

v

N

ρ

µ

M L T

1 1 −2

0 1 0

0 1 −1

0 0 −1

1 −3 0

1 −1 −1

Substituting the dimensions for the variables yields [MLT−2] = [L]m [LT−1]p [T−1]r [ML−3]q [ML−1T−1]s. To satisfy dimensional hom*ogeneity the mass, length and time dimensions must equate on either side of the equation, so it is possible to write three equations for M, L and T: [M], 1 = q + s;

(I)

[L], 1 = m + p − 3q − s; [T], −2 = −p − r − s.

(II) (III)

As there are five unknowns and only three equations, no complete solution is possible; however, it is possible to determine m, p, q in terms of r and s, hence q = 1 − s, p = 2 − r − s, m = 2 + r − s. The initial expression

F = K(d m, v p, N r, ρ q, µ s)

becomes F = K(d 2+r−s, v 2−r−s, N r, ρ1−s, µ s) Gathering terms, F = Kρ v 2d 2(ρ vdµ)−s (dNv)r. Since K, r and s are unknown this may be written as F = ρ v 2d 2φ (ρ vdµ , dNv), where φ means ‘a function of’. Note that the terms Fρ v 2d 2 , ρ vdµ and dNv are all non-dimensional groups, so that the initial relationship for F in terms of five other

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variables has been reduced to an F group defined in terms of two other groups – a clear advantage in experimentation. The grouping also removes the need to hold any particular variable constant while varying the others, a requirement that would in any case be impossible for viscosity. Assuming that the groups ρ vdµ and dNv were held constant then F = Cρ v 2d 2 where C is an experimentally derived constant. Similarly, this expression could be used to determine the thrust of a full-scale propeller provided both model and prototype had identical values of the two groups ρ vdµ and dNv, an important use of dimensional analysis to be returned to in Chapter 9.

8.10 DIMENSIONAL ANALYSIS BY THE GROUP METHOD The indicial method is rather lengthy if there are a large number of variables. It was therefore necessary to develop a more generalized methodology that would lead directly to a set of dimensionless groupings whose number could be determined in advance by a scrutiny of the matrix formed from the variables considered to be relevant to the investigation and the relevant dimensions necessary to describe those variables. Such a technique was developed in the early years of the twentieth century and is known as the Buckingham π method. The initial step in the application of the Buckingham π method is to list the variables considered to be significant and to form a matrix with their dimensions. Example 8.2 illustrated this technique with the six variables and three dimensions, M, L, T, forming the matrix illustrated. It is then necessary to determine the number of dimensionless groups into which the variables may be combined. This number may be found by application of the Buckingham π method, which states that The number of dimensionless groups arising from a particular matrix formed from n variables in m dimensions is n − r, where r is the largest non-zero determinant that can be formed from the matrix, and therefore the equation relating the variables will be of the form

φ (π1, π 2, π 3, . . . , πn−r) = 0. While this form of Buckingham’s theorem is correct, it is often simpler, in the treatment of fluid conditions that only involve the dimensions of mass, length and time, to state that the number of dimensionless groups formed from n variables in three dimensions is n − m, or n − 3. While this is also correct and widely quoted, care must be taken in applying this rule outside the confines of a strictly three-dimensional problem. The introduction of heat and temperature as dimensions in the study of thermodynamic and heat transfer phenomena increases the number of possible dimensions to five; however, in some cases the correct value of r will be less than this.

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As fluid mechanics and thermodynamicsheat transfer often share common courses, care should be taken in the acceptance of the simplified rule. The importance of determining the correct order for the largest non-zero determinant may be demonstrated by three examples taken from the area of fluid flow over a surface with and without heat transfer by either forced or natural convection.

EXAMPLE 8.3

Determine the number of dimensionless groups expected to be formed from the variables involved in the flow of fluid external to a solid body, extending this analysis to include both forced and natural convection from a surface.

Solution Case 1: no heat transfer condition. The force acting, F, may be expected to be a function of flow velocity v, density ρ, dynamic viscosity µ, and body characteristic length L. The matrix formed in the applicable dimensions of mass, length and time from the five applicable variables is as follows:

F v ρ µ L

M

L

T

1 0 1 1 0

1 1 −3 −1 1

−2 −1 0 −1 0

To determine the highest order of non-zero determinant it is necessary to refer to some simple laws governing determinants, for example: 1. 2. 3. 4. 5.

The order of a determinant is defined by the equal number of rows and columns displayed. The value of a determinant is unchanged if the order of its rows and columns is changed. The value is unchanged if the rows are changed to columns. If two rows or columns are identical then the value of the determinant is zero. If any row or column is multiplied by a constant then the value of the determinant is its previous value multiplied by that constant.

The above ‘rules’ allow the largest non-zero determinant to be recognized in any dimensional matrix. In the first case the highest order determinant possible would be third order, provided scrutiny of the rules above did not reduce this value. It will be seen by inspection that none applies and that the number of dimensionless groups expected in this case would be (5 − 3) or two. Case 2: forced convection. Considering the flow of fluid over a surface with a temperature difference between the fluid and the surface requires the introduction of

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two further dimensions, namely quantity of heat energy, H, and temperature, Θ, as set out in Table 8.2. The eight applicable variables may be formed into the dimensional matrix below:

Variable

M

L

T

H

Θ

Force, F Length, L Density, ρ Viscosity, µ Heat capacity, cp Thermal conductivity, k Velocity, v Heat transfer coefficient, h

1 0 1 1 −1 0 0 0

1 1 −3 −1 0 −1 1 −2

−2 0 0 −1 0 −1 −1 −1

0 0 0 0 1 1 0 0

0 0 0 0 −1 −1 0 −1

By inspection it is clear that the highest possible order would have been 5; however, inspection of the H and Θ columns indicates that these columns are −1 multiples of each other, thus any fifth order determinant formed from the matrix would be zero. Dropping either the H or Θ column and reapplying the rules set out above indicates that a fourth order non-zero determinant is possible and the number of dimensionless groups to be expected in this case is thus (8 − 4) or four. Case 3: natural convection. Here it is again necessary to introduce dimensions of heat and temperature. In this case, however, it is also necessary to introduce gravitational forces, represented by g, and a temperature difference, ∆T, to cater for the density– temperature buoyancy effects driving the process. The ten applicable variables may be formed into the dimensional matrix below:

Variable

M

L

T

H

Θ

Force, F Length, L Density, ρ Viscosity, µ Heat capacity, cp Thermal conductivity, k Coefficient of fluid thermal expansion, β Gravitational acceleration, g Temperature diff., ∆T Heat transfer coefficient, h

1 0 1 1 −1 0 0 0 0 0

1 1 −3 −1 0 −1 0 1 0 −2

−2 0 0 −1 0 −1 0 −1 0 −1

0 0 0 0 1 1 0 0 0 0

0 0 0 0 −1 −1 −1 0 1 −1

By inspection it is clear that the highest possible order would have been 5. In this case none of the rules indicated would reduce the order and, therefore, the number of dimensionless groups expected would be (10 − 5) or five.

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Taken together these three examples illustrate the importance of careful inspection of the dimensional matrix in order to determine the applicable number of dimensionless groups to be expected in any investigation.

Referring back to Example 8.2, it may be seen that as there are six variables, F, d, v, ρ, N, µ, and three fundamental dimensions, M, L, T, the value of r would be 3, and hence the number of dimensionless groups would be 3. The solution therefore contains the following groups:

π1 = Fρv 2d 2, π 2 = ρvdµ, π 3 = dNv. Independent dimensionless groups are defined as those which can be formed from any particular number of quantities, but are independent of each other in the sense that none of them can be formed by any combination of the others. In any particular problem, having determined the number of dimensionless groups as described above, the next step is to combine the variables to form the desired groupings. The following points are useful indicators of the best approach to follow: 1.

From the independent variables thought to describe the fluid flow condition select certain variables to act as repeating variables. These variables may appear in all or some groups. The number of repeating variables is therefore [n − (n − r)], i.e. the total number of variables minus the number of groups. The choice of these repeating variables is not arbitrary but should be guided by the following rules: (a) The repeating variables as a combination must include all the dimensions taken to describe the system: thus in Example 8.2 they must include M, L, T. This does not mean that each repeating variable includes all dimensions, but seen as a group this must be the case. (b) The repeating variables should be chosen with some regard for the practicality of any experimental investigation; they should be easily measurable or set by the investigator. Similarly, where the results of a dimensional analysis are to be the basis for a later design methodology, the repeating variable should be of prime interest to the designer. For example, it is more sensible to define pipe type in terms of pipe diameter than surface roughness as a repeating parameter, and density is perhaps better than viscosity as a descriptor of fluid type.

2.

3.

Combine the repeating variables with the remaining independent variables to form the required number of groups. It follows that each of the remaining variables now only appears in one group and these groups are often referred to by that variable name. A variable that is considered to be of minor significance will, as a result of (1) and (2) above, only appear in one group. The influence of this group will be negligible if this variable is truly inconsequential. This raises one of the interesting points in dimensional analysis, namely that there really are no ‘wrong’ answers, only answers that are more useful than others. The inclusion of a number of variables that have little or no effect on the flow phenomena will result in dimensionless groupings which will be shown by experimental investigation to be of no significance. Similarly, so long as the repeating variables chosen conform to the

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rule that they represent all the dimensions of the problem, the choice of repeating variable may also be arbitrary. This approach will result in the problem being defined in terms of a series of correctly dimensionless groups which, due to the poor choice of repeating variable, are of little use to the investigator. However, even in this case not all is lost, owing to the following rules that apply to the groups in an equation of the form

φ (π1, π 2, π 3, . . . , π n−r) = 0. (a) Any number of dimensionless groups may be combined by multiplication or division to form a new valid group. Thus π1 and π 2 may be combined to form π1′ = π1π 2, and the defining equation becomes

φ (π1′, π 2, π 3, . . . , π n−r) = 0. (b) The reciprocal of any dimensional group remains valid. An example of this will be met in the later treatment of fans and pumps where a reciprocal form of the Reynolds number will be recognizable. (c) Any dimensional group may be raised to any power and remain valid. (d) Any dimensional group may be multiplied by a constant and remain valid. This is useful in relating a particular group to an easily measured quantity, e.g. pressure coefficients. Groups often include the combination ρv 2 as the non-dimensioning denominator, while in fact the use of −12 ρv 2 would allow direct use of the flow kinetic energy term, in itself readily measured by use of a Pitot–static tube. Thus the general form of a dimensional relationship could appear as

φ [π1′, 1π 2, (π 3)i, . . . , −12 πn−r] = 0 and remain valid.

EXAMPLE 8.4

The variables controlling the motion of a floating vessel through the surrounding fluid are the drag force F, the vessel’s speed v, its length l, the density ρ and dynamic viscosity µ of the fluid and the acceleration due to gravity g. Derive an expression for F by dimensional analysis.

Solution The resistance to motion will be partly due to skin friction and partly due to wave resistance, thus involving both viscous and gravitational forces. The general form of the expression may be written as F = φ (v, l, ρ, µ, g). The dimensions of the variables involved may be tabulated as follows:

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Mass M Length L Time T

F

v

l

ρ

µ

g

1 1 −2

0 1 −1

0 1 0

1 −3 0

1 −1 −1

0 1 −2

As the number of variables n is six and the number of dimensions m is three, it follows that the number of dimensionless groups will be (n − m) or 3. Thus there will be three repeating variables chosen to non-dimensionalized groups that feature three variables of prime interest. From an experimental or design perspective it would be useful to relate the force needed to the design speed of a vessel of a given length in a particular fluid so it would be reasonable to choose v, l and ρ as the repeating variables, leaving force F, viscosity µ, and gravitational acceleration as the independent variables. (While gravity is sensibly a constant, it appears as an independent variable as it is unlikely to be a variable that could be controlled easily in an experiment.) The required solution will be

π1 = φ (π2, π3). The groups may therefore be defined as g F µ ------------- , ------------- , ------------a b c a b c a b c vl ρ vl ρ vl ρ For the force group, equating powers of M, L and T results in a set of three simultaneous equations that may be solved for the indices a, b and c: For M 1 = c For L 1 = a + b − 3c For T −2 = −a so that a = 2, b = 2, c = 1, and the force group becomes F ------------ . v 2l 2 ρ (Note that this could be rearranged as F -------------------, 0.5 ρ v 2l 2

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which has the form of a force coefficient incorporating a kinetic energy term and an area term – a form used in the treatment of lift and drag coefficients.) This example illustrates all the ‘rules’ set out in Section 8.10 and in addition it will be seen that the left-hand side of the M, L, T equations above will be identical for all groups; the right-hand side will vary for each group depending upon the unique variable. Therefore it is possible to express the M, L, T equation for each group in a matrix format to simplify and reduce repetition within the calculation:

Groups in

Left-hand side F µ g

Right-hand side

For M For L For T

1 1 −2

= +c = a + b − 3c = −a

1 −1 −1

0 1 −2

Solution F µ

g

a= b= c=

2 −1 0

2 2 1

1 1 1

Thus the dimensionless group expression becomes F/(v 2 l 2ρ) = φ 1(µvlρ, glv 2). However, it is normal to express these groups as F/(0.5ρ v 2l 2) = φ 2(vlρµ, v√(gl)), recognizable as a force coefficient, Reynolds number and Froude number. Therefore, the force necessary to propel the vessel may be expressed as a non-dimensional force coefficient, which includes kinetic energy and area terms, found to be dependent on the viscous and gravitational forces present, defined in terms of the Reynolds and Froude numbers respectively.

EXAMPLE 8.5

The variables governing the resistance to flow, or surface shear stress τ0, in a closed conduit are believed to include the flow mean velocity v, the conduit diameter D, its surface roughness k and the density ρ and dynamic viscosity µ of the fluid. In addition if the surface of the conduit is itself in motion then the surface velocity Vs may also be a factor. For both cases utilize a tabular format to determine the likely dimensionless groups.

Solution The general expression for the dependence of shear stress may be expressed as

τ0 = φ (v, D, k, ρ, µ, Vs). The dimensions of the variables involved may be tabulated as follows:

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Mass, M Length, L Time, T

τ0

v

D

k

ρ

µ

Vs

1 1 −2

0 1 −1

0 1 0

0 1 0

1 −3 0

1 −1 −1

0 1 −1

The number of variables is seven and the number of dimensions is three so the number of maximum groups, if Vs is included, is four, with three repeating variables. The choice of repeating variables is in this case straightforward as the investigator would wish to control the test variables of flow velocity, conduit diameter and fluid type, best described by density. With repeating variables of v, D and ρ it follows that it will be necessary to seek dimensionless groups featuring τ0, k, µ and Vs as follows:

τ0 -, π 1 = ---------------v aD b ρ c k -, π 2 = ---------------a b c vDρ

µ -, π 3 = ---------------v aD b ρ c Vs -. π 4 = ---------------a b c vDρ (Note that as before the index values a, b and c for each group will have different numerical values.) The following tabular layout for the M, L and T equations may be developed as in Example 8.4:

Groups in

Left-hand side τ0 k µ Vs

Right-hand side

For M For L For T

1 −1 −2

= +c = a + b − 3c = −a

0 1 0

1 −1 −1

0 1 −1

resulting in the following dimensionless groups:

τ0 -, π 1 = ---------1 −ρ v 2 2 k π 2 = ----, D

a= b= c=

Solution τ0 k µ

Vs

2 0 1

1 0 0

0 1 0

1 1 1

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µ ρ vD π 3 = ---------- = ---------- = Re, vDρ µ Vs π 4 = -----, v which include a stress coefficient, note the 12 added as before, a roughness ratio, the flow Reynolds number expressed as a reciprocal of π 3, and a velocity ratio. The wall roughness and the wall velocity groups could have been determined by inspection as both allow single variable non-dimensionality.

8.11 THE SIGNIFICANCE OF DIMENSIONLESS GROUPS Examples 8.4 and 8.5 have demonstrated the mathematical application of the rules of dimensional analysis. Application to real investigations and, most importantly, model testing and prototype design requires an understanding of the significance of the various groups identified, many of which will recur continuously in a wide range of fluid flow situations. These issues will be raised in Chapter 9 where the application of dimensional analysis along with the concepts of dynamic similarity will be developed. The application of dimensional analysis is also demonstrated by the development of friction factor relationships for pipe, duct and channel flows dependent on Reynolds number and relative roughness in Sections 10.4 and 10.5. As an introduction it is useful to consider the significance of the terms already introduced in this chapter. In Example 8.4, the force group Fρ v 2l 2 may be seen as a ratio of the shear force F on the hull to the inertia force represented by ρ v 2l 2, which may be recast in dimensional terms as ρ l 3 times lt 2, the product of a mass and an acceleration term. Similarly it will be seen that the Reynolds number ρ vlµ may be seen as a ratio of inertial and shear forces, as it may be expanded into {ρ v 2l 2µ (vl )l 2}, where the numerator has already been shown to be equivalent to an inertia force and the denominator has the form of the product of viscosity, velocity gradient and area, a shear force. The final group, v 2gl, known as the Froude number, applicable to free surface flow conditions may be expressed as a ratio of inertia to gravitational forces as it may be written as {ρ v 2l 2ρ gl 3}, where the numerator is an inertia force and the denominator is the product of mass and gravitational acceleration. Other groups will emerge, for example, Mach number, lift and drag coefficients, and various scale groups such as relative roughness. While these groups are suggested by the dimensional analysis techniques introduced here, their application requires consideration of the particular flow condition and the constraints of geometric and dynamic similarity, to be introduced in Chapter 9.

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Concluding remarks Dimensional analysis, as presented in this chapter, will be seen to be a mathematical technique that, in its simplified application to the three-dimensional system which is often sufficient in the study of fluid flows with no temperature-dependent or heat transfer effects, allows the enlightened design of experimental investigations. As mentioned, fluid mechanics depends heavily on empirical data, whether friction factors, life coefficients or machine performance data, and therefore a systematic empirical approach is essential. However, this chapter has also stressed that dimensional analysis alone cannot solve or define fluid flow problems; it can only suggest suitable groupings of variables that will allow the investigator to proceed. Chapter 9 will provide the basis for the use of dimensional analysis by introducing the laws of similarity. When combined with, or applied to, the groups suggested by dimensional analysis, similarity will allow the investigator to infer the performance of a prototype based upon the behaviour of a model, that behaviour being defined by the values and interrelationship of the variable groupings suggested by the dimensional analysis, and confirmed or modified by the experience of the investigator.

Summary of important equations and concepts 1.

Units are based on choice; dimensions are fundamental, Sections 8.2 and 8.3.

2.

Dimensional hom*ogeneity is a requirement of any equation, Section 8.4.

3.

Mass, length, time, heat energy and temperature are relevant dimensions for fluid and thermodynamic analysis, Section 8.6.

4.

Identification of the relevant dimensions for any variable requires a regression to the fundamental equations of motion, Section 8.6.

5.

The number of groups formed from n variables will be (n − r), where r is the highest order non-zero determinant formed from the dimensional matrix. Note that for most fluid mechanics applications that do not feature energy or temperature dimensions r is 3.

6.

Dimensional analysis is only a tool to guide an investigation – the choice of repeating variables is determined by the investigator to be the most suitable.

7.

A range of groupings will recur and these should be sought in any analysis, for example Reynolds number relating viscous to inertia forces, Froude number to represent gravitational forces. Similarly, pressure coefficients based on the flow kinetic energy and force coefficients incorporating kinetic energy and area.

Further reading Barr, D. I. H. (1983). A survey of procedures for dimensional analysis. International Journal of Mechanical Engeering & Education, 11(3), 147–159. Buckinkham, E. (1914, 1915). Model experiments and the form of empirical equations. Physics Review, 2, 345 (1915 Transactions of the ASME, 37, 263–96). Kline, S. J. (1965, reprinted 1986). Similitude and Approximation Theory. McGraw-Hill, New York. Langhaar, H. L. (1980). Dimensional Analysis and the Theory of Models. Robert E. Kreiger, Malabar, FL.

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Novak, P. and Cabelka, J. (1981). Models in Hydraulic Engineering. Pitman, London. Sedov, L. I. (1959, reprinted 1996). Similarity and Dimensional Methods in Mechanics. Academic Press, London.

Problems 8.1 Show that the frictional torque T required to rotate a disc of diameter d at an angular velocity ω in a fluid of density ρ is given by T = d 5ω 2ρφ ( ρd 2ω µ), and identify the Reynolds number group.

[ ρd(dω)µ ]

8.2 Develop an expression for the power P, developed by a hydraulic turbine, diameter d, at speed of rotation n, operating in a fluid of density ρ with available head h. [P = ρ n3d 5φ (n2d 2gh)] 8.3 Determine the dependence of the force F acting on a sphere moving at a constant velocity through a fluid on the fluid density and viscosity. [F = ρ D 2v 2φ (Re)] 8.4 If a circular cylinder of given length to diameter ratio, ld, is rotated about its geometric axis at an angular velocity ω at right angles to and in a uniform fluid stream of velocity u, show that the power required to rotate the cylinder is

given by P = ρ v 3dφ (udv, ω dv), where v is the fluid kinematic viscosity and ρ is the fluid density. 8.5 Show that the drag force on a body is a function of both Reynolds and Mach numbers in situations where viscous resistance and compressibility effects are major factors. 8.6 For a journal bearing of diameter d, length l, radial clearance c and eccentricity e, show that the load W that can be supported by the oil film of viscosity µ is given by Wµ nd 2 = φ (cd, ed, ld), when the speed of rotation of the bearing is n. 8.7 Show that the rate of flow Q over a vee-notch of included angle θ may be expressed as Q(gh5)0.5 = φ [(g h3)0.5v, gh2ρτ, θ ], where h is the head above the notch vertex, v is the fluid kinematic viscosity, τ is the fluid surface tension and g is acceleration due to gravity.

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Chapter 9

Similarity 9.1 9.2 9.3

9.4 9.5 9.6

Geometric similarity Dynamic similarity Model studies for flows without a free surface. Introduction to approximate similitude at high Reynolds numbers Zone of dependence of Mach number Significance of the pressure coefficient Model studies in cases involving free surface flow

Similarity applied to rotodynamic machines 9.8 River and harbour models 9.9 Groundwater and seepage models 9.10 Computer program GROUND, the simulation of groundwater seepage 9.11 Pollution dispersion modelling, outfall effluent and stack plumes 9.12 Pollutant dispersion in one-dimensional steady uniform flow 9.7

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The techniques of dimensional analysis introduced in Chapter 8 are of use to indicate the parameter groups that may determine flow under any particular set of prevailing conditions. Model testing, based upon a systematic variation of these groups, is necessary to determine their relative importance. Once this has been established the rules of geometric and dynamic similarity must be invoked to allow design decisions to be taken based upon test

results. This chapter defines both the most common dimensionless groupings found to determine flow conditions and their zones of influence, and introduces the laws of similarity necessary to translate a model into a prototype in each flow regime. Examples, drawn from a wide range of design cases, are presented, including internal pipe flows, external flows and fanpump design, and free surface flows, including river and harbour models. l l l

A floating semi-submersible drilling rig being tested in ‘design storm’ conditions in the Marintek wavebasin, image courtesy of Marintek

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FIGURE 9.1(a) Free surface waves in a laser wave tank experiment. The study of wave propagation and attenuation will be treated in Chapter 21. (Photograph courtesy of Professor Ian Grant, Fluid Loading and Instrumentation Centre, Heriot-Watt University, Edinburgh)

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FIGURE 9.1(b) Cavitation bubbles from a propeller under test in a cavitation tunnel. (Photograph courtesy of Professor Ian Grant, Fluid Loading and Instrumentation Centre, Heriot-Watt University, Edinburgh, and the Defence Research Agency, Haslar, UK)

Whenever the design engineer needs to take decisions at the design stage of a project, it will probably be necessary to initiate some form of model test programme. The basis of any such test series depends on the accurate use of instrumentation systems and the correct application of the theories of similarity. This, in turn, involves the application of dimensional analysis and the utilization of dimensionless groups such as the Reynolds, Froude or Mach numbers. Model testing occurs in all areas of engineering based on fluid mechanics. Wind tunnel tests on aircraft, cars and trains, towing tests on ships and submarines, riverharbour tests carried out using models of high levels of intricacy to simulate tidal flow: all serve to illustrate such use of models. A recent application has arisen from the problems of air flow around buildings, which may be studied in wind tunnels and by examining smoke generation and propagation through building models. Model tests, then, depend on two basic types of similarity, which may be considered separately: geometric and dynamic similarity. Figure 9.1(a) and (b) illustrates a range of model testing.

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GEOMETRIC SIMILARITY

9.1

The first requirement for model testing is a strict adherence to the principle of geometric similarity, i.e. the model must be an exact geometric replica of the prototype. Thus, for an aerofoil model of 110 scale, say, both the span and chord must be exactly 110 of the full-scale dimensions. This principle is, however, not fully applied to river models, where distortion of the vertical scale is necessary to obtain meaningful results because it is necessary to keep the relationship between wave properties and depth correct. Generally, it may be assumed that geometric similarity is achieved in model testing.

DYNAMIC SIMILARITY

9.2

The definition of dynamic similarity is that the forces which act on corresponding masses in the model and prototype shall be in the same ratio throughout the area of flow modelled. If this similarity is achieved, then it follows that the flow pattern will be identical for the model and the prototype flow fields. Before moving on to the consideration of particular flow situations, it is worthwhile to restate the derivation of the most common dimensionless groups whose respective values govern model testing. Consider a general, hypothetical flow situation where the pressure change ∆p between two points is dependent on mean velocity C, length l, density ρ and viscosity µ, bulk modulus K, surface tension σ and gravitational acceleration g: ∆p = f (C, l, ρ, µ, K, σ, g). With eight variables, defined by three dimensions, M, L and T, five dimensionless groups may be expected, ∆p −−−−−−2 = f 1 [ ρ vlµ , v ( Kρ ), ρ v 2σ , v ( lg ) ], ρv

1 − 2

recognizable as the pressure coefficient, Cp = ∆p( −12 ρ v 2 ), and the dimensionless groups Re (Reynolds number), Ma (Mach number), We (Weber number), Fr (Froude number), ∆p Cp = − −−−−− = f 1 ( Re, Ma, We, Fr ). 1 − ρ v2 2

(9.1)

Equation (9.1) indicates that the pressure coefficient is dependent upon the other dimensionless groups and is defined if the other groups are defined. Thus if the values of the Re, Ma, We and Fr groups were identical for a prototype and its scale model, i.e. thereby conforming to geometric similarity, it would follow that the pressure coefficient would also be equal for the model and the prototype. This equivalence would therefore allow the model-generated results to be utilized to predict, for

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example, the lift and drag forces on aerofoils. This mathematical result follows directly from the development of dimensional analysis already presented. However, it is also possible to confirm this conclusion by a parallel and independent analysis of the relative forces acting upon the model and prototype, remembering that the dimensionless groups included in equation (9.1) are in fact ratios of the applicable forces, the significance of each group depending upon the importance of those forces to the flow condition. Two examples will be considered, namely the forces acting on the airstream over an aerofoil and the forces acting on an element of free surface flow. In both cases the forces acting will be shown to be generated due to gravity, Fg, pressure, FP, and viscous, Fv, effects, although the significance of each will vary depending upon the example. The resultant force, FR, will act on the fluid element in each case and accelerate it in accordance with Newton’s second law. As the force polygons in the prototype, i.e. real flow situation, and the model will be similar, in a strictly geometric rather than a general sense, in both cases the magnitudes of the forces on the prototype fluid element will be in the same ratio to each other as the forces acting in the model flow. Thus, as the resultant force may be defined in terms of the fluid element mass and acceleration, it follows that (FR)p(FR)m = (ma)p(ma)m = (FP)p (FP)m = (Fg )p (Fg )m, where the suffixes m and p refer to the model and prototype flow condition, respectively. Now mass m ∝ ρl 3, acceleration a ∝ vt and Fg ∝ ρgl 3. Thus by substitution (vgt)m = (vgt)p and, as v ∝ lt, it can be arranged that tmtp ∝ (lmvm )(lp lvp), so that (v 2gl )m = (v 2gl)p. However, this relationship conforms to that already defined as the Froude number (note the squared form of the result compared with earlier definitions); thus the force conditions inherent in dynamic similarity have been shown to imply an equivalence of a variable group identified by the earlier dimensional analysis procedure. Considering the viscous forces, then, if Fv ∝ µ vl, it follows that (ma)p (ma)m = (FP)p (FP)m or

( ρl 3vt)p( ρl 3vt)m = ( µ vl )p( µ vl )m,

where further simplification yields ( ρl 2)( µt)p = ( ρl 2)( µt)m and, as t = lv, it follows that ( ρ vl)( µ)p = ( ρ vl )( µ)m.

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It will be appreciated that this equality is identical to the equivalence of Reynolds numbers already identified by dimensional analysis as a requirement for dynamic similarity. Thus a definition of dynamic similarity as requiring that the forces acting upon the model and prototype remain in the same ratio to each other may be developed to the point where the necessary equivalence of terms confirms the predictions of a dimensional analysis. Considering the pressure force illustrated for both cases in Fig. 9.2(a) and (b) it follows that (FP)p (FP)m = (ma)p (ma)m, where FP ∝ ∆pl 2, which leads to the conclusion that (Cp)m = (Cp)p. An inspection of the force polygons for both the air flow and free surface flow examples indicates that one of the forces could be determined if the other three were known. Thus the pressure force could be construed as dependent upon the viscous, gravitational and reaction forces shown. This would imply that the pressure coefficient depended upon the other parameters. Thus, if the Reynolds and Froude numbers are identical between the model and the prototype, it follows that the pressure coefficient will be equal for the model and the prototype. This is the same conclusion reached by the dimensional analysis approach, but arrived at independently by an analysis of the forces acting in each flow condition. While these two cases have shown that equivalence of the Reynolds and Froude numbers is necessary, it is also clear that had the flow conditions around the model and the prototype involved other forces, such as surface tension or elasticitycompressibility effects, then the analysis presented could have been extended to include additional variables that would have confirmed the importance of other dimensionless groups, such as the Weber or Mach numbers. Although the examples chosen are general it must be appreciated that in some flow conditions some forces predominate and the equivalence of the dimensionless groups representing these forces becomes imperative, while the equivalence of other groups becomes less significant. As was stressed in the development of the dimensional analysis approach, that methodology cannot itself inform the engineer as to the relative importance of each group; it was pointed out that in dimensional analysis, provided the mathematical rules are obeyed, there are no wrong answers, only more useful ones. Clearly for the case of air flow over an aerofoil the gravity forces become insignificant and the viscous forces predominate. Thus it is Reynolds number that must be held constant; the effect of Froude number may be neglected. Conversely, in the free surface flow case illustrated it is the gravity forces that determine the flow condition over the channel surface and thus Froude number equivalence is essential. In adopting this approach care must be taken to ensure that choice of model size takes account of other forces that might become important on the model. For example, if the spillway model is too small, i.e. the scale ratio is too small, then it is possible for viscous and surface tension forces to become important, thus rendering the dynamic similarity void as these forces do not appear in the prototype flow regime. As a result of the independent confirmation provided by this approach it is now possible to state that dynamic similarity between a model and prototype requires that the significant dimensionless groups have the same values for both. It will be found that the dimensionless groups necessary in a wide range of flow conditions have been identified by engineers. A brief listing follows:

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9.2

FIGURE 9.2 Forces acting on a fluid element. (a) Passing over an aerofoil. (b) In free surface flow

Dynamic similarity

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Reynolds number, Re: Reynolds number has already been defined as the ratio of inertia and viscous forces. It may be defined generally as Re = ρVLµ, where V is a characteristic velocity and L is a characteristic length. This general definition is important in understanding the wide influence of Reynolds number; in many cases Reynolds number is erroneously only associated with pipe flow where V is the mean flow velocity and L the pipe diameter. In open-channel flows the characteristic length L may be seen to be the hydraulic mean depth; in aerofoil theory it may be the wing chord; while in fan or pump analysis it may be the blade diameter. Froude number, Fr: In flows dominated by gravitational effects, notably free surface flows, it has been shown above that dynamic similarity requires that the ratio of inertia to gravitational forces remains constant, and this scales to the square of Froude number, defined as Fr = V(Lg)12, where V and L are characteristic values chosen as appropriate for the particular flow condition: for example, L could be depth in a rectangular cross-section channel but hydraulic mean depth in non-rectangular channels. Care should be taken with the definition of Froude number in texts as some authors utilize the squared term, which clearly differs for all values other than unity. A comparison of Froude and Reynolds numbers indicates immediately that if the same fluid is used for both prototype and model, it is impossible to have equivalence of both at the same time between a model and a prototype flow condition. Mach number, Ma: Mach number is defined as the ratio of fluid velocity to the local sonic velocity. If the flow results in compressibility effects, i.e. the assumption of constant density is no longer supportable, it becomes important to include the effects of the elastic forces acting, and therefore the appropriate force ratio would be between inertia and elasticity. This ratio would therefore be proportional to ρL2V 2KL2 or ρV 2K, where K is the bulk modulus of elasticity; this relationship is referred to as the Cauchy number. However, this is often simplified by noting that the wave propagation velocity in an isentropic gas or liquid may be expressed as (Kρ)12, and the ratio of inertia to elastic forces becomes (Vc)2, where c is the wave speed. This reduces the force ratio to the simpler and widely used Mach number, defined simply as Vc. This makes Mach number equivalence a necessity in modelling air flows where compressibility effects are important. It is also interesting to relate Mach number to an alternative definition of Froude number. The term (Lg)12 may be shown to be the speed of propagation of a surface wave in a free surface or open-channel condition. Thus this form of Froude number becomes a form of Mach number. At first sight this might appear to be interesting but not relevant. However, this is not the case, as in open-channel flows where the Froude number exceeds unity, i.e. V (Lg)12, surface waves cannot propagate upstream; these cases are known as supercritical flow. This is analogous to the supersonic definition where sound waves cannot propagate ahead of the object. Again the characteristic length L may be taken as the channel hydraulic mean depth in non-rectangular channels. Weber number, We: Weber number is defined as the ratio of inertia to surface tension forces. The definition of Weber number often varies between texts and care must be exercised in the use of tables. The expression following from the ratio quoted above and generally accepted is We = V( ρLσ )12, but the square and the reciprocal of this expression may be found in the literature. Pressure, stress and force coefficients: e.g. pressure Cp , lift CL, drag CD, skin friction Cf . As already mentioned in the development of the dimensional analysis methodology, it is useful to be able to relate pressure differences or flow-induced forces to the flow parameters via a series of non-dimensional groupings. It therefore follows that it is necessary to ‘non-dimensionalize’ the pressure or force in terms of readily measured

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9.3

Model studies for flows without a free surface. Introduction to approximate similitude at high Reynolds numbers

291

or defined variables. The pressure or stress coefficients may be ‘non-dimensionalized’ by use of the flow kinetic energy term. Thus Cp = ∆p −12 ρV 2,

(9.2)

where V is a chosen characteristic velocity and the strictly not required constant of −12 was historically included to allow direct use of experimental flow measurements. Force coefficients require that an area term be introduced, so that lift or drag coefficients are defined as CL = ∆p −12 ρV 2L2.

(9.3)

It will be appreciated that these coefficients are dependent upon other variable groupings as already illustrated. Power coefficients: Power is simply a rate of doing work, which is in turn definable as a force moving through a given distance. Thus it is possible to define power in the three dimensions of mass, length and time. Power coefficients, useful in the definition of pump, fan and turbine characteristics in terms of other flow variables included in the other common dimensionless groupings, may thus be expected to have a form PρV 2L3.

9.3

MODEL STUDIES FOR FLOWS WITHOUT A FREE SURFACE. INTRODUCTION TO APPROXIMATE SIMILITUDE AT HIGH REYNOLDS NUMBERS

Free surface effects are absent in bounded flows, this definition including both pipes or ducts flowing full and the flow around submerged bodies, such as aircraft, submarines or buildings. If the flows involved are low then compressibility effects may be ignored and the requirement to hold Mach number constant between the prototype and model may be relaxed. In these cases, therefore, the analysis presented would dictate an equivalence of Reynolds number. However, strict adherence to Reynolds number equivalence may prove inappropriate and unduly costly. The following examples will illustrate the difficulty.

EXAMPLE 9.1

A submarine-launched missile, 2 m in diameter and 10 m long, is to be tested in a water tunnel to determine the forces acting on it during its underwater launch. The maximum speed during this initial part of the missile’s flight is 10 m s−1. Determine the mean water tunnel flow velocity if a 120 scale model is employed and dynamic similarity is achieved.

Solution To comply with dynamic similarity the Reynolds numbers must be identical for both the model and the prototype:

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Rem = Rep, ( ρVLµ)m = ( ρVLµ)p. The model flow velocity is thus given by Vm = Vp(Lp Lm)( ρp ρm)( µm µp), but as ρp = ρm and µm = µp it follows that Vm = 10 × 20 × 1 × 1 = 200 m s−1. This is a high velocity and illustrates why few model tests are made with completely equal Reynolds numbers. At high Reynolds numbers, however, it will be shown that a relaxation of strict equivalence is acceptable.

EXAMPLE 9.2

An airship of 6 m diameter and 30 m length is to be studied in a wind tunnel. The airship speed to be investigated is at the docking end of its range, a maximum of 3 m s−1. Determine the mean model wind tunnel air flow velocity if the model is made to a 130 scale, assuming the same sea level air pressure and temperature conditions for the model and the prototype.

Solution Dynamic similarity requires the equivalence of Reynolds number; clearly there will be no compressibility effects on the prototype. Thus as before Vm = Vp(Lp Lm)( ρp ρm)( µm µp) and substituting ρp = ρm and µm = µp it follows that Vm = 3 × 30 × 1 × 1 = 90 m s−1. It is worth noting that at sea level the local sonic velocity may be taken as 340 m s−1 and therefore the prototype Mach number is 3340, approximately 0.01. The same calculation for the model indicates a Mach number of 90340, approaching 0.3, a value at which Mach number effects could be present.

The objective of dimensional analysis and the development of the theories relating to dynamic similarity have been to enable the establishment of testing techniques that would allow engineers to predict from model tests the flow condition to be encountered in a prototype; by definition these techniques should be both reliable and affordable. While the definition of affordable is obviously a variable dependent upon the particular application, it is clear from the examples above that strict adherence to Reynolds number equivalence can be problematic. In both cases the power required to drive the water and wind tunnels is considerable, and in the air flow case the flow velocity necessary would introduce compressibility effects that would not be present on the prototype. Therefore an alternative approach has to be found that conforms to the engineer’s understanding of both the forces acting on the model and the prototype and their relative importance. It has already been indicated

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9.4

Zone of dependence of Mach number

293

that for pipe flows, at high Reynolds number values, frictional effects tend to become independent of Reynolds number. This effect will also be demonstrated later for drag coefficients for cylinders and spheres. Thus for model tests to be applicable it may be sufficient to ensure that both the model and prototype flows possess Reynolds number values well into this range. The precise Reynolds number values at which this approach is acceptable are dependent upon the flow condition being investigated. However, checks may be carried out based around measurements of Cp values as model Reynolds number rises: once the Cp values become independent of Re the ‘safe’ model flow condition has been reached. Guidance can also be obtained by reference to current practice in that branch of wind or water tunnel testing. Alternative approaches involve retaining the Reynolds number equivalence by a change of fluid between the prototype and the model, e.g. the use of compressed air as a fluid in the model testing of hydroelectric turbines.

ZONE OF DEPENDENCE OF MACH NUMBER

9.4

Mach number becomes significant in flow situations where the ratio of flow velocity to sonic velocity exceeds about 0.25 to 0.3. It is normally difficult to satisfy both Reynolds number and Mach number equality simultaneously and so it is important that testing decisions are made based on previous experience of the type of flow to be investigated. For example, if the viscous motion of a fluid close to a boundary in supersonic flow is to be investigated, then Reynolds number equivalence would be the criterion, but if the object of the investigation is the flow conditions through the shock wave pattern around a body, then Mach number equivalence is paramount. Mach and Reynolds number equivalence may be achieved if it is possible to vary the fluid density conditions between the model and prototype, as illustrated below.

EXAMPLE 9.3

In order to undertake predictions of the lift and drag force on a scale model of an aircraft during a section of its operational envelope involving sea level flight at 100 m s−1, where the speed of sound may be taken as 340 m s−1, it is proposed to utilize a cryogenic wind tunnel with nitrogen at 5 atmospheres of pressure and a temperature of −90 °C, conditions at which the nitrogen density and viscosity may be taken as 7.7 kg m−3 and 1.2 × 10−5 N s, respectively. The speed of sound in nitrogen at this temperature is 295 m s−1. Determine the wind tunnel flow velocity, the scale of the model to ensure full dynamic similarity and the ratio of forces acting on the model and the prototype.

Solution In order to provide an equivalence of Mach number it follows that the wind tunnel velocity must be given by Vm = cm(Vpcp), where c is the appropriate local sonic velocity; hence Vm = 295 × 100340 = 86.76 m s−1. At this wind tunnel flow velocity it is possible, from the data given, to determine the necessary scale for which Reynolds number equivalence is achieved as

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Rem = Rep, ( ρVLµ)m = ( ρVLµ)p, ( ρm ρp)(Vm Vp)(Lm Lp)( µp µm) = 1. Thus (7.71.2)(86.76100)(Lm Lp)(1.8 × 10−51.2 × 10−5) = 1 Lm Lp = 1.0(6.4 × 0.868 × 1.5) = 0.12. The ratio of forces acting on the model and prototype may be determined by noting that the force coefficients will be equal if the Mach and Reynolds numbers are equal. Thus (Fm ρV 2L2)m = (Fp ρV 2L2)p , where F represents a typical force acting on the model and the prototype and L2 represents an area, for this case the projected wing area. Thus the ratio of forces becomes Fm Fp = ( ρm ρp)(Vm Vp)2(Lm Lp)2, Fm Fp = (7.71.2)(86.76100)2(0.12)2 = 0.0696 or 6.96 per cent.

9.5

SIGNIFICANCE OF THE PRESSURE COEFFICIENT

Referring back to equation (9.1), it will be seen that the pressure coefficient was defined by the other dimensionless groups judged to define the flow condition. Therefore it follows that, if dynamic similarity is achieved, the values of pressure coefficient measured in model tests will also apply to the prototype. This is essential in relating both pressure changes in the model and prototype flow conditions and determining the forces acting on the prototype. This second calculation involves multiplying the pressure coefficient by an appropriate area, e.g. the projected wing area in an aircraft model investigation where the model lift and drag forces would have been measured via the lift and drag component balances of the wind tunnel instrumentation. Thus in Example 9.1, if the pressure difference between two points on the surface of the missile had been 5.0 N m−2, then the pressure difference on the model would have been given by (Cp)m = (Cp)p, (∆p ρv 2)m = (∆p −12 ρv 2)p, −1 2

∆pm = ∆pp( ρmρp)(Vm Vp)2, ∆pm = 5 × 1 × (20010)2 = 2 kN m−2.

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Model studies in cases involving free surface flow

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The importance of the pressure coefficient may also be appreciated if the case of pipe flow modelling is considered.

EXAMPLE 9.4

Flow through a heat exchanger tube is to be studied by means of a 110 scale model. If the heat exchanger normally carries water, determine the ratio of pressure losses between the model and the prototype if (a) water is used in the model, (b) air at normal temperature and pressure is used in the model.

Solution For dynamic similarity the Reynolds numbers must be constant; hence Rem = Rep , Vm Vp = (Lp Lm)( ρp ρm )(µm µp). If the Reynolds numbers are equal, then so must be the pressure coefficients; therefore, (Cp )m = (Cp)p, (∆p ρ v 2)m = (∆p −12 ρ v 2)p, −1 2

∆pm = ∆pp( ρm ρp)(Vm Vp)2, ∆ pm −−−−− = ( ρm ρp)(Vm Vp)2 = ( ρp ρm )( µm µp)2(Lp Lm)2. ∆ pp (a) In the water model case, as the model and prototype fluid densities and viscosities are the same it follows that (∆p)m (∆p)p = 102 × 12 × 1 = 100. (b) If air is used as the model fluid then the full form of the pressure coefficient equivalence must be used:

ρp ρm = 10001.23, µm µp = 1.8 × 10−51 × 10−3 = 1.8 × 10−2, (∆p)m (∆p)p = 102 × (10001.23) × (1.8 × 10−2)2 = 26.34.

9.6

MODEL STUDIES IN CASES INVOLVING FREE SURFACE FLOW

In free surface model studies the effect of gravity becomes important and the governing parameter is Froude number. Generally the prototypes, i.e. large spillways, have Reynolds numbers large enough to be operating out of the range of dependence on Re; however, the model may be of such a size that, when Froude number

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equivalence is set up, the model Reynolds number is small enough to produce viscous effects not representative of the prototype. For this reason, the model must be large enough to place its Reynolds number above the viscous loss dependence level. One problem with free surface flow cases is that, generally, the same fluid is used for the model as for the prototype, so that the convenient expedient of substituting air for water in internal flows cannot be copied.

EXAMPLE 9.5

A 150 scale model of a proposed power station tailrace is to be used to predict prototype flow. If the design load rejection bypass flow is 1200 m3 s−1, what water flow rate should be used on the model?

Solution Equating Froude numbers, Frm = Frp, where Fr = C(lg). Therefore, Cm Cp = (lm lp). Flow rate may be determined by introducing the area ratio Am Ap = 12500 = (Scale)2. Hence, Q m A m C m l 2m −−−− = −−−−−−− = −− Qp Ap Cp l 2p

lm

⎛⎝ −l−⎞⎠ , p

Qm = Qp(lm lp)2.5 = 1200 × (150)2.5 = 0.067 m3 s−1. This relatively simple approach is complicated for the case of ship resistance testing, as the phenomenon is made up of two factors, namely the surface resistance of the hull, dependent on Reynolds number, and the wave resistance, which is Froude number dependent (see Section 12.4). Consider the case of a model to be towed at a speed such that the Froude number is satisfied: Frm = Frp, vm (lm g) = vp(lpg), vm vp = (lm lp). Now consider the same model and equate Reynolds numbers: Rem = Rep, ( ρ vlµ)m = ( ρ vlµ)p, vm vp = (lplm)( ρpρm)( µmµp) = lplm ,

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Similarity applied to rotodynamic machines

297

if ρm = ρp and µ m = µ p. Obviously, then, the two criteria cannot be satisfied simultaneously and the approach followed is to equate Froude number to model wave resistance forces as these are the more difficult to analyse. Viscous hull resistance is then calculated by analytical techniques and added to the wave resistance measured.

SIMILARITY APPLIED TO ROTODYNAMIC MACHINES

9.7

Application of the techniques of dimensional analysis to fans and pumps yields relationships of the form PN 3D 5ρ = f (QND 3, µρND 2, kD, aD, bD, cD),

(9.4)

Ps ρN 2D 2 = f (QND 3, µρND 2, kD, aD, bD, cD),

(9.5)

where P is shaft power, Q is volume flow rate, Ps is the pressure rise across the unit rotating at speed N, and of diameter D. The fluid type is defined by density ρ and viscosity µ, while the detail dimensions of the machine are a, b, c with surface roughness k. For geometrically similar machines operating at high Reynolds numbers, so that the term µND2ρ = Re becomes irrelevant, the expressions reduce to PN 3D 5ρ = f2(QND 3) and Ps ρN 2D 2 = f1(QND 3). Thus, for model testing to be valid, each of these groups should have identical values for the model and the prototype. Model testing is of particular value in the design and manufacture of the largerscale fans, pumps and turbines, to which these relationships also apply, except that the power terms relate to power generated rather than power supplied. Generally, the model scale is arranged so that the impeller diameters are less than 0.5 m and the same fluid is usually employed in the model tests as for the prototype. However, due to the lack of effect of Reynolds number, provided the flow is well into the fully turbulent region, it is possible to employ air or pressurized gas in order to obtain more manageable flow rates or machine scales. Denoting the model by m and the full-size machine by p, the following relations can be proved: 1.

Flow, Qm Qp = (ND 3)m (ND 3)p.

2.

(9.6)

Pressure rise (pumps) and pressure drop (turbines), (Ps)m (Ps)p = ( ρN 2D 2)m ( ρN 2D 2)p, where the density ratio may vary from unity.

(9.7)

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3.

Power supplied (pumps) and power generated (turbines), Pm Pp = (N 3D 5ρ)m (N 3D 5ρ)p.

(9.8)

It will be appreciated that these relations are all independent of operating pressure, so that, in theory, any convenient operating test rig head may be employed. In practice, this is not entirely true as the onset of cavitation is dependent on absolute pressures and, for pumps and turbines, its occurrence is of major importance, so that pressure levels should be as close as possible to the full-scale installation values. Theoretically, the efficiency of model and prototype should be the same. However, there will be some excess inefficiency in the model due to scale effects relating to leakage flow, roughness variations and manufacturing constraints (see Section 23.5).

EXAMPLE 9.6

A ventilation system fan is to be exported to a high-altitude region with an air density of 0.92 kg m−3 and is expected to deliver 2 m3 s−1 at a pressure differential of 200 N m−2. If the fan is to be driven at 1400 rev min−1 on installation, calculate the flow rate and pressure rise required on test at sea level, air density 1.3 kg m−3, and the appropriate fan test speed if conditions of dynamic similarity are to be achieved. (Assume no change in air viscosity with the change in altitude involved.)

Solution The fan test speed may be determined by equating the fan Reynolds numbers: (µρND 2 )S = ( µρND 2 )A, where the suffixes A and S refer to altitude and sea level conditions, respectively. Thus NS = ( ρND2 )A (ρD 2 )S = 1400 × (0.921.3) × 12 = 990 rev min−1. Note that the geometric scale is unity as the fan is its own model and that the viscosity ratio has been set to unity also. The sea level flow rate and pressure expected would thus follow as QS = QANSNA = 2 × 9901400 = 1.41 m3 s−1, ∆pS = ∆pA( ρS ρA)(NS NA )2(DS DA )2 = 200 × (1.30.92) × (9901400)2 × 12 = 141 N m−2.

EXAMPLE 9.7

A wind turbine used to generate electricity differs in a number of important ways from a water turbine operating within a conduit. Where as the air remotely upstream and downstream of the rotor remains at atmospheric pressure, the air flow velocity drops across the rotor while the cross-section of the affected airflow increases.

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9.8

River and harbour models

299

Using dimensional analysis determine the dependence of generated power on wind speed and turbine rotor diameter, and hence determine the increase in wind velocity or the increase in rotor diameter required to double the power generated by a wind turbine belonging to a dimensionally similar family of machines. Detail the factors determining wind turbine power generation.

Solution Identify the variables involved, neglecting for simplicity the geometric terms such as hub diameter, blade chord, roughness, which are all non-dimensionalized with respect to rotor diameter D. Variable

M

L

T

Power P Diameter D Rotational speed N Upstream wind velocity V1 Air density ρ Air dynamic viscosity µ

1 0 0 0 1 1

2 1 0 1 −3 −1

−3 0 −1 −1 0 −1

Six variables in three dimensions require three groups with ρ, D and N as the repeating variables. By inspection, the groups are Power coefficient PρN 3D5, Tip speed ratio V1 ND, and Reynolds number ρV1Dµ. Note that tip speed ratio replaces the flowrate coefficient for water turbines and pumpsfans due to the absence of an enclosing conduit. (ND is the tip speed.) Hence P depends on D2 and (ND)3 and therefore to double power output the wind speed would have to rise by a factor 20.33 = 1.28, or the rotor diameter would have to increase by 20.5 = 1.41. In summary, the output of wind generators depends upon: 1 2 3 4

the cube of the wind velocity through the rotor; the square of rotor diameter; the nature of the wind in terms of its inherent unsteady nature and its gust frequency; the overall efficiency of the mechanical and electrical components of the turbine, which when compounded with the max possible theoretical efficiency derived can reduce overall efficiencies to below 30 per cent.

9.8

RIVER AND HARBOUR MODELS

River and harbour engineering projects are costly undertakings and, as analytical techniques only provide approximate predictions of the likely effects of any river widening or harbour improvement, the use of models at an early stage in the design has many advantages. The problems arise when a suitable scale is to be chosen for the model; the adoption of a scale that will give reasonable channel depths will usually

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result in a model too large to be practical, while choosing a scale on areacost criteria yields channel depths that are very small. The problems of shallow model channels are: (1) accuracy in level and level change measurement becomes impossible to achieve; (2) the surface roughness of the channel beds would be impractically small and there is even a probability that channel flow would be laminar rather than turbulent, as normally found in practice. In order to provide a solution to these problems, distorted scaling is adopted, vertical scales of 1100 and larger being typical, while horizontal scales vary from 1200 to 1500. Distortion of this sort is suitable if the overall discharge characteristics of a long length of river are being studied. However, it should be appreciated that the micro-situation is not well modelled, and situations such as the effects of breakwater positioning should be studied on as large and as undistorted a scale as possible. In models of this type, strict geometric similarity is not achieved. However, if the mean flow velocity C and depth Z are arranged so that there is an equivalence of Froude number (CgZ) between model and river, then it will be expected that the flow type, i.e. fast or slow, will be the same at corresponding points on the model and river. Thus, C 2m Z m = C 2r Z r , and, hence, C m C r = ( Z m Z r ),

(9.9)

where m denotes model and r denotes full-scale river. The discharge Q through the model must depend on both vertical and horizontal scales; thus Q = ClZ and Qm Qr = (Zm Zr)32(lmlr),

(9.10)

where lm lr is the horizontal scale, 1:x, and Zm Zr is the vertical scale, 1: y. In order to manufacture models, it is necessary to have information on the effect of surface roughness, particularly the effects of model roughness size, which is often dictated by the manufacturing process chosen. From the Manning coefficient of surface roughness n and the Manning equation applied to both model and river, Cm n r Z m 23 Z m l m 1 2 −−− = ⎛ −−−⎞ ⎛ −−−−⎞ ⎛ −−−−−−−−−⎞ , C r ⎝ n m⎠ ⎝ Z r ⎠ ⎝ Z r l r ⎠ nm = nr x12y 23.

(9.11)

Full-scale river beds have values of surface roughness defined by n 0.03, and as the normal model surface finish, obtained by use of cement mortar for model surfaces, is of the order of n ≈ 0.012, artificial roughening is normally necessary. The maximum value of n obtainable by artificial means, i.e. adding wire mesh, gravel and even small rods to the bed, is of the order of 0.04. As mentioned, the micro-flow situation, i.e. eddies of local currents, will differ between model and river, so adjustment by surface roughness only is not a reasonable course of action. Figure 9.3(a)–(c) illustrates a typical model and the artificial channel bed roughening used. Before it is used to predict the effects of any modifications to the river channels, the model should first be checked for discharge and depth accuracy. Measurements of full-scale depths and discharges should be checked against model discharge and depth by use of equation (9.10), relating total flow rates, and the vertical scale. Model

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9.8

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FIGURE 9.3(a) A scaled 135 000 m3 LNG carrier moored at an undisclosed location in the shelter of a breakwater under multidirectional random wave attack (Image reproduced courtesy of HR Wallingford Group Ltd)

discharge rates are produced using recirculating pump circuits and orifice plate or notch flow measuring instrumentation. If the model values are acceptable, then testing can continue. If the depth and flow rates are in poor agreement with the full-scale results, then the model surface roughness should be adjusted or the scales altered. It should be noted that the above analysis is based on the premise that the Reynolds number of both model and full-scale river is such that the flow is totally turbulent and that variations in Reynolds number are not important. It is good practice, however, to check the values of model and channel Reynolds number to ensure that this simplification is valid, i.e. Re 2000. Estuary models may be constructed using the same general principles as outlined for river models; distorted scales are typically 150 to 1150 vertical and 1300 and 12500 horizontal. As the available data on tidal velocity distributions are likely to be sketchy, it is necessary to incorporate in the model the whole tidal channel system, as well as a substantial portion of adjacent coastline.

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FIGURE 9.3(b) The historic Flood Channel Facility at HR Wallingford (Image courtesy HR Wallingford Group Ltd)

Although surface roughness is not so critical in estuary models, the speed of propagation of the tide becomes an important design criterion as the tidal period governs the time available for result recording. The tidal period of the model is thus ( DistanceVelocity ) model Time in model −−−−−−−−−−−−−−−−−−−−−−− = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−. Time in estuary ( DistanceVelocity ) estuary Now, as the Froude numbers are equal (as for river models) it follows from equation (9.9) that (Time)m (Time)e = (lm le)(Ce Cm ) = (lm le)(Ze Zm ).

(9.12)

Thus, for a tidal period of 12.4 h and a model with 150, 1500 scales, the model tidal period is 10.5 min.

9.8

FIGURE 9.3(c) A bird’s eye view of a scale model of the entrance channel to Dar-es-Salaam harbour, showing flow patterns at ebb tide. The effect is obtained by scattering confetti on the water and then photographing it with a time exposure. Where the water is moving fastest, the confetti shows as a streak – the longer the streak, the faster the current. By studying results from a series of photographs representing various states of the tide, one is able to calibrate the model so that water flows in the model correspond closely to those recorded on site (Courtesy of UN Development Program, East African Harbours Corp., International Bank for Reconstruction and Development, Bertlin and Partners, Redhill, and The Model Laboratory, Wimpey Laboratory, Middlesex)

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While equation (9.12) refers to horizontal fluid velocity, the scale factor for vertical silt particle velocity is also of interest and may be derived in the same manner: Rate of fall in model Depth scale −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− = −−−−−−−−−−−−−−−−− Rate of fall in estuary Time scale = (Zm Ze)[(Zm Ze)(le lm )] = (Zm Ze)32(le lm ).

(9.13)

A further complication in estuary model studies, particularly of silting phenomena, is stratification effects due to density variations between salt and fresh water. If the use of saline solutions in the model testing is undesirable for corrosion reasons, then a stable clay solution may be employed. Estuary models are now commonly used to investigate the effects of discharge of power station or industrial cooling water flows and, here, density variations may again have to be modelled, based on temperature. Other uses of estuary models include silting and erosion studies and the spread and deposit of effluent discharged into the sea. Generally, the role of the estuary model should be seen more as a method of comparing the attributes of various design solutions, than as an accurate method of predicting the effects of one design. Harbour and coastal models require the inclusion of wave effects, and these are reproducible by means of mechanical wave-making devices. However, the type of wave motion encountered in coastal engineering is dependent on both water depth and wavelength for its propagation velocity, so that the degree of scale distortion acceptable in river and estuary models can no longer be applied. The best model studies are carried out with equal scaling; however, distortion of the vertical scale up to two or three times the vertical has been used.

EXAMPLE 9.8

It is proposed to construct a model of 18 km length of river, for which the first 8 km are tidal. The normal discharge of the river is known to be in the region of 300 m3 s−1, the average width and depth of the channel being 3 m and 65 m, respectively. Given a laboratory of 30 m length propose suitable scales and calculate the tidal period.

Solution (i) The largest scale possible would be 3018 × 1000 = 1600 for the horizontal distances. (ii) As the river is tidal, scale distortions of around 6 to 10 are acceptable, so a vertical scale of 160 could be employed. (iii) The model will be constructed to conform with these scales. However, in doing so, the effect of Reynolds number is assumed negligible. It is good practice to check the Reynolds number. Average river velocity = 300 m3 s−1(3 × 65) m2 = 1.54 m s−1. From equation (9.9), Cm = Cr(Zm Zr) =1.54 × (160) = 0.199 m s−1. Rem = ρ vmµ,

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m = Hydraulic mean depth = AreaPerimeter flow cross-section

where

−1−−− )( −65 −−−− −6−− = (3 × −601−− × 65 × −600 600 + 60 )

= 5.4 × 10−30.208 = 0.026 m. Thus,

Rem = 1000 × 0.199 × 0.0261.14 × 10 −3 = 4532,

which is sufficiently turbulent to allow Re effects to be ignored. (iv) The tidal period can be calculated from the time scale (equation (9.12)): ( Time ) m l m ⎛ Z r ⎞ 1 −−−−−−−−−−−− = −− −−−− = −−−−− × ( 60 ) = 0.0129. ( Time ) r l r ⎝ Z m ⎠ 600

Therefore, Tidal period of model = 12.4 × 60 × 0.0129 = 9.6 min.

9.9

GROUNDWATER AND SEEPAGE MODELS

While the area of groundwater flow and its implications within soil mechanics, reservoir design and runoff predictions are outside the scope of this text, the development of groundwater and seepage modelling dependent upon both the fundamentals of dimensional analysis and similarity, and the solution of the Laplace equation introduced in Chapter 7, are of interest. Water flow through the small passages or pores that exist within soils is known as seepage flow. The prediction of the forces consequent upon such flows is important as they can be of sufficient magnitude to be destructive. Similarly the prediction of seepage flow rate following rainstorms is essential in the design of reservoir catchments, the estimation of well yields and the provision of land drainage. In the study of flow through porous or granular media it is usual to exclude the effect of capillary action and to concentrate upon gravity-driven flow. Under this constraint the seepage may be investigated through a dimensional analysis based around the following parameters: V = φ (ρ, d, µ, g, n, S),

(9.14)

where V is the seepage velocity based on the flow divided by the seepage area, n is the void ratio, which in real situations will vary across the flow zone, d is the assumed particle size, which may also be variable in ‘real’ situations, ρ and µ are the fluid density and viscosity, and S is the hydraulic gradient driving the flow, −∂h∂x. A standard analysis yields the following group relationships: SgdV 2 = φ′(n, Re).

(9.15)

Manipulating equation (9.15) at low Reynolds numbers when inertial forces are negligible reduces to the Darcy law V = (ρgd 2µ) (1φ1n) S = −k∂h∂x,

(9.16)

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where k is the coefficient of permeability and is dependent upon soil porosity, particle size and orientation and the level of saturation. Equation (9.16) is only valid at Re values below 1, i.e. laminar filtration. When inertial forces cannot be ignored, e.g. for flow through coarse sand or gravel, the seepage flow may be non-linear. Again if viscous forces are ignored then equation (9.15) may be expressed as V 2 = Sgd φ2(n) = −C∂h∂x,

(9.17)

where C is dependent on Reynolds number. Equation (9.17) is valid under turbulent conditions at Re values above 104. Empirical results due to Yalin lead to an alternative formulation of equation (9.15): Sgρd(2ρV 2) = (1n6) (0.01 + 1Re).

(9.18)

Equation (9.15) identifies the dynamic similarity requirements for a model to determine seepage flows. The coefficient of permeability can be determined through laboratory tests under a fixed head difference ∆h, where the flow Q passing through a layer of thickness L and cross-section area A determines k as k = LQ(A∆h).

(9.19)

Alternatively in site operations the value of k is best determined through well pumping where it is assumed that the radial flow to the well identifies a k value as k = [Qln(r2r1)][π ( h 22 – h 21 ) ]

(9.20)

where h1,2 refers to the water table elevation above an assumed impermeable strata observed at radial distances r1,2 from the abstraction point under steady flow conditions in an unconfined hom*ogeneous layer. While both site measurement and model testing are options in the assessment of groundwater flows it is also attractive to consider numerical modelling of the flow by reference to the application of the stream function and velocity potential theory introduced in Chapter 7. The Laplace equations (7.26) and (7.27) apply to the flow conditions in a two-dimensional flow element in a soil of known permeability. As discussed in Chapter 7 the solution of the Laplace equations allows lines of constant stream function and velocity potential to be plotted in an x–y plane. The constant stream function lines represent streamlines, while equipotential lines define the distribution of pressure throughout the flow field. From the Darcy law for seepage flow the velocity potential may be expressed as Φ = −kh,

(9.21)

while the volumetric flow may be determined from the flow area represented by the distance between adjacent streamlines. The resultant mesh of equipotential and streamlines is known as a flow net, as discussed in Section 7.5. Returning to the well abstraction rate measurement of permeability it will be appreciated that the flow streamlines will be radial to the well while the lines of equipotential will form concentric circles, centred on the well.

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FIGURE 9.4 Intersection points for steam function values on grid represeting seepage flow zone below dam

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For flow through a porous medium assume for a unit thickness that adjacent streamlines are separated by a distance b, while adjacent lines of equipotential, differing by a head difference ∆h, are separated by a distance L. Thus the local pressure gradient is ∆hL and the flow is b × unit thickness. Hence from Darcy’s law ∆q = −bk∆hL

(9.22)

allowing a determination of the seepage flow. Section 7.5 details the steps necessary to develop flow nets and defines the significance of changes to the net mesh size as flow velocity changes. These processes will not be repeated here; however, it is useful to develop a numerical approach to the derivation of appropriate flow nets using the finite difference techniques, already introduced in Chapter 5, to solve the Laplace equations, drawing also upon the discussion of the importance of defining network boundary conditions and calculation step size. A general description will be presented followed by a particular example. The groundwater flow may be considered to occur in an x–y plane bounded by groundair or groundwater boundaries, or by boundaries representing impenetrable layers or assumed boundaries so remote from the source of the groundwater flow that the boundary may be considered impenetrable. The x–y zone may be considered to be represented by a grid of side dimensions ∆x, ∆y, so that any zone consists of a network of nodal points, some at the boundaries to the zone and some internal to it – as shown in Fig. 9.4. At each internal node, i.e. nodes not lying on a boundary, the Laplace equation for stream function, equation (7.26), must be satisfied:

∂ 2 ψ∂x 2 + ∂ 2ψ∂y 2 = 0.

(9.23)

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As introduced in Chapter 5, this equation may be expressed in finite difference form through the application of Taylor’s series, as the summation of the second order forwardbackward forms of the truncated Taylor series may be used to determine the second differential of variable ψ at x0 and y0 within the grid. Hence in the x direction:

ψ ( x 0 + ∆x ) – 2 ψ ( x 0 ) + ψ ( x 0 – ∆x ) -, ψ ″ ( x 0 ) = – ----------------------------------------------------------------------------------∆x 2

(9.24)

and similarly in the y direction

ψ ( y 0 + ∆y ) – 2 ψ ( y 0 ) + ψ ( y 0 – ∆y ) -. ψ ″ ( y 0 ) = – ---------------------------------------------------------------------------------∆y 2

(9.25)

As suggested in Chapter 7 it is appropriate to set values of ∆x and ∆y equal to unity so that substituting for the second differentials of stream function in the Laplace equation yields the following expression for the stream function at the internal node of interest, located at x0, y0: [ψ ( x 0 – ∆x ) + ψ ( x 0 + ∆x ) + ψ ( y 0 – ∆y ) + ψ ( y 0 + ∆y ) ] ψ x ,y = ----------------------------------------------------------------------------------------------------------------------------------- . 4 0

(9.26)

Clearly an identical relationship may be obtained for velocity potential. Boundaries formed by an impenetrable layer are clearly represented by zero flow streamlines while those between ground and air or ground and water are more difficult to define. However, it is clear that both of these may be represented by lines of equipotential, or velocity potential, as the pressure is constant. Also it is known from Chapter 7 that the streamlines, or lines of constant stream function, must intersect these boundaries at right angles: hence it is possible to represent these boundaries as the intersection between the zone of interest and a dummy symmetrical zone, see Fig. 9.4. Values of stream function at the nodes forming the boundary may then be determined from equation (9.26) with the values at the ‘imaginary’ node y0 + ∆y and ‘real’ node y0 − ∆y taken as equal. It is clearly possible to develop and write a simple computer program based on the discussion above to simulate the groundwater flow beneath a dam of arbitrary thickness built upon soil subject to seepage but contained within an impenetrable layer that extends at a constant depth to points remote from the dam walls. Values of stream function are assigned to the boundaries to the zone, and all internal nodes are assigned an arbitrary value. Equation (9.26) is used systematically to determine the appropriate nodal values on a square mesh until the difference between successive approximations is at an acceptable level. The solid boundary and impenetrable layer boundary values are retained throughout. The values at the groundwater and groundair interfaces are determined from equation (9.26), with the ‘dummy’ node having the same value as the corresponding node ∆y below the interface. A simple program may then be written to determine the intersection points on the grid for any series of stream function values; these values may then be sorted sequentially to allow graphical representation, Fig. 9.4. Once the streamlines are established it is traditional to draw onto the figure the location of lines of equipotential using two simple rules, namely that the equipotential lines cross the streamlines at right angles, and the zones enclosed by the intersection

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FIGURE 9.5 Streamlines representing seepage flow below dam with sheet piling added

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of adjacent streamlines and lines of equipotential are curvilinear squares. As it is assumed in the development of flow nets, see Chapter 7, that ∆Ψ = ∆Φ, it is possible to draw the lines of equipotential and hence to determine the number of equipotential drops along the streamline network. Taken together with the number of ‘flow passages’, defined as the number of zones between adjacent streamlines, a calculation of total seepage flow q may be undertaken, as from the Darcy law q = kHNf Nd,

(9.27)

where k is the coefficient of permeability, H is the pressure head difference across the system, Nf is the number of flow passages and Nd is the number of equipotential head drops estimated from drawing in the curvilinear squares between stream function and equipotential lines. Many of the criteria established in the discussion of flow modelling in Chapter 5 come into play in this example. The choice of grid size initially determines the accuracy of the stream function array. Numerical methods improve on the accuracy of the stream function array from which the streamlines may be plotted, and clearly a user graphical interface to introduce lines of equipotential would be advantageous. Figure 9.4 illustrates the flow under a simple dam. If a sheet pile curtain wall is introduced at the dam leading edge then the streamline contours change, as illustrated by Fig. 9.5. Here the sheet pile is included in the boundary representing the undersurface of the dam, namely a stream function value of unity. The resulting streamlines demonstrate the lack of symmetry to be expected. The inclusion of groundwater seepage modelling, even at this level, illustrates many of the advantages of numerical modelling as well as some of the pitfalls. The necessity to determine a suitable grid mesh is important, as is the clear definition of the boundary conditions to be used. A suitable computer model is presented below.

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9.10 COMPUTER PROGRAM GROUND, THE SIMULATION OF GROUNDWATER SEEPAGE Program GROUND predicts the seepage flow within a sub-surface section by solving the Laplace equation numerically for the stream function values within a predetermined grid. The program requires data concerning the grid size in the form of the limits of a double subscript array as well as the location of boundary surfaces where the stream function is set to a constant value. The data input comprises: Number of nodes in x direction, 1 to N. Number of nodes in y direction, 1 to M. Number of sections considered as making up a ground to air or ground to water interface. For each section input constant stream function value. Range of nodes to be held constant on upper surface of section, input first and last. Number of other nodes to be held constant. For each node to be held constant input x, y coordinates and constant value of stream function. The program generates data files defining the stream function value determined at each node.

9.10.1 Application example Reconsider the case illustrated in Fig. 9.4 if a vertical pile is placed at node 10 and extending down to node 6. The required data requested by the program consists of the following responses to screen questions: Grid has 19 ‘x’ direction nodes and nine ‘y’ direction nodes (I = 1–19 and J = 1–9). The surface is made up of three sub-sections: node 2–7, soilwater interface, initial value 0.5; node 8–10, base dam, initial value 1.0; node 11–18, soilair interface, initial value 0.5. Surface nodes to be held constant, 8–10. Five other nodes are to be held constant: x = 10, y = 2, initial value = 10, 3, 10, 4, 10, 5, 10, 6,

1.0 1.0 1.0 1.0 1.0

Note that the program sets the boundary nodes (1, 1–9), (19, 1–9) and (1–19, 9) to a zero unchanging value automatically. Figure 9.5 illustrates the modified stream function map.

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9.10.2 Additional investigations using GROUND The simulation may be used to investigate the effect of larger underground solid boundaries by extending the example given.

9.11 POLLUTION DISPERSION MODELLING, OUTFALL EFFLUENT AND STACK PLUMES Environmental concerns have led to increased interest in the modelling of pollution dispersion, either in applications such as sewage treatment outfalls or in airborne pollution. These studies are beyond the scope of the current text; however, it is useful to review some of the modelling considerations that make this a highly interesting area of research. In the case of effluent dispersion from sewer outfalls there are a number of distinct mechanisms and zones of application of the expected dimensionless groupings, such as the Reynolds and Froude numbers, with as might be expected some groups predominating within each zone. Figure 9.6 identifies these zones, based on the classification proposed by Ackers and Jaffrey in 1972, for a submerged effluent outfall in a tidal region. Sequentially from the outfall discharge the following zones may be considered: 1.

2.

3.

FIGURE 9.6 Dispersion of an effluent jet and identification of the relevant zones to be modelled, after Ackers and Jaffrey (1972)

Initially turbulent entrainment predominates, density differences are not important and the mechanism is governed by the discharge jet momentum. Successful modelling relies on geometric similarity, Reynolds number values above 2000 to avoid viscous effects and the Froude law. The effluent discharge jet level within the recipient flow will depend upon buoyancy effects, with mixing at the boundaries being dependent upon turbulence levels in the flow. The governing conditions again include geometric similarity and high Reynolds number but the applicable Froude number is now identified as the Froude densimetric number, defined as V( gh∆ ρρ)0.5. Once the jet breaks the surface of the recipient flow the resulting convective spread depends upon the density differences between the point at which the jet breaks the surface and the recipient flow. The densimetric Froude number is again the predominant term, with the characteristic length defined as the thickness of the buoyant layer. In addition Barr (1963) concludes that the model must be

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4.

5. 6.

subject to a limiting value of the densimetric Reynolds number, defined as (g∆ ρρ)0.5H 2.5(Lν ) 150, where L is the distance travelled by the convective effluent spread at a velocity that satisfies the Froude law – a restriction that leads to a vertically distorted model, Section 9.8. The modelling of the subsequent mass transport of the effluent via tidal or other currents requires the standard open channel conditions, namely Froude law, high Reynolds number and appropriate friction loss representation. Geometric similarity is required for the general dispersion of the effluent due to turbulence interaction. An undistorted model is appropriate. Finally, it may be necessary to model the rate of heat loss from the spreading effluent. Heat loss depends on area, time, temperature difference and a surface heat transfer coefficient, while temperature drop depends on mass and specific heat. Imposing equal model and prototype air temperatures leads to equal fluid temperatures between the model and the prototype to allow equal temperature drop so that the Froude number-dictated timescale leads to a distorted model scale.

To summarize these sequential mechanisms it may be seen that zones 1, 2 and 5 require an undistorted scale model, while zones 3 and 6 require a distorted model, as explained for river and harbour models in Section 9.8. Zone 4 may be undertaken with either. The actual bacterial and biological decay, in practice, is not normally considered. Overall the modelling of effluent outfalls has inherent difficulties, requiring at least two models with an interface. In addition, modelling the dispersion and diffusion of the effluent is problematic. While computational fluid dynamics offers apparent alternative approaches to this problem it is imperative to recall the importance of the definition of boundary conditions, no easier mathematically than described above through the zones identified by Ackers and Jaffrey. Analytical methods for the prediction of dispersion of pollution from gaseous plumes are based on the evaluation of the rise of the plume due to buoyancy and momentum and the treatment of the plume as an elevated source of non-buoyant pollution, effectively the model used for the effluent outfall, Fig. 9.6. However, analytical solutions for the effect of downwash, local terrain and temperature stratification are more problematic. Wind tunnel applications linked to the identification of the appropriate similarity conditions can contribute in these areas. The similarity constraints may be defined as: Geometric similarity, a scale ratio Froude number, Density ratio, Velocity scale, Reynolds number at the stack discharge,

Dm Dp, (DgU 2)m = (DgU 2)p ( ρs ρa )m = (ρs ρa )p (Ws U)m = (Ws U)p (DWs νs)m = (DWs νs)p

where D is the stack diameter, U is the ambient mean wind velocity, ρs and ρa are the stack discharge and atmospheric air densities and Ws is the jet velocity on exit from the stack. From these similarity considerations geometric scales are usually in the region 1400 to 11000; velocity scales vary from 120 to 130, resulting in scale velocities for winds up to 20 m s−1 of as low as 0.15 to 1.0 m s−1, which is very low and leads to modelling difficulties. Approximate solutions to deal with the low air speeds involve relaxing the density criterion and introducing the densimetric Froude

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number, already referred to in the discussion of effluent outfall studies, defined as Frd = ( ρs − ρa )Dg( ρaU 2). Use of a gas such as helium to represent the plume dispersion increases the velocity scale by approximately 50 per cent, thus making wind tunnel modelling practical. Accurate Reynolds number similarity in the stack is not possible, particularly with helium, as the model stack internal flow remains laminar while in practice the stack discharge will be turbulent. This exaggerates the plume exit velocity and momentum. Generally the problem of similar Reynolds number has little effect on the modelling of the exhaust as buoyancy forces predominate; however, it is of importance in the study of any downwash effects as the initial trajectory of the plume is momentum dominated. Maintenance of a densimetric Froude number leads to a correct relative scaling of the plume buoyancy and inertia forces. Relaxing the density ratio criterion results in differing mixing rates between the model and prototype and precludes the consistent scaling of the plume properties. The plume dispersion has been found to be highly sensitive to the presence of other buildings and local topography. The influence of building wakes in generating downwash forces on the plume has important consequences for the dispersion of stack plumes, leading to severe concerns as to the degree of ground level pollution. Figure 9.7 illustrates the likely effect of building wake on the dispersion of a stack plume and the effect of increasing stack height to limit the downwash experienced. In the first case, Fig. 9.7(a), the stack is relatively short compared with the height of the building. In this case the pollution is entrained within the wake and pollution levels are high downstream of the building, only decreasing due to natural buoyancy further downstream. The interaction of the dispersing plume with the downstream buildings has also been the subject of computational analysis, as illustrated by Fig. 5.21. In comparison increasing the stack height so that the plume is not entrained by the wake results in a naturally dispersing plume that limits ground level pollution. In terms of the modelling of gaseous plume dispersion in a wind tunnel the same fundamental groupings apply as for the effluent outfall case. The exact modelling of the stack discharge velocity will only become important if downwash is a consideration. The dispersion of the plume is highly dependent upon entrainment and the presence of adjacent buildings and topographical features. Modelling the density ratio FIGURE 9.7(a) Plume dispersion in presence of building wake-generated downwash, illustrating high pollution concentrations at ground level

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FIGURE 9.7(b) Plume dispersion with an increased height of stack effectively removing plume from the effect of the building wakegenerated downwash, illustrating lower pollution concentrations at ground level

correctly is important in the prediction of eventual dispersion and approximations, or a failure to do so adequately will lead to an under-estimate of the ground level pollution levels. The similarities between the air- and water-based examples chosen are self-evident, despite the obvious differences in fluid densities and the application sites.

9.12 POLLUTANT DISPERSION IN ONE-DIMENSIONAL STEADY UNIFORM FLOW The consequence of an insertion of a contaminant into an open channel flow, or into any bounded flow, will be that the local contamination will decrease downstream due to two identifiable flow processes. This dispersion will be dependent upon both advection – the contaminant is swept along with the flow – and diffusion – the contaminant will mix in all three dimensions due to the local action of flow turbulence. Figure 9.8 illustrates these processes. Contamination mass conservation may be described by a relationship of the form

∂C ∂P ------- + ------- + kC = 0, ∂t ∂x

(9.28)

where C is the local contaminant concentration, P is its flux and k is a rate coefficient included to allow for any modification to the contaminant during transport and is dependent upon concentration. Fick’s law, governing the conservation of contaminant flux, applied to the transport of a pollutant in either a quiescent fluid field or relative to the uniform motion of the fluid states that the transfer rate P of the pollutant per unit area normal to the flow direction is dependent upon the coefficient of molecular diffusion and the local concentration gradient, so that

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FIGURE 9.8 Pollutant dispersion due to advection and diffusion in steady uniform flow

∂C P = – Dm ------- . ∂x Relative to a flow of contaminant concentration C in a uniform stream of velocity U, Fick’s law may be re-cast as

∂C P – UC + D m ------- = 0. ∂x

(9.29)

It should be recognized that diffusion due to turbulence is several orders of magnitude greater than molecular diffusion and therefore molecular diffusion is normally neglected. However, by analogy, it is possible to apply equation (9.28) to turbulent mixing by replacing Dm with an appropriate and often empirical coefficient of turbulent diffusion or dispersion, D. It is stressed that this coefficient is often difficult to evaluate and is often determined by site trials to allow a degree of calibration for a particular stream (Wallis and Manson, 2003). It may well vary along a particular river or stream. In the case of steady uniform flow in a channel or duct, the analysis may be reduced to the one-dimensional problem illustrated below as an introduction to contaminant dispersion in a free surface flow. Combination of equations (9.28) and (9.29) yields the normally accepted combined advection–diffusion equation, including the optional first order rate term k,

∂C ∂ C ∂C ------- + U ------- – D ----------2- + kC = 0. ∂x ∂t ∂x 2

(9.30)

Numerical modelling of the combined process normally differentiates between the advection and diffusion process by splitting the process into two phases, namely advection described by

∂C ∂C ------- + U ------- = 0, ∂x ∂t

(9.31)

and diffusion, for a uniform cross-section channel, described by

∂ ∂C ∂C ------- = ------ ⎛D ------- ⎞ . ∂x ⎝ ∂x ⎠ ∂t

(9.32)

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D is known as the dispersion coefficient and may vary with location along a real stream or river. Measurements available in the literature show a wide variation dependent upon local conditions, ranging from 1 to 1000 m2 s−1. While prediction and measurement techniques have improved, these are still not wholly satisfactory (Wallis and Manson, 2003). Finite difference solutions may be developed for both equations (9.31) and (9.32). In the advection case a forward in time, central in space model would result in a relationship t + ∆t

t

t

t

C i +1/2 + C i −1/2 C i + Ci ------------------------- + U --------------------------------- = 0, ∆t ∆t

(9.33)

where the C it+12 + C it− 12 terms are found by interpolation between nodes i, i − 1 and i + 1. Interpolation between nodes at i, i + 1 and i + 2 can lead to rounding errors, as conditions at each node at a particular time are assumed to affect conditions midreach at that time, a problem addressed in Chapter 21 in the development of method of characteristics solutions to free surface flow problems. The choice of interpolation technique determines the accuracy of the model (Leonard, 1979), presented in order of increasing accuracy: 1.

The explicit upwind scheme, first-order accurate in space where C it−12 = C it−1

2.

The explicit central scheme, second-order accurate in space where t

C 3.

C it+12 = C it.

and

t i −12

t

t

C i −1 + C i = ---------------------2

and C

t i + 12

t

C i+1 + C i = ---------------------- . 2

The so-called explicit QUICK scheme, third-order accurate in space where t

t

t

t

t

C i −1 + C i C i −2 – 2C i −1 + C i C it−12 = ---------------------- – ----------------------------------------2 8 t

t

t

t

and

t

C i+1 + C i C i −1 – 2C i −1 + C i+1 C it+12 = ---------------------- – --------------------------------------------2 8 Schemes 1–3 are all first-order accurate in time. It will be appreciated that equation (9.31) may be reduced to dCdt = 0 for pure advection, effectively implying that concentration remains constant along a characteristic line drawn in the x–t plane representing the advection process in one-dimensional steady uniform free surface flow. In representing the diffusion equation, a finite difference representation forward in time and central in distance would yield t +∆t

t

t

t

t

C i −1 – 2C i + C i+1 C i + Ci -. ------------------------- = D ---------------------------------------2 ∆t ∆x

(9.34)

This reduces to Cit +∆t = α C it−1 + (1 − 2α)C it−1 + α C it− 1, where the diffusion number α = D∆t∆x must be < 0.5 for stability. This scheme, commonly known as the FTCS scheme, is second-order accurate in space and first-order accurate in time.

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9.12

Pollutant dispersion in one-dimensional steady uniform flow

317

Advection–diffusion numerical schemes rely on determining the advection concentration at the next time step and then determining the diffusion component to yield the overall contamination variation downstream in both time and space. Numerical difficulties include stability, the need to minimize numerical dispersion, and the estimation of a value for the diffusion coefficient. Stability is governed by the Courant number, defined in this context as C r = U∆xD, being equal to or less than unity. There is also the need to recognize that there will be an initial stage following contaminant injection, whether intentional or inadvertent, where the assumptions as to a fully mixed start condition will not be met (Fischer, 1967). This period, referred to as the mixing or advective period, has been suggested as having a duration (Burguete and Garcia-Navarro, 2002), defined as 2

b T A = t′ ----- , kz where b is the channel mean width, kz is the transverse mixing coefficient and t′ is taken as 0.1 for a mid-channel injection and 0.4 for a bank-side injection, yielding an advective zone length of LA = UTA. LA is typically 400b (Graf and Altinaker, 2000) for a bank-side injection and 100b to 300b for a mid-stream injection (Rutherford, 1994). An alternative to the established numerical techniques was proposed by Wiggert (2004). Referring to equations (9.28) and (9.29), the addition of a time derivative with respect to P yields an expression

∂P ∂C σ ------- + P – UC + D m ------- = 0 ∂t ∂x

(9.35)

where σ is defined as a temporal multiplier of a form previously used successfully in the development of solutions in the areas of natural gas transients, transient groundwater flows and unsteady heat transfer. Equations (9.28) and (9.35) may then be combined by the use of a method of characteristics approach (MoC), see Chapter 21, so that Equation (9.28) + λ Equation (9.35) = 0 dP dc ------ + λσ ------- + λ ( P – UC ) + kC = 0, dt dt provided that 1 dx ------ = λ D = -------- , Dσ dt 1 so that λ = ± ----------D σ and

D dx ------ = ± ----. σ dt

This leads to a pair of characteristic equations, dc ------ ± dt

σ dP

1

- ± ----------- ( P – UC ) + kC = 0, ⎛⎝ ---D-⎞⎠ -----dt D σ

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318

Chapter 9

Similarity

FIGURE 9.9 Characteristics drawn in an x–t plane for contaminant dispersion in steady uniform free surface flow

so that, for the x–t grid illustrated in Figure 9.9, the C + characteristic becomes CP – CR +

σ

1

- 0.5 ( P + P ---D- ( P – P ) + ---------D σ P

R

P

R

– UC P – UC R )∆t

+ 0.5k(CP + CR)∆t = 0 D dx when ------ = + ---- , and the C − characteristic becomes σ dt

CP – CS –

σ

1

- 0.5 ( P + P – UC ---D- ( P – P ) – ---------D σ P

S

P

S

P

– UC S )∆t

+ 0.5k(CP + CS)∆t = 0 D dx when ------ = − ----. σ dt In common with other MoC applications, boundary equations are required at entry, x = 0, and exit, x = channel length. The entry condition may be represented by a C = φ (t) relationship. (Note that this would assume stream width mixing.) The downstream boundary would be provided by setting either the local contamination gradient with respect to distance, or the local contamination concentration, to zero. (Note that this exit boundary should be set remote from any section where a prediction is required.) The Courant number required to satisfy stability in the MoC solution is effectively given by

D ∆t

- = 1, ---σ- -----∆x and is always satisfied. Figure 9.10 illustrates the attenuation of the input contamination profile as it is swept downstream for both a short and an extended input of contaminant. The downstream boundary is provided by an assumption of zero contamination gradient at the 10 km point. The dependence of the contaminant profile on both distance and initial dosing is demonstrated. It is stressed that this one-dimensional representation is subject to the limitations already outlined, in particular initial mixing distance and the difficulty in identifying a reasonable value for the dispersion coefficient.

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Summary of important equations and concepts

319

FIGURE 9.10 Contamination vs. time profiles at kilometre intervals along the stream following either a 0.25 or 1.0 hour contamination injection at x = 0

Concluding remarks Chapter 9 has introduced a range of applications of dimensional analysis and similarity in the design of fluid machinery, aerofoils, channels and harbours. These should be seen as merely examples of the application of similarity and dimensional analysis across the whole spectrum of fluid mechanics applications. While the examples given are necessarily constrained to fluid situations, the technique applies equally to flow conditions involving temperature change and heat transfer. The approach demonstrated will be returned to throughout the text, for example in the treatment of frictional losses in Chapter 10, the analysis of fluid machinery in Chapters 22 and 23, the study of free surface flows in Chapters 15 and 16 and elsewhere. In addition Chapter 9 has introduced both the concepts of groundwater flows and their modelling and aspects of pollution dispersion. The latter topic has been extended to include the initial modelling of contamination transport and dispersion in channel flows, a topic of concern in the event of flooding and inadvertent contamination of river flows. A thorough understanding of the application of dimensional analysis and similarity is essential to this text and to the general treatment of fluid mechanics.

Summary of important equations and concepts 1.

2.

3.

This chapter has defined the importance of both geometric and dynamic similarity. In particular it has reinforced the role of experience in the choice of defining dimensionless group where apparent incompatibilities arise. The zones of influence of the major dimensionless groupings, Reynolds, Froude and Mach, have been identified and the force ratios they represent defined, Section 9.2. The use of non-dimensional coefficients to define lift and drag and other forces has been introduced and the application of the similarity laws to a wide range of testing conditions introduced, from machinery to open channels and harbours; in the latter case the necessity to consider separate vertical and horizontal distance scales was discussed.

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320

Chapter 9

Similarity

4.

Numerical modelling drawing on the finite difference relationships introduced in Chapter 5 has been used to model both groundwater flows and contamination dispersion in free surface channel flows.

Further reading Allen, J. (1952). Scale Models in Hydraulic Engineering. Longman, London. Bain, D. C. et al. (1971). Wind Tunnels, An Aid to Engineering Structure Design. British Hydraulics Research Association, Cranfield. Bradshaw, P. (1964). Experimental Fluid Mechanics. Pergamon, Oxford. BS 848 Part I (1963). Fan Testing for General Purposes. Cermak, J. E. (ed.) (1979). Wind Engineering: Proceedings of the 5th International Conference, Fort Collins, Colorado Volumes 1 and 2, pp. 735–745. Kline, S. J. (1965). Similitude and Approximation Theory. McGraw-Hill, New York. Langhaar, H. L. (1951). Dimensional Analysis and Theory of Models. Wiley, New York. Novak, P. and Cabelka, J. (1981). Models in Hydraulic Engineering. Pitman, London. Sedov, L. I. (1959). Similitude and Dimensional Analysis in Mechanics. Academic Press, New York. Streeter, V. L. (ed.) (1961). Handbook of Fluid Dynamics. McGraw-Hill, New York. Wood, I. R., Bell, R. G. and Wilkinson, D. L. (1995). Ocean disposal of wastewater. Advanced Series on Ocean Engineering, Volume 8, World Scientific, London. Yallin, M. S. (1971). Theory of Hydraulic Models. Macmillan, London.

References Ackers, P. and Jaffrey, L. J. (1972). The application of hydraulic models to pollution studies. Symposium on Mathematical Modelling of Estuarine Pollution, Paper 16, Water Pollution Research Laboratory, Stevenage. Barr, D. I. H. (1963, 1967). Densimetric exchange flow in rectangular channels. La Houille Blanche, pp. 739– 66 (1963); pp. 619 – 32 (1967). Burguete, J. and Garcia-Navarro, P. (2002). Semi-Lagrangian conservative schemes vs. Eulerian schemes to solve advection in river flow transport. In: Proceedings of the 5th International Conference on Hydroinformatics, Cardiff, UK, paper no. 338. Fischer, H. B. (1967). The mechanics of dispersion in natural streams. Journal of Hydraulics, ASCE, 93, 187–216. Graf, W. H. and Altinaker, M. (2000). Hydraulique Fluvial Presses. Polytechniques et Universitaires Romandes, Lausanne, Switzerland. Leonard, B. P. (1979). A stable and accurate convective modeling procedure based on quadrartic upstream interpolation. Computational Methods in Applied Mechanics and Engineering, 19, 59 –98. Rutherford, J. C. (1994) River Mixing. J. Wiley & Sons, Chichester, UK. Wallis, S. G. and Manson, J. R. (2003). Methods for predicting dispersion coefficients in rivers. Proceedings of the ICE, Water Management, 157 (WM3), 131–141. Wiggert, D. C. (2004). Numerical solution of advection-diffusion using hyperbolic equations. 9th International Conference Pressure Surges, bHr Group, Chester, UK, March, Vol. 2, pp. 527–38.

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Problems

321

Problems 9.1 Water at 20 °C flows at 4 m s−1 in a 200 mm smooth pipe. Calculate the air velocity in a 100 mm pipe at 40 °C if the two flows are dynamically similar. [135.0 m s−1] 9.2 A spherical balloon to be used in air at 20 °C is tested by towing a 13 model submerged in a water tank. If the model is 1 m in diameter and the drag force is measured as 200 N at a model speed of 1.2 m s−1, what would be the expected prototype drag if the water temperature was 15 °C and dynamic similarity is assumed? [42.2 N] 9.3 Determine the relationship between model and prototype kinematic viscosity if both Reynolds number and Froude number are to be satisfied. [Linear scale to power 32] 9.4 A large venturi meter is calibrated by means of a 110 scale model using the same fluid as the prototype. Calculate the discharge ratio between model and prototype for dynamic similarity. [110] 9.5 The velocity at a point in a model spillway for a dam is 1 m s−1. For a scale of 110 calculate the corresponding velocity in the prototype. [3.16 m s−1] 9.6 If the scale ratio between a model spillway and its prototype is 125 what velocity and discharge ratio should apply between model and prototype? If the prototype discharge is 3000 m3 s−1 what is the model discharge? [15, 13125, 0.96 m3 s−1]

9.12 A model turbine employs 2 m3 s−1 water flow when simulating a full-scale prototype designed to be served by a 15 m3 s−1 flow. If the scale is 15 calculate the speed ratio and the shaft-delivered power ratio. [16.66, 1.48] 9.13 In a test on a centrifugal fan it was found that the discharge was 2.75 m3 s−1 and the total pressure 63.5 mm water column. The shaft power was 1.7 kW. If a geometrically similar fan having dimensions 25 per cent smaller but having twice the rotational speed was used, calculate the output, pressure generated and shaft power required. The air conditions are the same in both cases. [2.32 m3 s−1, 142.9 mm H2O, 3.2 kW] 9.14 A 15 scale model of the piping system of a water pumping station is to be tested to determine overall pressure losses. Air at 27 °C, 100 kN m−2 absolute, is available. For a prototype velocity of 0.45 m s−1 in a 4.25 m diameter duct section with water at 15 °C, determine the air velocity and quantity needed to model the situation. [31.39 m s−1, 17.83 m3 s−1] 9.15 The torque delivered by a water turbine depends upon discharge Q, head H, density ρ, angular velocity ω and efficiency η. Determine the form of the equation for torque. H3 ρ gH 4 ⋅ f ⎛ ω −−−−, η⎞ ⎝ Q ⎠

9.7 A 15 scale model of a missile has a drag coefficient of 3 at Mach number 2. What would the modelprototype drag ratio be in air at the same temperature and one-third the density for the prototype at the same Mach number? [125]

9.16 A 20 km length of river is to be modelled in a laboratory having only 12.5 m of available length. The river discharge is known to be in the range 400 – 500 m3 s−1 and the average length and width are 3.5 and 55 m, respectively. Propose suitable scales. [11600, 1100]

9.8 A ship model, scale 150, has a wave resistance of 30 N at its design speed. Calculate the prototype wave resistance. [3750 kN]

9.17 If the model in Problem 9.16 is tidal calculate the tidal period on the model. [4.65 min]

9.9 A ship having a length of 200 m is to be propelled at 25 km h−1. Calculate the prototype Froude number and the scale of a model to be towed at 1.25 m s−1. [0.157, 130.8] 9.10 A fan running at 8 rev s−1 delivers 2.66 m3 s−1 at a fan total pressure of 418 N m−2, the air having a temperature of 0 °C and 101.325 kN m−2 pressure. Given that the fan efficiency is 69 per cent, calculate the air quantity delivered, the fan total pressure and the fan power when the air temperature is increased to 60 °C and the barometric pressure [2.66 m3 s−1, 322 N m−2, 1.24 kW] falls to 95 kN m−2. 9.11 An axial flow water pump is to deliver 15 m3 s−1 against a head of 20 m water. Calculate the air flow delivery rate and pressure rise for a 13 scale model using air at 1.3 kg m−3 density if the model and prototype are driven at the same speed. [0.55 m3 s−1, 2.90 mm H2O]

9.18 Develop and write a simple computer program to simulate the groundwater flow beneath a dam of arbitrary thickness built upon soil subject to seepage but contained within an impenetrable layer that extends at a constant depth to points remote from the dam walls. Utilize the program to draw the groundwater flow streamlines beneath the structure. 9.19 Use the program listing provided in the solution to Problem 9.18 to investigate the effect of grid size and further investigate the streamline formation for the following cases: (a) rectangular seepage zone beneath a dam, cf. Fig. 9.5; (b) rectangular seepage zone beneath a dam with a centrally placed sheet pile; (c) seepage flow beneath a dam where the rectangular zone is stepped in from the left-hand side.

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Part IV

Behaviour of Real Fluids 10 Laminar and Turbulent Flows in Bounded Systems 324

12 Incompressible Flow around a Body 394

11 Boundary Layer 366

13 Compressible Flow around a Body 438

Weir and spillway flow from a reservoir, photo courtesy of West of Scotland Water

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In earlier chapters, the basic equations of continuity, energy and momentum were introduced and applied to fluid flow cases where the assumption of frictionless flow was made. The analysis presented in the following chapters will introduce concepts necessary to extend the previous work to real fluids in which viscosity is accepted, and hence leads to situations where frictional effects cannot be ignored. The concept of Reynolds number as an indication of flow type will be used extensively, and the fluid boundary layer, already introduced in Chapter 5, which lies between the free stream and the surface passed by the fluid and in which all the flow resistance is concentrated, will be expanded. It will be necessary to distinguish between two different situations: namely, that in which the fluid moves inside a pipe or duct or in a channel so that it is guided by a boundary surrounding the fluid; and that in which the fluid flows around a solid body. In the first case, the flow is sometimes referred to as bounded flow and in the second case as external flow. Examples of the latter are fluid flow around a bridge pier or flow of wind around a house. Also to this category belong all the cases of solid objects moving through a stationary fluid, because it is the relative velocity between the fluid and the object that really matters. Thus an aeroplane in flight or a sailing ship are examples of such situations. The bounded flow and the external flow around a body are both governed by the same basic principles. In all cases the fluid velocity at the boundary, i.e. where the fluid meets the solid surface, is equal to zero. This condition is sometimes referred to as the ‘no-slip’ condition. The velocity then increases with distance perpendicular to the boundary, the rate of increase being governed by the particular law applicable to the type of flow, which may be either laminar or turbulent. In the external flow, the fluid velocity at some distance away from the boundary reaches a free

stream velocity, which is the velocity of undisturbed (by the solid object) fluid, usually taken some distance upstream of the object. Thus for a bridge pier the fluid velocity at its surface will be zero and will increase away from it until it reaches the velocity of the undisturbed river. For a ship the velocity of the fluid at its surface will be equal to that of the ship and will diminish down to zero at some distance away from the ship as the water of the sea may be taken as stationary. For bounded flow, such as in a pipe, the velocity of the fluid is zero at the wall and increases to a maximum at the centre of the pipe where the boundary layers, starting from the diametrically opposite points on the wall, meet. In all the above cases, there is a velocity gradient and, thus, shear stresses in the fluid. In order to maintain flow this shear stress must also be maintained and this can only be achieved by additional forces doing work on the fluid. In other words, there must be a continuous supply of energy for the flow to exist. This energy, supplied solely to maintain flow in a bounded system, is usually expressed per unit weight of the fluid flowing and thus is in units of fluid head. Energy supplied per unit time −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Weight of fluid flowing Force × DistanceTime = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Specific weight × Discharge pQ p pa × st pav = −−−−−−−−−−− = −−−−−− = −−−−−− = −−− = h. ρ gQ ρ gQ ρ gQ ρ g This head (or energy) is considered as lost because it cannot be used for any other purpose than to maintain flow and hence it is called head loss. Such losses will be discussed in detail in Chapters 10 and 11. In external flows, the forces required to maintain the velocity gradient in the boundary layer and energy dissipation in separation wakes are called the drag and will be discussed fully in Chapters 12 and 13. l l l

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Chapter 10

Laminar and Turbulent Flows in Bounded Systems 10.1

Incompressible, steady and uniform laminar flow between parallel plates

10.2

Incompressible, steady and uniform laminar flow in circular cross-section pipes

10.3

Incompressible, steady and uniform turbulent flow in bounded conduits

10.4

Incompressible, steady and uniform turbulent flow in circular cross-section pipes

Steady and uniform turbulent flow in open channels 10.6 Velocity distribution in turbulent, fully developed pipe flow 10.7 Velocity distribution in fully developed, turbulent flow in open channels 10.8 Separation losses in pipe flow 10.9 Significance of the Colebrook– White equation in pipe and duct design 10.10 Computer program CBW 10.5

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The flow of real fluids exhibits viscous effects. This chapter will identify these effects for both laminar and turbulent incompressible flow conditions. The relationships defining fluid friction will be developed and will be shown to be applicable to laminar, turbulent, free surface or full-bore flow situations, provided that due consideration is given to the appropriate value of flow friction factor. The Hagen–Poiseuille, Darcy and Chezy equations linking flow velocity to frictional loss will be developed. The chapter will draw on earlier material describing both the application of the momentum equation, in order to identify frictional forces, and dimensionless analysis and similarity, in order to identify the dependence of flow frictional losses on other flow parameters. The

empirical nature of frictional loss prediction under turbulent flow conditions will be stressed and both the Moody chart and the Colebrook–White equation methodologies for loss determination introduced. Losses arising from flow disturbance caused by changes of direction, changes in flow cross-section or interaction with flow control devices will be introduced and characterized as separation losses, quantifiable via empirical coefficients based upon the flow kinetic energy. Expressions defining the velocity distribution across both laminar and turbulent fully developed pipe flow will be demonstrated. The calculation of flow friction factor based on the Colebrook–White equation is introduced as a computer program. l l l

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10.1 INCOMPRESSIBLE, STEADY AND UNIFORM LAMINAR FLOW BETWEEN PARALLEL PLATES Consider first the case of steady laminar flow between inclined parallel plates, one of which is moving at a velocity U (Fig. 10.1) in the flow direction. It is required to calculate the velocity profile between the plates and hence the flow through the system.

FIGURE 10.1 Laminar flow between parallel plates

This flow condition may be analysed by application of the momentum equation to an element of the flow – ABCD in Fig. 10.1 – and by consideration of the constraints imposed on the flow by limiting the analysis to steady, uniform, laminar flow. The momentum equation may be stated as Resultant force in flow direction =

Rate of change of momentum in flow direction.

However, as the flow is restricted to the steady uniform case, then the acceleration is zero. (If the acceleration of the flow is described by the equation dC ∂ C ∂ C ∂ x −−− = −−− + −−− ⋅ −−− , dt ∂ t ∂ x ∂ t

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10.1

Incompressible, steady and uniform laminar flow between parallel plates

327

then for steady flow ∂ C∂ t is zero and for uniform flow ∂ C∂ x is zero: hence the zero value of d Cdt.) Thus, the resultant force acting on the fluid element ABCD is zero and the flow is in a state of equilibrium under the action of the forces illustrated. If it is assumed that the plates are sufficiently wide to make edge effects negligible, then the resultant force, in the flow direction, on the fluid element may be expressed, for unit width of plate, as dτ dp p δ y – ⎛ p + −−− δ x⎞ δ y + W sin θ – τδ x + ⎛ τ + −−− δ y⎞ δ x = 0, ⎝ ⎝ dy ⎠ dx ⎠

(10.1)

where p is the static pressure of the flow, τ is the shear stress, θ is the plate inclination and W = ρgδxδy per unit width. Therefore, dτ dp – −−− δ x δ y + W sin θ + −−− δ y δ x = 0. dy dx If z is the elevation of the system above some horizontal datum, then dz sin θ = – −−− dx and, hence, by substitution for W and sin θ, dz dτ dp – −−− δ x δ y + ρ g δ x δ y ⎛ – −−− ⎞ + −−− δ x δ y = 0 , ⎝ dx ⎠ dy dx so that dτ d −−− = −−− ( p + ρ gz ), dy dx

(10.2)

where ( p + ρgz) is the piezometric pressure, denoted by p*. As previously stated, Section 1.9.1, the shear stress in laminar flow may be expressed in terms of the fluid viscosity and the velocity gradient as du τ = µ −−− . dy

(10.3)

Hence, integrating equation (10.2) and substituting for τ yields an expression in terms of the velocity gradient: du d τ = µ −−− = y −−− ( p + ρ gz ) + C 1 . dy dx

(10.4)

This integration was possible as ( p + ρgz) is assumed to vary only in the x direction. Integration of equation (10.4) with respect to y will yield an equation for the velocity distribution between the plates in the form u = f ( y), in terms of fluid viscosity,

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piezometric head and two constants of integration that may be evaluated by consideration of the system boundary conditions at y = 0 and y = Y: C1 1 d y2 u = − −−− ( p + ρ gz ) −− + y −−− + C 2 . µ dx 2 µ

(10.5)

At the interface between the fluid and the plates at y = 0 and y = Y, the relative velocity of the fluid to the plate is zero, i.e. the condition of no slip. Thus, at y = 0 it follows that the fluid velocity u = 0 as this plate is itself stationary. At y = Y the fluid velocity relative to the plate is zero; however, as the plate is moving at a velocity U in the flow direction, the value of u at y = Y must similarly be u = U. Substituting these two boundary conditions in turn into equation (10.5) yields: y = 0, u = 0; therefore C2 = 0; U Y d y = Y, u = U; therefore C 1 = µ −− – −− −−− ( p + ρ gz ); Y 2 dx or y 1 d u = −−U – −−− −−− ( p + ρ gz ) (Yy – y 2 ). Y 2 µ dx

(10.6)

Equation (10.6) represents the velocity profile across the gap between the two plates, and is a general equation from which a number of restricted cases may be considered. For example, 1.

Horizontal plates with no movement of the upper plates, i.e. U = 0, sin θ = 0; hence dzdx = 0 and 1 dp u = – −−− −−− (Yy – y 2 ). 2 µ dx

(10.7)

Note that equation (10.7) represents a parabolic velocity profile, and the negative sign recognizes that dpdx itself will be negative as the pressure drops in the flow direction. 2.

Horizontal plates with upper plate motion: y 1 dp u = −−U – −−− −−− (Yy – y 2 ). Y 2 µ dx

(10.8)

Equation (10.8) indicates that fluid flow may occur even if there is no pressure gradient in the x direction, provided the plates are in motion. In this case u = (yY )U, a straight line velocity distribution. This phenomenon is known as Couette flow. The volume flow rate Q may be calculated for any of the above cases by integrating the expression for δQ, the flow through an element δ y of the plate separation and of unit width, between the system boundary at y = 0 and y = Y.

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10.1

Incompressible, steady and uniform laminar flow between parallel plates

329

Generally, δ Q = uδ y per unit width; hence Q=

Y

u dy.

y=0

For the general case, illustrated in Fig. 10.1, Q per unit width becomes Q=

Y

y=0

y 1 d −−U – −−− −−− ( p + ρ gz ) (Yy – y 2 ) dy Y 2 µ dx

U y2 Y 1 d y2 y3 Y = ⎛⎝ −− −− ⎞⎠ – −−− −−− ( p + ρ gz ) ⎛⎝Y −− – −− ⎞⎠ . 2 3 0 Y 2 0 2 µ dx Therefore, 1 d Y3 UY Q = −−−−− – −−− −−− ( p + ρ gz ) −−− . 6 2 µ dx 2

(10.9)

For flow between stationary horizontal plates this reduces to 1 dp Q = −−−−− −−− Y 3 per unit width. 12 µ dx

EXAMPLE 10.1

Laminar flow of a fluid of viscosity µ = 0.9 N s m−2 and density ρ = 1260 kg m−3 occurs between a pair of parallel plates of extensive width, inclined at 45° to the horizontal, the plates being 10 mm apart. The upper plate moves with a velocity 1.5 m s−1 relative to the lower plate and in a direction opposite to the fluid flow. Pressure gauges, mounted at two points 1 m vertically apart on the upper plate, record pressures of 250 kN m−2 and 80 kN m−2, respectively. Determine the velocity and shear stress distribution between the plates, the maximum flow velocity and the shear stress on the upper plate (Fig. 10.2).

Solution Flow direction from direction of pressure gradient. 1260 At (1), p1 + ρgz1 = 250 + 9.81 × 1.0 × −−−−−− = 262.36 kN m−2. 1000 At (2), p2 + ρgz2 = 80 kN m−2 FIGURE 10.2

(10.10)

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as z = 0 if datum taken at (2). Flow is down slope, upper plate moves ‘up’ slope. Pressure gradient dp* ( 262.36 – 80 ) −−−−− = – −−−−−−−−−−−−−−−−−−− = – 182.36 2 dx 1 ⋅ 2 p* = (p + ρgz) = −128.95 kN m−2 per metre, i.e. z = 1. From equation (10.6), U 1 dp* u = y −− – −−− −−−−− (Yy – y 2 ) , Y 2 µ dx where U = −1.5 m s−1, Y = 0.01 m and u is the local velocity at a point y above the lower plate. Thus the velocity profile is – 1.5 128.95 × 10 3 u = −−−−−−y + −−−−−−−−−−−−−−−−−− ( 0.01y – y 2 ) 2 × 0.9 0.01 = −150y + 716.4y − 71.64 × 103y 2, u = 566.4y − 71.64 × 103y 2. Shear stress distribution is given by du τ y = µ ⎛ −−− ⎞ , ⎝ dy ⎠y du −−− = 566.4 – 143.28 × 10 3 y, dy

τy = 509.76 − 128.95 × 103y; u max occurs where dudy = 0, y = 566.4 × 10−3143.28 = 0.395 × 10−2. Hence, umax = 566.4 × 0.003 95 − 71.64 × 103 × 0.003 952 = 2.24 + 1.117 = 3.36 m s−1. Shear stress on upper plate is given by du τ Y = µ ⎛ −−−⎞ = 509.76 – 128.95 × 10 3 × 0.01 = 0.78 kN m−2. ⎝ dy⎠ y=Y This is the fluid shear at the plate; hence, the shear force on the plate is 0.78 kN per unit area resisting plate motion.

Equation (10.10) may be applied to laminar flow between concentric cylinders, provided that the annulus is of small dimensions compared with the cylinder diameter.

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10.2

Incompressible, steady and uniform laminar flow in circular cross-section pipes

331

FIGURE 10.3 Leakage flow past a piston within a cylinder

An example of this case would involve the leakage past a piston within a cylinder, as shown in Fig. 10.3. Hence, the leakage flow becomes 1 p1 – p2 Q = −−−−− −−−−−−−−− ( ∆R ) 3 2 π R 1 , 12 µ l

(10.11)

where ∆R = R1 − R2 and is the pistoncylinder separation, and the total width of the ‘parallel plates’ is given by the piston circumference. In this case, it will be seen that plate width edge effects can be ignored as the ‘parallel plates’ are effectively continuous.

10.2 INCOMPRESSIBLE, STEADY AND UNIFORM LAMINAR FLOW IN CIRCULAR CROSS-SECTION PIPES Steady, uniform, laminar flow in a circular cross-section pipe or annulus may be treated in the same manner as described for laminar flow between parallel plates. The analysis rests on the same basic principles, namely the application of the momentum equation to an element of flow within the conduit; the application of the shear stress– velocity gradient relationship (10.3); and the knowledge of the flow condition at the pipe wall, which allows the constants of integration to be evaluated, namely the no-slip condition. Consider an annular element in the flow of internal radius r and radial thickness δ r, as shown in Fig. 10.4, in an inclined tube, of radius R, carrying a fluid under laminar flow conditions. Applying the momentum equation to the situation illustrated in Fig. 10.4 yields an expression

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FIGURE 10.4 Forces acting on an annular element in a laminar pipe flow situation

dp p2 π rδ r – ⎛ p + −−− δ x⎞ 2 π rδ r + τ 2 π r dx ⎝ dx ⎠ d – 2 π r τδ x + −−− ( 2 π rτ dx ) δ r + W sin θ = 0, dr

(10.12)

where p is the flow static pressure, W = mg is the element weight and τ is the shear stress at radius r. Owing to the assumption of steady uniform conditions, the flow acceleration is zero and, hence, the resultant force on the element is zero. Putting W = 2π rδ rδxρg and sin θ = −dzdx, where z is the elevation of the pipe above some horizontal datum, reduces expression (10.12) to dz dp 1 d – −−− – − −−− (rτ ) – ρ g −−− = 0 dx dx r dr by dividing by 2π rδrδx. Rearranging, 1 d d −−− ( p + ρ gz ) + − −−− ( rτ ) = 0. r dr dx

(10.13)

The term (p + ρgz) is the flow piezometric pressure and is independent of r, enabling equation (10.13) to be integrated with respect to r. Hence, r2 d −− −−− ( p + ρ gh ) + rτ + C 1 = 0. 2 dx If conditions at the pipe centreline are substituted into the above expression, then C1 = 0 as r = 0.

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The shear stress–velocity gradient expression of equation (10.3) may be employed in a modified form to take note of the direction of measurement of distance r from the centre of the pipe, rather than the use of y measured from the pipe wall; hence du du τ = µ −−− = – µ −−− dy dr

(10.14)

and, by substituting for τ, above, du r2 d −− −−− ( p + ρ gz ) = r µ −−− – C 1 2 dx dr and

C1 r d du = −−− −−− ( p + ρ gz ) + −−− dr. rµ 2 µ dx

Integrating with respect to r yields an expression for the velocity variation across the flow in terms of r and known system parameters: C1 r2 d u = −−− −−− ( p + ρ gz ) + −−− log e r + C 2 . 4 µ dx µ

(10.15)

Values of C1 and C2 may be evaluated from boundary conditions at r = 0 and r = R. At r = 0 it has been shown that C1 = 0. At r = R, i.e. at the pipe wall, the local flow velocity u is zero; hence, R2 d C 2 = – −−− −−− ( p + ρ gz ) 4 µ dx and

( R2 – r2 ) d u = – −−−−−−−−−−−− −−− ( p + ρ gz ). 4 µ dx

(10.16)

Equation (10.16) describes the variation of local fluid velocity u across the pipe and, from the form of the equation, this velocity profile may be seen to be parabolic. The negative sign is again present due to the fact that the pressure gradient will be negative in the flow direction. The maximum velocity will occur on the pipe centreline, i.e. r = 0; hence R2 d u max = − −−− −−− ( p + ρ gz ). 4 µ dx

(10.17)

The volume flow rate through the pipe under these flow conditions may be calculated by integrating the incremental flow δQ through an annulus of radial width δ r at radius r across the flow from r = 0 to r = R (see Fig. 10.4):

δQ = u2π rδ r,

u2πr dr. R

Q=

(10.18)

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Substitution for u at general radius r yields an expression

π d Q = – −−− −−− ( p + ρ gz ) 2 µ dx

(R r – r ) dr R

2

3

r2 r4 R π d = – −−− −−− ( p + ρ gz ) ⎛⎝R 2 −− – −− ⎞⎠ 2 4 0 2 µ dx

π d = – −−− −−− ( p + ρ gz )R 4 8 µ dx or, in terms of a pressure drop ∆p over a length l of pipe of diameter d, Q = ∆pπ d 4128µ l.

(10.19)

The mean flow velocity is given by QA, where A is the pipe cross-sectional area π d 24. Hence,

π d 2 B = – −−− −−− ( p + ρ gz )R = −12 u max , 8 µ dx

(10.20)

as shown in Fig. 10.5.

FIGURE 10.5 Velocity distribution in laminar flow in a circular pipe

Equation (10.19) may be rearranged for the pressure loss, giving the well-known Hagen–Poiseuille equation: ∆p = 128µlQπd 4.

(10.21)

Alternatively, substituting for Q = (π d 24)B, ∆p = 32µlBd 2.

(10.22)

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10.3

EXAMPLE 10.2

Incompressible, steady and uniform turbulent flow in bounded conduits

335

Glycerine of viscosity 0.9 N s m−2 and density 1260 kg m−3 is pumped along a horizontal pipe 6.5 m long of diameter d = 0.01 m at a flow rate of Q = 1.8 litres min−1. Determine the flow Reynolds number and verify whether the flow is laminar or turbulent. Calculate the pressure loss in the pipe due to frictional effects and calculate the maximum flow rate for laminar flow conditions to prevail.

Solution 1.8 Mean velocity, B = QA = ⎛ −−−− ⎝ 60

πd 2 −−−−− ⎞ × 10 –3 m s –1 = 0.382 m s−1, 4 ⎠

Re = ρudµ = 1260 × 0.382 × 0.010.9 = 0.535. Therefore, flow is laminar as Re 2000 (see Section 4.10). Frictional losses may be calculated from the Hagen–Poiseuille equation (10.21): ∆p = 128µ lQπd 4 = 128 × 0.9 × 6.5 × 3 × 10−5(π × 0.014 ) = 715 × 103 N m−2. Upper limit of laminar flow conditions is reached when ReRecrit = QQcrit, Qcrit = (QRe)Recrit = (1.80.535) × 2000 litres min−1. Therefore,

Qcrit = 112 litres s−1.

10.3 INCOMPRESSIBLE, STEADY AND UNIFORM TURBULENT FLOW IN BOUNDED CONDUITS In the preceding sections expressions have been developed for the velocity distribution and pressure losses encountered during laminar flow. Reference to the Reynolds number of such flow, i.e. Re 2000 in closed circular pipes, shows that such flow is restricted to relatively low flow rate conditions for all gases and those liquids that do not possess a high viscosity. Thus, in general, turbulent flow conditions are far more likely in most engineering situations. In this section, expressions will be developed for the losses incurred in turbulent flow in both closed and open conduits. However, it will be seen that completely analytical solutions are not available and that empirical relationships are needed in order to produce the necessary expressions. Consider a small element of fluid within a conduit, as shown in Fig. 10.6. The flow is assumed to be uniform and steady so that the fluid acceleration in the flow direction is zero. Applying the momentum equation to the fluid element in the flow direction yields p1A − p2A − τ0 lP + W sin θ = 0,

(10.23)

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FIGURE 10.6 Turbulent flow in a bounded conduit

where P is the wetted perimeter of the element defined as that part of the conduit’s circumference which is in contact with the fluid. It will be seen that including the area over which the shear stress τ0 acts in the form of lP, as above, effectively renders the derivation applicable to both open or closed conduits. Putting W = ρgAl and sin θ = −∆zl yields A(p1 − p2) − τ0 lP − ρgA ∆z = 0, where p1, p2 are the static pressures in the flow at sections 1 and 2 (Fig. 10.6). Hence 1 P − [ ( p 1 – p 2 ) – ρ g ∆z ] – τ 0 −− = 0 l A where the first term represents a drop in piezometric head over a length l of the conduit, and the ratio AP is known as the hydraulic mean depth, normally denoted by m; thus, dp* τ 0 = m −−−−−, dx

(10.24)

where dp*dx is the rate of loss of piezometric head along the conduit and τ0 is the wall or boundary shear stress. In order to express τ0 in equation (10.24), the concept of a flow friction factor f is introduced, which is a non-dimensional, experimentally measured factor normally introduced in the form

τ0 = fρC 22,

(10.25)

where C is the mean flow velocity. Hence, dp* −−−−− = f ρ C 22m. dx

(10.26)

If the frictional head loss down a length l of the conduit is denoted by hf , then the rate of loss of piezometric pressure may be expressed as

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dp* −−−−− = f ρ C 22m = ρ gh f l dx and hf = f lC 22gm.

(10.27)

Now, as dp* d −−−−− = −−− ( p + ρ gz ), dx dx where z is the elevation of the conduit above some datum, then for open channels, as the static pressure p may be assumed to remain constant along the channel, it follows that dp* dz −−−−− = ρ g −−− = ρ g sin θ dx dx and, since for uniform flow the hydraulic gradient hf l is equal to the slope of the channel, hf −− = sin θ = i. l Then it follows, by equating equations (10.26) and (10.27), that f ρC 22m = ρgi so that C = (2gf ) × (mi). If now (2gf ) = C

(10.28)

is substituted, the expression known as the Chezy formula is obtained: C = C(mi),

(10.29)

yielding the flow rate through a given channel of a given slope and roughness. Various values of C are employed in open-channel design. For pipes running full of fluid, the wetted perimeter becomes the internal circumference of the pipeline: hence AP = m = πD 24πD = D4, so that equation (10.27) becomes for circular cross-sections 4f l C 2 h f = −−−− ⋅ −−− . d 2g

(10.30)

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This expression is in every way equivalent to the Chezy equation above and follows directly from the study of the general condition illustrated in Fig. 10.5. It is known as the Darcy–Weisbach equation for head loss in circular pipes. The Darcy equation is also the equivalent of the Poiseuille equation derived for laminar flow, with one important exception, namely the inclusion of an empirical factor f to describe the friction loss in turbulent flow, which was not necessary in the case of laminar flow. This fundamental difference arises from the complexity of turbulent flow, resulting in the fact that the relationship τ = µ(dudy) cannot be used and, therefore, an analytical solution is not possible.

10.4 INCOMPRESSIBLE, STEADY AND UNIFORM TURBULENT FLOW IN CIRCULAR CROSS-SECTION PIPES The head loss in turbulent flow in a closed section pipe is given by the Darcy equation (10.30), 4fl C 2 h f = −−−− ⋅ −−− . d 2g It will be seen from the above expression that all the parameters, with the exception of the friction factor f, are measurable. Results of extensive experimentation in this area led to the establishment of the following proportional relationships: 1. 2. 3. 4. 5. 6.

hf hf hf hf hf hf

∝ l; ∝ C 2; ∝ 1d; depends on the surface roughness of the pipe walls; depends on fluid density and viscosity; is independent of pressure.

The value of f must be selected so that the correct value of hf will always be given by the Darcy equation and so cannot be a single-value constant. The value of f must depend on all the parameters listed above. Expressed in a form suitable for dimensional analysis this implies that f = φ (C, d, ρ, µ, k, k′, α),

(10.31)

where k is a measure of the size of the wall roughness, k ′ is a measure of the spacing of the roughness particles, both having dimensions of length, and α is a form factor, a dimensionless parameter whose value depends on the shape of the roughness particles. In the general rough pipe case, dimensional analysis yields an expression f = φ2( ρ Cdµ, kd, k′d, α)

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339

or, in terms of Reynolds number, f = φ2(Re, kd, k′d, α).

(10.32)

Dimensional analysis can only indicate the best combination of parameters for an empirical solution; the actual algebraic format of the relation for friction factor in terms of the variables listed must be determined by experimentation. Blasius, in 1913, was the first to propose an accurate empirical relation for the friction factor in turbulent flow in smooth pipes, namely f = 0.079Re14.

(10.33)

This expression yields results for head loss to ±5 per cent for smooth pipes at Reynolds numbers up to 100 000. At this point it may be useful to note that the value of friction factor quoted in many American texts is 4f in the notation employed in this text, and so the value of the constant in Blasius’ equation will be changed. In this text the UK-recognized value of friction factor as defined by equation (10.30) will be used exclusively. For rough pipes, Nikuradse, in 1933, proved the validity of the f dependence on the relative roughness ratio kd by investigating the head loss in a number of pipes which had been treated internally with a coating of sand particles whose size could be varied. These tests in no way investigated the effect of particle spacing k′d, or of particle shape factor α, on the friction factor, but did show that, for one type of roughness, f = φ3(Re, kd).

(10.34)

It may well be argued that experimental problems would make it virtually impossible to hold k′d and α constant so that the effect of roughness size kd might be investigated in isolation. However, the accuracy of the results obtained by basing the value of f simply on Reynolds number and kd does suggest that the effects of particle spacing and shape are negligible compared with that of the relative roughness based solely on kd. Thus, the calculation of losses in turbulent pipe flow is dependent on the use of empirical results, and the most common reference source is the Moody chart, which is a logarithmic plot of f vs. Re for a range of kd values. This type of data presentation is commonly referred to as a Stanton diagram. A typical Moody chart is presented as Fig. 10.7, and a number of distinct regions may be identified and commented on. 1. The straight line labelled ‘laminar flow’, representing f = 16Re, is a graphical representation of the Poiseuille equation (10.19), i.e. Q = ∆pπ d 4128µl, hf = ∆pρg = 128µ lQρgπ d 4. Now,

Q = π (d 24)C.

Hence, from the Darcy equation, hf = 128µlπd 2C4ρgπd 4 = 4f lC 22gd

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FIGURE 10.7 Variation of friction factor f with Reynolds number and pipe wall roughness for ducts of circular cross-section

or

128 f = −−−−− µ ρ Cd 8 f = 16Re.

(10.35)

Equation (10.35) plots as a straight line of slope –1 on a log–log plot and is independent of pipe surface roughness. This relation also shows that the Darcy equation may be applied to the laminar flow regime provided that the correct f value is employed. 2. For values of kd 0.001 the rough pipe curves of Fig. 10.7 approach the Blasius smooth pipe curve due to the presence of the laminar sublayer (discussed in Chapter 11), which develops in turbulent flow close to the pipe wall and whose thickness decreases with increasing Reynolds number. Thus, for certain combinations of surface roughness and Reynolds number, the thickness of the laminar sublayer is sufficient to cover the wall roughness and the flow behaves as if the pipe wall were smooth. For higher Reynolds numbers the roughness particles project above the now decreased thickness laminar sublayer and contribute to an increased head loss. 3. At high Reynolds numbers, or for pipes having a high kd value, all the roughness particles are exposed to the flow above the laminar sublayer. In this condition, the head loss is totally due to the generation of a wake of eddies by each particle making up the pipe roughness. This form of head loss is known as ‘form drag’ and is directly proportional to the square of the mean flow velocity: thus hf ∝ C 2 and hence, from the Darcy equation, f is a constant, depending only on the roughness particle size. This condition is represented on the Moody chart by portions of the f vs. Re curves which are parallel to the Re axis and which occur at high values of Re and kd.

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10.4

EXAMPLE 10.3

Incompressible, steady and uniform turbulent flow in circular cross-section pipes

341

Calculate the loss of head due to friction and the power required to maintain flow in a horizontal circular pipe of 40 mm diameter and 750 m long when water (coefficient of dynamic viscosity 1.14 × 10−3 N s m−2) flows at a rate: (a) 4.0 litres min−1; (b) 30 litres min−1. Assume that for the pipe the absolute roughness is 0.000 08 m.

Solution (a) In order to establish whether the flow is turbulent or laminar it is first necessary to calculate the Reynolds number: Re = ρCdµ, but

Q = 4.0 × 10−360 = 66.7 × 10−6 m3 s−1

and

Pipe area, A = πd 24 = π (0.04)24 = 1.26 × 10−3 m2,

so that the mean velocity in the pipe is given by C = QA = 66.7 × 10−61.26 × 10−3 = 52.9 × 10−3 m s−1. 10 3 × 52.9 × 10 –3 × 0.04 −−−−−−−−− = 1856. Hence, Re = −−−−−−−−−−−−−−−−−−−−−−−–−− 1.14 × 10 3 Therefore, flow is laminar, since Re 2000. So, the loss due to friction may therefore be calculated by using either Poiseuille’s equation (i) or the Darcy equation and f = 16Re (ii): (i) Poiseuille’s equation: 128 µ lQ 128 × 1.14 × 10 –3 × 750 × 66.7 × 10 –6 ∆p = −−−−−−−−4−−− = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−−−−−−−−−−−−−−−−−−− πd π ( 0.04 ) 4 = 907.6 N m−2. Therefore, head lost due to friction is given by ∆p 907.6 h f = −−− = −−−−3−−−−−−−−−− = 92.4 × 10 –3 m of water. ρ g 10 × 9.81 (ii) Darcy equation: 4fl C 2 h f = −−−− −−−, d 2g Hence,

but

16 16 f = −−− = −−−−−− = 0.008 62. Re 1856

4 × 0.008 62 × 750 ( 52.9 × 10 –3 ) 2 h f = −−−−−−−−−−−−−−−−−−−−−−−−−− × −−−−−−−−−−−−−−−−−−− = 92.4 × 10 –3 m of water. 2 × 9.81 0.04

Power required to maintain flow, P = ρghfQ = 103 × 9.81 × 92.4 × 10−3 × 66.7 × 10−6 = 0.0605 W.

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(b)

30 × 10 –3 Q = −−−−−−−−−−−− = 0.5 × 10 –3 m 3 s –1 , 60

0.5 × 10 –3 C = −−−−−−−−−−−−−−–−3 = 0.4 m s –1 . 1.26 × 10

Therefore, 10 3 × 0.4 × 0.04 Re = −−−−−−−−−−−−−−−−−−–−3−−− = 14 035 1.14 × 10 and the flow is turbulent, so the Darcy equation must be used. To determine the value of friction factor: Relative roughness = kd = 0.000 080.04 = 0.002. From the Moody chart, for Re = 1.4 × 104 and relative roughness of 0.002, f = 0.008. Therefore, 4fl C 2 4 × 0.008 × 750 ( 0.4 ) 2 h f = −−−− −−− = −−−−−−−−−−−−−−−−−−−−−− × −−−−−−−−−−− = 4.89 m of water. 2 × 9.81 d 2g 0.04 Power required, P = ρghf Q = 103 × 9.81 × 4.89 × 0.5 × 10−3 = 24.0 W.

10.5 STEADY AND UNIFORM TURBULENT FLOW IN OPEN CHANNELS It was shown in Section 10.3 that the general equation for head losses in turbulent flow could be derived concurrently for both open and closed section conduits. The general equation (10.27) hf = f lC 22gm, reduces to the Chezy equation (10.29) C = C(mi), when it is realized that, for open channels, provided the flow is steady and uniform, hf l is equal to the slope of the channel. This, however, is not the case in non-uniform flow discussed in Chapter 16. Since C is the mean velocity in the channel of area A, it follows that the flow rate Q is given by Q = AC = AC(mi).

(10.36)

Although C is referred to as the Chezy coefficient, implying a dimensionless constant, this is not so; since C = 2gf, it has dimensions of L12T −1.

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Velocity distribution in turbulent, fully developed pipe flow

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It has been shown that, for pipe flow, the value of f depends both on the flow Reynolds number and on the surface roughness of the pipe material, so it would be reasonable to expect C to vary with Re and km, where m, the mean hydraulic depth, is employed as the characteristic length for the system. Generally, the dependence of C on Reynolds number is small and km is the predominant factor. For almost all open-channel work the flow may be assumed to be fully turbulent, with a high value of Reynolds number and, therefore, km may be taken as the sole factor affecting C values, provided the channel section shape remains simple. Finally, it will be seen that, as values of C depend only on Re and km and not on the Froude number of the open-channel flow, the Chezy equation applies equally to rapid or tranquil flow, defined in Chapter 15. However, in cases of non-uniform flow, an important distinction between the slope of the channel and the hydraulic gradient has to be made. This will be discussed in Chapter 16.

EXAMPLE 10.4

A rectangular open channel has a width of 4.5 m and a slope of 1 vertical to 800 horizontal. Find the mean velocity of flow and the discharge when the depth of water is 1.2 m and if C in the Chezy formula is 49.

Solution The mean velocity may be obtained using the Chezy formula: C = C(mi). For the channel, i = 1800 and m = AP. Now, A = 4.5 × 1.2 = 5.4 m2, P = 2 × 1.2 + 4.5 = 6.9 m, so that m = 5.46.9 = 0.783 m. Substituting into the Chezy formula, C = 49(0.783800) = 1.53 m s−1. The discharge is given by Q = CA = 1.53 × 5.4 = 8.27 m3 s−1.

10.6 VELOCITY DISTRIBUTION IN TURBULENT, FULLY DEVELOPED PIPE FLOW Owing to the nature of turbulent flow, there are difficulties in the derivation of expressions defining the distribution of velocity in pipe flow. The use of dimensional analysis, together with a series of assumptions based on the relative importance of the fluid viscosity and eddy viscosity terms in the laminar sublayer which is present in a

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fully developed turbulent boundary layer, does, however, allow the prediction of the form of the velocity distribution expressions. The algebraic format of these equations has been developed from experimental investigations and, although now well established and accepted, is empirical in nature. For fully developed turbulent pipe flow in a circular cross-section pipe, it would be reasonable to suppose that the local velocity u at a distance y from the pipe wall would be given by a general function of the form u = φ (ρ, µ, τ0, R, y, k),

(10.37)

where ρ, µ are the fluid density and viscosity, R is the pipe radius, k is the roughness particle size and τ0 is the wall shear stress. Dimensional analysis suggests an expression of the form

τ B ρR y k −−−−−−−−−−− = φ 1 ⎛ −−−− ⎞ ⎛ −−0⎞ , −−, −− . ⎝ ⎠ ⎝ ⎠ µ ρ R R ( τ 0 ρ )

The term (τ0ρ) has the dimensions of velocity and is referred to as the shear stress velocity u*. Bu* = φ1( ρ u*Rµ, yR, kR),

(10.38)

where ρu*Rµ is a form of the Reynolds number. To proceed beyond equation (10.38), it is necessary to make some assumptions about the importance of the various groups. Surface roughness, represented by kR, will affect the value of u*, but will only be a significant factor in the flow zone close to the wall. Similarly, the fluid viscosity will only be of major importance in the laminar sublayer close to the pipe wall. Thus, the velocity in the central turbulent core of the flow will be assumed to depend only on the positional group yR. It is customary to express this relationship in terms of the velocity defect, or the difference between the local velocity u at position y from the wall and the flow maximum velocity on the pipe centreline umax. Hence (umax − u)u* = φ2(yR).

(10.39)

This expression is referred to as the velocity defect distribution and is well supported by experimental work which shows that, for a wide range of flow Reynolds numbers, the velocity profiles only differ in the region close to the pipe wall. It is apparent from experimental results that, as the Reynolds number increases, the friction factor f and the shear stress τ0 terms become smaller, and the velocity profile across the central core of the flow becomes progressively more uniform. Prandtl proposed an empirical velocity distribution for this turbulent central core of the form uumax = ( yR)n,

(10.40)

where the value of n = 1--7- for Re 107 and decreases above this Reynolds number. This is well supported experimentally, but does break down at y = R as symmetry here demands that dudy = 0, which cannot be justified by the expression.

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If the case of the smooth pipe is considered, the kR group becomes unimportant and so, close to the pipe wall, the effect of pipe radius R is negligible, so long as y R; let y = y2 be the limit for this assumption so that uu* = φ3(Re*), 0 y y2,

(10.41)

where Re* = ρyu*µ, a group independent of pipe radius R. If equation (10.39) is applicable from y = y1 to R, and if experimental results which indicate that y2 y1 are accepted, then it becomes apparent that there is a region in the flow, close to the pipe wall, where both equations (10.39) and (10.41) apply simultaneously. For a smooth pipe, the relation uu* = φ1(Re, yR)

(10.42)

applies for y1 y R, where Re = ρRu*µ and, for y = R, umax u* = φ4(Re).

(10.43)

Adding equations (10.39) and (10.42) yields umax u* = φ4(Re) = φ1(Re, yR) + φ2( yR)

(10.44)

for y1 y R. Both equations (10.39) and (10.41) apply in the zone y1 y y2 and may be added to give umax u* = φ2(y*) + φ3(Re*) = φ4(Re),

(10.45)

where y* = yR. Differentiating (10.45) with respect to Re yields y* φ 3′ ( Re* ) = φ 4′ ( Re ),

(10.46)

as Re* = Re y*. Since φ 4′(Re) is independent of y*, so then is y*φ 3′(Re*), so that φ3′(Re*) has the form (1y*)φ(Re).

(10.47)

Similarly, differentiating equation (10.45) with respect to y* yields

φ 2′( y*) + Re φ 3′(Re*) = 0,

(10.48)

and so φ3′(Re*) has the form (1Re)φ (y*)

(10.49)

since φ 2′( y*) is independent of Re. In order for equations (10.49) and (10.47) to be satisfied simultaneously in the zone y1 y y2, it is necessary for φ (Re) = constantRe and φ (y*) = constanty*; therefore, φ3′(Re*) = ARe*, where A is a constant. From (10.46),

φ 4′ ( Re ) = y* φ 3′ ( Re ) = y*ARe* = ARe

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or

φ4(Re) = A log e Re + constant.

(10.50)

Similarly, from (10.48),

φ 2′ ( y*) = – Re φ 3′ ( Re* ) = – Re ARe* = −Ay* or, by integration,

φ2( y*) = −A log e y* + constant.

(10.51)

However, from equations (10.39), (10.45) uu* = φ4(Re) − φ2(y*). Thus, uu* = (A loge Re + constant) + (A loge y* + constant), uu* = A log e Re* + A1,

(10.52)

where A and A1 are constants to be determined experimentally. Equation (10.52) is known as the universal velocity distribution. However, it is to be noted that, due to the restriction placed on equations (10.39) and (10.45), the expression only applies in the central turbulent core of the pipeline, as shown in Fig. 10.8. FIGURE 10.8 Zones of application of empirical relations defining velocity distribution in turbulent pipe flow

Close to the pipe wall, within the laminar sublayer, the effect of fluid viscosity is predominant and so the expression τ = µ(dudy) may be integrated to describe the velocity distribution in this zone: u = τ ( yµ) + E, where E = 0 when u = 0 at y = 0. Hence, u = τ yµ or

u(τρ) = yρτ 12µρ12, uu* = Re*.

(10.53)

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Equation (10.53) may be plotted, as shown in Fig. 10.8, and intersects the straight line relation of equation (10.52) at a point which, theoretically, defines the upper limit of the laminar sublayer. In practice, the upper limit of the laminar sublayer is ill defined, and experimental results tend to smooth the intersection (Fig. 10.8). Nikuradse’s results for smooth pipes show that the constants in equation (10.52) may be taken as uu* = 2.5 log e Re* + 5.5 = 5.75 log10 Re* + 5.5.

(10.54)

As shown in Fig. 10.8, equation (10.53) applies accurately up to Re* 8 and (10.52) applies from Re* 30. By employing equation (10.52), it is possible to relate the friction factor to mean flow velocity and flow Reynolds number. From equation (10.25), f = 2τρB2, where B is the mean flow velocity. Mean velocity may be calculated by integration of equation (10.52) across the pipe, assuming the thickness of the laminar sublayer to be negligible, and dividing the result by the pipe cross-sectional area, as was done in the case of laminar flow (equation (10.20)). Substitution of friction factor for mean velocity through the relation above (equation (10.25)) yields an expression of the form 1f = F + G log e (Ref ),

(10.55)

where Re = ρdBµ and F, G are constants. The constants in equation (10.55) may be obtained from experimental investigations, and Nikuradse’s results for smooth pipes suggest an expression 1f = 4.07 log10 (Re f ) − 0.6,

(10.56)

although values of 4.0 and −0.4 for G and F do yield improved results. This expression has been verified for Reynolds numbers in the range 5000 Re 3 × 106. However, the Blasius expression for smooth pipes (equation (10.33)) gives reasonably accuracy for Re values up to 105. Similar relations for velocity distribution and friction factor may be determined for rough pipes, the form of the expressions being deduced by dimensional analysis techniques and the algebraic format of the relations being obtained empirically by extensive testing. The Moody chart (Fig. 10.7) showed that rough pipe turbulent flow falls into two regimes as far as friction factor calculation is concerned: first, a regime where friction factor is independent of Reynolds number and depends on surface roughness; second, a transitionary regime where the friction factor increases with Reynolds number for any particular pipe surface roughness, from the value appropriate for a smooth pipe up to the Reynolds-number-independent value mentioned. The mechanism responsible for this transition has already been explained in Section 10.4 in terms of the relation between roughness particle size and laminar sublayer thickness. Nikuradse showed that, by describing the flow Reynolds number in terms of the roughness particle size then the three identifiable flow regimes could be described as follows:

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1. 2. 3.

smooth pipe f results apply for Re = ρu*kµ 4; transition occurs for 4 Re 70; f is independent of Re for 70 Re. For the f independent zone, the velocity profile may be expressed as uu* = φ (yR, kR)

(10.57)

and experimental results verify an equation of the form uu* = 5.75 log10 (yk) + 8.48.

(10.58)

Integration to give mean velocity and the introduction of friction factor via equation (10.25) yields an expression similar to (10.55): 1f = 4 log10 (dk) + 2.28,

(10.59)

where d is the pipe diameter. For the transition regime, 4 ρu*kµ 70, an expression 1f = −4 log10 (k3.71d + 1.26Re f ),

(10.60)

where Re = ρudµ, has been shown to be applicable and may be seen to converge to equation (10.59) or (10.56) for fully rough pipes or smooth pipes characterized by Re → ∞ or k → 0. Equation (10.60) is known as the Colebrook–White equation and was employed by Moody in the preparation of the friction factor chart of Fig. 10.7 (see program CBW, Section 10.10). As mentioned in the derivation of the laminar flow equations, the results only apply to fully developed pipe flow and so do not cover the entry length of a pipeline. However, as this is normally short in comparison with the pipe length, no appreciable error arises.

EXAMPLE 10.5

Assuming the following velocity distribution in a circular pipe u = umax(1 − rR)17, where umax is the maximum velocity, calculate (a) the ratio between the mean velocity and the maximum velocity, (b) the radius at which the actual velocity is equal to the mean velocity.

Solution (a) The elementary discharge through an annulus dr is given by dQ = 2π ru dr = 2π umax(1 − rR)17 dr,

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and discharge through the pipe by

r(1 – rR) R

Q = 2 π u max

17

dr.

Let 1 − rR = x; then dx 1 −−− = – −− dr R

and

dr = – R dx

so that R − r = xR,

when r = 0, x = 1,

r = R − xR = R(1 − x), when r = R, x = 0. Therefore, substituting, Q = 2 π u max

(1 – x)x 1

R ( 1 – x )x 17 ( – R dx ) = 2 π R 2 u max 1

17

dx

7 87 7 157 1 = 2 π R 2 u max ⎛ −x – −−−x ⎞ ⎝8 ⎠0 15 7 7 105 – 56 49 = 2 π R 2 u max ⎛ − – −−− ⎞ = 2 π R 2 u max ⎛ −−−−−−−−−−−− ⎞ = π R 2 u max −−−, ⎝ 120 ⎠ ⎝ 8 15 ⎠ 60 and

2 −− B = QπR2 = πR 2umax −49 60 πR =

49 − −− 60 max

u

,

with the result that Bumax = 4960. (b)

u = B = 49umax 60 = umax(1 − rR)17.

Therefore, (4960)7 = 1 − rR and

rR = 1 − (4960)7 = 1 − 0.242 = 0.758.

Hence, r = 0.758R.

EXAMPLE 10.6

Assuming the universal velocity distribution for turbulent flow in a pipe, u ------ = 5.5 + 5.75 log10 Re*, u* determine the radius at which the point velocity is equal to the mean velocity and the ratio of mean velocity to maximum velocity.

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Solution The point velocity is given by u = u*(5.5 + 5.75 log10 Re*), but

u* = (τ0ρ) = constant and Re* = ρyu*µ.

Let ρ u*µ = a and, for a pipe, let y = r. Then,

u = u*(5.5 + 5.75 log10 ar).

To obtain the mean velocity, it is necessary first to calculate the discharge which, when divided by the cross-sectional area of the pipe, will give the mean velocity. Thus, the elementary discharge through an annulus dr is given by dQ = 2π ru dr = 2π u*(5.5r + 5.75r log ar) dr and discharge through the pipe by ⎛ Q = 2 π u* ⎜ 5.5 ⎝ Now,

R

r log R

r dr + 5.75 0

10

⎞ ar dr ⎟ . ⎠

R

r 2 R R2 r dr = ⎛⎝ −− ⎞⎠ = −−−, 2 0 2 0

and to obtain

R

1 r log 10 ar dr = −−−−−−−−− log e 10 0

r log ar dr, R

e

put log e ar = y. Hence, 1 dr dy = −−− a dr = −−− ar r and

r dr = dx.

Therefore, x = r 22. Integrating now by parts,

y dr = yx – r dy. Substituting

r log

10

1 1 1 ar dr = −r 2 log e ar – − r 2− dr 2 r 2 = −12 r 2 loge ar − r 24 = (r 22)(loge ar − −12 ).

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Therefore,

R

1 r log 10 ar dr = −−−−−−−−− log e 10 0

r log ar dr R

e

R

1 r2 1 = −−−−−−−−− −− ⎛ log e ar – −⎞ log e 10 2 ⎝ 2⎠ 0

r2 1 = −− ⎛⎝ log 10 ar – −−−−−−−−−−− ⎞⎠ 2 2 log e 10

R

0 2

R 1 = −−− ⎛ log 10 aR – −−−−−−−−−−− ⎞ 2 ⎝ 2 log e 10⎠ R2 = −−− ( log 10 aR – 0.217 ). 2 Substituting into the equation for Q, R2 R2 Q = 2 π u* 5.5 −−− + 5.75 −−− ( log 10 aR – 0.217 ) 2 2 = π u*R2(5.5 + 5.75 log10 aR − 1.248) = π u*R2(4.252 + 5.75 log10 aR). Therefore, mean velocity in the pipe B = QπR2 = u*(4.252 + 5.75 log10 aR). The radius at which the point velocity is the same as the mean velocity is now obtained by equating the two expressions: u*(5.5 + 5.75 log10 ar) = u*(4.252 + 5.75 log10 aR), 1.248 = 5.75(log10 aR − log10 ar). 0.217 = log10(Rr), 100.217 = Rr, so that, finally, r = 0.607R. The maximum velocity occurs at the centre of the pipe, where r = 0. Therefore, umax = 5.5u* and

Bumax = u*(4.252 + 5.75 log aR)5.5u* = 0.773 + 1.045 log10 aR = 0.773 + 1.045 log10( ρu*µ)R.

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10.7 VELOCITY DISTRIBUTION IN FULLY DEVELOPED, TURBULENT FLOW IN OPEN CHANNELS In the use of the Chezy and related equations for open-channel flow, the assumption is made that the flow is uniform across the channel. This is, in practice, never achieved, and, further, due to the lack of symmetry in open-channel flow, the accepted central position of maximum flow velocity for pipe flow is also not reproduced. The actual velocity profiles across the flow are influenced by the presence of the channel solid boundaries and the free surface. Irregularities in the solid boundaries are generally so large and random that each channel has its own individual velocity distribution and there is no direct equivalent to the velocity distribution expressions derived for pipe flow. Figure 10.9 illustrates a typical velocity distribution, the maximum velocity occurring at some depth below the free surface, usually between 5 and 25 per cent of flow depth, and the mean velocity, which is usually some 80 to 85 per cent of the free surface velocity, occurs at about 60 per cent of the flow depth below the free surface.

FIGURE 10.9 Velocity distribution in a simple open channel under fully developed, turbulent flow conditions

Normally, in open-channel calculations, the uncertainties involved in the flow parameters are so large as to render any variations of flow velocity away from the mean to be negligible, and these are neglected.

10.8 SEPARATION LOSSES IN PIPE FLOW Whenever the uniform cross-section of a pipeline is interrupted by the inclusion of a pipe fitting, such as a valve, bend, junction or flow measurement device, then a pressure loss will be incurred. The values of these losses, which are sometimes misleadingly referred to as ‘minor losses’, have to be included in a pipeline’s total resistance if errors in pump and system matching or flow calculations for a given pressure differential are to be avoided. In this treatment, the term ‘separation loss’ has been chosen to define pressure losses across such fittings, as it is felt that this term describes well the physical phenomena which occur at such obstructions in the pipeline. Generally, the flow separates from the pipe walls as it passes through the obstructing pipe fitting, resulting in the generation of eddies in the flow, with consequent pressure loss, as shown in Fig. 10.10 for the case of a sudden enlargement. For small, complex pipe networks such as those found in some chemical process plants, in aircraft fuel and hydraulic systems and in ventilation systems, the total effect

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FIGURE 10.10 Separation loss in a sudden enlargement

of separation losses may be the predominant factor in the system pressure loss calculation, exceeding the contribution of pipe friction at the design flow rate. Conversely, in large pipe systems, such as water distribution networks or overland oil pipelines, the losses due to pipe fitting may be negligible compared with the friction loss and may often be ignored.

10.8.1 Losses in sudden expansions and contractions Generally, the losses due to pipe or duct fittings are determined experimentally. However, the case of a sudden expansion in a pipe or duct may be determined analytically. Figure 10.1l illustrates a sudden enlargement; consider a control volume ABCDEF as shown and let p1 and p2 be the pressures at sections 1 and 2, respectively, where the mean flow velocities are related by the continuity equation FIGURE 10.11 Calculation of the loss coefficient for a sudden enlargement

A1B1 = A2B2

(10.61)

and represent the duct cross-sectional areas A1, A2. By application of the momentum equation between sections 1 and 2 the following relation may be derived: Resultant force in Rate of change of momentum = flow direction in flow direction, p1A1 + p′(A2 − A1) − p2A2 = ρQ(B2 − B1),

(10.62)

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where Q = A2B2 and p′ is the pressure acting on the annulus represented by AB and CD, of cross-sectional area (A2 − A1). It may be assumed that p1 = p′, owing to the small radial acceleration at entry to the larger-diameter duct at section ABCD, a result which is well supported experimentally. Hence, equation (10.62) reduces to (p1 − p2)A2 = ρQ(B2 − B1) = ρB2A2(B2 − B1), p1 − p2 = ρ B2 (B2 − B1).

(10.63)

If Bernoulli’s equation is now applied between sections 1 and 2, with a term h included to represent the separation loss, then an expression for the pressure differential p1 − p2 may be derived: p1ρg + B 21 2g + Z1 = p2 ρg + B 22 2g + Z2 + h, where Z1 = Z2 if the enlargement is situated in a horizontal pipe or duct; thus, h = (p1 − p2 )ρg + ( B 21 − B 22 )2g and, substituting for p1 − p2 from equation (10.63)

ρ B 2 ( B 2 – B 1 ) B 21 – B 22 1 h = −−−−−−−−−−−−−−−−− + −−−−−−−−− = −−− ( 2B 22 – 2B 2 B 1 + B 21 – B 22 ) ρg 2g 2g ( B1 – B2 )2 1 = −−− ( B 21 – 2B 1 B 2 + B 22 ) = −−−−−−−−−−−−−. 2g 2g Thus, the loss due to sudden enlargement is given by h = (B1 − B2)22g.

(10.64)

Alternatively, since, from equation (10.61) B2 = B1(A1A2), 2 B 22 A 2 2 h = ( B 21 2g ) ( 1 – A 1 A 2 ) = −−− ⎛ −−− – 1⎞ . ⎝ ⎠ 2g A 1

(10.65)

This expression is sometimes referred to as the Borda–Carnot relationship and is usually within a few per cent of the experimental result for the separation loss incurred by sudden enlargement in coaxial pipelines. The loss at exit from a pipe into a reservoir may be obtained by considering equation (10.65). It will be seen that as A2 → ∞, so B2 → 0 and h → B 21 2g, i.e. the kinetic energy of the approaching flow. This case is obviously representative of a pipe discharging into a large tank or a duct discharging to atmosphere and is the accepted expression for conduit exit loss. Sudden contractions in a duct or pipe may also be dealt with in this way, provided that there is little or no loss between the upstream large-section conduit and the vena contracta formed within the smaller conduit just downstream of the junction, as shown in Fig. 10.12.

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FIGURE 10.12 Sudden contraction. Loss approximated by consideration of sudden enlargement between the vena contracta and section 2

It is not possible to apply the momentum equation between sections 1 and 2 in Fig. 10.12, owing to the uncertain pressure distribution across the face ABCD. However, it has been shown experimentally that the majority of the pressure loss occurs as a result of the eddies formed as the flow area expands from the vena contracta area up to the full cross-section of the downstream pipe. If the area of the vena contracta is Ac then accurate results may be achieved by applying the sudden enlargement expression between Ac and A2 at section 2; thus 2 2 B 22 A 2 B 22 1 h ≈ −−− ⎛ −−− – 1⎞ = −−− ⎛ −−− – 1⎞ , 2g ⎝ A c ⎠ 2g ⎝ C c ⎠

(10.66)

where Cc is the coefficient of contraction for the junction based on the smaller-pipe entry diameter BC. The above equation indicates, since the expression in brackets is constant for any given area ratio, that it may be generalized into the form h = KB 22 2g,

(10.67)

where K is known as the loss coefficient. Table 10.1 shows some experimental values of Cc and the corresponding values of K obtained with sharp pipe edges.

TABLE 10.1 Loss coefficients for sudden contraction

EXAMPLE 10.7

A2 A1

0.1

0.3

0.5

0.7

1.0

Cc K

0.61 0.41

0.632 0.34

0.673 0.24

0.73 0.14

1.0 0

In a water pipeline there is an abrupt change in diameter from 140 mm to 250 mm. If the head lost due to separation when the flow is from the smaller to the larger pipe is 0.6 m greater than the head lost when the same flow is reversed, determine the flow rate.

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Solution When the flow is from the smaller to the larger pipe the loss is due to sudden enlargement and is given by equation (10.65): h = ( B 21 2g)(1 − A1A2)2 = ( B 21 2g)(1 − 0.314)2 = 0.47 B 21 2g, where B1 is the velocity in the smaller pipe. When the flow is reversed, the loss is due to sudden contraction. Area ratio, A2A1 = (140250)2 = 0.314. From Table 10.1, K = 0.33 (say). Therefore, the loss, h′ = 0.33 B 22 2g, where B2 is again the velocity in the smaller pipe. Since the flow rate is the same in both cases, then B1 = B2 = B and

h − h′ = 0.6,

so that (0.47 − 0.33)(B22g) = 0.6 and

2 = 9.17 m s−1.

10.8.2 Losses in pipe fittings, bends and at pipe entry Losses in pipe fittings are usually expressed in the form already suggested for the loss at sudden contraction, namely h = K(B 22g), where K is the fitting loss coefficient. It is a non-dimensional constant and its value is obtained experimentally for any pipe fitting. Table 10.2 sets out some typical values. The major advantage of expressing losses due to separation in the above form is that it can easily be incorporated into the steady flow energy equation, Section 6.4, as will be shown in Chapter 14. Figure 10.13 illustrates the flow in a pipe bend, demonstrating the area of flow separation which results in the loss coefficients for bends listed in Table 10.2. As the bend becomes sharper, so the areas of separation become more extensive and the loss coefficient increases. Losses due to flow control devices are also illustrated in Figure 10.3. Loss at entry to a pipe from a reservoir is a special case of sudden contraction, in which the velocity in the reservoir is considered to be zero. Owing to the fact that the fluid enters the pipe from all directions, a vena contracta is formed downstream of the pipe inlet and, consequently, the loss is associated with enlargement from the vena contracta to the full-bore pipe. This is the same situation as in the case of sudden contraction.

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10.8

TABLE 10.2 Head loss coefficients for a range of pipe fittings

Fitting Gate valve (open to 75 per cent shut) Globe valve Spherical plug valve (fully open) Pump foot valve Return bend 90° elbow 45° elbow Large-radius 90° bend Tee junction Sharp pipe entry Radiused pipe entry Sharp pipe exit

Separation losses in pipe flow

357

Loss coefficient K 0.25 → 25 10 0.1 1.5 2.2 0.9 0.4 0.6 1.8 0.5 →0.0 0.5

FIGURE 10.13 Separation at bends and valves

Figure 10.14 illustrates various types of pipe entry conditions. The results tabulated in Table 10.2 and indicated in Fig. 10.14 may be explained by reference to the flow separation at entry to the pipe explained above, so that the sharper the entry corner, the smaller is the vena contracta, and, hence, the greater the flow separation and the higher the value of K.

FIGURE 10.14 Pipe entry losses

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10.8.3 Equivalent length for pipe fitting loss calculations Separation loss coefficients K may also be defined in terms of an equivalent length of straight pipe, of the same diameter as that including the fitting, that would result in the same frictional loss as that incurred by flow separation through the fitting. This is justified by consideration of the Darcy equation and equation (10.67), hf = 4f leB2d2g = KB 22g, where le is the equivalent length of pipe, diameter d, that would yield a friction loss equivalent to the particular fitting. Thus, le = Kd4f

(10.68)

and so le is normally calculated as a number of pipe diameters. le may be the equivalent length for a single fitting or the summation of all the separation loss coefficients for a particular system. Hence, for the total pressure loss through a pipeline of length l and diameter d, the expressiom hf = 4f(l + le )B 22dg

(10.69)

may be employed.

10.8.4 Diffusers In order to avoid the head losses incurred by the installation of sudden enlargements into pipe and duct flow, diffusers are commonly employed. Figure 10.15 illustrates a FIGURE 10.15 Loss of pressure in a conical diffuser

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typical conical diffuser and the variation in pressure loss across it with diffuserincluded angle. The loss experienced depends on the area ratio between which the diffuser operates and the included angle of the diffuser θ. The total loss across the diffuser is made up of two components, the first due to fluid friction along the length of the diffuser, which, therefore, increases as θ decreases for a given area ratio, i.e. the diffuser length l increases as θ decreases and results in an increase in friction loss. The second contribution to the total loss is dependent on θ and is the separation loss, which increases with increasing included angle for a given area ratio, reaching a maximum when the diffuser approaches a sudden enlargement. The minimum loss for any particular area ratio will, therefore, be a compromise, where the angle of the diffuser is sufficiently small to limit separation, or flow eddy, losses, but not so small as to increase the length of the diffuser to the point where the frictional losses become predominant. Normally 6° to 7° included angle is the minimum acceptable. Diffusers are found in a wide range of applications where it is necessary to reduce the flow velocity by means of an area change, without undue pressure loss. For example, wind tunnel return circuits are at one end of the size spectrum and venturi meter discharge diffusers at the other.

10.9 SIGNIFICANCE OF THE COLEBROOK–WHITE EQUATION IN PIPE AND DUCT DESIGN While the friction and separation loss equations already defined are essential to determine the values of these terms in a fluid network, the designer is often faced with the need to link delivered flow, whether liquid or gas, to the overall ‘cost’ in terms of the pressure loss to be overcome by a fan or pump, or by gravity if the terrain allows that option. Thus an expression linking flow rate Q to pressure loss per unit length of system ∆pL and incorporating the roughness of the pipe material is required. It will be appreciated that use of the equivalent length approach to represent separation losses is particularly attractive within this methodology. By combining the Darcy equation for the pressure loss due to friction with the Colebrook–White friction factor relationship it is possible to eliminate the friction factor and arrive at a relationship linking, for any given fluid in a given duct or pipe, the flow rate to the conduit diameter and pressure loss per unit length. This technique was of considerable interest to network designers prior to the advent of readily available computer models. While the methodology remains of interest it should be seen as the basis for a computerized approach; clearly the design curves to be demonstrated could be made available within a program. From the Darcy equation ∆p ρ fv 2 −−− = −−−−−−, L 2m where m = D4 for circular cross-section conduits and v = QA, A being the crosssectional area of the conduit. Thus

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∆p 4 ρ fQ 2 −−− = −−−−−−−−−−−−−−−−−−2 2D ( π D 24 ) L ∆p 64 ρ fQ 2 −−− = −−−−−−2−−−−−. 2 π D5 L Solving for the friction factor f, yields 2D 5 π 2 ∆pL f = −−−−−−−−−−−−−−2−−−−. 64 ρ Q From the definition of Reynolds number, for a circular cross-section conduit, Re = ρVDµ = ρQDµA = 4ρQµπD. (Note that a similar approach could be followed for any cross-sectional shape based upon the hydraulic mean depth m.) Substituting for both f and Re in the Colebrook–White equation yields the required relationship between Q, ∆pL and the conduit surface roughness: 1 k 1.255 −− = – 4 log ⎛ −−−−−−−−− + −−−−−−− ⎞ ⎝ 3.71D Ref ⎠ f Q −−−−−−−−−−5−−−−2−−−−−−−−−−−−−−−−−−− [ ( 2D π ∆pL )64 ρ ] 1.255 ⎧ k ⎫ −−−−−−−−−−−−−−−−−−−−−2− ⎬. = – 4 log ⎨ −−−−−−−−− + −−−−−−−−−−−−−−−−−−−−−−−−−−−−−5−−−− 2 ⎩ 3.71D ( 4 ρ Qµπ D ) [ ( 2D π ∆pL )64 ρ Q ] ⎭

(10.70)

For a given fluid, i.e. density and viscosity constant and assuming a circular crosssection conduit, this expression reduces to

D ⎧ k ⎫ Q = – C 1 [ ( ∆pL )D 5 ] log ⎨ −−−−−−−−− + C 2 −−−−−−−−−−−−−−−−−−5− ⎬. 3.71D [ ( ∆pL )D ] ⎭ ⎩

(10.71)

This expression may be solved for any circular cross-section conduit or may be presented in graphical form as illustrated in Fig. 10.16. The curves illustrate a number of fundamental relationships that govern pressure loss in duct and pipe flow: 1.

For a given flow rate the pressure loss rises as the conduit diameter reduces. The Darcy equation has already indicated that this is a fifth-power relationship.

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FIGURE 10.16 Schematic of the circular duct sizing charts for air at 20 °C that may be constructed from the Colebrook–White loss equations

2.

It is possible to retain the same flow velocity as flow rate rises by increasing the conduit diameter. The schematic chart in Fig. 10.16 illustrates this option, which might be important if acoustic considerations are important in the design of air ducting.

The inclusion of fittings in a pipe or duct network can be the main source of pressure loss. In order to represent such separation losses the addition of the fitting equivalent length to the length of the conduit allows the overall pressure loss to be calculated from the chart. It is stressed that this technique would in all probability now form the basis for computer-aided design calculations, but it does serve to illustrate some of the fundamental principles involved.

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10.10 COMPUTER PROGRAM

CBW

Program CBW provides a calculation of friction factor in a circular cross-section conduit flowing full of a given fluid by application of the Colebrook–White equation (10.60). The calculation requires pipe diameter D (m), pipe roughness k (mm), either steady flowrate Q (m3 s−1) or mean velocity V (m s−1), fluid density ρ (kg m−3) and either the dynamic viscosity µ (N s m−2), or kinematic viscosity υ (m2 s−1), for the fluid. The program output simply consists of a screen-displayed value for friction factor f.

10.10.1 Application example Using the following data for a water carrying pipeline: D = 0.1 m; k = 0.15 mm; Q = 0.01 m3 s−1; ρ = 1000.0 kg m−3; µ = 1 × 10−3 N s m−2, yields a value of friction factor f = 0.005 76 at Reynolds number = 127 324 and mean velocity V = 1.273 m s−1. For an identical air flow the value of friction factor becomes f = 0.008 3 at Reynolds number = 9 708.

10.10.2 Additional investigations using CBW The program may be used to investigate: 1. 2. 3.

the influence of pipe properties on friction factor, including diameter and roughness; the effect of fluid properties on Reynolds number and friction factor; the accuracy of Colebrook–White in the laminar region.

Concluding remarks Chapter 10 has drawn upon earlier material both to differentiate between laminar and turbulent flows and to investigate the dependence of flow frictional losses on conduit and fluid properties. While the laminar flow study indicated that it was possible to develop theoretical expressions for both frictional losses and the velocity profiles within a bounded fluid stream, the equivalent expressions for turbulent flow require an empirical friction factor, defined either graphically by the Moody chart or by the Colebrook–White equation. The Darcy equation for frictional loss was developed and shown to be the equivalent of the Chezy expression for open channels. This understanding that the fundamental equations apply across the boundary between full-bore and free surface flow, and are independent of conduit cross-section so long as the flow remains uniform, is important and will be shown later to apply to both steady flows (Chapters 14 and 15) and unsteady flows (Chapters 20 and 21). Frictional losses have been shown not to be the sole cause of resistance to flow in conduits. The concept of separation loss was introduced and shown to be dependent

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Further reading

363

upon flow kinetic energy. The introduction of separation losses will enable a later treatment of flow balancing within networks (Chapter 14) and will also be important in the study of unsteady flow, particularly the effects of valve closure, in Chapter 20. The definition of separation losses in terms of the equivalent pipe length that would be necessary to generate the same effect via frictional loss was introduced and shown to be particularly helpful in system design via the Colebrook–White-based charts introduced in this chapter. A treatment of velocity profiles within fully developed pipe flow, and similarly within free surface flows, introduced the concept of the boundary layer and further differentiated the flow regimes present that will be referred to in later material. The definitions of frictional and separation loss introduced in this chapter will be utilized throughout the remainder of the text.

Summary of important equations and concepts 1.

2.

3.

4.

This chapter emphasizes the differences between laminar and turbulent flow regimes, deriving the Hagen–Poiseuille equation (10.21) for laminar flows and contrasting this to the empirical approach necessary in turbulent flows, e.g. the Darcy and Chezy equations, (10.30) and (10.29) respectively, relying on the Colebrook–White expression for friction factor, equation (10.60). The comparison of the open-channel Chezy and full-bore flow Darcy equations is important, relying upon the definition of hydraulic mean depth, i.e. (cross-sectional area A)(wetted perimeter P). While the velocity profile in laminar flow is derived, equations (10.5) and (10.20), velocity profiles in turbulent flow are dependent upon empirical results. Typical velocity distributions for full-bore flows and free surface channel flows are presented, in the latter case the observation that the free surface has a centre line velocity less than the maximum for the profile is important. The application of the ‘no-slip’ condition is stressed. The expression of fitting separation losses in a form compatible with frictional losses, is essential to the easy application of the steady flow energy equation, Section 6.4.

Further reading CIBSE (2001). Guide to Current Practice, Vol. C, Chartered Institution of Building Services Engineers. Colebrook, C. F. (1939). Turbulent flow in pipes with particular reference to the transition region between smooth and rough pipe laws. Journal of the Institution of Civil Engineers, 11 (4), 133–56. Jepson, R. W. (1976). Analysis of Flow in Pipe Networks. Ann Arbor Science, Ann Arbor. Moody, L. F. (1944). Friction factors for pipe flow. Transactions of the ASME, 66, 671–84. Ward-Smith, A. J. (1980). Internal Fluid Flow: The Fluid Dynamics of Flow in Pipes and Ducts. Oxford University Press.

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Laminar and Turbulent Flows in Bounded Systems

Problems 10.1 Show that, for laminar flow between two infinite, moving flate plates, distance z apart, the flow rate Q is given by an expression of the form

10.8 For laminar flow in a tube calculate the position of the average cross-sectional velocity. [0.293 × radius from tube wall]

1 dp z 3 z Q = – − −−− −−− + − ( U + V ), µ dl 12 2

10.9 An oil having a viscosity of 0.048 kg m−1 s−1 flows through a 50 mm diameter tube at an average velocity of 0.12 m s−1. Calculate the pressure drop in 65 m of tube and the velocity 10 mm from the tube wall. [4.8 kN m−2, 0.154 m s−1]

where µ is the fluid viscosity, dpdl is the pressure gradient in the flow direction and U and V are the absolute velocities of the two plates. 10.2 For the case set out in Problem 10.1 above derive the shear stress expressions for each plate surface: U′ z dp τ 0 = µ ⎛ −−− – −−− −−− ⎞ , ⎝ z 2 µ dl ⎠ U′ z dp τ z = µ ⎛ −−− + −−− −−− ⎞ , ⎝ z 2 µ dl ⎠

where U′ = ( U + V ).

10.3 A thin film of oil, thickness z and viscosity µ, flows down an inclined plate. Show that the velocity profile is given by

ρg u = −−− ( z 2 – y 2 ) sin θ , 2µ where u is the local velocity at a depth y below the free surface, θ is the plate inclination to the horizontal and ρ is the fluid density. 10.4 For the case in Problem 10.3 above calculate the flow rate per unit plate width if the fluid has a viscosity of 0.9 N s m−2, a density of 1260 kg m−3, the plate is inclined at 30° and the depth of flow is 10 mm. [0.137 litres min−1 m−1] 10.5 A film of fluid, density 2400 kg m−3, flows down a vertical plate with a free surface velocity of 0.75 m s−1. If the film is 20 mm thick determine the fluid viscosity. [6.28 N s m−2] 10.6 Fluid of density 1260 kg m−3 and viscosity 0.9 N s m−2 passes between two infinite parallel plates, 2 cm separation. If the flow rate is 0.5 litres s−1 per unit width calculate the pressure drop per unit length if both plates are stationary. [0.68 kN m−2 m−1] 10.7 The radial clearance between a hydraulic plunger and the cylinder wall is 0.15 mm, the length of the plunger 0.25 m and the diameter 150 mm. Calculate the leakage rate past the plunger at an instant when the pressure differential between the two ends of the plunger is 15 m of water. Viscosity of hydraulic fluids is 0.9 N s m−2. [5.2 × 10−3 litres min−1]

10.10 Oil of specific gravity 0.9 and kinematic viscosity 0.000 33 m2 s−1 is pumped over a distance of 1.5 km through a 75 mm diameter tube at a rate of 25 × 103 kg h−1. Determine whether the flow is laminar and calculate the pumping power required, assuming 70 per cent mechanical efficiency. [Re = 397, laminar, 48.8 kW] 10.11 For the flow conditions set out in Problem 10.10 above calculate the shear stress at the tube walls. [55.3 N m−2] 10.12 Air at 20 °C is drawn through a 0.5 m diameter duct by a fan. If the volume flow rate is 4.5 m3 s−1 and the duct is 12 m long, with a friction factor of 0.005, determine the fan shaft power necessary, assuming 80 per cent mechanical efficiency. Take air density as 1.2 kg m−3 and viscosity as 1.8 × 10−5 N s m−2. [0.85 kW] 10.13 Water at a density of 998 kg m−3 and kinematic viscosity 1 × 10−6 m2 s−1 flows through smooth tubing at a mean velocity of 2 m s−1. If the tube diameter is 30 mm calculate the pressure gradient per unit length necessary. Assume that the friction factor for a smooth pipe is given by 16Re for laminar flow and 0.079Re14 for turbulent flow. [1.34 kN m−2 m−1] 10.14 In a laboratory the water supply is drawn from a roof storage tank 25 m above the water discharge point. If the friction factor is 0.008, the pipe diameter is 5 cm and the pipe is assumed vertical, calculate the maximum volume flow achievable, if separation losses are ignored. [0.01 m3 s−1] 10.15 For the case set out in Problem 10.14 above calculate the relative roughness of the pipe used, if the water is at 0 °C. [0.006] 10.16 The friction factor applicable to turbulent flow in a smooth glass pipe is given by f = 0.079Re14. Calculate the pressure loss per unit length necessary to maintain a flow of 0.02 m3 s−1 of kerosene, specific gravity 0.82, viscosity 1.9 × 10−3 N s m−2, in a glass pipe of 8 cm diameter. If the tube is replaced by a galvanized steel pipeline, wall roughness

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Problems 0.15 mm, calculate the increase in pipe diameter to handle this flow with the same pressure gradient. [1332 N m−2 m−1, 6.75 per cent] 10.17 Define static regain along a diffuser and show that it may be calculated as Static regain = −12 ρ ( V 21 – V 22 ) – −12 K ρ V 21 . 10.18 A 150 mm diameter pipe reduces in diameter abruptly to 100 mm. If the pipe carries water at 30 litres s−1

365

calculate the pressure loss across the contraction and express this as a percentage of the loss to be expected if the flow was reversed. Take the coefficient of contraction as 0.6. [3.2 kN m−2, 143 per cent] 10.19 An air duct, carrying a volume Q of air per second, is abruptly changed in section. Deduce the diameter ratio for the two duct sections if the pressure loss is to be independent of flow direction. Assume a value of 0.6 for the contraction coefficient. [1.732]

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Chapter 11

Boundary Layer 11.1 11.2

11.3 11.4

11.5 11.6 11.7

Qualitative description of the boundary layer Dependence of pipe flow on boundary layer development at entry Factors affecting transition from laminar to turbulent flow regimes Discussion of flow patterns and regions within the turbulent boundary layer Prandtl mixing length theory Definitions of boundary layer thicknesses Application of the momentum equation to a general section of boundary layer

Properties of the laminar boundary layer formed over a flat plate in the absence of a pressure gradient in the flow direction 11.9 Properties of the turbulent boundary layer over a flat plate in the absence of a pressure gradient in the flow direction 11.10 Effect of surface roughness on turbulent boundary layer development and skin friction coefficients 11.11 Effect of pressure gradient on boundary layer development 11.8

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At the interface between a fluid and a surface in relative motion a condition known as ‘no slip’ dictates an equivalence between fluid and surface velocities. Away from the surface the fluid velocity rapidly increases; the zone in which this occurs is known as the boundary layer and its definition is fundamental to all calculations of surface drag and viscous forces. This chapter will present both a qualitative and quantitative treatment of the development of boundary layers, both laminar and turbulent, including the laminar sublayer, and will introduce the velocity profiles appropriate to each. The effect of the boundary layer on the velocity

profiles already discussed in terms of bounded flows, together with definitions of the boundary layer in terms of its physical thickness and its effect upon the flow, quantified in terms of displacement and momentum thickness, are presented. The dependence of boundary layer effects on shear stress, Reynolds number and surface roughness will be discussed and the application of the momentum equation in the determination of skin friction demonstrated. The sensitivity of a boundary layer to pressure gradients imposed upon the flow will be discussed, with particular reference to flow separation over aerofoil sections and the loss of lift. l l l

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The drag on a body passing through a fluid may be considered to be made up of two components: the form drag, which is dependent on the pressure forces acting on the body; and the skin friction drag, which depends on the shearing forces acting between the body and the fluid. Form drag will be dealt with in detail in Chapter 12, while the mechanics of skin friction will be covered in this chapter. In Chapter 10, it was shown that in both laminar and turbulent flow in pipes the fluid velocity is not uniform but varies from zero at the wall to a maximum at the pipe centre. It was further shown that, in general, the velocity distribution is dependent upon the Reynolds number, which defines the type of flow. This chapter will be concerned with the analysis of the effects the fluid viscosity has on the velocity gradient near a solid boundary and, hence, how it affects the skin friction. Such analysis is most conveniently carried out by the consideration of flow over a flat plate of infinite width. The shear stress on a smooth plate is a direct function of the velocity gradient at the surface of the plate. That a velocity gradient should exist in a direction perpendicular to the surface is evident, because the particles of fluid adjacent to the surface are stationary whilst those some distance above the surface move with some velocity. The condition of zero fluid velocity at the solid surface is referred to as ‘no slip’ and the layer of fluid between the surface and the free stream fluid is termed the boundary layer. Thus, it will be appreciated that any calculations of surface resistance or skin friction forces will obviously involve the integration of the shear stress at the surface over the whole fluid immersed area and will be directly concerned with the patterns of flow within the boundary layer. Within this context, the importance of Reynolds number becomes self-evident as, with the dramatic change in particle motion consequent upon a transition from a laminar to a turbulent type of flow, considerable changes in boundary layer flow patterns and velocity gradients must be expected that will materially affect any calculations of surface resistance.

11.1 QUALITATIVE DESCRIPTION OF THE BOUNDARY LAYER As mentioned above, the boundary layer is taken as that region of fluid close to the surface immersed in the flowing fluid. Figure 11.1 illustrates such a flat plate in a free fluid stream. Only the top surface boundary layer is shown but there will, in practice, be symmetry between the upper and lower surface boundary layers, provided both surfaces are identical in nature. The fluid in contact with the plate surface has zero velocity, ‘no slip’, and a velocity gradient exists between the fluid in the free stream and the plate surface. Now, shear stress may be defined (equation (10.3) ) as

∂u τ = µ −−− , ∂y

(11.1)

where τ is the shear stress, µ the fluid viscosity and dudy the velocity gradient. This shear stress acting at the plate surface sets up a shear force which opposes the fluid motion, and fluid close to the wall is decelerated. Further along the plate, the shear force is effectively increased owing to the increasing plate surface area affected,

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11.1

Qualitative description of the boundary layer

369

FIGURE 11.1 Development of the boundary layer along a flat plate, illustrating variations in layer thickness and wall shear stress

so that more and more of the fluid is retarded and the thickness of the fluid layer affected increases, as shown in Fig. 11.1. Returning to the Reynolds number concept, if the Reynolds number locally is based on the distance from the leading edge of the plate, then it will be appreciated that, initially, the value is low, so that the fluid flow close to the wall may be categorized as laminar. However, as the distance from the leading edge increases so does Reynolds number, until a point must be reached where the flow regime becomes turbulent. For smooth, polished plates the transition may be delayed until Re equals 500 000. However, for rough plates or for turbulent approach flows, transition may occur at much lower values. Again, the transition does not occur in practice at one well-defined point but, rather, a transition zone is established between the two flow regimes, as shown in Fig. 11.1. The random particle motion characterizing turbulent flow results in a far more rapid growth of the boundary layer in the turbulent region, so that the velocity gradient at the wall increases, as does the corresponding shear force opposing motion. Figure 11.1 also depicts the distribution of shear stress along the plate in the flow direction. At the leading edge, the velocity gradient is large, resulting in a high shear stress. However, as the laminar region progresses, so the velocity gradient and shear stress decrease with thickening of the boundary layer. Following transition the velocity gradient again increases, and the shear stress rises. Theoretically, for an infinite plate, the boundary layer goes on thickening indefinitely. However, in practice, the growth is curtailed by other surfaces in the vicinity. This is particularly the case for boundary layers within ducts, as will be

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described in Section 11.2, where the growth is terminated when the boundary layers from opposite duct surfaces meet on the duct centreline. It must be appreciated that the thickness of the boundary layer, δ in Fig. 11.1, is much smaller than x.

11.2 DEPENDENCE OF PIPE FLOW ON BOUNDARY LAYER DEVELOPMENT AT ENTRY The types of fluid flow considered in Chapter 10 were confined to steady and uniform flow. The assumption of steady, uniform flow conditions led to the simplifying condition that the fluid elements were under no acceleration, either spatial or temporal, and allowed the development of the equations set out there. This assumption implies that the flow conditions are fully established and are not subject to any changes. This is, obviously, not the case in the initial length of a pipe where the boundary layer is still developing and growing in thickness up to its maximum, which for a closed pipe will be the pipe radius. Initially, as the boundary layer develops, it will be laminar in form. However, as described earlier, the boundary layer will become turbulent, depending upon the ratio of inertial and viscous forces acting on the fluid, this condition being normally monitored by reference to the value of the flow Reynolds number. For pipe flow, it is normal practice to base the Reynolds number, Re, on the mean flow velocity and the pipe diameter. Generally, for values of Re 2000, the flow may be assumed to be laminar, although it has been shown possible to maintain laminar flow at higher values of Reynolds number under specialized laboratory conditions. Above Re = 2000 it is, however, reasonable to suppose that the flow will be turbulent and that the boundary layer development will include a transition and a turbulent region, as described for the flat plate. The only major difference is that, in the pipe flow case, there is a limit to the growth of the boundary layer thickness, namely the pipe radius. If, therefore, this limit is reached before transition occurs, i.e. if laminar boundary layers meet at the pipe centre, the flow in the remainder of the pipe will be laminar. On the other hand, if transition within the boundary layer occurs before they fill the pipe, the flow in the rest of the pipe will be turbulent. These two cases are illustrated in Fig. 11.2. Once the boundary layer, whether laminar or turbulent in nature, has grown to fill the whole pipe cross-section, the flow may be said to be fully developed and no further changes in velocity profile are to be expected downstream, provided that the pipeline characteristics (i.e. diameter, surface roughness) remain constant. Theoretically, the entry length for a particular pipe (i.e. the distance from entry at which a laminar or turbulent boundary layer ceases to grow) is infinite. However, it is normally assumed that the flow has become fully developed when the maximum velocity, at the pipe centreline, becomes 0.99 of the theoretical maximum. Using this approximation, typical entry lengths for the establishment of fully developed laminar or turbulent flow may be taken as 120 and 60 pipe diameters, respectively. The entry length characteristic of turbulent flow is the shorter owing to the higher growth rate of the turbulent boundary layer. Thus, the assumption of steady, uniform flow restricts the application of the equations derived for pipe flow to that part of a conduit beyond the entry length.

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11.3

Factors affecting transition from laminar to turbulent flow regimes

371

FIGURE 11.2 Development of fully developed laminar and turbulent flow in a circular pipe. (a) Laminar flow conditions, Re 2000. (b) Turbulent flow conditions, Re 2000.

Normally, this is not a serious restriction as the entry length is usually small compared with the total length of the pipeline. Similarly, all the equations derived for laminar flow have depended on the shear stress–viscosity relation of equation (11.1), τ = µ dudy. In turbulent flow, owing to the random nature of the motion of the fluid particles, the apparent shear stress may be expressed as du τ = ( µ + ε ) −−− , dy

(11.2)

where ε is the eddy viscosity and is often much larger than µ. Since eddy viscosity is difficult to determine, equations dealing with the calculations of pressure loss associated with turbulent flow in pipes, established in Chapter 10, were developed, introducing the concept of an empirical friction factor. However, it will be shown that the friction factor is related to the skin friction coefficient defined later in this chapter.

11.3 FACTORS AFFECTING TRANSITION FROM LAMINAR TO TURBULENT FLOW REGIMES As mentioned above, the transition from laminar to turbulent boundary layer conditions may be considered as Reynolds number dependent, Re = ρUs xµ, and a figure of 5 × 105 is often quoted. However, this figure may be considerably reduced if the surface is rough. For Re 105, the laminar layer is stable; however, at Re near 2 × 105 it is difficult to prevent transition. The presence of a pressure gradient d pd x can also be a major factor. Generally, if d pd x is positive, then transition Reynolds number is reduced, a negative d pd x increasing transition Reynolds number. This effect forms the basis of suction high-lift devices designed for aircraft wings. Figure 11.3 illustrates typical velocity profiles through the boundary layer in both the laminar and turbulent regions, the increased velocity gradient dudy being apparent. As mentioned, the growth of the boundary

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FIGURE 11.3 Typical velocity profiles in the laminar and turbulent boundary layer regions

layer thickness is more rapid in the turbulent region, roughly varying as x 0.8 here compared with x 0.5 in the laminar region. In calculations involving long plates, it is often reasonable to suppose that transition occurs close to the leading edge and, in such cases, the presence of the laminar section may be ignored. The study of the turbulent boundary layer is the more important as in most engineering applications the flow Reynolds number is sufficiently high to ensure transition and the establishment of a turbulent boundary layer. However, it will be appreciated that the random motion of the fluid particles must die out very close to the surface to maintain the condition of ‘no slip’ at the planefluid interface. To accommodate this, the presence of a laminar sublayer in the turbulent region has been established, the thickness of this being small compared with the local boundary layer thickness, as shown in Fig. 11.1. The velocity profile across this sublayer is assumed linear and tangential to the velocity profile up through the turbulent boundary layer.

11.4 DISCUSSION OF FLOW PATTERNS AND REGIONS WITHIN THE TURBULENT BOUNDARY LAYER Figure 11.4 illustrates the velocity distribution through one particular section in a turbulent boundary layer. As mentioned above, very close to the plane surface the flow remains laminar and a linear velocity profile may be assumed. In this region, the velocity gradient is governed by the fluid viscosity (equation (11.1)): du −−− = τ 0 µ . dy Rearranging this expression yields, after integration,

or

u = (τ0µ)y

(11.3)

( τ 0 ρ ) u −−−−−−−−−− = −−−−−−−−−− y, ν ( τ 0 ρ )

(11.4)

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11.4

Discussion of flow patterns and regions within the turbulent boundary layer

373

FIGURE 11.4 Eddy formation in the boundary layer

where v is the fluid kinematic viscosity µρ. The term (τ 0ρ) is common in boundary layer theory and is termed the shear stress velocity due to the units of velocity applicable to the combination (see Chapter 10) and is denoted by u*. Thus, u y −−− = −−−−−− . u* vu*

(11.5)

Experimentally, it has been shown that the laminar sublayer occurs in flows where equation (11.5) has a value less than approximately 5, so that the thickness of the laminar sublayer y = δ ′ becomes

δ ′ = 5vu*,

(11.6)

indicating that the sublayer thickness will be small for large shear stress flows, i.e. u* large, and that it will increase in the downstream direction as shear stress decreases in this direction. Above the laminar sublayer, the flow regime is turbulent and equation (11.1) no longer adequately represents the shear forces acting. It is appropriate here to describe the mechanism of flow within this upper region. Owing to the random motion of the fluid particles, eddy patterns are set up in the boundary layer which sweep small masses of fluid up and down through the boundary layer, moving in a direction perpendicular to the surface and the mean flow direction. Owing to these eddies, fluid from the upper higher-velocity areas is forced into the slower-moving stream above the laminar sublayer, having the effect of increasing the local velocity here relative to its value in the laminar sublayer. This increase in velocity of fluid close to the wall is shown in Fig. 11.3. Conversely, slow-moving fluid is lifted into the upper levels, slowing down the fluid stream and, by doing so, effectively thickening the boundary layer, explaining the more rapid growth of the turbulent boundary layer compared with the laminar one. The process described is, effectively, a momentum transfer phenomenon. However, the effect is analogous to a shear stress applied to the fluid as the overall deceleration is increased as boundary layer thickness increases. In order to explain this process, and to retain the useful form of equation (11.1), a new viscosity term may be introduced, the eddy viscosity, ε, and equation (11.1) rewritten as the relationship mentioned earlier (equation (11.2)),

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FIGURE 11.5 Velocity fluctuations in the mean flow direction and normal to the surface at a point in the turbulent boundary layer

du τ = ( ε + µ ) −−− . dy Figure 11.5 illustrates the likely output from a velocity-measuring device positioned within the turbulent boundary layer. Here it will be seen that, although the mean velocity is in the flow direction, there are fluctuations in velocity corresponding to the random particle motion. If the fluid velocity is made up of a mean value B and fluctuating components u′ and v′ in the flow direction and perpendicular to it, respectively, then it may be assumed that the apparent shear stress required to duplicate the eddy effects discussed above would be

τ = −ρB′C′,

(11.7)

i.e. the shear stress opposing motion is given by the product of fluid density and the average product of the normal velocity fluctuations over an incremental time period.

11.5 PRANDTL MIXING LENGTH THEORY In the form of equation (11.7), little further may be done. However, Prandtl (1875– 1953) – who, in 1904, was responsible for stating the basics of boundary layer theory and proposing that all viscous effects are concentrated within it – developed the necessary theory to relate the apparent shear stress to mean velocity distribution through the boundary layer. This theory, summarized below, is known as the Prandtl mixing length theory. Prandtl defined the mixing length as that distance l in which a particle loses its excess momentum and assumes the mean velocity of its surroundings, an idea in some respects similar to the mean free path. In practice, the loss or transfer of momentum would be gradual over the length l. Assuming that the changes of velocity u′ and v ′ following from this particle motion would be equal – a not unreasonable assumption – it may be seen that v′ = u′ = l dudy (Fig. 11.6) and, from equation (11.7),

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11.5

Prandtl mixing length theory

375

FIGURE 11.6 Concept of mixing length in Prandtl’s theory

du 2 τ = ρ l 2 ⎛ −−−⎞ . ⎝ dy⎠

(11.8)

Close to the surface, Prandtl assumed that l became dependent on the distance from the surface, or l = ky. This allows for l to have zero value at the boundary where y = 0. Hence du 2 τ = ρ k 2 y 2 ⎛ −−−⎞ , ⎝ dy⎠ where k was proposed as a universal constant having a value around 0.4. More recent work has shown distinct limitations in this approach, and values varying from 0.4 have been recorded. However, the Prandtl mixing length theory was a major advance at the time and may still be of use in particular situations. Close to the surface it may be assumed that the shear stress equals the surface value, so du 2 τ 0 = ρ k 2 y 2 ⎛ −−−⎞ ⎝ dy⎠ or

(11.9)

( τ 0 ρ ) dy du = −−−−−−−−−−− −−− , k y

which, on integration, yields uu* = (1k) log e y + C.

(11.10)

Values of the integration constant C have been experimentally determined in the form C = 5.56 − (1k) log e(vu*), so that a velocity distribution uu* = (1k) log e [y(u*v)] + 5.56

(11.11)

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FIGURE 11.7 Overlap of velocity distributions

is obtained. Substituting 0.4 for k yields uu* = 5.75 log10 [y(u*v)] + 5.56

(11.12)

in terms of log base 10. Comparison of equation (11.12), which applies for 30 yu*v 500, with equation (11.5) shows that there is a major change in velocity profile between the laminar sublayer and the turbulent boundary layer region. However, both profiles are related through the yu*v term. Figure 11.7 illustrates these profiles. However, above yu*v = 500, experimental results indicate that a better fit is obtained by a velocity defect law of the form (Us − u)u* = f(yδ ).

(11.13)

Thus, three zones of application of velocity distribution equations are apparent. These zones do not possess sharp boundaries; rather they merge into each other. In the intersection zones, experimental results straddle the predictions of each equation; however, the general boundaries of 30 and 500 for yu*v are adequate in this treatment. In terms of experimental results, a simplified velocity profile, which applies to 90 per cent of the boundary layer thickness but not to the 10 per cent close to the plane surface, was proposed by Prandtl: uUs = (yδ )n.

(11.14) 1

The value of n for Reynolds numbers in the region 105 Re 107 may be taken as −7 and the expression (11.14) is known as the seventh-power law. In addition, it is usually assumed that the velocity profile through the laminar sublayer is linear and tangential to the seventh-power law.

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11.6

Definitions of boundary layer thicknesses

377

11.6 DEFINITIONS OF BOUNDARY LAYER THICKNESSES So far the boundary layer thickness has been referred to only in physical terms; namely, boundary layer thickness is defined as that distance from the surface where the local velocity equals 99 per cent of the free stream velocity:

δ = y(u = 0.99Us ) ,

(11.15)

where Us is the free stream velocity. It is possible, however, to define boundary layer thickness in terms of the effect on the flow.

11.6.1 Displacement thickness δ * Owing to the presence of the boundary layer, the flow past a given point on the surface is reduced by a volume equivalent to the area ABC in Fig. 11.8. This volume reduction is given by the integral 1(Us − u)dy. If the area ABC is equated to an area ABDE, whose volume may be calculated as δ *Us, then the displacement thickness for the boundary layer may be defined as the distance the surface would have to move in the y direction to reduce the flow passing by a volume equivalent to the real effect of the boundary layer:

FIGURE 11.8 Displacement thickness

∞

δ* =

(1 – uU ) dy. s

(11.16)

11.6.2 Momentum thickness θ The fluid passing through the element δy carries momentum at a rate ( ρuδy)u per unit width, whereas, in the absence of the boundary layer, u would equal Us, so that the total reduction in momentum flow is ∞

ρ(U – u)u dy, s

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FIGURE 11.9 Control volume applied to a general section of boundary layer over a flat plate

which may be equated to the momentum carried through a section θ deep per unit width at the free stream velocity (Fig. 11.9): ∞

( ρ U s θ )U s =

ρ(U – u)u dy, s

∞

θ=

−Uu−− ⎛⎝1 – U−u−− ⎞⎠ dy. 0

s

(11.17)

s

11.7 APPLICATION OF THE MOMENTUM EQUATION TO A GENERAL SECTION OF BOUNDARY LAYER Von Kármán first applied the momentum equation to a general section of a boundary layer. Regardless of the position of the section in either the laminar or turbulent boundary layer regions, it is possible to equate the skin friction drag force to the product of rate of change of momentum and mass of fluid affected by the boundary layer. Figure 11.9 illustrates a control volume ABCD around a general section of boundary layer. It will be assumed that the flow continues to be incompressible and that as dpdx, the pressure gradient in the flow direction, is also zero so will be dUs dx, i.e. any change in free stream velocity. Flow enters the control volume through AB and BC as shown and leaves via CD. Assuming a unit width of surface, then the momentum equation applied to this control volume in the flow direction becomes – τ 0 ∆x =

δ2

ρ u 22 dy –

δ1

ρu dy – ρU (δ – δ ). 2 1

2 s

2

1

(11.18)

Taking the terms in order: τ0 ∆x represents the shear force, opposing motion in the positive flow direction, over the immersed area ∆x;

δ2 0

ρ u 22 dy =

ρu dyu δ2

2

2

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is the rate of momentum transfer through a section dy on CD at a height y above the surface, where the local velocity is u2; the third term is identical in form to the preceding integral except that it applies to a section dy on AB; the last term is a measure of the momentum, in the flow direction, carried into the control volume across the boundary layer upper surface BC. But Us ( δ2 – δ1 ) =

δ2

δ1

u 2 dy –

u dy,

(11.19)

1

i.e. the difference in flow rates past AB and CD. Hence, – τ 0 ∆x =

δ2

ρ u 22 dy –

=ρ

⎛ ρ u 21 dy – ρ U s ⎜ ⎝ 0 δ1

δ2

( u 22 – U s u 2 ) dy – 0

⎧ ⎪ = ρU ⎨ ⎪ ⎩ 2 s

δ2

⎞

δ1

u 2 dy – 0

u dy⎟⎠ 1

δ1

(u – U u ) dy 2 1

s 1

δ2 0

2 ⎛ −u−−2 ⎞ – −u−−2 dy – ⎝ U s⎠ U s

δ1 0

⎫ 2 ⎛ −u−−1 ⎞ – −u−−1 dy ⎪⎬. ⎝ U s⎠ U s ⎪ ⎭

As ∆x approaches zero in the limit, and multiplying both sides by −1, the equation above reduces to d τ 0 = ρ U 2s −−− dx

δ

−Uu−− ⎛⎝1 – U−u−− ⎞⎠ dy 0

s

(11.20)

s

or, by reference to the momentum thickness defined in equation (11.17), dθ τ 0 = ρ U 2s −−− . dx

(11.21)

Equation (11.20) is the momentum equation applied to a general boundary layer section and is of general use in deriving further relations in the boundary layer. It is also of use in the area of heat transfer through boundary layers, although these applications are outside the scope of this text. In the following sections, more detailed relations applying to the laminar and turbulent boundary layers individually will be presented.

11.8 PROPERTIES OF THE LAMINAR BOUNDARY LAYER FORMED OVER A FLAT PLATE IN THE ABSENCE OF A PRESSURE GRADIENT IN THE FLOW DIRECTION In practice, the laminar section of the boundary layer formed as a result of flow over a surface is short; however, it will always exist, even in flows that are nominally turbulent. For example, consider the inlet section of a circular cross-section duct. As discussed in Chapter 10, the flow in the duct will be considered turbulent if the

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Reynolds number based on mean fluid velocity and duct diameter Re = ρBdµ 2000. However, as far as the boundary layer is concerned, the transition from laminar to turbulent occurs at Reynolds numbers above 105 based on mean fluid velocity and distance measured from the entry to the duct, Re = ρUs xµ, so that there will always be a finite length of laminar boundary layer. Blasius developed a series of analytical solutions for the laminar boundary layer, which will be quoted in this section and compared with approximate results that may be derived from the assumptions already discussed: namely, a linear relation between shear stress and vertical distance to the surface and the absence of a pressure gradient across the flat surface. In all laminar flow, equation (11.1) applies, i.e. du τ 0 = µ ⎛ −−−⎞ , ⎝ dy⎠ y=0

(11.22)

and, from equation (11.20), the momentum equation, du d µ ⎛ −−−⎞ = ρ −−− ⎝ dy⎠ y=0 dx

δ

u(U – u) dy.

(11.23)

s

If the assumption is made that velocity profiles through the boundary layer are geometrically similar along the whole length of the laminar section, then this may be expressed as u = Us f(η),

(11.24)

where η = yδ, u = 0 at y = 0, or η = 0, u = Us at y = δ, or η = 1. Substituting into (11.23) yields

µ df (η) − U s −−−−−−− δ dη

η =0

d ⎧ = ρ −−− ⎨ U 2s δ dx ⎩

[1 – f (η)] f (η) dη ⎬⎭, 1

⎫

(11.25)

where the limits of integration have been changed as η = 1 at y = δ. Owing to the geometric similarity of velocity profiles, f (η) is independent of position along the laminar section x, so that

[1 – f (η)] f (η) dη 1

is independent of x and may be regarded as a constant C1 and [∂ f(η)∂η] 0 as a constant C2. As a result, equation (11.25) becomes

µ dδ − U s C 2 = ρ U 2s C 1 −−− δ dx or

C2 dδ µ −−− = ρ U s δ −−− . C1 dx

Integrating,

δ 22 = (µρUs)(C2 C1)x + constant, so

δ = [(2C2C1)( µρUs)]x = (2C2C1)x(Re)

(11.26)

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if δ = 0 at x = 0, i.e. zero boundary layer thickness at the leading edge, and Rex = ρUs xµ. From equation (11.25), 2C 2 µ d ( x 12 ) dδ τ 0 = ρ U 2s C 1 −−− = ρ U 2s C 1 ⎛ −−−−− −−−−− ⎞ −−−−−−−−−−, ⎝ C 1 ρ U s ⎠ dx dx

2C 2 µ x –1 2 C1 C2 τ 0 = ρ U 2s C 1 ⎛ −−−−−−−−− ⎞ −−−−−− = ρ U 2s ⎛ −−−−−−− ⎞ ⎝ C1 ρ Us ⎠ 2 ⎝ 2Re x ⎠

(11.27)

and, from the integral of shear stress at the wall over the length l of the laminar boundary layer, the total skin friction force per unit width of surface may be written as F=

l

ρU C −dxd−−δ dx = (ρU C δ ) l

τ 0 dx =

2 s

1

2 s

1

l 0

2C 1 C 2 = ρ U 2s l ⎛ −−−−−−−−− ⎞ , ⎝ Re l ⎠

when equations (11.27) and (11.26) are employed to substitute for τ0 and δ. Simplifying yields F = ρ U 2s (2C1C2 µρUsl)l = (2C1C 2 ρµ U 3s l ),

(11.28)

and the skin friction coefficient Cf may then be calculated as Cf = F −12 ρ U 2s l per unit width.

(11.29)

However, equation (11.28) is of little value in the form shown owing to the presence of C1 and C2. If these constants could be evaluated, then the skin friction for a flat plane would be known. The values of C1, C2 depend on the assumptions made with respect to the variation of shear stress with distance above the plane, i.e. τ = f( y) or f(η). Now, we have already mentioned that this function may be assumed to be linear and the boundary conditions at y = 0 and y = δ are known to be τ = τ0 and τ = 0, respectively. This may be expressed by a relation of the form

τ = C3(δ − y),

(11.30)

du µ −−− = C 3 ( δ – y ) dy

µu = C3(yδ − y 22) + C4. Now u = 0 at y = 0. Therefore, C4 = 0 and so

µu = C3(yδ − y 22). As u = Us when y = δ, C3 = 2µUs δ 2. Hence,

µu = 2µ(Us δ 2)(yδ − y 22), uUs = 2(yδ − y 22δ 2) = 2(η − η 22), and so uUs = 2η − η2

(11.31)

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is the resulting velocity profile from the linear shear stress vs. distance above surface assumption. This allows C1 and C2 to be evaluated as C1 =

1

[1 – (2η – η )](2η – η ) dη 1

[ 1 – f (η) ] f (η) dη = 0

2

2

= 215 and

∂ C 2 = −−− f (η) ∂η

η =0

∂ = −−− ( 2 η – η 2 ) ∂η

= ( 2 – 2 η ) η =0 η =0

= 2. Substituting back, yields −−− ) [ x ( Re ) ] = 5.48x ( Re ) δ = ( 2 × 2 × 15 x x 2

and

(11.32)

( 2C 1 C 2 ρµ U l ) C f = −−−−−−−−1−−−−−−−2−−−−−−−−− −ρU l s 2 3 s

⎛ 2 × −152−− × 2 × ρµ U 3s l ⎞ ⎜ −−−−−−−−1−−−2−−−−−4−−2−−−−−−− ⎟ −ρ U l ⎝ ⎠ s 4

( ) = −−−−− per plate side Re

=

32 −−− 15

x

Cf = 1.4 Re −12 . x

(11.33)

Blasius was able, by reference to the general equations of motion for boundary layers, to plot the velocity distribution up through the laminar boundary layer in a form uUs = f(y Re −12 x) x

(11.34)

and, from this plot, again making the assumptions that y = δ when u = 0.99Us and that the velocity profiles are geometrically similar along the surface, Blasius was able to show that, approximately,

δ = 5x Re −12 , x

(11.35)

which is comparable with equation (11.32) above. Similarly, by taking the slope of the curve (11.34) at y = 0, Us ⎛ du −−−⎞ = 0.332 −−− Re 12 x , ⎝ dy⎠ y=0 x which, when substituted into du τ 0 = µ ⎛ −−−⎞ ⎝ dy⎠ y=0

(11.36)

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and integrated from x = 0 to l, gives a skin friction coefficient, Cf = 1.33Re −12 , l

(11.37)

which is comparable with equation (11.33) above. A number of alternative velocity profiles have been suggested to replace equation (11.31). However, the results do not differ substantially from those shown above. Values for the displacement and momentum thicknesses of the boundary layer may also be calculated in terms of δ, initially from equation (11.31) substituted into equations (11.16) and (11.17), respectively. Hence,

δ* = δ

[1 – f (η)] dη = δ (1 – 2η + η ) dη, 1

1

2

δ * = δ3 = 1.86x Re −12 x

(11.38)

and

f (η)[1 – f (η)] dη 1

θ=δ

(2η – η )(1 – 2η + η ) dη 1

=δ

2

2

θ = −152−− δ = 0.73x Re –x1 2 .

(11.39)

Experimental results verify the Blasius solution except close to the leading edge of the surface, where the assumption of a zero velocity component normal to the surface is not strictly valid. However, the results above are usable for the calculation of skin friction forces and boundary layer thicknesses. Typical laminar boundary layer thicknesses are of the order of 0.75 mm in air at 100 m s−1, Re = 106, and typical lengths, for a smooth flat plate, would be around 160 to 200 mm. Measurement of boundary layer velocity profiles is difficult and requires specialized information. The advent of the hot-wire anemometer has made life a lot easier here, but great care is still necessary to ensure that the results obtained are not a direct function of the experimental set-up. Again, it may be appreciated that the laminar section of the boundary layer is, generally, of secondary importance to the turbulent section, which will be dealt with in the next section.

EXAMPLE 11.1

Oil with a free stream velocity of 3.0 m s−1 flows over a thin plate 1.25 m wide and 2 m long. Determine the boundary layer thickness and the shear stress at mid-length and calculate the total, double-sided resistance of the plate ( ρ = 860 kg m−3, ν = 10−5 m2 s−1, ν = µρ).

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Solution Calculate the Reynolds number at x = 1 m: Rex = Us xν = 3x10−5. Therefore Re 12 = 5.48 × 10 2. x Note that Re is low enough to allow the laminar boundary layer to survive over the whole plate. From equation (11.36):

τ 0 = 0.332µ (Usx) Re 12 x 3 = 0.332 × 10−5 × 860 × − × 5.48 × 102 = 4.7 Nm2 1 The skin friction force is given by, double sided, F = 2 × −12 ρ U 2s l × b × Cf , where l is plate length and b is plate width, F=2×

1 − 2

× 860 × 32 × 2 × 1.25 × Cf ,

where (from equation (11.37) ) Cf = 1.33Re −12 = 1.33(6 × 105)12. l Therefore, F = 860 × 18 × 1.25 × 1.33[(60) × 102 ] = 33.224 N.

11.9 PROPERTIES OF THE TURBULENT BOUNDARY LAYER OVER A FLAT PLATE IN THE ABSENCE OF A PRESSURE GRADIENT IN THE FLOW DIRECTION The majority of boundary layers met in engineering practice are turbulent over most of their length, and so the study of this section of the development of the boundary layer is usually regarded as of greater fundamental importance than that of the laminar section. In many cases, the laminar section of the boundary layer is short enough, compared with the total length of the surface, to be ignored in calculations of skin friction forces. The momentum equation (11.20) may be applied to the turbulent boundary layer as no limiting assumptions were made in its derivation. However, as mentioned in Chapter 5, a new relation for the velocity profile up through the boundary layer will have to be found and the shear stress will no longer be obtained simply from the product of fluid viscosity and the gradient of the velocity profile.

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Owing to the basic similarity between the development of boundary layers within circular cross-section pipes and over flat pipes, Prandtl suggested that the results from the pipe case be applied to the analysis of flat-plate turbulent boundary layers. As was mentioned in Section 11.2, the boundary layer growth in pipes is limited to the pipe radius R, so that u = Us at y = R, and the mean flow velocity in turbulent pipe flow is known to be about 0.8Us. The velocity distribution in such flow is adequately represented by the Prandtl power law, uUs = (yδ )n,

(11.40)

where n = −17 for Rex 107. Obviously, this profile breaks down at the wall, where y = 0. However, the presence of a laminar sublayer has already been discussed (Section 11.2) where the velocity decreases linearly to zero at the wall, this profile being tangential to the power law (Fig. 11.7). To develop the analogy between flat plates and pipe flow, it is necessary to appreciate that δ = R in the fully developed region and to develop some relation for τ0 to replace equation (11.1), which no longer applies. Blasius proposed that, for smooth pipes, the shear stress at the wall could be expressed by

τ0 = f −12 ρB 2,

(11.41)

where B is the mean fluid velocity equal to 0.8Us and f is an empirical constant known as the friction factor, which is a function of flow Reynolds number (Re = ρBdµ, d = pipe diameter) and the ratio of wall roughness to pipe diameter. Friction factors are covered in more detail in Chapter 10. Thus, τ0 = −12 ρ ( 0.8U 2s ) f and, as Blasius developed the expression f = 0.079Re14 = 0.079( ρBdµ)14 to apply to smooth pipes, substitution yields an expression

τ0 = −12 ρ (0.8Us ) 2 0.079( µρ 0.8Us 2R)14 and, if δ = R, then,

τ0 = 0.0225 ρ U 2s ( µρUsδ )14.

(11.42)

As the assumption of zero pressure gradient has been made, equation (11.20) can be applied. Thus,

d u u τ 0 = ρU 2s −−− −−− ⎛ 1 – −−− ⎞ dy ⎝ dx U s Us⎠ dδ = ρU 2s −−− dx

(1 – η 1

17

where uUs = (yδ )17 = η17.

) η 17 dη ,

(11.43)

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Therefore, dδ 7 τ 0 = −−− ρ U 2s −−− . dx 72

(11.44)

Equating these two expressions (11.42) and (11.44) for τ0 yields

δ 14 dδ = 0.234( µρUs )14 dx. Integrating yields 4 − 5

δ 54 = 0.234( µρUs)14x + C5.

Now, if the turbulent boundary layer is assumed to extend to the plate leading edge, which is reasonable if the plate is long compared with the length of the laminar layer, then δ = 0 at x = 0 and C5 = 0. Hence,

δ 54 = 0.292( µρUs)14x, δ = 0.37x( ρUs xµ)15 = 0.37x Re x−15.

(11.45)

Comparing equation (11.45) to (11.32), it may be seen that the turbulent boundary layer grows more rapidly than the laminar layer, the proportional to distance along the plate being to the power x 45 and x 12, respectively. The skin friction force on the flat surface may be determined by eliminating δ between equations (11.42) and (11.45). Hence,

τ0 = 0.029ρ U 2s ( µρUs x)15 and

τ dx per unit width, l

F=

where l is plate length. F = 0.036ρU 2s l( µρUs l)15 = 0.036ρ U 2s l Re −15 l

(11.46)

and the skin friction coefficient, Cf = F −12 ρU 2s l per unit width Cf = 0.072 Re −15 . l

(11.47)

The expression above is valid for Reynolds numbers up to 107, but experimental results indicate that a better approximation is given by Cf = 0.074 Re −15 . l

(11.48)

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Prandtl has suggested subtracting the length of the laminar layer, resulting in an expression Cf = 0.074 Re −15 − 1700 Re −1 l l to apply from Rel = 5 × 105 to 107. To extend the Reynolds number range further, Schlichting employed the logarithmic velocity distribution for pipes under turbulent flow conditions, which have already been mentioned in Chapter 10, resulting in a semi-empirical relation, Cf = 0.455(log10 Rel )−2.58,

(11.49)

applying from 106 Rel 109. Comparison of equation (11.47) with equation (11.37) shows that the skin friction is proportional to the −95 power of velocity of the main stream and the −45 power of plate length for the turbulent layer, compared with the −32 and −12 powers, respectively, for the laminar layer. Generally, then, it may be seen that retention of a laminar boundary as long as possible is desirable from a drag viewpoint. Figure 11.10, a plot of Cf vs. Rel , illustrates the variations in skin friction coefficient.

FIGURE 11.10 Variation of skin friction coefficient with Reynolds number

EXAMPLE 11.2

A smooth flat plate 3 m wide and 30 m long is towed through still water at 20 °C at a speed of 6 m s−1. Determine the total drag on the plate and the drag on the first 3 m of the plate.

Solution For the whole plate: Rel = 1000 × 6 × 3010−3 = 180 × 106,

ρ = 1000 kg m−3, µ = 10−3 N s m−3, Cf = 0.455[log10(1.8 × 108)]2.58 = 0.001 96.

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Drag on both sides of the plate, F = 2( −12 ρU 2s Cf) × b × l = 2 ×

1 − 2

× 1000 × 36 × 0.001 96 × 90

= 6.36 kN. Considering the point at which the boundary layer becomes turbulent, assume transition at Rel = 105: 105 = ρUs l t µ, where l t is the transition length, l t = 105µρUs = 105 × 10−3103 × 6 = 0.0167 m. Thus, it is reasonable to ignore the laminar layer compared with the 30 m plate length. Drag on the first 3 m is then calculated in the same way as shown above for the full plate length.

11.10 EFFECT OF SURFACE ROUGHNESS ON TURBULENT BOUNDARY LAYER DEVELOPMENT AND SKIN FRICTION COEFFICIENTS Initially, the effect of surface roughness is to cause transition from laminar to turbulent conditions closer to the leading edge of the surface. Indeed, the method commonly used to trigger a turbulent boundary layer over model surfaces in wind tunnels is to fix a trip wire or a band of sandpaper or rough material along the surface leading edge to ensure correct drag readings from the models tested. Following transition, where the boundary layer is still thin, the value of kδ may be significant and all the surface roughness protrudes above the boundary layer. In this case, all the drag is due to the eddies caused by the flow passing over the surface roughness and the drag is proportional to the square of the free stream velocity. As the boundary layer continues to develop so the layer depth increases, and the laminar sublayer eventually becomes thick enough to cover all the surface roughness, so that the eddy-related losses mentioned above do not occur. In this case, which occurs for high Reynolds numbers, the degree of roughness of the surface becomes unimportant, i.e. a change in roughness height k would not affect the drag force. Under these special conditions the surface is said to have become hydraulically smooth.

11.11 EFFECT OF PRESSURE GRADIENT ON BOUNDARY LAYER DEVELOPMENT So far, the assumption made of zero pressure gradient in the flow direction across the flat surfaces considered has been unquestioned. The presence of a pressure gradient ∂p∂x effectively means a ∂u∂x term, i.e. the flow stream velocity changes across the

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FIGURE 11.11 Variation of pressure and velocity over a curved surface

surface. If, for example, a curved surface is considered, then the velocity is seen to vary as shown in Fig. 11.11. If the pressure decreases in the downstream direction, then the boundary layer tends to be reduced in thickness, and this case is termed a favourable pressure gradient. If the pressure increases in the downstream direction, then the boundary layer thickens rapidly; this case is referred to as an adverse pressure gradient. This adverse pressure gradient, together with the action of the shear forces described in the boundary layer, if they act for a sufficient length, will bring the boundary layer to rest and the flow separates from the surface. This flow separation has serious consequences in the design of aerofoils, as, once the flow breaks away from the surface, all lift is lost. Owing to the continuing action of the adverse pressure gradient downstream of the separation point, reversed flow eddies are formed which act to increase drastically the drag force acting on the surface. Figure 11.11 illustrates the changes in boundary layer velocity profile under the conditions described above. Generally, then, for the design of aerofoils or other lift-producing surfaces, such as pump and fan blades, the onset of separation should be avoided by design. In the particular case of aerofoil design, this has led to a number of ingenious lift-sustaining devices which act either to revitalize the slow-moving air layer by the introduction of a faster moving jet or to remove the surface layer prior to separation by sucking away this lowvelocity layer. One of the earliest devices of the first type was the Handley Page leading edge slot, which passed high-velocity air from below the wing into the upper wing surface boundary layer prior to separation, thus preventing the change of shape of the velocity profile shown in Fig. 11.12. More recently, a French short takeoff and landing (STOL) transport relied on exhaust air from the turbines ducted and discharged along the wing leading edge to prevent separation and loss of lift at slow speed and high wing angles of attack. The second method, sucking away the boundary layer, has been employed in the study of laminar flow wings for long-range transport aircraft where the marked reduction in skin friction drag that would follow from an entirely laminar boundary layer covering the wing would have obvious range andor lifting capacity advantages. The effects of wake formation are not solely concerned with aerofoils, but have resonant failure results in bridge design. Given certain wind speeds over and under a bridge span, the alternate breaking away of the flow from the upper and lower surfaces can impose cyclic loads which, under special conditions, can correspond to the structure’s natural frequency. The examples above have all dealt with the formation of the boundary layer external to a flat or curved surface. However, as mentioned for the pipe case, boundary

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FIGURE 11.12 Effect of an adverse pressure gradient on boundary layer development. Flow separation of this type is illustrated for a leading edge slat study in Fig. 11.13

FIGURE 11.13 Surface flow patterns seen during an investigation into the effect of sealing slat tracks on a modern commercial aircraft, photo courtesy of Aerodynamics Laboratory, British Aerospace Systems, Filton, Bristol. Note air flow from bottom right to top left and the flow separation indicated in the surface oil layer due to the unsealed slat– leading edge join line

layers form within any duct and grow to fill the duct – imposing a velocity profile of some sort across the duct cross-section. Generally, this is of little concern. However, in the special case of aircraft engine intakes leading to the engine first-stage compressors, the development of the boundary layer can be adverse to the performance of the

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engine. For the best output, the velocity profile at entry to the compressor, normally now of an axial design, should be as uniform as possible, which cannot occur with a fully developed boundary layer. To prevent this it is now quite common to bleed or suck the boundary layer away down the length of the intake. A particular complication arises in the case of an aircraft engine designed to operate at supersonic speed. (With two historical exceptions these are all military aircraft.) It is necessary to decelerate the air prior to entry to the compressors. However, this requires the generation of a shock wave pattern in the intake. If this shock wave pattern is represented as a step increase in pressure, then it will be seen as a con-centrated adverse pressure gradient which could cause boundary layer separation in the intake. This, in turn, would cause the formation of an eddy wake which would be likely to stall the compressor – with obvious consequences of loss of engine power. To avoid this boundary layer, bleeding is again employed. With the few examples given above, the study of the boundary layer can be seen to be of fundamental importance in the understanding of fluid flow phenomena.

Concluding remarks This chapter has provided both a qualitative and a quantitative view of the importance of the boundary layer formed at the interface between a fluid and a solid boundary. The treatment has included reference to the historically important development of this aspect of fluid flow analysis, including the work of Prandtl and von Kármán. The development of expressions defining the velocity profiles within both laminar and turbulent boundary layers has been presented, together with the importance of the laminar sublayer. The effect of surface roughness and applied pressure gradient upon boundary layer development has been explained. The presence of the boundary layer and its reaction to an applied pressure gradient will be returned to in the treatment of aerofoil lift and stall conditions (Chapter 12) together with elements of the ideal flow theory developed in Chapter 7. The influence of boundary layer growth on the pipe or duct length necessary to produce fully developed, steady, uniform flow was discussed. This has a direct implication for flow monitoring by velocity measurement, as only when the flow has become fully developed can any local velocity measurement be used to calculate cross-sectional flow rate. In fluid flow networks the boundary layer is a determinant of the flow development and resistance. It is essential in the possible heat transfer that can occur between a surface and a fluid. In the study of aerofoil lift and drag the boundary retaining the boundary layer in contact with the surface is essential to the continuance of lift; boundary layer separation leads to stall conditions. Therefore it may be seen that an understanding of the mechanism of boundary layer growth and retention will be indispensable to later sections of this text.

Summary of important equations and concepts 1.

The condition of ‘no slip’ between a fluid and a surface or between two fluids is essential in the analysis of unseparated flow conditions.

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2.

3. 4. 5.

6.

The emphasis on the dependence of shear stress on both velocity gradient and viscosity, the identification of the random nature of fluid particle motion in turbulent flows and the introduction of eddy viscosity, equation (11.2), are all essential to the study of boundary layer development. Sections 11.1 to 11.4 introduce and discuss the development of the boundary layer and represent important concepts to be drawn on later. Section 11.5 introduces the Prandtl mixing length theory, leading to a definition of the velocity profile up through a boundary layer and boundary layer thickness. Sections 11.8 and 11.9 apply the momentum concepts first introduced in Chapter 5 to both laminar and turbulent boundary layers, in each case leading to definitions of velocity profile and skin frictional drag. Section 11.11 introduces the effect of pressure gradient on ‘real’ boundary layer development.

Further reading Schlichting, H. and Gersten, K. (2000). Boundary Layer Theory, 8th edn (translated by K. Mayes). Springer, Berlin.

Problems 11.1 Air at 20 °C and with a free stream velocity of 40 m s−1 flows past a smooth thin plate which is 3 m wide and 10 m long in the flow direction. Assuming a turbulent boundary layer from the leading edge determine the shear stress, laminar sublayer thickness and the boundary layer thickness 6 m from the leading edge. Take density = 1.2 kg m−3 and kinematic viscosity as 1.49 × 10−5 m2 s−1. [2.015 N m−2, 0.06 mm, 80 mm] 11.2 Determine the ratio of momentum and displacement thickness to the boundary layer thickness δ when the layer velocity profile is given by uUs = ( yδ ) 12, where u is the velocity at a height y above the surface and the flow free stream velocity is Us. [0.166, 0.333] 11.3 Repeat Problem 11.2 above if the velocity profile is given by u πy −−− = sin ⎛ − − ⎞ . ⎝2 δ ⎠ Us

Take density as 860 kg m−3, kinematic viscosity as 10 m2 s−1. [13.69 mm, 2.085 N m−2, 35.44 N] −5

11.5 A flat plate is drawn submerged through still water at a velocity of 9 m s−1. If the plate is 3 m wide and 20 m long determine the position of the laminar to turbulent transition and the total drag force acting on the plate. Take water temperature as 20 °C. [0.01 m, 9.5 kN] 11.6 An open rectangular box section, sides 3 m × 20 m and 1.5 m × 20 m, is drawn submerged through still water, at 20 °C, at a velocity of 9 m s−1. Determine the overall drag force, neglecting any edge effects. [28.62 kN] 11.7 Estimate the skin friction drag on an airship 92 m long, average diameter 18 m, being propelled at 130 km h−1 through air at 90 kN m−2 absolute pressure and 27 °C. [6.7 kN] 11.8 Assuming a velocity distribution defined by

[0.136, 0.36] 11.4 Oil with a free stream velocity of 2 m s−1 flows over a thin plate 2 m wide and 3 m long. Calculate the boundary layer thickness and the shear stress at the mid-length point and determine the total surface resistance of the plate.

uUs = sin (πy2δ ) determine the general expressions for growth of the laminar boundary layer and for the surface shear stress for a smooth 3 flat plate. [δ = 4.8xRe12 x , τ 0 = 0.33 ( µ U s ρx)]

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11.9 Air at 20 °C and 760 mm Hg absolute pressure flows past a smooth wind tunnel wall, with a free stream velocity of 160 km h−1. Determine the position along the wall, in the flow direction, at which the boundary layer becomes turbulent and the distance to a boundary layer thickness of 25 mm. All wall measurements may be assumed to be taken from the working section entrance and edgecorner effects may be ignored. [33.6 mm, 1.4 m]

laminar, the ratio of towing speeds so that the drag force remains constant regardless of whether a or b is in the flow direction is given by

11.10 Show that, if a flat plate, sides a, b in length, is towed through a fluid so that the boundary layer is entirely

11.11 Repeat Problem 11.10 above if the boundary layer is considered fully turbulent. [UaUb = 9(ab)]

Ua Ub = 3(ab), where Ua is the free stream velocity if side a is in the flow direction and Ub is the corresponding fluid velocity if b is in the flow direction.

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Chapter 12

Incompressible Flow around a Body 12.1

Regimes of external flow

12.2

Drag

12.3

Drag coefficient and similarity considerations

12.4

Resistance of ships

12.5

Flow past a cylinder

Flow past a sphere Flow past an infinitely long aerofoil 12.8 Flow past an aerofoil of finite length 12.9 Wakes and drag 12.10 Computer program WAKE 12.6 12.7

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Chapters 7 and 11 contribute elements to the treatment of incompressible flow around a body in that the likely flow patterns were established, the effect of pressure gradient identified and the importance of fluid viscosity in the formation of boundary layers established. This chapter extends this previous material to allow the determination of the forces acting upon a body moving in a fluid field. Drag, including pressure and skin friction effects, will be defined, together with lift. These effects will be treated for both fully and partially submerged bodies. Dimensional analysis and similarity will be utilized to determine the zones of dependence of these forces on Reynolds number, Froude number and Mach

number. The flow over cylinders and spheres will be treated as a precursor to the consideration of flow over aerofoil sections, where the ideal flow treatment of Chapter 7 will be invoked in the discussion of lift generation. The importance of boundary layer separation will be stressed and the dependence of aerofoil lift on angle of incidence identified, together with the onset of stall conditions. The development of aerofoil trailing vortices and induced drag will be discussed and the momentum equation will be applied to wakes in order to determine pressure drag. A computer program to determine the drag on a body by means of wake traverse data is introduced. l l l

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12.1 REGIMES OF EXTERNAL FLOW In Chapter 7, flow around a cylinder was discussed and expressions enabling the calculation of velocity and pressure in the flow field around the cylinder were derived. Clearly, such a flow may be described as external as it is concerned with the pattern of streamlines surrounding a solid body immersed in a moving fluid. However, the treatment of Chapter 7 excluded any effects which viscosity may have on the flow pattern because that chapter was concerned with ideal flow only. Chapter 11 introduced the concept of a boundary layer and dealt with the effects viscosity has on a fluid adjacent to a solid surface and with the calculation of forces acting on the surface due to fluid friction. We are, therefore, now in a position to consider the external flows of real fluids, namely taking into account viscous effects. The knowledge of potential flow and of boundary layer theory makes it possible to treat an external flow problem as consisting broadly of two distinct regimes: that immediately adjacent to the body’s surface, where viscosity is predominant and where frictional forces are generated, and that outside the boundary layer, where viscosity is neglected but velocities and pressures are affected by the physical presence of the body together with its associated boundary layer. In this outside zone, the theories of ideal flow may be used. In addition, there is the stagnation point at the front of the body (which may stretch into a stagnation region if the body is very blunt) and there is the flow region behind the body (which is known as the wake). These flow regimes are shown in Fig. 12.1.

FIGURE 12.1 Flow regimes around an immersed body

The wake, which starts from points S at which the boundary layer separation occurs, deserves a fuller description. It will be remembered from Chapter 11 that separation occurs due to adverse pressure gradient (∂p∂x 0), which, combined with the viscous forces on the surface, produces flow reversal, thus causing the stream to detach itself from the surface. The same situation exists at the rear edge of a body as it represents a physical discontinuity of the solid surface. In both cases the flow reversal produces a vortex, as shown in Fig. 12.2. The flow in the wake is thus highly turbulent and consists of large-scale eddies. High-rate energy dissipation takes place there, with the result that the pressure in the wake is reduced. A situation is thus created whereby the pressure acting on the front

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FIGURE 12.2 Formation of a vortex in a wake

of the body (the stagnation pressure) is in excess of that acting on the rear of the body, so that a resultant force acting on the body in the direction of the relative fluid motion exists. This force, arising from the pressure difference, or more generally from the nonuniform pressure distribution on the body, is called pressure drag. It is worth while to recollect the findings of Chapter 7 dealing with ideal flow. There, in the absence of viscosity, the flow pattern over the rear part of a body, such as a cylinder, was symmetrical with respect to that over the front half of the body. There were two stagnation points – at the front and at the rear – and the pressure at these points was the same. Thus, there was no resultant force due to pressure acting on the body in the direction of the relative motion (see Section 7.10). A situation very similar to this exists in the case of flow of the real fluids around streamlined bodies, but only at very low Reynolds numbers. There is no wake then and the pressure at the rear stagnation point is nearly equal to that at the front stagnation point. The degree to which the rear stagnation pressure approaches the front stagnation pressure is sometimes called pressure recovery. In the ideal flow, the pressure recovery is complete. When the flow separates from the surface and the wake is formed, the pressure recovery is not complete. The larger the wake, the smaller is the pressure recovery and the greater the pressure drag. The art of streamlining a body lies, therefore, in shaping its contour so that separation, and hence the wake, is eliminated, or at least in confining the separation to a small rear part of the body and, thus, keeping the wake as small as possible. Such bodies are known as streamlined bodies. Otherwise a body is referred to as bluff and a significant pressure drag is associated with it.

12.2 DRAG Pressure drag was described in the preceding section, but asymmetry of pressure distribution, which is responsible for it, is not necessarily the only cause for the existence of a force acting on an immersed body in the direction of relative motion. Thus, in general, when a body is immersed in a fluid and is in relative motion with respect to it, the drag is defined as that component of the resultant force acting on the body which is in the direction of the relative motion. The force component perpendicular to the drag, i.e. acting in the direction normal to the relative motion, is called lift and was defined in Section 7.10. Both lift and drag components of the resultant force are shown in Fig. 12.3. In Chapter 11, frictional drag was discussed in connection with the boundary layer theory. It is the force on the body acting in the direction of relative motion due to fluid shear stress at the surface. Thus, in external flow, the immersed body is

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FIGURE 12.3 Lift and drag on a body

subjected to frictional drag over its entire surface. Total drag on the body, often called profile drag, is, therefore, made up of two contributions, namely the pressure (or form) drag and the skin friction drag. Thus, Profile drag = Pressure drag + Skin friction drag.

(12.1)

FIGURE 12.4

The relative contribution of pressure drag and friction drag to the profile drag depends upon the shape of the body and its orientation with respect to the flow. Take, for example, a small rectangular flat plate. If it is held in a fluid stream ‘edge on’, as shown in Fig. 12.4(a), the pressure drag will be negligible because even though the pressure recovery is incomplete, the resulting pressure difference will act on a very small frontal area (that perpendicular to the flow). The skin friction drag, however, will be substantial, owing to the formation of the boundary layer on both sides of the plate. If, however, the plate is held perpendicularly to the flow (as in Fig. 12.4(b)) the drag will be almost entirely due to pressure difference, whereas the skin friction drag will be negligible. The foregoing description of the two kinds of drag can now be formalized mathematically as follows. If (Fig. 12.5) ps is the fluid pressure acting on the surface element ds, and it acts in the direction perpendicular to the surface, then the force on that part of the body due to the pressure is ps ds. This may be resolved into FIGURE 12.5

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Drag

399

components parallel and perpendicular to the relative direction of motion. The parallel component responsible for the pressure drag is ps cos θ ds. If this component is now integrated around the whole contour of the body, the pressure drag is obtained. Thus, pressure drag, Dp =

p cos θ ds. s

(12.2)

Similarly, the friction force on the body is manifested by the existence of the shear stress at the surface S. This also acts on the element ds and gives rise to a tangential force τ0 ds, whose component in the direction of the motion is τ0 sin θ δ s. Performing, again, the integration around the body’s contour, the skin friction drag is obtained. Thus, skin friction drag, Df =

τ sin θ ds. 0

(12.3)

Both contributions to the profile drag can, therefore, be theoretically calculated, but the first requires knowledge of the pressure distribution around the body and the other knowledge of the shear stress distribution on the surface. The determination of these could be very laborious and it is, therefore, usually simpler to measure the profile drag experimentally, as a force component in a wind tunnel. It is customary to relate the measured drag to the projected area of the body A, the fluid density ρ, and the free stream velocity U0 by the expression D = −12 C D ρ U 20 A,

(12.4)

where CD is known as the drag coefficient and A is the area of the body’s projection on a plane perpendicular to the relative direction of motion. A similar exercise of summation may be carried out for the force components normal to the direction of motion to give lift. This is also related to ρ, U0 and A by an analogous expression, L = −12 C L ρ U 20 A.

(12.5)

The resultant force on the body is, of course, obtained by compounding lift and drag: F = ( L 2 + D 2 ) = −12 ρ U 20 A ( C 2L + CD2 ).

EXAMPLE 12.1

(12.6)

A kite, which may be assumed to be a flat plate of face area 1.2 m2 and mass 1.0 kg, soars at an angle to the horizontal. The tension in the string holding the kite is 50 N when the wind velocity is 40 km h−1 horizontally and the angle of the string to the horizontal direction is 35°. The density of air is 1.2 kg m−3. Calculate the lift and the drag coefficients for the kite in the given position indicating the definitions adopted for these coefficients.

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Solution Since the wind is horizontal, the drag, by definition, will also be horizontal and the lift vertical. The kite is in equilibrium and, therefore, lift and drag must be balanced by the string tension and the weight of the kite. Resolving forces into horizontal and vertical components (Fig. 12.6),

FIGURE 12.6

L = T sin 35° + mg = 50 sin 35° + 1.0 × 9.81 = 38.49 N, D = T cos 35° = 50 cos 35° = 40.95 N. But lift L = −12 C L ρ U 20 A and, therefore, 2 × 38.49 2L C L = −−−−−−2−− = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−2−−−−− ρ U 0 A 1.2 ( 40 × 10003600 ) 1.2 = 0.432. Similarly, the drag coefficient, 2 × 40.95 2D C D = −−−−−−2−−− = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−2−−−−− ρ U 0 A 1.2 ( 40 × 10003600 ) 1.2 = 0.460. Both coefficients have been based on the full area of the kite, because the projected area varies with incidence. This is also the accepted practice in the case of aerofoils.

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Drag coefficient and similarity considerations

401

12.3 DRAG COEFFICIENT AND SIMILARITY CONSIDERATIONS In order to obtain some idea of the nature of the drag coefficient, it is informative to carry out a dimensional analysis exercise (see Chapter 8) in which the drag on an immersed body is considered to be the dependent variable while the following are included as independent variables: the fluid density ρ, its viscosity µ, free stream velocity U0, a linear dimension of the body l, the weight of the fluid per unit mass g (acceleration due to gravity), surface tension σ, and bulk modulus K. Thus, D = f( ρ , µ, U0, l, σ, g, K) or

D = ρ aµ b U c0 l dg eσ fK h,

and, substituting the dimensions, ML M −−−−2− = −−−3 T L

a

M −−−− LT

b

c

L d L −− [ L ] −−−2 T T

e

M −−−2 T

f

h

M −−−2−− . TL

Equating indices: [M]

1 = a + b + f + h;

therefore, a = 1 − b − f − h; [T]

−2 = −b − c − 2e − 2 f − 2h.

Thus, c = 2 − b − 2e − 2 f − 2h; [L]

1 = −3a − b + c + d + e − h,

from which, d = 1 + 3a + b − c − e + h = 1 + 3(1 − b − f − h) + b − 2 + b + 2e + 2f + 2h − e + h = 2 − b − f + e. Therefore, D = ρ(1−b− f−h)µ bU (02 – b –2 e –2 f –2 h ) l (2−b−f +e)g eσ fK h f µ b σ gl e K h = ρ U 20 l 2 ⎛ −−−−−−− ⎞ ⎛ −−−−−−2− ⎞ ⎛ −−−−2 ⎞ ⎛ −−−−−−2 ⎞ . ⎝ ρ U0 l ⎠ ⎝ ρ U 0 l ⎠ ⎝ U 0 ⎠ ⎝ ρ U 0 ⎠

But

ρU0lµ = Re (Reynolds number),

U 20 gl = Fr 2 (Froude number),

ρlU 20 σ = We2 (Weber number),

U 20 (Kρ) = Ma2 (Mach number),

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so that D = ρ U 20 l 2φ (Re, Fr, We, Ma).

(12.7)

Comparing this expression for drag with that of equation (12.4), 1 − 2

CD ρ U 20 A = ρ U 20 l 2φ (Re, Fr, We, Ma).

Since, for a body of a fixed shape, A = λ l 2, where λ is a numerical constant, and incorporating the constant −12 as well as λ into the function φ ′, such that

φ′( ) = φ ( ) −12 λ, we obtain 1 − 2

λCD ρ U 20 l 2 = ρ U 20 l 2φ (Re, Fr, We, Ma)

and, finally, CD = φ′(Re, Fr, We, Ma).

(12.8)

Equation (12.8) demonstrates that the drag coefficient is not a numerical constant, but a proportionality coefficient whose numerical value is a function of a series of dimensionless groups. These groups, and also others which, for simplicity, were not incorporated into the analysis (such as relative roughness, free stream turbulence level, cavitation number), come into play if the kind of forces represented by them are of significance. For example, Re will predominate in cases where viscous forces are dominant, Fr will only be significant in the presence of gravity waves (wavemaking drag), Ma will dominate at high compressibility rates associated with highspeed gas flow, cavitation number will not be important unless cavitation occurs, etc. It may, therefore, be said that, in general, the drag coefficients (and lift coefficients as well, since an analogous expression may be derived for them) for two geometrically similar situations will be the same if the other parameters are the same. For example, the drag on a smooth sphere in an incompressible fluid without cavitation will be such that CD = f (Re), which means that so long as Re is the same the drag coefficient for any sphere of any size in any fluid will be the same provided other parameters are insignificant or irrelevant. For instance, if the free stream level of turbulence is not the same CD will vary. This is why the value of CD for a sphere falling in a stationary liquid (zero turbulence) may be different from that obtained when the sphere is stationary and the fluid is moving past it. Similarly, the boundary layer transition and separation affect both lift and drag and, hence, their values for two situations may not be the same unless all the parameters involved are the same.

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Resistance of ships

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It must also be remembered that if one effect is absent from (say) the model situation of a dynamically simpler system, it must also be absent from the prototype situation. For example, aerofoils tested in water must not cavitate if their performance in air is required, or submarine hulls when tested in a wind tunnel should not be subjected to high velocities to avoid Mach number effects. Although the values of drag coefficient vary with Re and other parameters described, they depend primarily upon the shape of the body and its orientation with respect to the fluid flow. Appendix 2 gives values of CD at specified Re for a variety of commonly encountered shapes.

12.4 RESISTANCE OF SHIPS So far in this chapter, it was assumed that the body which is in relative motion with respect to the fluid is totally immersed in it. An important case, however, exists when the body is partly immersed in a liquid, an example being a ship. When it travels on the surface of water, two main sets of waves are produced, one originating at the bow and the other at the stern of the ship, both diverging from each side of the hull. Energy is required to generate these waves, and this energy originates from the propulsion system of the ship, which must therefore overcome not only the skin friction drag and the form drag but also the additional resistance in generating the waves. This additional resistance is known as wave-making drag or wave drag. (Note, however, that the term ‘wave drag’ is also used to describe the compressibility effects at supersonic velocities. See Section 13.1.) It is not possible to measure the wave resistance directly. It is, therefore, normally obtained by measurement of the total drag and subtracting from it the calculated value of the skin friction drag: Wave-making resistance = Total drag − Skin friction drag.

(12.9)

In this equation the form drag (due to the wake at the stern) is included in the wavemaking resistance. The application of dimensional analysis to the problem carried out in Section 12.3 indicated that the friction drag is dependent upon Reynolds number and the wave-making resistance upon Froude number. The latter is the ratio of inertia forces to the gravity forces and, in the present context, is defined as Fr = v (gl),

(12.10)

where v is the velocity of the ship and l is its length. It may also be shown by dimensional analysis that the velocity of propagation of surface waves, sometimes called celerity, is given by c = (gL)φ (dL, hL),

(12.11)

where L is the wavelength, h is the height of the waves, d is the depth of the water. If the ratio hL is small, the celerity is not affected by it and is given by c = [( gL2π) tan h(2πdL)],

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which, for deep-water waves, where d L, reduces to c = (gL2π).

(12.12)

Experiments which involve towing model ships indicate that the bow and stern waves produced by them travel at the same speed as the ship. This may be demonstrated by suddenly stopping the ship and measuring the wave velocity. Thus, v = C = ( gL2π). But, from (12.10), v = Fr(gl ), so that, equating the two expressions, the Froude number may be written as Fr = 0.4 (Ll ).

(12.13)

This expression is important because it indicates two things. First, it shows that the Froude number describes completely the interrelation between the ship’s length and the wavelength produced by it and, hence, determines the wave-making flow pattern at a given speed; it, therefore, also demonstrates that, for dynamic similarity of the wave-making resistance, the Froude number must be the same for the model and for the prototype. Second, it indicates how many wavelengths there are in a ship’s length and, hence, describes the interaction between the bow and the stern systems of waves, which may be beneficial or detrimental to the ship’s resistance. For example, at certain speeds the waves superpose in such a way that a travelling mound of water is built at the stern. The hydrostatic pressure of this mound acts on the ship pushing it forward and, hence, diminishing its wave-making resistance. However, at other speeds the superposition may produce a travelling trough at the stern, thus increasing the resistance. This is very pronounced at Fr of about 0.6, when the ship rides on the back of its first bow wave crest with the stern in the trough. This ‘uphill’ ride means a very much increased wave-making resistance. The most dramatic situation occurs, however, at Fr = 1, attained by planing speed boats, at which the boat rides on top of the wave crest and the wave-making resistance is then reduced very considerably. Figure 12.7

FIGURE 12.7 Wave-making resistance of a ship

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Resistance of ships

405

shows the ‘ups and downs’ of the wave-making resistance due to the interaction of the wave systems as the Froude number is increased. A ‘mean’ resistance curve is also shown. A more extensive ‘mean’ wave-making resistance curve is drawn, together with the frictional resistance curve, in Fig. 12.8. The two curves add up to the total resistance of a ship. Note the drop of the wave-making resistance at Fr = 1 and the consequent effect upon the total resistance.

FIGURE 12.8 Ship’s resistance

The procedure for predicting the total resistance of a ship during its design stage is based on towing model tests which are aimed at determination of the wave-making resistance (including form drag) and on calculations of frictional drag related to the mean wetted area of the ship. Model towing tests are carried out at a corresponding speed based on Froude number as the criterion. Thus, if suffix ‘m’ refers to the model and suffix ‘p’ to the prototype, (Fr)m = (Fr)p or

vm (glm ) = vp (glp),

from which the corresponding speed vm = vp(lmlp).

(12.14)

The total drag of the model Dm is measured at this speed. The frictional drag of the model Df m is calculated using the boundary layer theory or empirical formulae determined by towing thin plates. Hence, the model wave-making resistance Rm is obtained: Rm = Dm − Df m. This is then scaled up to predict the wave-making resistance of the prototype Rp using the general drag relationship (equation (12.7)), but neglecting all parameters except Froude number, so that Rp = ρp v 2p l 2p φ (Fr)p

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and Rm = ρm v 2m l 2m φ (Fr)m. Dividing one equation by the other, and since (Fr)m = (Fr)p by design of the towing tests (corresponding speed), it follows that Rp = Rm( ρpρm)(vpvm)2(lplm )2.

(12.15)

By adding to this the calculated skin friction drag for the prototype Df p, the total drag is obtained: Dp = Rp + Df p .

EXAMPLE 12.2

(12.16)

A ship is to be built having a wetted hull area of 2500 m2 to cruise at 12 m s−1. A 140 full-size scale model is tested at the corresponding speed and the measured total resistance is found to be 32 N. From separate tests, the skin friction resistance for the model was found to be 3.7v1.95 (newtons per square metre of wetted area) whereas for the prototype this is estimated to be 2.9v1.8, where v is the velocity in metres per second. Find the expected total resistance of the full-size ship if it operates in sea water of density 1025 kg m−3 whereas the model is tested in fresh water.

Solution The corresponding speed at which the model must be tested is given by equation (12.14), obtained by equating the Froude number for the model and for the prototype: vm = vp(lmlp) = 12(140) = 1.90 m s−1. Now, the skin friction drag of the model at this test speed will be 1.95 Df m = 3.7 v 1.95 (2500402) = 20.2 N m Am = 3.7(1.90)

and, hence, the model’s wave-making resistance will be Rm = Dm − Df m = 32 − 20.2 = 11.8 N. Now, using equation (12.15), the wave-making resistance of the ship is obtained:

ρp lp 2 vp 2 1025 12 2 R p = R m −−− ⎛ −− ⎞ ⎛ −−− ⎞ = 11.8 −−−−−− ( 40 ) 2 ⎛ −−−−−− ⎞ = 771.9 kN. ⎝ 1.90 ⎠ ρm ⎝ lm ⎠ ⎝ vm ⎠ 1000 Now the skin friction drag for the ship is calculated from 1.8 −3 Df p = 2.9 v 1.8 p × Ap = 2.9(12) 2500 × 10 = 635.13 kN

and, hence, the total drag of the ship, Dp = Rp + Df p = 771.9 + 635.13 = 1407.06 kN.

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Flow past a cylinder

407

12.5 FLOW PAST A CYLINDER In this section a thin circular cylinder of infinite length, placed transversely in a fluid stream, will be used to discuss in greater detail the changes in flow pattern and in the drag coefficient which accompany the variation of Reynolds number. It must be remembered that for a given cylinder of a given diameter immersed in a given fluid the Reynolds number is directly proportional to the velocity and, therefore, the variation with Reynolds number could be imagined as the variation with velocity for a given cylinder. We assume also that there are no end effects and therefore that the flow is two dimensional. At very small values of Re, say below 0.5, the inertia effects are negligible and the flow pattern is very similar to that for ideal flow, the pressure recovery being nearly complete. Thus, pressure drag is negligible and the profile drag is nearly all due to skin friction. Figure 12.9(a) shows the flow pattern and associated pressure distribution for such a case. Figure 12.10 indicates a straight line relationship between CD and Re in this range, from which we conclude that the drag D is directly proportional to velocity U0. At increased Re, say between 2 and 30, separation of the boundary layer occurs at points S as indicated in Fig. 12.9(b). Two symmetrical eddies, rotating in opposition to one another, are formed. They remain fixed in position and the main flow closes behind them. The separation of the boundary layer is reflected in the variation of CD graph by the curvature of the line indicating that the drag is now proportional to U n0 , where n → 2. Further increase of Re tends to elongate the fixed eddies, which then begin to oscillate until at about Re = 90, depending upon the free stream turbulence level, they break away from the cylinder as shown in Fig. 12.9(c). The breaking away occurs alternately from one and then the other side of the cylinder, the eddies being washed away by the main stream. This process is intensified by a further increase of Re, whereby the shedding of eddies from alternate sides of the cylinder is continuous, thus forming in the wake two discrete rows of vortices, as shown in Fig. 12.9(c). This is known as a vortex street or von Kármán vortex street. At this stage the contribution of pressure drag to the profile drag is about three-quarters. Von Kármán showed analytically, and confirmed experimentally, that the pattern of vortices in a vortex street follows a mathematical relationship, namely hl = (1π) sinh−1(l) = 0.281,

(12.17)

where h and l are indicated in Fig. 12.11. It will be seen that shedding of each vortex produces circulation and, hence, gives rise to a lateral force on the cylinder. Since these forces are periodic following the frequency of vortex shedding, the cylinder may be subjected to a forced vibration. The familiar ‘singing’ of telephone wires is due to this phenomenon, caused by a lateral wind, whereas the collapse of suspension bridges and the ‘flutter’ of aerofoils are the result of a resonance between the natural frequency of the body and the frequency of forced vibration due to vortex shedding. The frequency of such forced vibration, sometimes called self-induced vibration, may be calculated from an empirical formula due to Vincent Strouhal: fdU0 = 0.198(1 − 19.7Re),

(12.18)

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FIGURE 12.9 Flow past a cylinder

in which fdU0 = Str

(12.19)

and is known as Strouhal number. The formula applies to 250 Re 2 × 105. It is fortunate that at higher values of Reynolds number the vortices disappear because of high rates of shear and are then replaced by a highly turbulent wake. This produces an increase in the value of CD at about Re = 3 × 104. Pressure drag is now responsible for nearly all the drag.

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FIGURE 12.10 Drag coefficient for a sphere and a cylinder

FIGURE 12.11 (a) Von Kármán vortex street. (b) Smoke visualization showing von Kármán vortex street

(b)

Flow past a cylinder

409

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Up to Re 2 × 105 the boundary layer on the cylinder is laminar, but at approximately that value, depending upon the intensity of free stream turbulence, it changes to turbulent before separation, as indicated in Fig. 12.9(d). The effect of this is that separation points move further back and, hence, there is a marked drop in the value of CD . At Re 107 the value of CD appears to be independent of Re, but there are insufficient experimental data available for this end of the range. Figure 12.12 compares pressure distributions around a cylinder for ideal flow with the real flow at low and high values of Reynolds number.

FIGURE 12.12 Pressure distribution around a cylinder

EXAMPLE 12.3

Electrical transmission towers are stationed at 500 m intervals and a conducting cable 2 cm in diameter is strung between them. If an 80 km h−1 wind is blowing transversely across the wires, calculate the total force each tower carrying 20 such cables is subjected to. Assume there is no interference between the wires and take air density as 1.2 kg m−3 and viscosity 1.7 × 10−5 N s m−2. Also establish whether the wires are likely to be subjected to self-induced vibrations and if so what the frequency would be.

Solution Drag on one wire, D = −12 ρ C D U 20 A. In order to establish the value of CD , it is necessary to calculate the value of Re first.

ρU 0 d 1.2 × 80 × 1000 × 0.02 Re = −−−−−−− = −−−−−−−−−−−−−−−−−−−−−−−−−−−–−5−−−− = 3.14 × 10 4 . µ 3600 × 1.7 × 10 Now, from Fig. 12.10, for the above value of Re the drag coefficient is CD = 1.2.

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The projected area of a single wire between towers, A = 0.02 × 500 = 10 m2. Hence, drag on wire, 80 × 1000 2 D = −12 × 1.2 × 1.2 ⎛ −−−−−−−−−−−−−− ⎞ × 10 = 3556 N. ⎝ 3600 ⎠ Therefore, the force on each tower due to 20 cables is F = 20 × 3556 = 71.11 kN. Since 250 Re 105, ‘singing’ may occur. Using equation (12.18), U0 19.7 f = 0.198 −−− ⎛ 1 – −−−−−− ⎞ d ⎝ Re ⎠ 80 × 1000 19.7 = 0.198 −−−−−−−−−−−−−−−− ⎛ 1 – −−−−−−−−−−−−−−4⎞ ⎝ 3600 × 0.02 3.14 × 10 ⎠ = 219.9 Hz.

12.6 FLOW PAST A SPHERE So far our discussion of drag has been confined to two-dimensional flow. We will now examine the flow past the simplest of all three-dimensional bodies, the sphere. There is a great similarity in the development of drag at increasing Re between the sphere and the cylinder, except that the vortex street associated with the latter and twodimensional bodies such as aerofoils is not formed in the case of three-dimensional bodies. Instead, a vortex ring occurs, which for a sphere is formed at about Re = 10 and becomes unstable at 200 Re 2000 when it tends to move downstream of the body, to be immediately replaced by a new ring. This process is not periodic, however, and does not give rise to vibrations of the sphere. The study of flow past a sphere is of great practical importance because it is the foundation of a branch of fluid mechanics, namely particle mechanics. This subject concerns itself with all problems associated with the flow of solid particles in a fluid or liquid particles in a gas and encompasses practical problems such as pneumatic conveying, particle separation, sedimentation, filtration, etc. In practice, most particles are not spheres and there are ways of classifying them in accordance with shape, but in the end they are always related to the sphere as the simplest theoretical shape and, therefore, most amenable to both analytical as well as experimental investigations. At very low values of Re, during so-called ‘creeping’ flow around a sphere, it may be assumed that the inertial effects are negligible and, hence, the steady flow Navier–Stokes equations may be greatly simplified by omission of the inertia term, thus enabling the calculation of viscous drag. Stokes obtained the solution for drag by expressing the simplified Navier–Stokes equation together with the continuity equation in polar coordinates and using the

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boundary conditions that all velocity components are zero at the surface of the sphere. His solution is the well-known equation, D = 6πµRU0 ,

(12.20)

in which R is the radius of the sphere, U0 is the free stream velocity of the fluid and µ is its dynamic viscosity. This relationship holds true for Re 0.1, but may be used with negligible error up to Re = 0.2. In this range, often referred to as Stokes flow, the drag coefficient may be calculated by equating the general drag equation (12.4) to the Stokes solution, 1 − 2

but

C D ρ U 20 A = 6πµRU0,

A = πR2,

so that CD = 12µρU0 R = 24µρU0 d, where d = 2R is the diameter of the sphere. But ρU0dµ = Re, based on the sphere diameter, and, therefore, CD = 24Re for Re 0.2.

(12.21)

At larger values of Re, separation of the boundary layer occurs and the Navier– Stokes equations cannot be used. It is, therefore, necessary to rely on empirical expressions. One such formula extends Stokes’ law to Re 100 and is as follows: CD = (24Re)(1 +

3 −−− 16

Re)12.

(12.22)

Beyond Re = 100 it is necessary to use values of CD as a function of Re from the graph, as in Fig. 12.13. Stokes’ formula forms the basis for the determination of viscosity of oils, which consists in allowing a sphere of known diameter to fall freely in the oil. After initial acceleration, the sphere attains a constant velocity known as terminal velocity which FIGURE 12.13 Drag coefficient for a sphere

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is reached when the external drag on the surface and buoyancy, both acting upwards and in opposition to the motion, become equal to the downward force due to gravity. At this equilibrium condition, 6πµUtR +

4 − 3

π R3ρg

= −43 π R3ρp g,

(Drag) + (Buoyancy) = (Gravity) where ρ = density of the fluid, ρp = density of the sphere material, Ut = terminal velocity. Thus, 6µUt = −43 R2( ρp − ρ)g and

Ut = (2R 29µ)( ρp − ρ)g

or, in terms of the sphere’s diameter d, Ut = (d 218µ)( ρp − ρ)g.

(12.23)

By timing the rate of fall of the sphere, the terminal velocity is measured and, hence, the viscosity of the fluid may be determined using the above equation. Alternatively, the method may also be used to determine the mean diameter of spherical particles by allowing them to settle freely in a liquid of known viscosity. At large values of Reynolds number, the flow over the front half of a sphere may be divided into a thin boundary layer region, where viscosity effects are dominant, and an outer region, in which the flow corresponds to that of an inviscid fluid. The pressure is decreasing over the front half of the sphere from the stagnation point onwards, thus having a stabilizing effect on the boundary layer, which remains laminar up to about Re = 5 × 105. Beyond the minimum pressure point on the sphere (at about 80°) the boundary layer is subjected to an adverse pressure gradient and separation occurs. At low Re it begins at the rear stagnation point and with increasing Re it moves forward, reaching the 80° point from the front stagnation at a Reynolds number of about 1000. Pressure drag begins to dominate and CD becomes independent of Re until, at about Re = 5 × 105, transition in the boundary layer occurs, becoming turbulent before separation. This moves the separation point to the rear, making the wake smaller and abruptly reducing the value of CD from about 0.5 to 0.2. The experimental determination of CD for a sphere is difficult because, first, the method of supporting the sphere in the wind tunnel affects the results and, second, the results depend upon the free stream turbulence level, which is difficult to control, and upon the roughness of the sphere, which is difficult to reproduce. It is not surprising, therefore, that the early experimenters produced conflicting results. Any data which do not specify the method of support and free stream turbulence level should be viewed with caution. However, for the purposes of particle mechanics the flow past a sphere is subdivided into three regimes as follows: Re 0.2,

CD = 24Re;

1.

Stokes flow,

2.

Allen flow,

0.2 Re 500, CD = f (Re);

3.

Newton flow,

500 Re 105,

CD = constant = 0.44.

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These regimes are shown in Fig. 12.13. The calculation of terminal velocity is of great importance in particle mechanics because it forms the basis of such operations as settling or sorting. In sorting, for example, solid particles are introduced into a vertical stream of fluid, as shown in Fig. 12.14. If the fluid were stationary, that is B = 0, the particle would attain a constant terminal descending velocity vt. If, however, the fluid is moving vertically up with a velocity B there are three distinct possibilities: 1. FIGURE 12.14 Spherical particle falling into a vertical fluid stream

2. 3.

when B vt the particle will be falling down with an absolute velocity v = vt − B, where vt is now the relative velocity between the fluid and the particle and governs the drag on the particle; when B = vt the particle will be suspended, having an absolute velocity of v = 0; when B vt the particle will move upwards with the fluid with an absolute velocity v = B − vt.

Thus, if particles of different size or weight are introduced into a vertical fluid stream, some will be carried over and some will descend, thus enabling sorting to be carried out. The difficulty in deciding the correct sorting velocity B lies in the fact that it usually corresponds to the particle terminal velocity in the Allen flow regime, where CD is a function of Reynolds number. This cannot be calculated until the velocity is known, which is precisely the variable we are trying to establish. To demonstrate the difficulty let us examine a general case for terminal velocity. It occurs when Drag + Buoyancy = Gravitational force, but drag D = −12 ρCDAv 2. For a spherical particle, A = π d 24 and

D = −18 ρCDπd 2v 2,

so that 1 − 8

ρCDπ d 2v 2 + (π d 36)ρ g = (π d 36)ρp g.

Simplifying and rearranging, 1 − 8

and

ρCDv 2 = (d6)( ρp − ρ)g

v = vt = [ −43 dg( ρp − ρ)CDρ].

Thus, v = f(CD) = f1(Re) = f2(v). For Stokes flow, CD = 24Re, and the substitution gives vt = d 2( ρp − ρ)g18µ, which is the equation (12.23) already derived. For Allen flow two alternatives are possible:

(12.24)

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1. The CD = f (Re) curve may be approximated to a straight line, giving CD = 18.5Re 0.6.

(12.25)

This yields a cumbersome and inaccurate equation for vt. 2. A more satisfactory procedure is to eliminate vt from equation (12.24) and to replace it by Re in the following manner. From (12.24), CD = 4d( ρp − ρ)g3 v 2t ρ; multiplying both sides by Re2 = (vt dρµ)2, CD Re2 = 4d 3( ρp − ρ)ρg3µ2.

(12.26)

The right-hand side of this equation can be calculated for any given fluid and particle combination, since all relevant parameters are known. Thus, the value of CD Re 2 becomes known. This is then referred to a graph relating CD Re 2 to Re, shown in Fig. 12.15. This graph is simply a replot of the Allen part of the CD = f (Re) graph.

FIGURE 12.15 The CDRe 2 vs. Re graph

EXAMPLE 12.4

A particle of 1 mm diameter and density 1.1 × 103 kg m−3 is falling freely from rest in an oil of 0.9 × 103 kg m−3 density and 0.03 N s m−2 viscosity. Assuming that Stokes’ law applies, how long will the particle take to reach 99 per cent of its terminal velocity? What is the Reynolds number corresponding to this velocity?

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Solution The equation of motion for the particle is Mass × Acceleration = Resultant force on the body in the direction of motion = Gravity − Buoyancy − Drag, dv m −−− = mg – m 0 g – D , dt where m = mass of particle, m0 = mass of oil displaced by the particle, D = drag on the particle. Thus m0 dv D −−− = g – −−− g – −− , dt m m but

D = 3πµdv and m = −16 πd 3ρp.

Therefore, Dm = 18µ vd 2ρp; also

m0m = ρ0 ρp,

so that

ρ0 dv 18 µ v −−− = g ⎛ 1 – −− ⎞ – −−−2−−−−. ⎝ ⎠ d ρp dt ρp To facilitate integration, let A = g(1 − ρ0ρp) and B = 18µd 2ρp, so that dv −−− = A – B v. dt Hence, t=

0.99v t

1 dv −−−−−−−−− = – −− log e ( A – Bv ) B A – Bv

0.99v t

1 1 1 A = – −− log e ( A – 0.99Bv t ) + −− log e A = −− log e ⎛ −−−−−−−−−−−−−−−−− ⎞ . ⎝ B B B A – 0.99Bv t ⎠ But

vt = d 2( ρp − ρ)g18µ ;

therefore, ( ρ p – ρ )g 18 µ d 2 ( ρ p – ρ )g 0.99Bv t = 0.99 −−− −−− × −−−−−−−−−−−−−−−− = 0.99 −−−−−−−−−−−−− 2 d ρp 18 µ ρp = 0.99(1 − ρρp)g = 0.99A. Hence, 4.60 1 A 1 1 1 t = −− log e ⎛ −−−−−−−−−−−−−−⎞ = −− log e ⎛ −−−−−−−−−−−⎞ = −− log e 100 = −−−−−−. ⎝ A – 0.99A⎠ B ⎝ 1 – 0.99⎠ B B B

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But B = 18µd 2ρp = 18 × 0.03[(0.1 × 10−2)2 × 1.1 × 103] = 490 and

t = 4.60490 = 0.0094 s.

Terminal velocity, vt = d 2( ρp − ρ0)g18µ = 10−6(1.1 − 0.9)103 × 9.81(18 × 0.03) = 3.63 × 10−3 m s−1. Reynolds number at this velocity, Re = ρ vt dµ = 0.9 × 10 3 × 3.63 × 10−3 × 10−30.03 = 0.1089.

EXAMPLE 12.5

A solid particle of specific gravity 2.4, when settling in oil of specific gravity 0.9 and viscosity 0.027 P, attains a terminal velocity of 3 × 10−3 m s−1. What should be the velocity of an air stream (density 1.3 kg m−3), blowing vertically up, in order to carry the particle at a velocity of 0.5 m s−1? Viscosity of air may be taken as 1.7 × 10−5 N s m−2.

Solution It is first necessary to determine the diameter of the particle. This may be done from the settling data in oil, assuming Stokes flow. This assumption will have to be checked. From equation (12.23), vt = d 2( ρp − ρ)g18µ. Therefore, d=

18 µ v t −−−−−−−−−−−−− = ( ρ p – ρ )g

18 × 0.027 × 10 –1 × 3 × 10 –3 −−−−−−−−−−−−−−−−−−−−−−−−−3−−−−−−−−−−−−−− ( 2.4 – 0.9 )10 × 9.81

= 0.0995 × 10−3 m. To check the flow regime, vd ρ 3 × 10 –3 × 0.0995 × 10 –3 × 0.9 × 10 3 Re = −−−−− = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− µ 0.0027 = 0.0995. Therefore Re 0.1 and so the assumption of Stokes flow was correct. Now, for the particle to move vertically upwards with absolute velocity v in an air stream of absolute velocity u, the relative velocity vt is vt = u − v.

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It is this relative velocity which is responsible for the drag on the particle. Therefore, vt = [4dg( ρp − ρair)3CDρair ]. Since we do not know CD, we calculate CD Re2 = 4d 3(ρp − ρair)gρair3µ2 = 4(0.0995 × 10−3)3(2.4 × 103 − 1.3)9.81 × 1.3[3 × (1.7 × 10−5)2] = 139 and from Fig. 12.15 we read that Re = 5 and hence the relative velocity vt may be obtained from the expression for Re: Re = ρvdµ; therefore, vt = µ Reρd = 1.7 × 10−5 × 5[1.3 × 0.0995 × 10−3] = 0.657 m s−1. The upward air velocity required: u = vt + v = 0.657 + 0.5 = 1.157 m s−1.

12.7 FLOW PAST AN INFINITELY LONG AEROFOIL An aerofoil may be defined as a streamlined body designed to produce lift. There are other lift-producing surfaces such as hydrofoils or circular arcs. In general the following elementary aerofoil theory also applies to these surfaces. There is an accepted terminology concerning aerofoils, and familiarization with it is necessary in order to understand the discussion of flow past aerofoils. Figure 12.16 shows an aerofoil section, and the following are some of the most important terms relating to it:

FIGURE 12.16 An aerofoil

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Leading edge Trailing edge Chord line Chord, c Camber line Camber, δ Percentage camber Span, b Plan area, A

Mean chord, I Aspect ratio, AR Deviation, θ Angle of attack (incidence) Pressure coefficient, Cp

Flow past an infinitely long aerofoil

419

the front, or upstream, edge, facing the direction of flow; the rear, or downstream, edge; a straight line joining the centres of curvature of the leading and trailing edges; the length of chord line between the leading and trailing edges; the centreline of the aerofoil section; the maximum distance between the camber line and the chord line; = 100δc per cent is a measure of aerofoil curvature; the length of the aerofoil in the direction perpendicular to the cross-section; the area of the projection of the aerofoil on the plane containing the chord line. If the aerofoil is of constant cross-section, A = c × b; = Ab; = (Span)(Mean chord) = bc = b 2A; angle between the tangent to camber line at trailing edge and the tangent to camber line at leading edge; the angle between the direction of the relative motion and the chord line; = (p − p0) −12 ρU0 , where p is the local pressure and p0 is the pressure far upstream of the aerofoil where velocity is V0.

The primary purpose of an aerofoil is to produce lift when placed in a fluid stream. It will, of course, experience drag at the same time. In order to minimize drag, an aerofoil is a streamlined body. A measure of its usefulness as a wing section of an aircraft or as a blade section of a pump or turbine is the ratio of lift to drag. The higher this ratio is, the better the aerofoil, in the sense that it is capable of producing high lift at a small drag penalty. In an aircraft it is the lift on the wing surfaces which maintains the plane in the air. At the same time it is the drag which absorbs all the engine power necessary for the craft’s forward motion. Similarly, in pumps, the head generated is due to the lift produced by the impeller blades, whereas the torque necessary to rotate the blades overcomes the drag on them. Thus, the liftdrag ratio, 1 −ρC U 2A Lift CL 2 L 0 −−−−−−− = − −. 1−−−−−−−−−−−−− = −−− Drag −2 ρ C D U 20 A C D

(12.27)

The creation of lift is, therefore, of primary importance. How does an aerofoil produce lift? How does it start when the motion of the aerofoil begins and how is it maintained during the motion? We will attempt to answer these questions by reference to potential flow theory, expounded in Chapter 7, and by reference to the boundary layer theory. The Kutta–Joukowsky law, derived for a cylinder with circulation, relates lift to circulation. It is not limited to cylinders, but may be shown to apply to any twodimensional section. The important point, however, is that lift exists only if there is circulation around the section. A rotating cylinder placed in a real fluid produces circulation by viscous action of its rotating surface on the fluid. Aerofoils, however, do not rotate and, hence, there must be a different mechanism of producing and maintaining circulation. Let us first consider how the circulation starts. It was shown earlier how a vortex is formed, either due to separation of a stream or at the rear of a blunt body. Similarly, if two parallel

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FIGURE 12.17 Starting vortex

streams of unequal velocity meet, there is a discontinuity of velocity at their interface and that produces a vortex. In the same manner, when a slightly inclined aerofoil starts motion it splits the flow into two streams: one over the upper surface and one over the lower surface. The velocities in these streams are not equal due to the inclination of the aerofoil and, therefore, when they meet at the trailing edge a starting vortex is produced as shown in Fig. 12.17. This vortex is cast off soon after the beginning of the motion. It does, however, give rise to circulation around the aerofoil, which is equal in strength (but opposite in sign) to the circulation of the starting vortex. Let us now consider the situation during the motion of the aerofoil. For the circulation to exist there must be vorticity in the stream which cuts across the circulation contour. The flow around the aerofoil may be considered as potential outside the boundary layer and, therefore, it is irrotational there. Hence, there is no vorticity and there cannot be any circulation associated with it. Within the boundary layer, however, the flow is viscous and owing to the velocity gradient vorticity exists there.

FIGURE 12.18 Circulation in the boundary layer

Consider an element of fluid ABCD in the boundary layer as shown in Fig. 12.18. Taking δx as small, the change of velocity in the x direction ( (∂ vx∂x) dx) may be neglected. The circulation for the element becomes ΓABCD = −(vx + δ vx)δx + vxδx = −δ vxδx and vorticity,

ζABCD = δ vxδxδxδy = δ vxδ y.

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FIGURE 12.19 (a) Flow around an aerofoil, with circulation. (b) Helium bubble in air flow visualization around an aerofoil

(b)

Thus, there is vorticity in the boundary layer and its value depends upon the velocity gradient. Figure 12.19 shows the flow around an aerofoil with an exaggerated boundary layer and vorticity within it. It will be noticed that, because the velocity gradients are of opposite sign on the top and bottom surfaces, the vorticity in the upper boundary layer is clockwise whereas the vorticity in the lower boundary layer is anticlockwise. If these two vorticities are equal in strength, they cancel each other and the resultant circulation around the contour is zero. This occurs, for example, in the case of a symmetrical aerofoil without camber placed in a fluid stream at zero angle of incidence. Because of complete flow symmetry, the growth of the boundary layer at the top and bottom is identical and hence vorticities are the same in strength and of opposite rotation. However, if the vorticity over the top surface exceeds that over the bottom, the resultant circulation around the aerofoil will be clockwise, as shown in Fig. 12.19(a). The circulation contour may be drawn arbitrarily around the aerofoil and, provided it contains the whole of the boundary layer, the value of circulation will not be affected because the irrotational flow outside the boundary layer makes no contribution to it. We therefore deduce that the circulation around the aerofoil, Γa = vds, will be clockwise if the velocities over the upper surface are greater than the velocities over the lower surface, or, more exactly, if

2

1

v ds. 1

v ds >

3

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FIGURE 12.20 Pressure distribution around an aerofoil

Such velocity distributions must, in accordance with Bernoulli’s equation, be accompanied by higher pressures on the bottom surface and lower pressures on the top surface. Such a pressure distribution, as shown in Fig. 12.20, gives rise to a resultant upward force, namely the lift. Summarizing, at a small angle of incidence the fluid flowing over the bottom surface of an aerofoil is slowed down, thus increasing the pressure, which means that the pressure gradient there is favourable, the boundary layer thickness is small and the anticlockwise vorticity in it is also small. Over the upper surface the vorticity is greater, the pressure gradient adverse, the boundary layer thicker and the clockwise vorticity in it greater. Thus, the resulting pressure difference gives rise to lift, which may be related to the circulation around the aerofoil. By the same argument, a negative lift (downward force) may exist for negative values of angles of incidence. The foregoing discussion indicates a strong dependence of lift upon the incidence angle. Let us consider this in greater detail by referring to the potential flow around a cylinder as our model. It will be remembered from our discussion of this topic in Chapter 7 that the increase of circulation around the cylinder alters the position of stagnation points, as shown in Fig. 7.29. Since lift (by Kutta–Joukowsky, L = ρU0Γ) depends upon circulation, it may therefore be related to the position of stagnation points. The analogy between the cylinder and the aerofoil is illustrated in Fig. 12.21. The stream function for the flow around a cylinder with circulation is given by equation (7.64): Ψ = U0(r − a2r) sin θ + (Γ2π) log e r,

FIGURE 12.21 Relationship between the zero-lift line on the aerofoil and the position of stagnation points on a cylinder with circulation

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where U0 = upstream velocity; a = radius of the cylinder; Γ = circulation around the cylinder; r, θ = cylindrical coordinates. The velocity on the cylinder surface is the tangential velocity vθ (since there is no velocity into or out of the cylinder) at r = a. Thus, since (from equation (7.13))

∂Ψ v θ = – −−−− , ∂r it follows that vθ = −U0(1 + a 2r 2) sin θ − Γ2π r and, making r = a, vθ = −2U0 sin θ − Γ2πa.

(12.28)

At stagnation points, vθ = 0, and, therefore, Γ2πa = −2U0 sin θ, so that the location of the stagnation points is given by

θ = sin−1(−Γ4πU0a). This equation results in two solutions, namely

θ1 = −α and θ2 = −(180° − α). The corresponding angle of incidence α for the aerofoil is, therefore, measured from the position of the aerofoil in the stream such that there is zero lift. The axis of the aerofoil parallel to the direction of flow under this condition and drawn through the trailing edge is known as the zero-lift axis and is shown in Fig. 12.22. Thus, α 0 is the negative angle of incidence corresponding to no lift.

FIGURE 12.22 The zero-lift axis of an aerofoil

Returning now to our analogy with the cylinder, it is possible to express the circulation in terms of α (equation (7.65)): Γ = −4πU0 a sin α and, using Kutta–Joukowsky’s expression, which is defined as negative for clockwise circulation, the lift becomes L = ρU0Γ = 4π a ρ sin α U 20 .

(12.29)

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Comparing this with equation (12.5) for lift, namely L = −12 CL ρ U 20 A, the following relationship is obtained: 4πaρ U 20 sin α = −12 CLρ U 20 A, so that CL = (8π aA) sin α,

(12.30)

indicating that the coefficient of lift is directly proportional to sin α, which, for small angles of incidence, means that it is proportional to the angle of incidence. This theory is in good agreement with experimental results. Figure 12.23 shows the calculated values of CL for a given aerofoil together with the measured values, as functions of the angle of attack. The drag coefficient is also shown.

FIGURE 12.23 Calculated and experimental values of the coefficient of lift for an aerofoil

The good agreement at small angles of attack is related to the fact that there is no separation of the boundary layer at these small angles. As the angle of incidence is increased, however, separation occurs at the top surface near the trailing edge, thus reducing slightly the rate of increase of the lift with the angle of attack. As the

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FIGURE 12.24 Separation due to increased angle of incidence of an aerofoil

incidence is increased, the point of separation moves forward, as shown in Fig. 12.24. It will be seen that the wake widens and, hence, the drag increases until at some stage the separation point moves to a position such that any further increase of incidence no longer produces an increase of lift. This position is called stall and constitutes a critical angle of attack above which the lift drops rapidly, as indicated in Figs 12.23 and 12.25. The stall is accompanied by a rapidly increasing drag, which is mainly due to the increasing wake. Typical aerofoil characteristics are shown in Fig. 12.25. In addition to curves of lift and drag coefficients the diagram also shows the liftdrag ratio and pressure coefficient Cp.

FIGURE 12.25 Typical characteristics of an aerofoil

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12.8 FLOW PAST AN AEROFOIL OF FINITE LENGTH The previous section dealt with the flow past an infinitely long aerofoil or one which is bounded by parallel plates at the ends. Such conditions or assumptions mean that the flow is truly two-dimensional and there is no spanwise variation of flow patterns and forces for a constant chord aerofoil. In three-dimensional flow, the aerofoil is of finite length (span) b and without walls at the ends, so that they extend freely into the surrounding fluid. This has a considerable effect on the spanwise distribution of lift.

FIGURE 12.26 End effects on an aerofoil of finite span

When an aerofoil is subjected to lift force, the pressure on its underside is greater than that on the top. This pressure difference between the upper and the lower surface causes flow around the tips of the aerofoil from the underside to the upper surface, as indicated in Fig. 12.26. This end flow affects the rest of the flow pattern in the following manner. The flow on the underside is deflected towards the tips of the aerofoil in order to supply the necessary end flow, whereas the flow at the top of the aerofoil is deflected from the tips towards the centre. This produces unstable flow at the trailing edge, causing a vortex sheet which rolls up into two vortices emanating from somewhere near the tips. It is the condensation of water vapour due to low pressure in these tip vortices that is sometimes seen at the tips of aircraft wings. Since there is end flow at the tips, the pressure difference between the top and bottom surfaces of an aerofoil must decrease from a maximum at the mid-span towards the tips where it is zero. Consequently, the circulation around the aerofoil of finite span must also decrease from its maximum value Γa at the centreline down to zero at the tips. The lift is, of course, affected in the same way. The distribution may be approximated to an ellipse, as shown in Fig. 12.27.

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FIGURE 12.27 Distribution of lift along a wing’s span

A further consequence of the tip vortices is that they induce a downward velocity component which is known as downwash velocity Ci. Its presence means that the relative velocity of motion between the fluid and the aerofoil is no longer the free stream velocity U0 but velocity U, deflected from U0 by an angle ε known as the induced angle of incidence. The resulting geometry is shown in Fig. 12.28. What follows is that, in accordance with the definition of lift, which stipulates that it is perpendicular to the relative direction of motion, the true lift is normal to U. However, since it is more convenient and customary to relate lift and drag to the direction of the free stream relative to the aerofoil, the true lift L0 is resolved into L, the component perpendicular to U0, and Di, the component parallel to U0. This latter component, which is in the same direction as drag, is known as induced drag and is added to pressure drag and the skin friction drag to give the total drag on an aerofoil. The expression for induced drag is derived as follows. The true lift per unit length of span is given by L0 = ρU Γ; hence, the induced drag per unit span, D′i = L0 sin ε = ρU Γ sin ε. But sin ε = vi U and, using Prandtl’s approximation for elliptical spanwise lift distribution that vi = Γ02b, where Γ0 is the maximum circulation at the centreline, sin ε = Γ02bU and D′i = ρU Γ(Γ02bU ) = ρΓ(Γ02b).

FIGURE 12.28 Induced drag

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Now, for the elliptic spanwise distribution of Γ, Γ = Γ0 [1 − (2xb)2 ]12, where x is the distance from the centreline. Thus, the induced drag for the total span,

ρ D i = −−−Γ 20 2b

+b2

– b2

12

2x 2 1 – ⎛ −−− ⎞ ⎝ b⎠

ρ bπ dx = −−−Γ 20 −−− 2b 4

= ρπ Γ 20 8,

(12.31)

is obtained by substituting 2xb = sin θ. But L0 =

+b2

ρ U Γ dx = ρ U Γ 0

– b2

+b2

– b2

12

2x 1 – ⎛ −−−⎞ ⎝b⎠

2

π dx = ρU Γ 0 b −, 4

from which Γ0 = 4L0ρUbπ and, substituting into (12.31), Di = ( ρπ8)(4L0ρUbπ)2 = 2 L 20 ρπU 2b2. However, from similar triangles, L0U = LU0 and, hence, Di = (2ρπ b2)(LU0) 2. If the coefficient of induced drag is defined as C D = Di −12 ρU 2A i

and, since CL = L −12 ρU 2A, by substitution L Di C L 2L 2 C D = −−−−−−− = −−−−−−−−−2−−−−2 = 2C L −−−−−−−2−−−−2 ρπ b U LC L L ρπ b U i

− C ρU 2A 2 L 2 A 2 cb = 2C L −−−−−−−−2−−−−2−− = C L −−−−2 = C L −−−−2 ρπ b U πb πb 1

C 2L c = −−−− × − . π b But c 1 − = −−−−−−−−−−−−−−−−−−−−−, b ( Aspect ratio )

(12.32)

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so that C 2L C D = −−−−−−−−−−−−−−−−−−−−−−− . π ( Aspect ratio ) i

(12.33)

This equation shows that a large aspect ratio minimizes the induced drag, as would be expected.

EXAMPLE 12.6

A wing of an aircraft of 10 m span and 2 m mean chord is designed to develop a lift of 45 kN at a speed of 400 km h−1. A 120 scale model of the wing section is tested in a wind tunnel at 500 m s−1 and ρ = 5.33 kg m−3. The total drag measured is 400 N. Assuming that the wind tunnel data refer to a section of infinite span, calculate the total drag for the full-size wing. Assume an elliptical lift distribution and take air density as 1.2 kg m−3.

Solution Wing area, A = 2 × 10 = 20 m2. Coefficient of drag from the model data, 400 D CD = − − = 0.012. 1−−−−−−−− = − 1−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− − ρ U 2A − × 5.3 ( 500 ) 2 20( 20 ) 2 2 2 For the prototype, U = 400 km h−1 = 111.1 m s−1 and the lift coefficient, 45 000 L CL = − −=− − = 0.304. 1−−−−−−−− 1−−−−−−−−−−−−−−−−−−−−−−− − ρ U 2A − × 1.2 ( 111.1 ) 2 20 2 2 Now, assuming an elliptical distribution, the coefficient of induced drag, C D = C 2L π (AR) = (0.304)2π ( −102−− ) = 0.0059. i

Hence, the total drag coefficient, C D = CD + C D = 0.012 + 0.0059 = 0.0179 t

i

and the total drag on the wing, D = −12 C D ρU 2A = t

Therefore, D = 2.65 kN.

1 − 2

× 0.0179 × 1.2(111.1)2 × 20 = 2648.9 N.

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12.9 WAKES AND DRAG It was explained in Section 12.1 that pressure drag is closely related to boundary layer separation and the formation of a wake at the rear of the body. The size of the wake and the pressure within it are the two factors which determine the magnitude of the pressure drag. The wider the wake the greater is the area over which the pressure difference between the front and the rear of the body acts and hence the greater is the drag. Equally, the lower the pressure within the wake, the greater is the pressure difference acting on the body and hence the drag. The two effects are in fact related: as the width of the wake is reduced the pressure within it increases and approaches the free stream pressure. This interdependence was first shown theoretically by Helmholtz, who assumed a stagnant wake region behind the body. For such theoretical flow (known as Helmholtz flow) the separation points move towards the rear of the body and the wake is reduced as the pressure within the wake is increased. Thus, when pressure recovery is assumed to be complete, that is when the pressure within the wake is the same as the free stream pressure, the wake disappears completely and there is no pressure drag. Empirical evidence of flow past cylinders, spheres and other bodies supports the above principles. In particular, the work of Eisenberg and Reichardt provided strong evidence of linear correlation between the drag of various bodies and the pressure coefficient of the cavity behind them. It follows, therefore, that in order to minimize pressure drag it is important to reduce the width of the wake as much as possible. This is achieved by preventing or delaying boundary layer separation from the surface of the body. In Chapter 10 it was shown that the turbulent boundary layer separates less easily than the laminar boundary layer and therefore in the former case the separation points are always further to the rear of the body, the wake is narrower and the drag coefficient is considerably smaller than for the laminar boundary layer. For example, for a cylinder and for a specific Re, laminar separation before transition occurs at θ = ±98° (measured from the rear stagnation point) and the drag coefficient CD = 1.2, whereas turbulent separation after transition occurs at θ = ±60° and CD = 0.3. Similarly for a sphere: for laminar separation θ = ±100° and CD = 0.44, but for turbulent separation θ = ±60° and CD = 0.22. (Figure 12.10 shows that CD varies by up to 10 per cent with Re over small Re ranges.) The level of free stream turbulence has little effect on the value of the drag coefficient as such, but it does affect the Reynolds number at which transition from a laminar to a turbulent boundary layer takes place (see Section 12.5). Generally higher levels of turbulence cause earlier transition. The fluid velocity in the wake is greatly reduced compared with that upstream of the body. It is in general not uniform, unsteady and sometimes oscillatory. Much work has been done on the nature of this very complicated unsteady flow which can give rise to significant body forces, particularly on bluff bodies at subcritical Reynolds number. For most flows, however, it is sufficient to use time-averaged velocities, as measured by a Pitot–static tube for example. The drag force, and hence the drag coefficient of a body, due to the relative motion of a fluid over it, may be determined either by direct force measurement or by calculation from detailed velocity and pressure distributions in the wake. The latter method is based on the application of Newton’s second law of linear motion to an immersed body which causes fluid deceleration in the wake and hence a change of fluid linear momentum, but also a difference of pressure between the front and rear of the body.

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FIGURE 12.29

Consider flow past a two-dimensional body mounted in a wind tunnel as shown in Fig. 12.29. Let the free stream velocity and static pressure in front of the body be U0 and p0 , respectively, and the drag force on the body per unit span be D′. Let also the velocity and pressure profiles downstream of the body be as shown, so that at any distance y from the centreline the velocity is u1 = f (y) and the pressure is p1 = f ′(y). Now, if the half-width of the wake w is defined as that distance from the centreline at which the velocity becomes constant, then u1 varies within the band −w y +w and becomes equal to U1 outside the wake where it is constant. Taking the control volume ABCD such that AB is far upstream of the body where both the velocity and the pressure are constant, CD is downstream of the body and cuts through the wake, and BC and AD are coincident with the tunnel walls, the forces acting on the fluid in the x direction are: pressure forces due to pressure difference, the negative drag force D with which the body acts on the fluid, and forces due to shear stresses Fτ along the tunnel walls resulting from the boundary layer there. The sum of all these forces must be equal to the rate of change of linear momentum in the x direction. Now, Pressure force on fluid element dy, per unit span = (p0 − p1) dy, Resultant force on the fluid within the control volume, per unit span =

+b

( p 0 – p 1 ) dy – D′ – F τ ,

–b

Mass flow rate through fluid element dy, per unit span = ρu1 dy, Change of velocity = (u1 − U0). Therefore, Rate of change of fluid momentum =

+b

ρ u 1 ( u 1 – U 0 ) dy,

–b

and equating the forces to the rate of change of momentum:

+b

–b

ρ u 1 ( u 1 – U 0 ) dy =

+b

–b

( p 0 – p 1 ) dy – D′ – F τ ,

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and the drag force on the body per unit span: D′ =

+b

( p 0 – p 1 ) dy –

–b

+b

ρ u 1 ( u 1 – U 0 ) dy – F τ .

(12.34)

–b

The drag force includes both the skin friction drag and pressure drag because both produce the overall change of momentum. The force Fτ may be calculated using boundary layer analysis, but if the width of the tunnel b is large compared with the frontal width of the body d, then Fτ is small and may be neglected. Thus, changing the order of velocities in the second integral: D′ =

+b

( p 0 – p 1 ) dy +

–b

+b

ρ u 1 ( U 0 – u 1 ) dy .

(12.35)

–b

However, the typical velocity and pressure distributions indicate that at distances greater than w from the centreline the fluid is unaffected by the body, and the velocity and pressure there are constant. If the wake is enclosed, that is within the walls of a tunnel, the constant velocity U1 outside the wake is not equal to the free stream velocity, because continuity demands that the velocity defect within the wake is made up outside it. From the continuity equations, therefore, U0 b = U1 ( b – w ) +

+w

u 1 dy,

–w

from which U0 b 1 U 1 = −−−−−−−−−− – −−−−−−−−−− (b – w) (b – w)

+w

u 1 dy.

(12.36)

–w

Similarly, the constant pressure p′1 outside the wake may not be equal to p0 , so that in such a case the integration must include the whole tunnel width. So, to evaluate drag both the velocity and the pressure distributions are necessary if the traversing is carried out close behind the body. However, some simplifications and approximations are possible in appropriate cases. First, for a free wake, such as that forming behind an aircraft wing during flight, there is no restriction of x direction mass flow continuity imposed by the tunnel walls and therefore the velocity outside the wake U1 is equal to U0. Similarly the pressure outside the wake p′1 is equal to the free stream pressure p0. Thus, both the integrals of equation (12.35) may have their limits changed to ±w because they become equal to zero outside these limits. Therefore, equation (12.35) for a free wake becomes D′ =

+w

–w

( p 0 – p 1 ) dy +

+w

ρ u 1 ( U 0 – u 1 ) dy.

(12.37)

–w

Now, since the drag is obviously independent of any traverse which may be carried out at any distance downstream of the body, it follows that the sum of the two integrals of equation (12.37) must remain constant and independent of x. So, as the wake diffuses with increasing distance from the body, its width will increase and the velocity defect (U0 − u) will decrease, as shown in Fig. 12.30. At some distance,

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FIGURE 12.30

sufficiently far away downstream of the body, the static pressure across the wake and flow will be constant and equal to the upstream free stream static pressure p0. Thus the pressure integral in equation (12.37) becomes equal to zero, and the drag becomes D′ =

+w 2

–w2

ρ u 2 ( U 0 – u 2 ) dy 2 ,

(12.38)

where suffix 2 refers to the faraway downstream section. Using the definition of the coefficient of drag of equation (12.4), D = −12 ρCD U 20 A, and remembering that D′ = DL, we obtain D′ D 2 −=− CD = − 1−−−−−−−−−− = − 1−−−−−−− − ρ U 2 Lc −ρU 2c c 2 2 0 0

+w 2

–w2

u2 u2 −−− ⎛ 1 – −−− ⎞ dy 2 . U0 ⎝ U0 ⎠

(12.39)

Note that in the above equation the value of CD would be based on the span area, since it was taken that A = cL, which is usual for lifting surfaces such as plates or aerofoils. However, for bluff bodies the frontal area A = dL is commonly used, in which case equation (12.39) would take the form 2 CD = − d

+w 2

–w2

u2 u2 −−− ⎛ 1 – −−− ⎞ dy 2 . U0 ⎝ U0 ⎠

(12.39a)

To be consistent, the numerical values of Reynolds number at which values of CD are determined and quoted are customarily also based on c for lifting surfaces and on d for bluff bodies. Equations (12.38) and (12.39) are convenient to use provided it is practicable to carry out the traversing at such a distance that p2 = p0. If this is not possible equation (12.37) must be used unless an approximation is acceptable. One such approximation was developed by B. M. Jones and has been used extensively together with a Pitot rake behind a body. The method assumes that within the wake along any given streamtube between sections 1 and 2 the total pressure remains constant. It is then possible to modify equation (12.38) so that it refers to section 1 using also continuity to account for the larger wake width at section 2.

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Thus: pT = p1 + −12 ρ u 21 , and also pT = p2 + −12 ρ u 22 . Thus the velocities are given by u1 =

2 − ( pT – p1 ) ρ

and

u2 =

2 − ( pT – p2 ) , ρ

but since p2 = p0, u2 =

2 − ( pT – p0 ) . ρ

Similarly the free stream velocity U0 may be expressed in terms of the total pressure there: U0 =

2 − ( pT – p0 ) . ρ 0

Continuity: u1 dy1 = u2 dy2, from which u1 dy 2 = −− dy 1 . u2 Substituting into equation (12.38), D′ =

+w 1

–w1

u1 ρ u 2 ( U 0 – u 2 ) −− dy 1 = u2

+w 2

ρ u 1 ( U 0 – u 2 ) dy 1 ,

–w1

and replacing the velocities, D′ = 2

+w 1

( p T – p 1 ) [ ( p T – p 0 ) – ( p T – p 0 ) ] dy 1 . 0

(12.40)

–w1

Also, equation (12.39) becomes 2 CD = − c

+w 1

–w1

p T – p 0 12 p 1 ⎞ 12 ⎛ −p−−T−−–−−− 1 – ⎛ −−−−−−−−−− ⎞ −− dy 1 . ⎝ pT – p0 ⎠ ⎝ pT – p0 ⎠ 0

(12.41)

In order to evaluate the drag of a body and its drag coefficient using the above equations a traverse of total pressures and static pressures at some section downstream of the body is required together with the reading of total and static pressures upstream of the body. The integration may be performed graphically – the method was devised prior to the widespread availability of computers. It must be remembered,

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435

however, that the method is approximate and therefore it is preferable to use equation (12.37) in conjunction with a suitable numerical integration method and a computer.

12.10 COMPUTER PROGRAM

WAKE

Program WAKE calculates the drag per unit span for a body in an airstream and the associated drag coefficient from a static and stagnation pressure traverse carried out at right angles to the wake downstream of the body. The calculation assumes a twodimensional incompressible flow and requires data detailing the flow conditions, air temperature and traverse experimental results. Equations (12.37) and (12.39) are invoked, together with the equation of state (1.13) and Bernoulli’s equation, with the integration of equation (12.37) being undertaken by use of the trapezoidal rule.

12.10.1 Application example The calculation requires the following data: 1. 2. 3. 4. 5. 6. 7.

test site barometric pressure, B mm of Hg; static pressure upstream of the body, pSu in mm of H2O gauge; temperature upstream of the body, T °C; body characteristic dimension, i.e. frontal width or chord, C mm; upstream free stream velocity, U0 m s−1, or flow stagnation pressure, pT in mm of H2O gauge; number of traverse readings, N, maximum 50, and their spacing, H mm (it is assumed that readings are taken at equispaced locations across the wake); up to 50 static pressure readings, ps , and a stagnation pressure reading, pT, all in mm of H2O gauge, or the value of the constant static pressure, ps, at the traverse section and up to 50 values of the stagnation pressure, pT, as recorded at each traverse point, all in mm of H2O gauge. ( Note that static pressure will always be less than the stagnation pressure due to the kinetic pressure term.) 0

For the following data: B = 770 mm Hg; pSu = 78.5 mm H2O; T = 20 °C; C = 20 mm; U0 = 45.0 m s−1; N = 15; H = 5 mm; for a constant traverse section static pressure ps = 48.8 mm H2O. The 15 stagnation pressure values pT in mm H2O are 176, 176, 170, 150, 136, 110, 85, 70, 82, 111, 135, 155, 173, 175, 176, illustrating the expected velocity distribution across the wake. As a result of the program calculation the drag per unit span was 41.08 kN m−1 and the drag coefficient CD was 1.65, with a reference free stream velocity of 45 m s−1, air flow density of 1.23 kg m−3 and a body characteristic dimension of 20 mm.

12.10.2 Additional investigations using WAKE The computer program calculation may also be used to investigate 1. 2.

the influence of traverse intervals on the predicted drag; the effect of local air conditions, pressure and temperature on the drag coefficient.

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Concluding remarks The application of earlier material defining flow regimes, together with the dimensional analysis and similarity techniques, was utilized in the treatment of drag forces acting on fully and partially submerged bodies and the definition of zones of influence for each of the major dimensionless groupings, Reynolds, Froude and Mach numbers. The earlier treatment of ideal flow was seen to have some application in the flow external to the body. Similarly the effect of an adverse pressure gradient on the boundary layer at the fluidsurface interface was referred to. The importance of the momentum equation was again emphasized by its use in developing drag forces from wake traverses. The flow over a cylinder was utilized as a means of illustrating separation effects, together with vortex generation, and the dependence of both these effects upon flow conditions, while the flow over a sphere was utilized to discuss the measurement of fluid viscosity, Stokes method. The detail flow experienced over an aerofoil was considered, with particular attention being paid to the generation of lift and the minimization of drag. The material covered in this chapter will be utilized in the treatment of rotodynamic machinery where the mechanisms of energy transfer are dependent upon the forces acting between the moving blades and the fluid passing through the machine.

Summary of important equations and concepts 1.

2.

3.

4. 5.

This chapter introduces the concepts of lift and drag forces and their respective non-dimensional coefficients, equations (12.4) and (12.5), in the case of drag also differentiating between skin friction and pressure drag, equation (12.1). The dependence of drag forces on a range of non-dimensional groups is emphasized and the zone of influence of each identified. In particular Section 12.4 emphasizes the role of Froude number in ship resistance. With reference to flow past cylinders and spheres the applicable flow regimes are identified based on Reynolds number; in particular the Stokes law is identified, together with Allen and Newton flow conditions for a sphere, and the concept of terminal velocity is highlighted, equation (12.23). Lift and drag on an aerofoil section is introduced, together with the definition of angle of attack and stall conditions. Flow conditions in the wake downstream of a body in a fluid stream is considered in Section 12.9 and a computer calculation is presented in Section 12.10.

Problems 12.1 A wing of a small aircraft is rectangular in plan having a span of 10 m and a chord of 1.2 m. In straight and level flight at 240 km h−1 the total aerodynamic force acting on the wing is 20 kN. If the liftdrag ratio is 10 calculate the coefficient of lift and the total weight the aircraft can carry. Assume air density to be 1.2 kg m−3. [0.622, 1990 kg] 12.2 A screen across a pipe of rectangular cross-section 2 m by 1.2 m consists of well-streamlined bars of 25 mm

maximum width and at 100 mm centres, their coefficient of total drag being 0.30. A water stream of 5.5 m3 s−1 passes through the pipe. What is the total drag on the screen? If a rectangular block of wood 1 m by 0.3 m and about 25 mm thick is held by the screen, making suitable assumptions, estimate the increase of the drag. [449 N, 3759 N] 12.3 A parachute of 10 m diameter when carrying a load W descends at a constant velocity of 5.5 m s−1 in atmospheric air at a temperature of 18 °C and pressure of

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437

1.0 × 105 N m−2. Determine the load W if the drag coefficient for the parachute is 1.4. [1.992 kN]

velocity of the balloon. Take viscosity of air to be 1.8 × 10−5 N s m−2. [0.932 m s−1]

12.4 Prove that the viscous resistance F of a sphere of diameter d moving at constant speed v through a fluid of density ρ and viscosity µ may be expressed as

12.9 A small spherical water droplet falls freely at a constant speed in air. An air bubble of the same diameter rises freely at a constant rate in water. Derive an expression for the ratio of the distances travelled by the droplet and the air bubble during the same time and calculate this ratio if

µ 2 ρ vd F = k −− f ⎛ −−−−− ⎞ , ρ ⎝ µ ⎠

where k is a constant.

Two balls made of steel and aluminium are allowed to sink freely in an oil of specific gravity 0.9. Determine the ratio of their diameters if dynamic similarity must be obtained when the balls attain their terminal sinking velocities. The specific gravities of steel and aluminium are 7.8 and 2.7, respectively. [0.639] 12.5 The drag and bending moment on a structure in a 40 km h−1 wind is to be studied using a 120 scale model in a pressurized wind tunnel. If the tunnel and ambient temperatures are the same but the air density in the tunnel is eight times that of the ambient air, calculate the air speed in the tunnel and the bending moment for the structure if that measured on the model is 30 N m. [100 km h−1, 4800 N m] 12.6 A submarine periscope is 0.15 m in diameter and is travelling at 15 km h−1. What is the frequency of the alternating vortex shedding and the force per unit length of the periscope? Take the density of water as 1.03 × 103 kg m−3 and kinematic viscosity as 1.25 mm2 s−1. [5.5 Hz, 805 N m−1] 12.7 A 120 model of a cargo ship 120 m long is towed in fresh water at a velocity of 2.5 m s−1. The measured total drag is 105 N. The skin friction drag coefficient is 0.002 72 and the wetted area is 6.5 m2. The estimated skin friction drag coefficient for the prototype is 0.0018. Determine: (a) the wave drag for the model, (b) the wave drag coefficient for the model, (c) the wave drag and the total drag for the ship, and (d ) the power required to tow the model and to propel the ship at its design cruising speed. [(a) 49.75 N, (b) 0.002 45, (c) 408 kN, 435 kN, (d ) 262.5 W, 4863 kW] 12.8 A spherical weather balloon of 2 m diameter is filled with hydrogen. The total mass of the balloon skin and the instruments it carries is 29.682 kg. If at a certain altitude the density of air is 1.0 kg m−3 and is ten times the density of hydrogen in the balloon, determine the steady upward

ρair = 1.2 kg m−3, µair = 1.7 × 10−5, µwater = 10−3 N s m−2. [58.8 for d 0.0446 mm; depends on d for 0.0446 d 2.04 mm; 28.9 for d 2.04 mm] 12.10 (a) A submarine is deeply submerged and moving along a straight course. Describe the physical phenomena that give rise to resistance to its motion. The submarine now comes to the surface and continues on course. What changes occur in the resistance phenomena? (b) The following data refer to a 120 scale model of a cargo vessel under test in a model basin: Model speed Total resistance Model length Wetted surface area Basin water density Kinematic viscosity

1.75 m s−1 34.25 N 6.20 m 5.91 m2 998 kg m−3 0.1010 × 10−5 m2 s−1

The ITTC coefficients may be calculated from CF = 0.075(log10Re − 2)2, where Re is the Reynolds number. What will be the total resistance for the smooth ship at the corresponding speed in sea water of kinematic viscosity 0.1188 × 10−5 m2 s−1? [244.5 N] 12.11 When a slender body held transversely is tested in a wind tunnel it is found that the decrease in velocity in the wake is approximately linear. It decreases from the undisturbed velocity u0 at double the solid width to 0.2U0 at the axis, the pressure in the wake being constant throughout and the same as that in the undisturbed stream. If such a body of 1.5 m width moves, under dynamically similar conditions, through still air at 150 m s−1, calculate the drag on the solid per unit length and the drag coefficient. The air is at 5 °C and a pressure of 510 mm of mercury. Take the density of mercury as 13.6 × 103 kg m−3 and the gas constant for air as R = 287 J kg−1 K−1. [21.49 kN, 1.493]

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Chapter 13

Compressible Flow around a Body 13.1 Effects of compressibility 13.2 Shock waves 13.3 Oblique shock waves

13.4 Supersonic expansion and compression 13.5 Computer program NORSH

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The dimensional analysis introduced in chapter 12 indicated that the external flow Mach number becomes a determinant of the forces acting on a body in supersonic flows. The formation of shock waves, effectively reducing fluid velocity to less than the speed of sound, leads to sudden changes in pressure, generating wave drag. This becomes the predominant factor in defining supersonic body profiles. This chapter introduces these concepts, defining the changes in pressure and temperature across shock

waves by reference to both the energy equation (Chapter 5) and the equation of state for gases (Chapter 1). Normal and oblique shock waves are treated, together with a discussion of the supersonic flow expansion and compression to be found at changes in surface orientation, leading to the propagation of Mach waves that can coalesce into a shock wave at concave corners. The calculation of flow conditions across a shock is included in the form of a computer program. l l l

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13.1 EFFECTS OF COMPRESSIBILITY In the previous chapter, the discussion of drag in external flow was limited to the influence of Reynolds number and Froude number. The former involves the relative influence of two fluid properties, namely the density and the viscosity, whereas the latter is concerned with the effects of gravity. However, in Section 12.3 it was shown by dimensional analysis that Mach number, which is a measure of the importance of elastic forces in the fluid, may be of significance. This occurs when changes of density are appreciable, and the flow is then called compressible. Mach number is also the ratio of the free stream velocity and the velocity of propagation of pressure waves, called the velocity of sound (see Section 5.13): Ma = U0c.

(13.1)

Since very significant changes occur at Ma = 1, the flows are classified into subsonic for Ma 1 and supersonic for Ma 1. In subsonic flow, at relatively low velocities the viscous forces and, hence, Re are of predominant importance. The density changes are small, Ma is also small and its influence is negligible. As the velocity is increased we know from previous paragraphs that at some value of Re the drag coefficient becomes independent of it. This is, however, accompanied by a simultaneous increase of Mach number, whose influence becomes more and more pronounced, and cannot be neglected. In supersonic flow, shock waves are formed. They not only affect the boundary layer and, hence, the skin friction drag and the position of separation, which controls the form drag, but also produce an abrupt change of pressure. This gives rise to additional drag known as wave drag. Since the wave drag is not related to viscosity, but to pressure change across the shock wave, it would be present in an ideal fluid at supersonic flow. At supersonic flow, the wave drag constitutes the largest contribution to total drag and, therefore, streamlining the rear part of the body, which is so important in subsonic flow, has little effect. As will be shown later, in order to reduce the wave drag in supersonic flow the nose of the body must be sharp and pointed. This confines the shock wave to only a small region. Thus, the streamlining requirements for supersonic flow are completely the reverse of those for subsonic flow. Whereas the latter requires a rounded nose and long, gradually pointed tail, the former requires a sharp, pointed nose and rounded, blunt tail. The effect of Mach number on the coefficient of drag for projectiles is shown in Fig. 13.1, which also indicates the considerable reduction in CD achieved by a pointed nose. The effects of compressibility in external flow are not confined to drag. Since, fundamentally, they take into account the variations of fluid density, all parameters are affected. As an illustration, let us consider the very important conditions at the front stagnation point on a body. Let the pressure, temperature and density at the stagnation point be denoted by suffix T and those in the free stream some distance upstream of the body by a suffix 0, as indicated in Fig. 13.2. The conditions at the stagnation point may be expressed in terms of those upstream by the application of Bernoulli’s equation and by remembering that the velocity at the stagnation point is zero. First, assuming incompressible flow, we obtain

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441

FIGURE 13.1 Effect of Mach number on the coefficient of drag for projectiles

pT = p0 + −12 ρ U 20

(13.2)

and, since ρ0 = ρ T = ρ, it follows from Boyle’s law ( pρ = f (T )) that TT T0 = pTp0 . Hence, the stagnation temperature is obtained: TT = T 0 ( pTp0) = T 0 [( p0 + −12 ρ U 20 )p0] TT = T 0 [1 + −12 ( ρp0) U 20 ].

(13.3)

But the equation of state (equation (1.13)) gives p0 = ρRT0,

(13.4)

from which ρp0 = 1RT0 , which, on substitution into (13.3), gives TT = T 0 (1 + −12U 20 RT0) = T0 + −12U 20 R.

FIGURE 13.2

(13.5)

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Now, assuming that the flow is compressible and the process by which it is brought to rest at the stagnation point is frictionless and adiabatic (no heat exchange) and, therefore, isentropic, the appropriate form of Bernoulli’s equation, derived from equations (5.21) and (1.17) (see Example 13.1) gives

γ pT γ p 0 U 20 −−−−−− −−− = −−−−−− −− + −−−−. γ – 1 ρT γ – 1 ρ0 2

(13.6)

But, from the equation of state, pρ = RT, so that [γ (γ − 1)]RTT = [γ (γ − 1)]RT0 + U 20 2 and the stagnation temperature, TT = T0 + [(γ − 1)γ R](U 20 2).

(13.7)

However, it was also shown in Section 5.13 that the velocity of sound is given by equation (5.31), c = (γ RT) = [(γ − 1)cpT ] = (γ pρ),

(13.8)

from which γ R = c 2T0 may be substituted into equation (13.7), giving TT = T0 + [(γ − 1)2](U 20 c2 )T0. But

U0c = Ma0 ,

so that, finally, TT = T0 [1 + −12 ( γ − 1) Ma 20 ].

(13.9)

Now, since, for isentropic processes, TTT0 = ( pTp0)(γ −1)γ, the stagnation pressure may be obtained by pT = p0(TTT0)γ (γ −1) = p0[1 + −12 (γ − 1) Ma 20 ]γ (γ −1).

(13.10)

In order to compare this expression with equation (13.2) for incompressible flow, it may be rearranged as a pressure ratio and then the expression in square brackets expanded using the binomial theorem ( justified because −12 (γ − 1) Ma 20 1 for subsonic flow). Thus, γ ( γ –1 ) pT ⎛ γ –1 −−− = 1 + −−−−−− Ma 20⎞ ⎠ p0 ⎝ 2

γ γ γ (2 – γ ) = 1 + − Ma 20 + − Ma 40 + −−−−−−−−−−−Ma 60 + … . 2 8 48

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443

Rearranging and taking (γ2) Ma 20 outside the bracket, pT γ Ma 2 ( 2 – γ ) −−− – 1 = − Ma 20 1 + −−−−−−0 + −−−−−−−−−Ma 40 + … 2 4 24 p0 and

γ Ma 2 ( 2 – γ ) p T – p 0 = − p 0 Ma 20 1 + −−−−−−0 + −−−−−−−−−Ma 40 + … . 2 4 24

But

γ γ U2 γ 1 U2 − p 0 Ma 20 = − p 0 −−−20 = − p 0 −−−−−−0−−− = − p 0 U 20 , γ p ρ 2 2 c 2 2 0 0

so that, finally, Ma 2 ( 2 – γ ) p T – p 0 = −12 ρ 0 U 20 1 + −−−−−−0 + −−−−−−−−−Ma 40 + … . 24 4

(13.11)

Now, if the flow is considered incompressible, the corresponding pressure difference ( pT − p0)ρ=constant given by equation (13.2) is less than the correct pressure difference given above. The ratio between the two is called the compressibility factor. Thus pT – p0 pT – p0 Compressibility factor = −−−−−−−−−−−−−−−−−−−−−−− = −1−−−−−−−−2 ( p T – p 0 ) ρ =constant −2 ρ 0 U 0 Ma 2 ( 2 – γ ) Compressibility factor = 1 + −−−−−−0 + −−−−−−−−−Ma 40 + … . 4 24

(13.12)

Figure 13.3 shows the variation of compressibility factor with Ma, indicating that for Ma 0.2 the error in assuming the flow to be incompressible amounts to less than 1 per cent, but for Ma 0.5 exceeds 5 per cent and becomes 27.6 per cent at Ma = 1. FIGURE 13.3 Variation of compressibility factor with Mach number

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EXAMPLE 13.1

Show that for horizontal isentropic flow Bernoulli’s equation takes the form

γ p C2 −−−−−− − + −− = constant. γ –1 ρ 2 Calculate, working from the above equation, the stagnation pressure, temperature and density for an airstream at Ma = 0.7 and density ρ = 1.8 kg m−3 and temperature of 75 °C. Take R = 287 J kg −1 K−1 and γ = 1.4.

Solution Euler’s equation (5.21) states 1 dp dC dz − −−− + C −−− + g −−− = 0, ρ ds ds ds which, upon integration, becomes C −−− + −− + gz = constant . dp ρ 2 2

Now, for horizontal flow, z = 0, and for isentropic flow, pργ = constant (equation (1.17)). Therefore, p ρ = ⎛ −−−−−−−−−−−− ⎞ ⎝ constant ⎠ and

−−− = G p dp ρ

– 1γ

1γ

p 1γ = −−−− G

G dp = −−−−−−−−− p ( γ –1 ) γ + C . 1 – 1γ

But G = p1γρ and so, substituting, Gγ −−− = −−−−−− p dp ρ γ –1

( γ – 1 ) γ

γ p 1 γ γ p + C = −−−−−− −−−−− p ( γ –1 ) γ + C = −−−−−− − + C. γ –1 ρ γ –1ρ

Therefore, Euler’s equation, after integration for isentropic conditions, becomes

γ p C2 −−−−−− − + −− = constant. γ –1ρ 2 At stagnation point, v = 0; therefore, applying Bernoulli’s equation to a point in the free stream and the stagnation point (suffix T),

γ pT γ p 0 C 20 −−−−−− −−− = −−−−−− −− + −− , γ – 1 ρT γ – 1 ρ0 2 p T p 0 γ – 1 C 20 −−− = −− + −−−−−− −− . ρT ρ0 γ 2 But, from equation (1.17),

ρT = ρ0(pTp0)1γ.

(I)

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Therefore p T ⎛ p 0 ⎞ 1γ p 0 γ – 1 C 20 −−− −−− = −− + −−−−−− −− , ρ0 ⎝ pT ⎠ ρ0 γ 2

γ – 1 C 20 p T( γ –1 )γ = p 0( γ –1 )γ + ρ 0 ⎛ −−−−−−⎞ −−−−−− ⎝ γ ⎠ 2p1γ 0

(II)

Now, from the equation of state, p0 = ρ0RT = 1.8 × 287(273 + 75) = 179.8 kN m−2; velocity of sound, c = (γ RT)12 = [1.4 × 287(273 + 75)] = 373.9 m s−1; stream velocity, C0 = Ma c = 0.7 × 373.9 = 261.7 m s−1. Substituting into (II), 0.4 ( 261.7 ) 2 p 0.41.4 = ( 179.8 × 10 3 ) 0.41.4 + 1.8 ⎛ −−−− ⎞ × −−−−−−−−−−−−−−−−−−−−−−0.714 −−−− T ⎝ 1.4 ⎠ 2 ( 179.8 × 10 3 ) = 31.72 + 3.12 = 34.84. Hence, pT = (34.84)3.5 = 249.6 kN m−2. Now, from (I), 249.6 0.714 ρ T = 1.8 ⎛ −−−−−−− ⎞ = 2.275 kg m –3 ⎝ 179.8 ⎠ and, therefore, pT 249.6 × 10 3 T T = −−−−−− = −−−−−−−−−−−−−−−− = 382.3 K ρ T R 2.275 × 287 = 109.3 °C.

13.2 SHOCK WAVES Weak pressure change in the fluid is propagated through the fluid continuum with the velocity of sound, which is a function of the elastic properties of the fluid. Thus, if a periodic pressure disturbance occurs at a point S in Fig. 13.4 in a stationary fluid, the resulting pressure waves will travel radially outwards from point S as concentric spheres. If the period of the disturbance is ∆t, then the distance travelled by a wave

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FIGURE 13.4 Wave propagation in a stationary fluid

between the first and second disturbance will be c∆t. By the time the second wave has covered the distance c∆t, the first wave, being c∆t ahead of the second, will have travelled a total distance of 2c∆t. Thus, all the successive waves are equidistant from each other in all directions, the distance being c∆t.

FIGURE 13.5 Wave propagation in a fluid moving with a velocity smaller than that of sound

Now, consider a situation (as shown in Fig. 13.5) in which the source of periodic disturbance S is placed in a moving fluid whose velocity U0 is less than the velocity of sound. The waves are still concentric spheres, but are being swept away by the moving fluid. The lateral distance over which each sphere moves during the periodic time ∆t is U0 ∆t. Thus, the absolute velocity with which the disturbance is now propagated depends upon the direction, being (U0 + c) in the direction of fluid motion but only (U0 − c) in the opposite direction. The source remains within the spheres, but the distance between the consecutive waves will be large downstream of S and small upstream of it. This concentration of spheres’ surfaces upstream will increase as the velocity U0 approaches the velocity of sound c, until, when U0 = c and Ma = 1, all the spherical waves become tangential to each other at S. If the fluid velocity is increased further so that U0 c and Ma 1, the spheres are swept away faster than they are generated, the distance U0 ∆t being greater than c∆t. Such a situation is shown in Fig. 13.6.

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FIGURE 13.6 Wave propagation in a fluid moving with a velocity greater than that of sound

The surface tangential to all the spherical waves is, of course, a cone, known as the Mach cone, which contains within itself the subsonic region called the action zone. Outside the cone, in the silent zone, the flow is supersonic and hence the disturbance generated at S is not ‘communicated’ to any part of the zone. This is the reason for it to be called silent. It follows from the geometry of the situation that the greater is U0 the greater will be the distances travelled by the spheres (U0 ∆t) and, since c∆t remains constant, the Mach angle α, defined as sin α = c∆tU0 ∆t = cU0 = 1Ma or

α = sin−1(1Ma),

(13.13)

will decrease. If the source of disturbance S is replaced by a thin wedge, as shown later in Fig. 13.11, every point on the body becomes a source of disturbance and generates weak Mach waves. The pattern resulting from the superimposition of these waves yields a shock wave across which finite changes of flow parameters occur. If the plane of the shock wave is perpendicular to the direction of flow the shock wave is known as a normal shock wave. Consider such a shock wave: let the parameters upstream of the shock wave, in the silent zone where Ma 1, be denoted by a suffix 1 and those downstream of the shock, in the action zone where Ma 1, be denoted by suffix 2, as indicated in Fig. 13.7. The flow is considered to be adiabatic and frictionless, but not isentropic. This is because there is dissipation of mechanical energy across the shock which results in an increase of entropy. The process is, thus, an irreversible one. A perfect gas is also stipulated. The derivation of the relationship between the upstream and downstream Mach numbers, and, hence, between the remaining parameters, is based on the four fundamental relationships: the continuity equation, the steady flow energy equation, the momentum equation and the equation of state. They are applied to a horizontal streamtube of constant cross-section.

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FIGURE 13.7 Normal shock wave

1. The continuity equation

ρ1U1A = ρ2U2A; therefore,

ρ1U1 = ρ2U2. But, for adiabatic flow, Ma = Uc = U(γ RT), so that U = Ma(γ RT), which gives

ρ1 Ma1(γ RT1) = ρ2 Ma2(γ RT2 ). Therefore,

ρ1Ma1T1 = ρ2 Ma2T2.

(13.14)

However, from the equation of state, p1ρ1T1 = p2ρ2T2 , so that ρ1ρ2 = (p1p2)(T2 T1). Substituting into (13.14) in order to eliminate the density ratio: ( p1p2 )(T2T1) Ma1T1 = Ma2T2, which gives p1 Ma1 T1 = p2 Ma2 T2.

(13.15)

2. Steady flow energy equation (6.10): U 21 2 + H1 = U 22 2 + H2 = HT, where H1 and H2 are the enthalpies and HT is the stagnation enthalpy, which remains constant across the shock waves. But HT = cpTT and, by equation (13.9),

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γ –1 T T = T 0 ⎛ 1 + −−−−−− Ma 20⎞ , ⎝ ⎠ 2 γ –1 γ –1 so that T 1 ⎛ 1 + −−−−−− Ma 21⎞ = T 2 ⎛ 1 + −−−−−− Ma 22⎞ . ⎝ ⎠ ⎝ ⎠ 2 2

(13.16)

3. Momentum equations: p1A − p2A = ρ1AU1(U2 − U1), p1 − p2 = ρ1U1U2 − ρ1 U 21 . But ρ1U1 = ρ2U2, so that p1 − p2 = ρ2 U 22 − ρ1 U 21 or

p1 + ρ1 U 21 = p2 + ρ2 U 22 .

However, U 2 = γ RTMa2, so that p1 + γ RT1 Ma 21 ρ1 = p2 + γ RT2 Ma 22 ρ2 and

p1(1 + γ Ma 21 RT1ρ1p1) = p2(1 + γ Ma 22 RT2 ρ2 p2).

In addition RTρp = 1, which gives p1(1 + γ Ma 21 ) = p2(1 + γ Ma 22 ).

(13.17)

In order to obtain the relationship between Ma1 and Ma2, it is necessary to eliminate pressures and temperatures from equations (13.15), (13.16) and (13.17). The former objective is realized by dividing equation (13.17) by equation (13.15), which gives 1 + γ Ma 2 1 ⎛ 1−−−+−−−γ−−Ma −−−−−− ⎞ T 1 = ⎛ −−−−−−−−−−−−−− ⎞ T 2 . ⎝ Ma 1 ⎠ ⎝ Ma 2 ⎠ 2

2

This equation is now divided by the square root of equation (13.16). The result is 1 + γ Ma 21 1 + γ Ma 22 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−12 −− = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−12 −− = f ( Ma, γ ). Ma 1 [ 1 + −12 ( γ – 1 )Ma 21 ] Ma 2 [ 1 + −12 ( γ – 1 )Ma 22 ]

(13.18)

It shows that the above particular function of Mach number and γ is constant across the shock and determines the relationship between the upstream and downstream values of the Mach number. It is plotted in Fig. 13.8 for air (γ = 1.4). By solving equation (13.18), the expression for Ma2 is obtained: 12

2 ⎧ Ma 1 + 2( γ – 1 ) ⎫ Ma 2 = ⎨ −−−−−−−−−−−−−−−−−−−−−−−−2−−−−− ⎬ . ⎩ [ 2 γ ( γ – 1 ) ]Ma 1 – 1 ⎭

(13.19)

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FIGURE 13.8 Change of Mach number across a shock wave

This equation can now be substituted into equations (13.15), (13.16) and (13.17) to give the following ratios across the shock wave:

γ (γ – 1) γ –1 2γ T2 −−− = −−−−−−−−−−−−−−−−−−2 ⎛ 1 + −−−−−− Ma 21⎞ ⎛ −−−−−− Ma 21 – 1⎞ , ⎠ ⎝γ – 1 ⎠ T 1 ( γ + 1 ) 2 Ma 1 ⎝ 2

(13.20)

p2 2γ γ –1 −− = −−−−−− Ma 21 – −−−−−− , p1 γ + 1 γ +1

(13.21)

ρ2 U1 γ + 1 Ma 2 −− = −−− = −−−−−− −−−−−−−−−−−−−−−−−1−−−−−−−−−−−. ρ1 U2 2 1 + [ ( γ – 1 )2] Ma 21

(13.22)

The above three ratios are functions of Ma and γ only and are plotted in Fig. 13.9 for γ = 1.4. The strength of a shock wave is defined as the ratio of the pressure rise across the shock to the upstream pressure. Thus, Shock strength = ( p2 − p1)p1 = p2 p1 − 1.

(13.23)

By substitution from equation (13.21), Shock strength = 2γ ( Ma 21 − 1)(γ + 1).

(13.24)

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FIGURE 13.9 Changes of parameters of state across a shock wave

It is also useful to have an expression for the ratio of the stagnation pressures, which may be obtained using equation (13.10), pT = p0{1 + [(γ − 1)2] Ma 20 }γ (γ −1), together with (13.21). This procedure, although using an isentropic equation, is justified because the stagnation pressure is defined as resulting from a reversible adiabatic and, hence, isentropic process and, in any case, would take place either upstream or downstream of the shock. Thus, from (13.10), γ ( γ −1 )

pT p 1 ⎧ 1 + [ ( γ – 1 )2 ] Ma 21 ⎫ −−− = −− ⎨ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−2 ⎬ pT p 2 ⎩ 1 + [ ( γ – 1 )2 ] Ma 2 ⎭ 1

.

2

Substituting for p1p2 from (13.21), γ ( γ −1 )

pT ⎛ 2 γ γ – 1 –1 ⎧ 1 + [ ( γ – 1 )2 ] Ma 21 ⎫ −−− = −−−−−− Ma 21 – −−−−−− ⎞ ⎨ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−2 ⎬ pT ⎝ γ + 1 γ + 1 ⎠ ⎩ 1 + [ ( γ – 1 )2 ] Ma 2 ⎭

.

1

2

Now, eliminating Ma 22 using equation (13.19) and simplifying gives pT ( γ + 1 ) Ma 21 −−−−− −−− = −−−−−−−−−−−−−−−−− ( γ – 1 ) Ma 21 + 2 pT 1

2

γ ( γ −1 )

γ +1 −−−−−−−−−−2−−−−−−−−−−−−− 2 γ Ma 1 – ( γ – 1 )

1( γ −1 )

.

(13.25)

It is now possible to show that the flow across the shock is irreversible, and (hence) accompanied by an increase of entropy, by obtaining the relationship between pressure and density and comparing it with the isentropic relationship (equation (1.17)), namely pργ = constant.

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Eliminating Ma 21 from equations (13.21) and (13.22), the following relationship, known as the Rankine–Hugoniot relation, is obtained:

ρ2 γ + 1 p2 −− = ⎛ −−−−−−⎞ −− + 1 ⎝ γ – 1⎠ p 1 ρ1

p2 ⎛ γ + 1⎞ −− + −−−−−− . p1 ⎝ γ – 1 ⎠

(13.26)

It is evidently different from pρ γ = constant or ρ2 ρ1 = ( p2 p1)1γ. Figure 13.10 demonstrates the deviation of the Rankine–Hugoniot relation from the isentropic equation for γ = 1.4.

FIGURE 13.10 Comparison of Rankine– Hugoniot and isentropic curves for γ = 1.4

The increase of entropy across a shock is obtained from: dT p 1 −−−− = c −−−− + −− d ⎛ − ⎞ dQ T T T ⎝ ρ⎠ T 1 = c log ⎛ −−−⎞ + R ρ d ⎛ −⎞ ⎝T ⎠ ⎝ ρ⎠ 2

2

S2 – S1 =

2

v

1

1

1

2

2

v

e

1

1

T2 ρ2 = c v log e ⎛ −−− ⎞ – R log e ⎛ −− ⎞ . ⎝ T1 ⎠ ⎝ ρ1 ⎠

(13.27)

An alternative expression for the specific entropy in terms of pressure ratio may be obtained as follows:

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453

R S2 – S1 T2 ρ2 −−−−−−−−− = log e ⎛ −−− ⎞ – −− log e ⎛ −− ⎞ , ⎝ T1 ⎠ cv ⎝ ρ1 ⎠ cv but

R cp – cv −− = −−−−−−−− = ( γ – 1 ) cv cv

and

ρ2 T1 p2 −− = −−− × −−, ρ1 T2 p1

T1 p2 T2 S2 – S1 so that −−−−−−−−− = log e ⎛ −−− ⎞ – ( γ – 1 ) log e ⎛ −−− ⎞ + log e ⎛ −− ⎞ ⎝ ⎠ ⎝ ⎝ ⎠ T2 p1 ⎠ cv T1 T2 T1 T1 p2 = log e ⎛ −−− ⎞ – γ log e ⎛ −−− ⎞ + log e ⎛ −−− ⎞ – ( γ – 1 ) log e ⎛ −− ⎞ ⎝ T1 ⎠ ⎝ T2 ⎠ ⎝ T2 ⎠ ⎝ p1 ⎠ T2 T2 T2 p2 = log e ⎛ −−− ⎞ + γ log e ⎛ −−− ⎞ – log e ⎛ −−− ⎞ – ( γ – 1 ) log e ⎛ −− ⎞ . ⎝ T1 ⎠ ⎝ T1 ⎠ ⎝ T1 ⎠ ⎝ p1 ⎠ Finally,

S2 – S1 T2 p2 −−−−−−−−− = γ log e ⎛ −−− ⎞ – ( γ – 1 ) log e ⎛ −− ⎞ . ⎝ T1 ⎠ ⎝ p1 ⎠ cv

EXAMPLE 13.2

(13.27a)

A Pitot–static tube is inserted into an airstream of velocity U0, pressure 1.02 × 105 N m−2 and temperature 28 °C. It is connected differentially to a mercury U-tube manometer. Calculate the difference of mercury levels in the two limbs of the manometer if the velocity U0 is (a) 50 m s−1, (b) 250 m s−1 and (c) 420 m s−1. Take the specific gravity of mercury as 13.6 and for air γ = 1.4 and R = 287 J kg−1 K−1.

Solution The two limbs of the manometer are connected one to the total (or stagnation) connection of the Pitot–static tube and the other to the static connection. Thus, the manometer ‘reads’ the difference between the two so that pT − p = ρHg gh, where ρHg is the density of mercury and h is the difference between the mercury levels. Thus, h = ( pT − p)ρHg g. It is, therefore, necessary to obtain (pT − p) for the three cases. First, the value of the Mach number must be calculated in order to establish the type of flow taking place, which will govern the choice of appropriate equations. (a)

50 50 U U Ma = −−−0 = −−−−−−−0−−−− = −−−−−−−−−−−−−−−−−−−−−−−−−− = −−−−−−−−− = 0.14. c ( γ RT ) ( 1.4 × 287 × 301 ) 347.77

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Therefore the flow may be considered as incompressible and equation (13.2) may be used: p T − p0 = −12 ρ U 20 . But,

pρ = RT,

from which

ρ = pRT = 1.02 × 105(287 × 301) = 1.18 kg m−3 and

pT − p = −12 ρ U 20 =

1 − 2

× 1.18(50)2 = 1475 N m−2,

h = 1475(13.6 × 103 × 9.81) = 11.06 × 10−3 m of mercury = 11.06 mm of mercury. (b)

Ma = 250347.77 = 0.719.

Compressibility effects must be taken into account and, therefore, either (i) equation (13.10) is used or (ii) the value of the compressibility factor is obtained from Fig. 13.3. (i) pT = p0{1 + [(γ − 1)2] Ma 20 }γ (γ −1) = 1.02 × 105[1 + (0.42)(0.719) 2 ]1.40.4 = 1.44 × 105 N m−2. Therefore, pT − p0 = (1.44 − 1.02)105 = 0.42 × 105 N m−2. (ii) From Fig. 13.3, for Ma = 0.719, ( pT − p0) −12 ρ0 U 20 = 1.135. Therefore, 1.18 pT − p0 = 1.135 × −−−−−− (250)2 = 0.418 × 105 N m−2. 2 Taking 0.42 × 105 as more accurate, 0.42 × 10 5 h = −−−−−−−−−−−−−−3−−−−−−−−−− = 315 × 10 –3 m of mercury 13.6 × 10 × 9.81 = 315 mm of mercury. (c)

Ma = 420347.77 = 1.208.

The flow is supersonic and, therefore, a shock wave will be formed owing to the disturbance created by the Pitot–static tube. As the nose of the tube is rounded, it is reasonable to assume that the shock will be detached and a section of it just upstream of the Pitot–static tube will be normal to it. Thus, the pressure downstream of the shock and upstream of the tube will be given by equation (13.21): 2γ 2.8 γ –1 0.4 p 2 = p 1 ⎛ −−−−−− Ma 21 – −−−−−−⎞ = 1.02 × 10 5 −−−− ( 1.208 ) 2 – −−−− ⎝γ + 1 2.4 γ + 1⎠ 2.4 = 1.567 × 105 N m−2.

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13.3

Oblique shock waves

455

Now, in order to calculate the stagnation pressure there, using equation (13.10), it is necessary first to determine the Mach number in the action zone between the shock wave and the Pitot–static tube. This may be obtained from equation (13.19): 2 ⎧ Ma 1 + 2 ( γ – 1 ) ⎫ Ma 2 = ⎨ −−−−−−−−−−−−−−−−−−−−−−−−−2−−−−− ⎬ ⎩ [ 2 γ ( γ – 1 ) ]Ma 1 – 1 ⎭ ( 1.208 ) 2 + 20.4 Ma 22 = −−−−−−−−−−−−−−−−−−−−−−−−−−2−−−−− = 0.70. ( 2.80.4 ) ( 1.208 ) – 1 12

or

γ –1 Hence, p T = p 2 ⎛ 1 + −−−−−− Ma 22⎞ ⎝ ⎠ 2

γ ( γ −1 )

2

1.40.4 0.4 = 1.567 × 10 5 ⎛ 1 + −−−− × 0.70 ⎞ ⎝ ⎠ 2

= 2.479 × 105 N m−2. Therefore, ( 2.479 – 1.567 )10 5 h = −−−−−−−−−−−−−−−−3−−−−−−−−−−− = 683.4 × 10 –3 m = 683 mm of mercury. 13.6 × 10 × 9.81

13.3 OBLIQUE SHOCK WAVES When a shock wave is not perpendicular to the direction of flow, it is called an oblique shock wave (Fig. 13.11). It occurs during flow past a wedge or sharp object or when the supersonic flow is forced to change direction by a solid boundary, as shown in Fig. 13.12. One way of treating an oblique shock wave is to consider its normal and tangential components. The normal component undergoes changes associated with the normal shock wave, whereas the tangential component remains unchanged. Thus, only the normal velocity component is reduced, causing the deflection of the flow. It is important to note that although u2n must be subsonic, being downstream of the normal shock, the resultant velocity downstream of the oblique shock, namely U2 = ( u 22n + u 22t ), may be supersonic, provided u2t is large enough. Thus, Ma2 is always smaller than Ma1, but it may be greater than 1. The equations derived for the normal shock wave are valid provided they are applied to the normal velocity components. Since u1n = U1 sin β and

u2n = U2 sin( β − θ ),

where β = shock angle (with respect to upstream flow direction) and θ = deflection angle, it is sufficient to substitute these expression – as well as Ma1 sin β for Ma1 and Ma2 sin( β − θ ) for Ma2 – in the normal shock equations. The angles β and θ are related by

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FIGURE 13.11 Oblique shock wave

FIGURE 13.12 Flow deflection due to an oblique shock wave

u 2n tan( β – θ ) ρ 1 p 1 T 1 −−−− = −−−−−−−−−−−−−− = −− = −−−−−−. u 1n tan β ρ2 p2 T2

(13.28)

Using equations previously derived, it may be shown that

and

tan ( β – θ ) 2 + ( γ – 1 ) Ma 21 sin 2 β −−−−−−−−−−−−−−− = −−−−−−−−−−−−−−−−−−−2−−−−−−− −−−− ( γ + 1 ) Ma 1 sin 2 β tan β

(13.29)

2 cot β ( Ma 21 sin β – 1 ) −−−−−−−−−−−−−−−−−−−−−−−−−, tan θ = −−−−−−−− Ma 21 ( γ + cos 2 β ) + 2

(13.30)

from which the deflection angle may be determined. This equation has two real roots, giving two values of β for each value of θ and Ma1 as shown by the plot in Fig. 13.13. The two values correspond to a strong and a weak wave, respectively. For the strong shock wave, the downstream flow is always subsonic and the shock angle is large; for the weak wave, the downstream flow is usually supersonic and the shock angle is smaller. The chain curve in Fig. 13.13 separates the region of Ma2 1 from that in which Ma2 1. The heavy line, however, joins the maximum values of θ (= θmax ) and thus separates the weak shock from the strong.

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13.4

Supersonic expansion and compression

457

FIGURE 13.13 Oblique shock angles for γ = 1.4

The plot also indicates that, for the shock to occur, θ must be smaller than θmax for a given value of the upstream Mach number. If the physical situation is such that this condition is not satisfied – for example, if the wedge angle is greater than θmax for the flow Mach number – the shock will detach itself from the wedge, thus creating a subsonic space just in front of the wedge. Such a shock wave is always curved, as shown in Fig. 13.14. It, thus, extends further and significantly increases the wave drag on the body. Hence a sharp, pointed nose is a better shape for supersonic flow.

FIGURE 13.14 Detached shock wave

13.4 SUPERSONIC EXPANSION AND COMPRESSION Consider supersonic flow round an infinitesimal corner, which may be convex or concave as shown, with the angle δθ greatly exaggerated (Fig. 13.15). The corner constitutes a disturbance and, since δθ is very small, the disturbance is small, thus generating a very weak shock wave. Such a wave of infinitesimal strength is called a Mach wave. It may be represented by a Mach line whose angle µ is given by sin µ = 1Ma.

(13.31)

When the supersonic flow is forced to negotiate a corner, the flow remains parallel to both the upstream and the downstream solid boundary surfaces. Since the

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FIGURE 13.15 Supersonic expansion and compression at a corner

tangential velocity component remains unaltered, it follows from the velocity triangles of Fig. 13.15 that there must be a change in the resultant velocity. This change is positive, i.e. there is an increase of velocity for the convex corner and there is a negative change or a decrease of velocity for a concave corner. These changes must be accompanied by the corresponding changes in pressure. Thus, there is a pressure drop or expansion at a convex corner and a pressure rise or compression at a concave corner. Any convex corner of finite deflection θ may be regarded as a series of consecutive infinitesimal corners of deflections δθ. This gives rise to a fan of Mach waves (or characteristics), as shown in Fig. 13.16, through which smooth and isentropic expansion takes place. This is known as Prandtl–Mayer flow. The evaluation of changes of pressure and Mach number is carried out in steps through each successive Mach line. Such a step-by-step method is called the method of characteristics and is beyond the scope of this book.

FIGURE 13.16 Prandtl–Mayer expansion

A concave corner of finite deflection gives rise to a series of Mach lines that converge into an envelope and thus form a shock wave as shown in Fig. 13.17. Such a compression process is, therefore, not isentropic, since the changes occur across a shock wave.

FIGURE 13.17 Shock wave at a concave corner

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Concluding remarks

13.5 COMPUTER PROGRAM

459

NORSH

Program NORSH calculates the Mach number, celerity, gas velocity and the parameters of state downstream of a normal shock, together with the entropy change across the shock. The program also determines the upstream parameters not input as data and presents the output in tabular form. The calculation invokes equations (1.3), (5.31), (13.1), (13.10), (13.19), (13.21), (13.22) and (13.27).

13.5.1 Application example The calculation requires the following data: 1. 2. 3.

the values of gas constant, R J kg−1 K−1, and the ratio of specific heats, γ ; the values of any two of the following parameters of state upstream of the shock: static pressure, p1 kN m−2, temperature, T1 K, and density, ρ kg m−3; the value of either the Mach number or the gas velocity, u1 m s−1, upstream of the shock.

For the following data R = 287 J kg−1 K−1; γ = 1.4; p1 = 102 kN m−2; T1 = 301 K; u1 = 420 m s−1, the output table details both the downstream conditions and the unstated upstream conditions: Upstream Mach number Static pressure Stagnation pressure Temperature Density Celerity Flow velocity Entropy change

kN m−2 kN m−2 K kg m−3 m s−1 m s−1

1.21 102 249.85 301 1.181 347.77 420

Downstream 0.84 156.57 247.86 340.98 1.6 370.14 309.96 2.29 J kg−1 K−1

13.5.2 Additional investigations using NORSH The computer program may be used to investigate the dependence of downstream conditions on systematic changes in one upstream parameter for the range of input data possible.

Concluding remarks The effects of fluid compressibility have been discussed in this chapter, with particular reference to the development of shocks within a flow external to a body. The effect of shock waves on the drag forces acting on the body was discussed and solutions aimed at minimizing these forces introduced. The equation of state and the energy equation,

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introduced in Chapters 1 and 5, were essential in the development of relationships linking flow parameters across the shock. The discussion featured both normal and oblique shocks, together with the generation of a detached shock ahead of a body.

Summary of important equations and concepts 1.

2.

3.

The effects of compressibility are emphasized and a compressibility factor, equation (13.12), introduced following the development of expressions for stagnation temperature and pressure, equations (13.3) and (13.10). The concept of a shock wave is introduced, Section 13.2, and equations defining temperature, pressure, and density ratios across normal shocks derived, equations (13.20), (13.21) and (13.22). Entropy change across the shock is also derived as equation (13.27). These equations are utilized in a computer program NORSH, presented in Section 13.5. The application of normal shock equations to an oblique shock is emphasized in Section 13.3.

Problems 13.1 A Pitot–static tube is inserted into an airstream and the mercury manometer connected differentially to it shows a difference in levels of 300 mm. The free stream temperature and pressure are 40 °C and 150 kN m−2 absolute. Calculate the air velocity and the percentage error which would have been committed if the flow was considered as incompressible. (Specific gravity of mercury = 13.6.) [219 m s−1, 4.6 per cent]

pressure is shown as 120 mm of mercury. The barometric pressure is 760 mm of mercury and the stagnation temperature is 40°C. Calculate the Mach number and the air velocity. [0.35, 123 m s−1]

13.2 If the difference between static and stagnation pressure in standard air ( p = 101.3 kN m−2, T = 288 K) is 600 mm of mercury, compute the air velocity assuming (a) the air is incompressible, (b) the air is compressible, and hence calculate the compressibility factor. [(a) 361 m s−1, (b) 316 m s−1, 1.31]

show that for air (R = 287 J kg−1 K−1; γ = 1.4), assuming horizontal, isentropic flow, the difference between stagnation temperature and free stream temperature is approximately given by

13.3 An airstream issues from a nozzle into the atmosphere where the barometric pressure is 750 mm of mercury and the temperature is 20 °C. Assuming that for air the difference between the stagnation temperature and the free stream temperature is given by V0 T T – T 0 = ⎛ −−− ⎞ °C, ⎝ 45 ⎠ 2

where V0 = 250 m s−1 is the free stream velocity, calculate the stagnation temperature, pressure density and the Mach number of the flow. For air R = 287 J kg−1 K−1 and γ = 1.4. [324 K, 144.1 kN m−2, 1.55 kg m−3, 0.729] 13.4 A Pitot–static tube is inserted into the test section of a subsonic wind tunnel. It indicates a static pressure of 80 kN m−2, while the difference between stagnation and static

13.5 Starting from the differential form of Euler’s equation 1 dp dv dz − −−− + C −−− + g −−− = 0, dx dx ρ dx

V0 2 T T – T 0 = ⎛ −−− ⎞ °C, ⎝ 45 ⎠ where V0 is the free stream velocity in metres per second. Calculate also the percentage error involved in the above approximation. [0.8 per cent] 13.6 An airstream with velocity 500 m s−1, static pressure 60 kN m−2 and temperature −18 °C undergoes a normal shock. Determine the air velocity and the static and stagnation conditions after the wave. [255 m s−1, 160.8 kN m−2, 255 kN m−2] 13.7 Given that the Mach number downstream of a normal shock is expressed in terms of the Mach number upstream of the shock as follows: Ma21 + 2 ( γ – 1 ) Ma 22 = −−−−−−−−−−−−−−−−−−−−−−−−2−−−−−− [ 2 γ ( γ – 1 ) ]Ma 1 – 1

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Problems derive an expression for the pressure ratio across the shock wave and hence an expression for the density ratio in terms of pressure ratio and Ma1. 13.8 A normal shock moves into still air with a velocity of 1500 m s−1. The still air is at 10 °C and 80 kN m−2. Calculate the stagnation pressure and temperature behind the wave. [12.8 kN m−2, 700 °C ] 13.9 The measured Mach angle for a bullet has a magnitude of 30°. Estimate the speed of the bullet if the temperature and pressure of the atmo-sphere were 5 °C and 90 kN m−2, respectively. What Mach angle would indicate the same velocity in air at 15 °C and 101 kN m−2? [668.4 m s−1, 30.6°] 13.10 Show that for a normal shock wave p1(1 + γ Ma 21 ) = p2(1 + γ Ma 22 ), where p1 and p2 are pressures upstream and downstream of a shock wave, respectively, Ma1 and Ma2 are the Mach numbers upstream and downstream of the shock, respectively, and γ is the ratio of specific heats. A projectile with a rounded nose moves through still air at Ma = 5. The air pressure is 60 kN m−2 and the temperature is −10 °C. Assuming that the shock wave formed at the nose of the projectile is detached and normal to it determine the stagnation pressure and temperature at the nose. Take γ = 1.4 and use the relationship given in Problem 13.8. [1959 kN m−2, 1525 K]

461

13.11 A two-dimensional wedge is used to measure the Mach number of the flow in a supersonic wind tunnel using air. If the total wedge angle is 20° and the shock wave angle is 60° calculate the Mach number in the tunnel and downstream of the shock. [1.36, 1.11] 13.12 (a) Starting from the momentum considerations and given that the Mach number downstream of a normal shock Ma2 is related to the Mach number upstream of the shock Ma1 by the equation Ma 21 + 2 ( γ – 1 ) Ma 22 = −−−−−−−−−−−−−−−−−−−−−−−−−2−−−−−, [ 2 γ ( γ – 1 ) ]Ma 1 – 1 show that for air the shock strength is given by p2 – p1 −−−−−−−−− = 1.167 (Ma 21 – 1 ). p1 (b) A supersonic aircraft flies horizontally overhead at 3000 m through still air. The time interval between the instant the aircraft is directly overhead an observer on the ground and the instant the shock wave is detected by him is 7.0 s. If the velocity of sound in air is 335 m s−1 calculate the velocity of the aircraft and the stagnation pressure on its nose. Take the atmospheric pressure at 3000 m to be 70 kN m−2. Note that for normal shock in air Ma 21 + 5 Ma 22 = −−−−−−−−2−−−−−. 7Ma 1 – 1 Take γ = 1.4 for air.

[538.4 m s−1; 188.8 kN m−2]

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Part V

Steady Flow in Pipes, Ducts and Open Channels 14 Steady Incompressible Flow in Pipe and Duct Systems 464

16 Non-uniform Flow in Open Channels 528

15 Uniform Flow in Open Channels 508

17 Compressible Flow in Pipes 560

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In the previous part, the behaviour of real fluids has been examined and, in particular, the energy losses which occur due to friction and other causes. In the following chapters, consideration is given to the practical design of pipelines and channels. It is usual to treat liquids under steady flow conditions as if they were incompressible, since the changes of pressure are not large enough to produce significant changes of density. This permits the use of the simple constant

density form of the continuity and energy equations, as shown in Chapter 14, which also covers the analysis of pipe networks under such conditions. Pipes and ducts can have a number of different functions; the most common is to convey fluids from point to point, in which case almost the whole of the head available to produce flow is used in overcoming resistance in the pipeline. Power from a pump, pressure vessel or high-level reservoir may also be transmitted along a pipeline if the fluid travelling through the pipeline arrives at the point of use under pressure or at high velocity. The flow of liquids through open channels is dealt with in two parts. In Chapter 15 we consider uniform flow and the design of channel cross-sections for optimum performance, while Chapter 16 is concerned with non-uniform flow phenomena and the water surface profiles which can occur under these conditions. When gases flow through pipelines it is, usually, necessary to take changes of density and temperature along the length of the pipe into account. In Chapter 17, the basic equations of compressible flow are considered and first applied to frictionless flow through orifices, venturi contractions and nozzles. The formation of a normal shock wave in a diffuser is discussed. For pipelines of constant cross-section with frictional resistance, an analysis is made for both adiabatic and isothermal conditions.

Opposite: Internal views of the London ring main, photo courtesy of Thames Water Left: Supply to a hydroelectric power station in the Scottish Highlands, photo courtesy of Scottish and Southern Energy plc

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Chapter 14

Steady Incompressible Flow in Pipe and Duct Systems 14.1 14.2 14.3 14.4 14.5 14.6

14.7

General approach Incompressible flow through ducts and pipes Computer program SIPHON Incompressible flow through pipes in series Incompressible flow through pipes in parallel Incompressible flow through branching pipes. The threereservoir problem Incompressible steady flow in duct networks

14.8

14.9 14.10 14.11 14.12 14.13 14.14

Resistance coefficients for pipelines in series and in parallel Incompressible flow in a pipeline with uniform draw-off Incompressible flow through a pipe network Head balance method for pipe networks Computer program HARDYC The quantity balance method for pipe networks Quasi-steady flow

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The concepts of continuity of mass flow and energy are utilized in this chapter to develop the steady flow energy equation and to demonstrate its application to both pipe and duct flows and flows possessing a free surface. A computer program designed to illustrate the application of the steady flow energy equation to flow in pipes and ducts is discussed. The definitions of frictional and separation losses introduced in Chapter 10 are

included to allow the determination of system losses and the dependence of the flow in networks on the relative resistance of the alternative flow paths available. Network analysis fundamentally based on Kirchhoff’s laws is applied to introduce the Hardy–Cross technique for the prediction of system flow distribution, and a computer program to analyse network flow distributions is introduced. l l l

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14.1 GENERAL APPROACH This section is concerned with the analysis of the steady flow of a fluid in closed or open conduits. A closed conduit is a pipe or duct through which the fluid flows while completely filling the cross-section. Since the fluid has no free surface, it can be either a liquid or a gas, its pressure may be above or below atmospheric pressure, and this pressure may vary from cross-section to cross-section along its length. An open conduit is a duct or open channel along which a liquid flows with a free surface. At all points along its length the pressure at the free surface will be the same, usually atmospheric. An open conduit may be covered providing that it is not running full and the liquid retains a free surface; a partly filled pipe would, for example, be treated as an open channel. In either case, as the fluid flows over the solid boundary a shear stress will be developed at the surface of contact (as discussed in Chapter 11) which will oppose fluid motion. This so-called frictional resistance results in an energy transfer within the system, experienced as a ‘loss’, measurable in a fluid flow by changes in fluid pressure or head. In addition to the losses attributable to friction, separation losses due to the flow disruption at changes in section, direction or around valves and other flow obstructions also contribute to the overall energy transfers to be accounted for. The first approach to the analysis of bounded systems is therefore to consider the energy balance between two chosen locations along the flow. In Fig. 14.1, for flow across the control volume boundaries represented by the conditions at A and B, the energy audit may be expressed, in terms of energy per unit volume, as FIGURE 14.1 Energy change

pA + −12 ρ v 2A + ρgzA + ∆ppump = pB + −12 ρ v 2B + ρgzB + −12 ρKu 2, where all terms are defined in the dimensions of pressure and hence are amenable to direct experimental measurement for any particular flow condition. The pressure loss experienced as a result of friction and separation of the flow from the walls of the conduit has been shown to be defined by a term of the form 1 − ρKu 2, where u is the local flow velocity and K is a constant dependent upon the 2 conduit parameters, i.e. length, diameter, roughness or fitting type, utilized here to represent both frictional and separation losses. This form of the steady flow energy equation is particularly suited to the study of steady flow conditions in air duct systems as the constituent terms are all amenable to measurement by pressure transducers, or, more simplistically, by manometers. Traditionally in the study of water conduits the steady flow energy equation has been cast

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14.2

Incompressible flow through ducts and pipes

467

in its form of energy per unit weight, resulting in all the terms having the dimensions of head: hA + v 2A 2g + zA + ∆ hpump = hB + v 2B 2g + zB + Ku 22g. While this format is correct and accepted for water-carrying systems, care should be taken in its application in general as all too often it is forgotten that the ‘head’ term is measured in ‘metres of flowing fluid’. Hence pump characteristic data in metres of water will, for example, not apply without modification if the fluid is oil of a given specific gravity. In general the head form of the equation will only be used for water examples; the pressure form is generally applicable for all systems and is recommended. Also, for steady flow to be maintained it is necessary that Mass per unit time entering Mass per unit time leaving = the control volume at A the control volume at B. For incompressible flow the density remains constant and hence the continuity of mass flow equation above reduces to Volume per unit time entering Volume per unit time leaving = the control volume at A the control volume at B. Analysis of all steady flow problems in pipes and channels is based on the application of the steady flow energy equation and the continuity of volumetric flow equation, applied between suitable points in the system.

14.2 INCOMPRESSIBLE FLOW THROUGH DUCTS AND PIPES For incompressible flow, since there is no change of density with pressure, the steady flow energy equation reduces to a form of Bernoulli’s equation with the addition of terms for the energy losses due to friction and separation, for work done by the fluid in driving turbines or for work done on the fluid by the introduction of a pump or fan. All these terms represent energy per unit volume, measured in pressure units, or energy per unit weight, measured in terms of the head of the fluid concerned. The pressure loss, ∆ p, or energy lost per unit volume due to friction, may be conveniently expressed via the Darcy equation ∆ p = 4 f Lρ v 22D

(14.1)

for a circular cross-section conduit flowing full. In terms of head this expression becomes ∆ h = 4 f Lv 22gD.

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Both forms of the Darcy equation may be applied to either laminar or turbulent flow provided the correct form of friction factor, f, is introduced. It should be noted that in laminar flow f = 16Re and hence depends only on flow velocity, v. This second form of the Darcy equation may also be utilized in the study of steady, uniform, free surface flow, provided that the conduit diameter hydraulic mean depth, m = D4, is replaced by an appropriate value of m. Separation losses may be expressed either as a pressure term, K −12 ρ v 2, or as a head term, Kv 22g, where the value of K depends on the type of fitting encountered. Alternatively, fitting losses may be included by the addition of an equivalent length of pipe or duct that would generate the same friction loss as the separation of flow around the fitting generates; this extra equivalent length is simply added to the conduit length and is normally expressed as so many conduit diameters. Tabular values exist for a wide range of fittings and partial valve-opening settings. Often the engineer is more concerned with the flow deliverable rather than the flow velocity in the conduit, although this too can be of prime interest acoustically or where scouring is a concern. An alternative form of the Darcy equation may be obtained by writing Q Q −−−. v = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− = −−−−−− Pipe cross-section area π D 24 Substituting in the Darcy equation yields 64 fL ρ Q 2 ∆ p = −−−−−−−−−−−−−−−, 2D ( π D 2 ) 2

64 fLQ 2 ∆ h = −−−−−−−−−−−−−−−−−. 2gD ( π D 2 ) 2

Both these expressions indicate that the dependence of frictional loss on conduit diameter is a fifth-power relationship, making the reduction of pipe diameter a potentially costly exercise. In SI units, g = 9.81 m s−2, these expressions reduce to ∆p = 3.24 f LρQ 2D 5, ∆h = f LQ 23.03D 5 or, within an error of 1 per cent for the head definition, ∆h = f LQ 23D 5.

(14.2)

In general, for all pipes, ducts and fittings, the loss of pressure or head may be expressed as either ∆p or ∆h = KQ 2, where K is a resistance coefficient.

EXAMPLE 14.1

Water discharges from a reservoir A (Fig. 14.2) through a 100 mm pipe 15 m long which rises to its highest point at B, 1.5 m above the free surface of the reservoir, and discharges direct to the atmosphere at C, 4 m below the free surface at A. The length of pipe l1 from A to B is 5 m and the length of pipe l2 from B to C is 10 m. Both the entrance and exit of the pipe are sharp and the value of f is 0.08. Calculate (a) the mean velocity of the water leaving the pipe at C and (b) the pressure in the pipe at B.

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FIGURE 14.2 Flow through a siphon

Solution (a) To determine the velocity C, first apply the steady flow energy equation between point A on the free surface and point C at the exit from the pipe, since the pressure and elevation of these points are known: Total energy per unit Total energy per unit = + Losses. weight at A weight at C

(I)

Since the entrance to the pipe is sharp, there will be a loss of 0.5C 22g (see Section 10.8.2). The loss due to friction in the length of pipe AC is given by the Darcy formula as 4f ( l 1 + l 2 ) C 2 −−−−−−−−−−−−−− −−−. d 2g There will be no loss of energy at the exit because, although the pipe exit is sharp, the water emerges into the atmosphere without any change of the cross-section of the stream. At both A and C the pressure is atmospheric, so that pA = pC = zero gauge pressure. Also, if the area of the free surface of the reservoir is large, the velocity at A is negligible. Thus, zA = zC + C 22g + ∑ Losses. Substituting in (I), C2 C 2 4f ( l 1 + l 2 ) C 2 z A = ⎛ z C + −−− ⎞ + 0.5 −−− + −−−−−−−−−−−−−− −−−, ⎝ 2g ⎠ 2g d 2g 4f ( l 1 + l 2 ) C2 z A – z C = −−− 1 + 0.5 + −−−−−−−−−−−−−− . 2g d Putting zA − zC = 4 m, l1 = 5 m, l2 = 10 m, d = 100 mm = 0.1 m, f = 0.08, C2 4 × 0.08 × 15 4 = −−−−−−−−−−− 1 + 0.5 + −−−−−−−−−−−−−−−−−− m, 2 × 9.81 0.1 4 × 2 × 9.81 C 2 = −−−−−−−−−−−−−−−− = 1.585, 49.5 C = 1.26 m s−1.

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(b) To find the gauge pressure pB at B, apply the steady flow energy equation between A and B: C 2A pB C 2 C 2 4f l 1 C 2 A ⎛ −p−− + −−− + z A⎞ = ⎛ −−− + −−− + z B⎞ + 0.5 −−− + −−−−− −−−. ⎝ ρ g 2g ⎠ ⎝ ρ g 2g ⎠ 2g d 2g Since A is at atmospheric pressure, pA = 0 (gauge) and, if the reservoir is large, CA = 0, so that pB 4f l 1 C2 z A = −−− + z B + −−− ⎛ 1 + 0.5 + −−−−−⎞ , ρg 2g ⎝ d ⎠ 4f l 1 C2 p B = ρ g ( z A – z B ) – ρ −− ⎛1.5 + −−−−− ⎞ . 2⎝ d ⎠ Substituting (zA − zB) = −1.5 m, C = 1.26 m s−1, f = 0.08, l1 = 5 m, d = 100 mm = 0.1 m, ρ = 103 kg m−3, 4 × 0.08 × 5 10 3 × 1.26 2 p B = 10 3 × 9.81 × ( – 1.5 ) – −−−−−−−−−−−−−−− ⎛1.5 + −−−−−−−−−−−−−−−− ⎞ ⎝ ⎠ 0.1 2 = −14.71 × 103 − 13.87 × 103 N m−2 = −28.58 × 103 N m−2 = 28.58 kN m−2 below atmospheric pressure.

14.3 COMPUTER PROGRAM

SIPHON

This program uses the steady flow energy equation, along with the separation and frictional loss equations already introduced, to investigate the flow of a fluid between two reservoirs or pressurized tanks, via a series pipe network of up to five pipes. The simulation presents pressure and flow data along the series pipe system and is capable of dealing with the possibility of a high point in the pipeline profile, similar to that addressed in Example 14.1. The program calculates the maximum flow rate between the supply and collection reservoirstanks and calculates the local pressure profile along the system to check against a violation of the vapour pressure limit. A maximum flow is identified to avoid cavitation. The program accepts data on the absolute pressure level ‘above’ the fluid in the two reservoirs or tanks. Fluid surface level in both reservoirs is required, together with the entryexit depth of the pipeline connection to both. General data are also required for fluid density, vapour pressure and the number of pipes in the series. For each pipe in the line, data defining entry level above a datum (synonymous with the exit from the upstream pipe), pipe length, diameter and friction factor are required and data concerning separation losses per pipe are also requested with a location expressed as a percentage of pipe length from entry. The discharge level for the final system pipe is also required.

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14.3.1 Application example Consider a two-pipe system leading from one tank to an open reservoir. The pressure in the upstream tank is 300 kN m−2 absolute and the fluid in the downstream reservoir is open to atmosphere, i.e. 100 kN m−2. The fluid level in the upstream tank is 10 m above a datum, that in the reservoir 6 m. Fluid density is 1000 kg m−3 and its vapour pressure is 20 N m−2. Pipe data are as follows: Entry Pipe elevation length (m) (m) Pipe 1 9 Pipe 2 11.5

5 10

Diameter (m)

Friction factor

Separation Location loss factor (% length)

0.1 0.1

0.08 0.08

0.5 1.0

0.0 100.0

Pipe 2 discharges into the downstream reservoir at an elevation of 5 m above datum. The simulation indicates that there is no violation of the vapour pressure lower limit, the flow velocity in each pipe was 3.11 m s−1 and the flow rate was 0.024 5 m3 s−1.

14.3.2 Additional investigations using SIPHON The simulation may be used to investigate: 1. 2. 3.

the effect of changes in the pressure ‘above’ the fluid in each tank or reservoir; the effect of variations in pipe diameter or friction factor; the influence of separation losses, up to five per pipe, on the flow conditions, analogous to the introduction of valves along the pipeline.

Following Example 14.1 the influence of pipeline profile may also be considered, the simulation visual graphical output illustrating both this and the associated pressure profile.

14.4 INCOMPRESSIBLE FLOW THROUGH PIPES IN SERIES When pipes of different diameters are connected end-to-end to form a pipeline, so that the fluid flows through each in turn, the pipes are said to be in series. The total loss of energy, or pressure loss, over the whole pipeline will be the sum of the losses for each pipe together with any separation losses such as might occur at the junctions, entrance or exit.

EXAMPLE 14.2

Two reservoirs A and B (Fig. 14.3) have a difference level of 9 m and are connected by a pipeline 200 mm in diameter over the first part AC, which is 15 m long, and then 250 mm diameter for CB, the remaining 45 m length. The entrance to and exit from the pipes are sharp and the change of section at C is sudden. The friction coefficient f is 0.01 for both pipes.

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FIGURE 14.3 Pipes in series, showing head losses and the total energy line and hydraulic gradient

(a) List the losses of head (energy per unit weight) that occur, giving an expression for each. (b) Use program SIPHON (Section 14.3) to calculate the system flow rate and hydraulic gradient.

Solution (a) The losses of head which will occur are as follows: (i) Loss at entrance to pipe AC. This is a separation loss and, since the entrance is described as sharp and is below the free surface of the reservoir (from Section 10.8.2), the value of k will be 0.5: Loss of head at entry, h1 = 0.5 v 21 72g. (ii) Friction loss in AC. Using the Darcy formula, we have 4 f l 1 v 21 Loss of head in friction in AC, = h f = −−−−− −−− . d 1 2g 1

(iii) Loss at change of section at C. There will be a separation loss at the sudden change of section. From Section 10.8.1, the loss at a sudden enlargement will be Loss of head at sudden enlargement, h2 = (v1 − v2 )272g. (iv) Friction loss in CB. Using the Darcy formula, 4f l 2 v 22 Loss of head in friction in CB, = h f = −−−−− −−− . d 2 2g 2

(v) Loss of head at exit. Since the exit is described as sharp and is beneath the surface of the reservoir B, there will be a separation loss as explained in Section 10.8: Loss of head at exit, h 3 = v 22 72g. (b) Volume flow rate, Q = 0.158 m3 s−1.

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Incompressible flow through pipes in parallel

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14.5 INCOMPRESSIBLE FLOW THROUGH PIPES IN PARALLEL When two reservoirs are connected by two or more pipes in parallel, as shown in Fig. 14.4, the fluid can flow from one to the other by a number of alternative routes. The difference of head h available to produce flow will be the same for each pipe. Thus, each pipe can be considered separately, entirely independently of any other pipes running in parallel. For incompressible flow, the steady flow energy equation can be applied for flow by each route and the total volume rate of flow will be the sum of the volume rates of flow in each pipe.

FIGURE 14.4 Pipes in parallel

EXAMPLE 14.3

Two sharp-ended pipes of diameter d1 = 50 mm, and d2 = 100 mm, each of length l = 100 m, are connected in parallel between two reservoirs which have a difference of level h = 10 m, as in Fig. 14.4. If the Darcy coefficient f = 0.008 for each pipe, calculate: (a) the rate of flow for each pipe, (b) the diameter D of a single pipe 100 m long that would give the same flow if it was substituted for the original two pipes.

Solution (a) Since the two pipes are in parallel, we can deal with each pipe independently and apply the steady flow energy equation between points A and B on the free surfaces of the upper and lower reservoirs, respectively. For flow by way of pipe 1, C A2 p B C B2 C 21 4f l C 21 C 21 A ⎛ −p−− + −−− + z A⎞ = ⎛ −−− + −−− + z B⎞ + ⎛ 0.5 −−− + −−−− −−− + −−− ⎞ . ⎝ ρ g 2g ⎠ ⎝ ρ g 2g ⎠ ⎝ 2g d 1 2g 2g ⎠ Since pA = pB = atmospheric pressure and, if the reservoirs are large, CA and CB will be negligible, 4f l C 21 z A – z B = ⎛ 1.5 + −−−− ⎞ −−−. ⎝ d 1 ⎠ 2g

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Putting zA − zB = h = 10 m, f = 0.008, l = 100 m, d1 = 50 mm = 0.05 m, 4 × 0.008 × 100 C 21 10 = ⎛ 1.5 + −−−−−−−−−−−−−−−−−−−−−−⎞ −−−, ⎝ ⎠ 2g 0.05 C 21 = 2g × 10(1.5 + 64), C1 = 1.731 m s−1. Volume rate of flow through pipe 1, Q1 = (π4) d 21 C1 = (π4) × 0.052 × 1.731 = 0.0034 m3 s−1. For flow by way of pipe 2, C A2 p B C B2 C 22 4f l C 22 C 22 A ⎛