An_Introduction to Computational Fluid Dynamics 2nd

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ANIN_A01.qxd 29/12/2006 09:53 AM Page iii An Introduction to Computational Fluid Dynamics THE FINITE VOLUME METHOD Second Edition H K Versteeg and W Malalasekera

ANIN_A01.qxd 29/12/2006 09:53 AM Page i An Introduction to Computational Fluid Dynamics Supporting resources Visit www.pearsoned.co.uk/versteeg to ﬁnd valuable online resources For instructors • PowerPoint slides that can be downloaded and used for presentations For more information please contact your local Pearson Education sales representative or visit www.pearsoned.co.uk/versteeg

ANIN_A01.qxd 29/12/2006 09:53 AM Page ii 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

ANIN_A01.qxd 29/12/2006 09:53 AM Page iv 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 1995 Second edition published 2007 © Pearson Education Limited 1995, 2007 The rights of H K Versteeg and W Malalasekera to be identiﬁed as authors of this work have been asserted by them 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, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any afﬁliation with or endorsement of this book by such owners. ISBN: 978-0-13-127498-3 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 A catalog record for this book is available from the Library of Congress 10 10 9 09 7 8 08 6 07 5 06 4 3 2 1 Typeset by 35 in 10/11pt Ehrhardt MT Printed and bound by Bell & Bain Limited, Glasgow The publisher’s policy is to use paper manufactured from sustainable forests.

ANIN_A01.qxd 29/12/2006 09:53 AM Page v Contents Preface Acknowledgements 1 Introduction 1.1 What is CFD? 1.2 How does a CFD code work? 1.3 1.4 Problem solving with CFD Scope of this book 2 Conservation laws of fluid motion and boundary conditions 2.1 Governing equations of ﬂuid ﬂow and heat transfer 2.1.1 Mass conservation in three dimensions 2.1.2 Rates of change following a ﬂuid particle and for a ﬂuid element 2.1.3 Momentum equation in three dimensions 2.1.4 Energy equation in three dimensions Equations of state Classiﬁcation of physical behaviours Conservative form of the governing equations of ﬂuid ﬂow 2.2 2.3 Navier–Stokes equations for a Newtonian ﬂuid 2.4 2.5 Differential and integral forms of the general transport equations 2.6 2.7 The role of characteristics in hyperbolic equations 2.8 2.9 2.10 Auxiliary conditions for viscous ﬂuid ﬂow equations 2.11 Problems in transonic and supersonic compressible ﬂows 2.12 Summary Classiﬁcation method for simple PDEs Classiﬁcation of ﬂuid ﬂow equations xi xiii 1 1 2 4 6 9 9 10 12 14 16 20 21 24 24 26 29 32 33 35 36 38 40 44 49 3 Turbulence and its modelling 40 3.1 What is turbulence? 3.2 Transition from laminar to turbulent ﬂow 3.3 Descriptors of turbulent ﬂow

ANIN_A01.qxd 29/12/2006 09:53 AM Page vi vi CONTENTS 3.4 Characteristics of simple turbulent ﬂows 3.4.1 Free turbulent ﬂows 3.4.2 Flat plate boundary layer and pipe ﬂow 3.4.3 Summary 3.5 The effect of turbulent ﬂuctuations on properties of the mean ﬂow 3.6 Turbulent ﬂow calculations 3.7 3.8 Reynolds-averaged Navier–Stokes equations and classical turbulence models 3.7.1 Mixing length model 3.7.2 The k–εmodel 3.7.3 Reynolds stress equation models 3.7.4 Advanced turbulence models 3.7.5 Closing remarks – RANS turbulence models Large eddy simulation 3.8.1 Spacial ﬁltering of unsteady Navier–Stokes equations 3.8.2 Smagorinksy–Lilly SGS model 3.8.3 Higher-order SGS models 3.8.4 Advanced SGS models 3.8.5 3.8.6 LES applications in ﬂows with complex geometry 3.8.7 General comments on performance of LES Initial and boundary conditions for LES 3.9 Direct numerical simulation 3.9.1 Numerical issues in DNS 3.9.2 Some achievements of DNS 3.10 Summary 52 53 57 61 61 65 66 69 72 80 85 97 98 98 102 104 105 106 108 109 110 111 113 113 4 The finite volume method for diffusion problems 115 Introduction Finite volume method for one-dimensional steady state diffusion 4.1 4.2 4.3 Worked examples: one-dimensional steady state diffusion 4.4 4.5 4.6 Finite volume method for two-dimensional diffusion problems Finite volume method for three-dimensional diffusion problems Summary 115 115 118 129 131 132 5 The finite volume method for convection---diffusion problems 134 Introduction Steady one-dimensional convection and diffusion 5.1 5.2 5.3 The central differencing scheme 5.4 Properties of discretisation schemes 5.4.1 Conservativeness 5.4.2 Boundedness 5.4.3 Transportiveness Assessment of the central differencing scheme for convection– diffusion problems 5.5 5.6 The upwind differencing scheme 5.6.1 Assessment of the upwind differencing scheme 5.7 The hybrid differencing scheme 5.7.1 Assessment of the hybrid differencing scheme 134 135 136 141 141 143 143 145 146 149 151 154

ANIN_A01.qxd 29/12/2006 09:53 AM Page vii CONTENTS 5.7.2 Hybrid differencing scheme for multi-dimensional convection–diffusion 5.8 The power-law scheme 5.9 Higher-order differencing schemes for convection–diffusion problems 5.9.1 Quadratic upwind differencing scheme: the QUICK scheme 5.9.2 Assessment of the QUICK scheme 5.9.3 Stability problems of the QUICK scheme and remedies 5.9.4 General comments on the QUICK differencing scheme 5.10 TVD schemes 5.10.1 Generalisation of upwind-biased discretisation schemes 5.10.2 Total variation and TVD schemes 5.10.3 Criteria for TVD schemes 5.10.4 Flux limiter functions 5.10.5 Implementation of TVD schemes 5.10.6 Evaluation of TVD schemes 5.11 Summary 6 Solution algorithms for pressure---velocity coupling in steady flows Introduction Assembly of a complete method 6.1 6.2 The staggered grid 6.3 The momentum equations 6.4 The SIMPLE algorithm 6.5 6.6 The SIMPLER algorithm 6.7 The SIMPLEC algorithm 6.8 The PISO algorithm 6.9 General comments on SIMPLE, SIMPLER, SIMPLEC and PISO 6.10 Worked examples of the SIMPLE algorithm 6.11 Summary vii 154 155 156 156 162 163 164 164 165 167 168 170 171 175 176 179 179 180 183 186 190 191 193 193 196 197 211 7 Solution of discretised equations 212 7.1 Introduction 7.2 The TDMA 7.3 7.4 7.5 7.6 Application of the TDMA to two-dimensional problems Application of the TDMA to three-dimensional problems Examples 7.5.1 Closing remarks Point-iterative methods 7.6.1 7.6.2 Gauss–Seidel iteration method 7.6.3 Relaxation methods Jacobi iteration method 7.7 Multigrid techniques 7.7.1 An outline of a multigrid procedure 7.7.2 An illustrative example 7.7.3 Multigrid cycles 7.7.4 Grid generation for the multigrid method Summary 7.8 212 213 215 215 216 222 223 224 225 226 229 231 232 239 241 242

ANIN_A01.qxd 29/12/2006 09:53 AM Page viii viii CONTENTS 8 The finite volume method for unsteady flows 243 Introduction 8.1 8.2 One-dimensional unsteady heat conduction 8.2.1 Explicit scheme 8.2.2 Crank–Nicolson scheme 8.2.3 The fully implicit scheme Illustrative examples 8.3 8.4 Implicit method for two- and three-dimensional problems 8.5 Discretisation of transient convection–diffusion equation 8.6 Worked example of transient convection–diffusion using QUICK 8.7 differencing Solution procedures for unsteady ﬂow calculations 8.7.1 Transient SIMPLE 8.7.2 The transient PISO algorithm Steady state calculations using the pseudo-transient approach A brief note on other transient schemes 8.8 8.9 8.10 Summary 243 243 246 247 248 249 256 257 258 262 262 263 265 265 266 9 Implementation of boundary conditions 267 Introduction Inlet boundary conditions 9.1 9.2 9.3 Outlet boundary conditions 9.4 Wall boundary conditions 9.5 The constant pressure boundary condition 9.6 9.7 9.8 Symmetry boundary condition Periodic or cyclic boundary condition Potential pitfalls and ﬁnal remarks 267 268 271 273 279 280 281 281 10 Errors and uncertainty in CFD modelling 285 10.1 Errors and uncertainty in CFD 10.2 Numerical errors 10.3 Input uncertainty 10.4 Physical model uncertainty 10.5 Veriﬁcation and validation 10.6 Guidelines for best practice in CFD 10.7 Reporting/documentation of CFD simulation inputs and results 10.8 Summary 11 Methods for dealing with complex geometries Introduction 11.1 11.2 Body-ﬁtted co-ordinate grids for complex geometries 11.3 Catesian vs. curvilinear grids – an example 11.4 Curvilinear grids – difﬁculties 285 286 289 291 293 298 300 302 304 304 305 306 308

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