ESSENTIAL COMPUTATIONAL FLUID DYNAMICS ESSENTIAL COMPUTATIONAL FLUID DYNAMICS Oleg Zikanov JOHN WILEY & SONS, INC. This book is printed on acid-free paper. Copyright © 2010 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken,New Jersey Published simultaneouslyin Canada No part of this publicationmay be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the priorwritten permission of the Publisher, or authorization through payment of the appropriateper-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923,(978) 750-8400,fax (978)646-8600, or on the Web at www.copyright.com. Requests to the Publisherfor permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030,(201) 748-6011, fax (201) 748-6008,or online at www.wiley.com/go/permissions. Limit of Liability/Disclaimerof Warranty: While the publisherand theauthor have used theirbest efforts in preparing thisbook,they make no representationsor warranties with respect to the accuracy or completeness of thecontents of this book and specifically disclaimanyimpliedwarrantiesofmerchantabilityorfitnessforaparticularpurpose.No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitablefor your situation.You should consult with a professional where appropriate. Neither thepublishernor the author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental,consequential, or other damages. For general information about our otherproducts and services, please contact our Customer Care Department within the United States at (800) 762-2974,outsidethe United States at (317) 572-3993 or fax (317)572-4002. Wiley also publishesits books in a variety of electronic formats. Some content that appears in print may not be availablein electronic books.For more information about Wiley products, visit our Web siteat www.wiley.com. Libraryof Congress Cataloging-in-PublicationData: Zikanov,Oleg. Essential computational fluid dynamics / Oleg Zikanov. p. cm. Includes bibliographicalreferences and index. ISBN 978-0-470-42329-5 (cloth) 1. Fluid dynamics–Mathematics. I. Title. QA911.Z55 2010 532(cid:2).0501515–dc22 2009042596 ISBN: 978-0-470-42329-5 Printed in theUnited States of America 10 9 8 7 6 5 4 3 2 1 To Elena CONTENTS PREFACE xv 1 What Is CFD? 1 1.1. Introduction / 1 1.2. Brief History of CFD / 4 1.3. Outline of the Book / 6 References and Suggested Reading / 7 I Fundamentals 9 2 Governing Equations of Fluid Dynamics and Heat Transfer 11 2.1. Preliminary Concepts / 11 2.2. Mass Conservation / 14 2.3. Conservation of Chemical Species / 15 2.4. Conservation of Momentum / 16 2.5. Conservation of Energy / 19 2.6. Equation of State / 21 2.7. Equations in Integral Form / 21 2.8. Equations in Conservation Form / 24 2.9. Equations in Vector Form / 25 2.10. Boundary Conditions / 26 2.10.1. Rigid Wall Boundary Conditions / 27 2.10.2. Inlet and Exit Boundary Conditions / 29 2.10.3. Other Boundary Conditions / 29 vii viii CONTENTS References and Suggested Reading / 30 Problems / 30 3 Partial Differential Equations 32 3.1. Model Equations; Formulation of a PDE Problem / 33 3.1.1. Model Equations / 33 3.1.2. Domain, Boundary, and Initial Conditions / 35 3.1.3. Equilibrium and Marching Problems / 36 3.1.4. Examples / 37 3.2. Mathematical Classification of PDE of Second Order / 40 3.2.1. Classification / 40 3.2.2. Hyperbolic Equations / 42 3.2.3. Parabolic Equations / 45 3.2.4. Elliptic Equations / 46 3.3. Numerical Discretization: Different Kinds of CFD / 46 3.3.1. Spectral Methods / 47 3.3.2. Finite Element Methods / 49 3.3.3. Finite Difference and Finite Volume Methods / 49 References and Suggested Reading / 52 Problems / 52 4 Basics of Finite Difference Approximation 55 4.1. Computational Grid / 55 4.1.1. Time Discretization / 55 4.1.2. Space Discretization / 56 4.2. Finite Differences and Interpolation / 57 4.2.1. Approximation of ∂u/∂x / 57 4.2.2. Truncation Error, Consistency, Order of Approximation / 58 4.2.3. Other Formulas for ∂u/∂x: Evaluation of the Order of Approximation / 60 4.2.4. Schemes of Higher Order for First Derivative / 62 4.2.5. Higher-Order Derivatives / 63 4.2.6. Mixed Derivatives / 64 4.2.7. Truncation Error of Linear Interpolation / 66 4.3. Approximation of Partial Differential Equations / 67 4.3.1. Approach and Examples / 67