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Volume 1

Fundamentals of Computational Fluid Dynamics

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Numerical Computation of Internal and External Flows Volume 1 Fundamentals of Computational Fluid Dynamics Second edition

Charles Hirsch

AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Butterworth-Heinemann is an imprint of Elsevier

Butterworth-Heinemann is an imprint of Elsevier Linacre House, Jordan Hill, Oxford OX2 8DP 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA First published by John Wiley & Sons, Ltd Second edition 2007 Copyright © 2007. Charles Hirsch. All rights reserved The right of Charles Hirsch to be identified as the authors of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 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 the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-7506-6594-0 For information on all Butterworth-Heinemann publications visit our web site at http://books.elsevier.com Typeset by Charon Tec Ltd (A Macmillan Company), Chennai, India www.charontec.com Printed and bound in Great Britain 07 08 09 10

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To the Memory of Leon Hirsch and Czypa Zugman my parents, struck by destiny

Contents

Preface to the second edition Nomenclature

xviii

Introduction: An Initial Guide to CFD and to this Volume I.1 The position of CFD in the world of virtual prototyping I.1.1 The Definition Phase I.1.2 The Simulation and Analysis Phase I.1.3 The Manufacturing Cycle Phase I.2 The components of a CFD simulation system I.2.1 Step 1: Defining the Mathematical Model I.2.2 Step 2: Defining the Discretization Process I.2.3 Step 3: Performing the Analysis Phase I.2.4 Step 4: Defining the Resolution Phase I.3 The structure of this volume References

1 1 2 3 5 11 11 13 15 16 18 20

Part I The Mathematical Models for Fluid Flow Simulations at Various Levels of Approximation

21

1 The Basic Equations of Fluid Dynamics Objectives and guidelines 1.1 General form of a conservation law 1.1.1 Scalar Conservation Law 1.1.2 Convection–Diffusion Form of a Conservation Law 1.1.3 Vector Conservation Law The Equations of Fluid Mechanics 1.2 The mass conservation equation 1.3 The momentum conservation law or equation of motion 1.4 The energy conservation equation 1.4.1 Conservative Formulation of the Energy Equation 1.4.2 The Equations for Internal Energy and Entropy 1.4.3 Perfect Gas Model 1.4.4 Incompressible Fluid Model A1.5 Rotating frame of reference A1.5.1 Equation of Motion in the Relative System A1.5.2 Energy Equation in the Relative System A1.5.3 Crocco’s Form of the Equations of Motion A1.6 Advanced applications of control volume formulations A1.6.1 Lift and Drag Estimations from CFD Results A1.6.2 Conservation Law for a Moving Control Volume Summary of the basic flow equations vi

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Contents Conclusions and main topics to remember References Problems

vii 63 63 63

2 The Dynamical Levels of Approximation 65 Objectives and guidelines 65 2.1 The Navier–Stokes equations 70 2.1.1 Non-uniqueness in Viscous Flows 73 2.1.1.1 Marangoni thermo-capillary flow in a liquid bridge 73 2.1.1.2 Flow around a circular cylinder 77 2.1.2 Direct Numerical Simulation of Turbulent Flows (DNS) 83 2.2 Approximations of turbulent flows 86 2.2.1 Large Eddy Simulation (LES) of Turbulent Flows 87 2.2.2 Reynolds Averaged Navier–Stokes Equations (RANS) 89 2.3 Thin shear layer approximation (TSL) 94 2.4 Parabolized Navier–Stokes equations 94 2.5 Boundary layer approximation 95 2.6 The distributed loss model 96 2.7 Inviscid flow model: Euler equations 97 2.8 Potential flow model 98 2.9 Summary 101 References 101 Problems 103 3 The Mathematical Nature of the Flow Equations and Their Boundary Conditions Objectives and guidelines 3.1 Simplified models of a convection–diffusion equati