Joel H. Ferziger I Milovan Peric

Computational Methods for Fluid Dynamics

Springer-V erlag Berlin Heidelberg GmbH

Joel H. Ferziger / Milovan Peric

Computational Methods for Fluid Dynamics

third, rev. edition

With 128 Figures

Springer

Professor Joel H. Ferziger Stanford University Dept. of Mechanical Engineering Stanford, CA 94305 USA

Dr. Milovan Perie Computational Dynamics Diirrenhofstra6e 4 D-90402 Niirnberg

ISBN 978-3-540-42074-3

Library of Congress Cataloging-in-Publication Data

Ferziger, Joel H.: Computational Methods for Fluid Dynamics / Joel H. Ferziger / Milovan Peric. - 3., rev. ed.­Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Tokyo: Springer, 2002

ISBN 978-3-540-42074-3 ISBN 978-3-642-56026-2 (eBook) DOI 10.1007/978-3-642-56026-2

This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of iIlustrations, re citation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution act under German Copyright Law.

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Preface

Computational fluid dynamics, commonly known by the acronym 'CFD', is undergoing significant expansion in terms of both the number of courses offered at universities and the number of researchers active in the field. There are a number of software packages available that solve fluid flow problems; the market is not quite as large as the one for structural mechanics codes, in which finite element methods are well established. The lag can be explained by the fact that CFD problems are, in general, more difficult to solve. However, CFD codes are slowly being accepted as design tools by industrial users. At present, users of CFD need to be fairly knowledgeable, which requires education of both students and working engineers. The present book is an attempt to fill this need.

It is our belief that, to work in CFD, one needs a solid background in both fluid mechanics and numerical analysis; significant errors have been made by people lacking knowledge in one or the other. We therefore encourage the reader to obtain a working knowledge of these subjects before entering into a study of the material in this book. Because different people view numeri­cal methods differently, and to make this work more self-contained, we have included two chapters on basic numerical methods in this book. The book is based on material offered by the authors in courses at Stanford Univer­sity, the University of Erlangen-Niirnberg and the Technical University of Hamburg-Harburg. It reflects the authors' experience in both writing CFD codes and using them to solve engineering problems. Many of the codes used in the examples, from the simple ones involving rectangular grids to the ones using non-orthogonal grids and multigrid methods, are available to interested readers; see the information on how to access them via Internet in the ap­pendix. These codes illustrate the methods described in the book; they can be adapted to the solution of many fluid mechanical problems. Students should try to modify them (e.g. to implement different boundary conditions, interpo­lation schemes, differentiation and integration approximations, etc.). This is important as one does not really know a method until s/he has programmed and/or run it.

Since one of the authors (M.P.) has just recently decided to give up his pro­fessor position to work for a provider of CFD tools, we have also included in the Internet site a special version of a full-featured commercial CFD package

VI

that can be used to solve many different flow problems. This is accompanied by a collection of prepared and solved test cases that are suitable to learn how to use such tools most effectively. Experience with this tool will be valu­able to anyone who has never used such tools before, as the major issues are common to most of them. Suggestions are also given for parameter variation, error estimation, grid quality assessment, and effiCiency improvement.

The finite volume method is favored in this book, although finite difference methods are described in what we hope is sufficient detail. Finite element methods are not covered in detail as a number of books on that subject already exist.

We have tried to describe the basic ideas of each topic in such a way that they can be understood by the reader; where possible, we have avoided lengthy mathematical analysis. Usually a general description of an idea or method is followed by a more detailed description (including the necessary equations) of one or two numerical schemes representative of the better meth­ods of the type; other possible approaches and extensions are briefly de­scribed. We have tried to emphasize common elements of methods rather than their differences.

There is a vast literature devoted to numerical methods for fluid mechan­ics. Even if we restrict our attention to incompressible flows, it would be impossible to cover everything in a single work. Doing so would create con­fusion for the reader. We have therefore covered only the methods that we have found valuable and that are commonly used in industry in this book. References to other methods are given, however.

We have placed considerable emphasis on the need to estimate numerical errors; almost all examples in this book are accompanied with error analysis. Although it is possible for a qualitatively incorrect solution of a problem to look reasonable (it may even be a good solution of another problem), the consequences of accepting it may be severe. On the other hand, sometimes a relatively poor solution can be of value if treated with care. Industrial users of commercial codes need to learn to judge the quality of the results before believing them; we hope that this book will contribute to the awareness that numerical solutions are always approximate.

We have tried to cover a cross-section of modern approaches, including di­rect and large eddy simulation of turbulence, multigrid methods and parallel computing, methods for moving grids and free surface flows, etc. Obviously, we could not cover all these topics in detail, but we hope that the informa­tion contained herein will provide the reader with a general knowledge of the subject; those interested in a more detailed study of a particular topic will find recommendations for further reading.

While we have invested every effort to avoid typing, spelling and other errors, no doubt some remain to be found by readers. We will appreciate your notifying us of any mistakes you might find, as well as your comments and suggestions for improvement of future editions of the book. For that

VII

purpose, the authors' electronic mail addresses are given below. We also hope that colleagues whose work has not been referenced will forgive us, since any omissions are unintentional.

We have to thank all our present and former students, colleagues, and friends, who helped us in one way or another to finish this work; the complete list of names is too long to list here. Names that we cannot avoid mentioning include Drs. Ismet Demirdzic, Samir Muzaferija, Zeljko Lilek, Joseph Oliger, Gene Golub, Eberhard Schreck, Volker Seidl, Kishan Shah, Fotina (Tina) Katapodes and David Briggs. The help provided by those people who created and made available 'lEX, 9-'lEX, Linux, Xfig, Ghostscript and other tools which made our job easier is also greatly appreciated.

Our families gave us a tremendous support during this endeavor; our special thanks go to Anna, Robinson and Kerstin Peric and Eva Ferziger.

This collaboration between two geographically distant colleagues was made possible by grants and fellowships from the Alexander von Humboldt Foundation and the Deutsche Forschungsgemeinschaft (German National Re­search Organization). Without their support, this work would never have come into existence and we cannot express sufficient thanks to them.

Milovan Peric milovan@cd.co.uk Joel H. Ferziger ferziger@leland.stanford.edu

Contents

Preface ................................... ".................... V

1. Basic Concepts of Fluid Flow ............................ 1 1.1 Introduction........................................... 1 1.2 Conservation Principles ................................. 3 1.3 Mass Conservation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Momentum Conservation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Conservation of Scalar Quantities. . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 Dimensionless Form of Equations. . . . . . . . . . . . . . . . . . . . . . . .. 11 1.7 Simplified Mathematical Models. . . . . . . . . . . . . . . . . . . . . . . . .. 12

1.7.1 Incompressible Flow. . . .. .. . . .. .. . . . . . . .. . . . . .. . .. 12 1.7.2 Inviscid (Euler) Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13 1.7.3 Potential Flow. . . . . . . . . . .. .. . . . . .. .. . . . . . . . . .. . .. 13 1.7.4 Creeping (Stokes) Flow ........................... 14 1.7.5 Boussinesq Approximation. . . . . . . . . . . . . . . . . . . . . . . .. 14 1.7.6 Boundary Layer Approximation. . . . . . . . . . . . . . . . . . .. 15 1.7.7 Modeling of Complex Flow Phenomena. . . . . . . . . . . .. 16

1.8 Mathematical Classification of Flows. . . . . . . . . . . . . . . . . . . . .. 16 1.8.1 Hyperbolic Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17 1.8.2 Parabolic Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17 1.8.3 Elliptic Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17 1.8.4 Mixed Flow Types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18

1.9 Plan of This Book ..................................... , 18

2. Introduction to Numerical Methods. . . . . . . . . .. . . . . .. . . . .. 21 2.1 Approaches to Fluid Dynamical Problems. . . . . . . . . . . . . . . .. 21 2.2 What is CFD? .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23 2.3 Possibilities and Limitations of Numerical Methods. . . . . . . .. 23 2.4 Components of a Numerical Solution Method. . . . . . . . . . . . .. 25

2.4.1 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25 2.4.2 Discretization Method ............................ 25 2.4.3 Coordinate and Basis Vector Systems. . . . . . . . . . . . . .. 26 2.4.4 Numerical Grid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 26 2.4.5 Finite Approximations. . . . . . . . . . . . . . . . . . . . . . . . . . .. 30

X Contents

2.4.6 Solution Method ...... . . . . . . . . . . . . . . . . . . . . . . . . . .. 30 2.4.7 Convergence Criteria ............................. 31

2.5 Properties of Numerical Solution Methods. . . . . . . . . . . . . . . .. 31 2.5.1 Consistency..................................... 31 2.5.2 Stability........................................ 32 2.5.3 Convergence..................................... 32 2.5.4 Conservation.................................... 33 2.5.5 Boundedness..................................... 33 2.5.6 Realizability..................................... 33 2.5.7 Accuracy........................................ 34

2.6 Discretization Approaches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35 2.6.1 Finite Difference Method. . . . . . . . . . . . . . . . . . . . . . . . .. 35 2.6.2 Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36 2.6.3 Finite Element Method. . .. . . . . . . . . . . . . . . . . . . . . . .. 36

3. Finite Difference Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39 3.1 Introduction........................................... 39 3.2 Basic Concept ......................................... 39 3.3 Approximation of the First Derivative. . . . . . . . . . . . . . . . . . . .. 42

3.3.1 Taylor Series Expansion. . . . . . . . . . . . . . . . . . . . . . . . . .. 42 3.3.2 Polynomial Fitting. . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. 44 3.3.3 Compact Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45 3.3.4 Non-Uniform Grids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47

3.4 Approximation of the Second Derivative.. . . . . . . . . . . . . . . . .. 49 3.5 Approximation of Mixed Derivatives. . . . . . . . . . . . . . . . . . . . .. 52 3.6 Approximation of Other Terms. . . . . . . . . . . . . . . . . . . . . . . . . .. 53 3.7 Implementation of Boundary Conditions. . . . . . . . . . . . . . . . . .. 53 3.8 The Algebraic Equation System . . . . . . . . . . . . . . . . . . . . . . . . .. 55 3.9 Discretization Errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58 3.10 An Introduction to Spectral Methods. . . . . . .. . . . . . . . . . . . .. 60

3.10.1 Basic Concept ................................... 60 3.10.2 Another View of Discretization Error. . . . . . . . . . . . . .. 62

3.11 Example .............................................. 63

4. Finite Volume Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 71 4.1 Introduction........................................... 71 4.2 Approximation of Surface Integrals ....................... 72 4.3 Approximation of Volume Integrals . . . . . . . . . . . . . . . . . . . . . .. 75 4.4 Interpolation and Differentiation Practices. . . . . . . . . . . . . . . .. 76

4.4.1 Upwind Interpolation (UDS) . . . . . . . . . . . . . . . . . . . . . .. 76 4.4.2 Linear Interpolation (CDS) . . . . . . . . . . . . . . . . . . . . . . .. 77 4.4.3 Quadratic Upwind Interpolation (QUICK). . . . . . . . . .. 78 4.4.4 Higher-Order Schemes ............................ 79 4.4.5 Other Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 81

4.5 Implementation of Boundary Conditions. . . . . . . . . . . . . . . . . .. 81

Contents XI

4.6 The Algebraic Equation System . . . . . . . . . . . . . . . . . . . . . . . . .. 82 4.7 Examples.............................................. 82

5. Solution of Linear Equation Systems. . . . . . . . . . . . . . . . . . . . .. 91 5.1 Introduction........................................... 91 5.2 Direct Methods ........................................ 92

5.2.1 Gauss Elimination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92 5.2.2 LU Decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 94 5.2.3 Tridiagonal Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95 5.2.4 Cyclic Reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 96

5.3 Iterative Methods ...................................... 97 5.3.1 Basic Concept ................. .'................. 97 5.3.2 Convergence..................................... 98 5.3.3 Some Basic Methods .............................. 100 5.3.4 Incomplete LU Decomposition: Stone's Method ...... 101 5.3.5 AD! and Other Splitting Methods .................. 105 5.3.6 Conjugate Gradient Methods ...................... 107 5.3.7 Biconjugate Gradients and CGSTAB ............... 110 5.3.8 Multigrid Methods ............................... 112 5.3.9 Other Iterative Solvers ........................... , 116

5.4 Coupled Equations and Their Solution .................... 116 5.4.1 Simultaneous Solution ............................ 117 5.4.2 Sequential Solution ............................... 117 5.4.3 Under-Relaxation ................................ 118

5.5 Non-Linear Equations and their Solution .................. 119 5.5.1 Newton-like Techniques ........................... 119 5.5.2 Other Techniques ................................ 121

5.6 Deferred-Correction Approaches .......................... 122 5.7 Convergence Criteria and Iteration Errors ................. 124 5.8 Examples .............................................. 129

6. Methods for Unsteady Problems. . . . . . . . . . . . . . . . . . . . . . . . .. 135 6.1 Introduction ........................................... 135 6.2 Methods for Initial Value Problems in ODEs ............... 135

6.2.1 Two-Level Methods ............................... 135 6.2.2 Predictor-Corrector and Multipoint Methods ........ , 138 6.2.3 Runge-Kutta Methods ............................ 140 6.2.4 Other Methods ................................... 142

6.3 Application to the Generic Transport Equation ............ , 142 6.3.1 Explicit Methods ................................. 143 6.3.2 Implicit Methods ................................. 148 6.3.3 Other Methods ................................... 151

6.4 Examples .............................................. 152

XII Contents

7. Solution of the Navier-Stokes Equations .................. 157 7.1 Special Features of the Navier-Stokes Equations ............ 157

7.1.1 Discretization of Convective and Viscous Terms ...... 157 7.1.2 Discretization of Pressure Terms and Body Forces .... 158 7.1.3 Conservation Properties ........................... 160

7.2 Choice of Variable Arrangement on the Grid ............... 164 7.2.1 Colocated Arrangement ........................... 165 7.2.2 Staggered Arrangements .......................... 166

7.3 Calculation of the Pressure .............................. 167 7.3.1 The Pressure Equation and its Solution ............. 167 7.3.2 A Simple Explicit Time Advance Scheme ............ 168 7.3.3 A Simple Implicit Time Advance Method ............ 170 7.3.4 Implicit Pressure-Correction Methods ............... 172

7.4 Other Methods ......................................... 178 7.4.1 Fractional Step Methods .......................... 178 7.4.2 Streamfunction-Vorticity Methods .................. 181 7.4.3 Artificial Compressibility Methods .................. 183

7.5 Solution Methods for the Navier-Stokes Equations .......... 188 7.5.1 Implicit Scheme Using Pressure-Correction and a Stag-

gered Grid ....................................... 188 7.5.2 Treatment of Pressure for Colocated Variables ....... 196 7.5.3 SIMPLE Algorithm for a Colocated Variable Arrange-

ment ............................................ 200 7.6 Note on Pressure and Incompressibility .................... 202 7.7 Boundary Conditions for the Navier-Stokes Equations ....... 204 7.8 Examples .............................................. 206

8. Complex Geometries ..................................... 217 8.1 The Choice of Grid ..................................... 217

8.1.1 Stepwise Approximation Using Regular Grids ........ 217 8.1.2 Overlapping Grids ................................ 218 8.1.3 Boundary-Fitted Non-Orthogonal Grids ............. 219

8.2 Grid Generation ........................................ 219 8.3 The Choice of Velocity Components ...................... 223

8.3.1 Grid-Oriented Velocity Components ................ 224 8.3.2 Cartesian Velocity Components .................... 224

8.4 The Choice of Variable Arrangement ...................... 225 8.4.1 Staggered Arrangements .......................... 225 8.4.2 Colocated Arrangement ........................... 226

8.5 Finite Difference Methods ............................... 226 8.5.1 Methods Based on Coordinate Transformation ....... 226 8.5.2 Method Based on Shape Functions ................. 229

8.6 Finite Volume Methods ................................. 230 8.6.1 Approximation of Convective Fluxes ................ 231 8.6.2 Approximation of Diffusive Fluxes .................. 232

Contents XIII

8.6.3 Approximation of Source Terms .................... 238 8.6.4 Three-Dimensional Grids .......................... 239 8.6.5 Block-Structured Grids ........................... 241 8.6.6 Unstructured Grids ............................... 244

8.7 Control-Volume-Based Finite Element Methods ............ 245 8.8 Pressure-Correction Equation ............................ 247 8.9 Axi-Symmetric Problems ................................ 252 8.10 Implementation of Boundary Conditions ................... 254

8.10.1 Inlet ............................................ 255 8.10.2 Outlet '" ....................................... 255 8.10.3 Impermeable Walls ............................... 256 8.10.4 Symmetry Planes ................................. 258 8.10.5 Specified Pressure ................................ 258

8.11 Examples .............................................. 259

9. Turbulent Flows . ......................................... 265 9.1 Introduction ........................................... 265 9.2 Direct Numerical Simulation (DNS) ....................... 267

9.2.1 Example: Spatial Decay of Grid Turbulence .......... 275 9.3 Large Eddy Simulation (LES) ............................ 277

9.3.1 Smagorinsky and Related Models ................... 279 9.3.2 Dynamic Models ................................. 281 9.3.3 Deconvolution Models ............................. 283 9.3.4 Example: Flow Over a Wall-Mounted Cube .......... 284 9.3.5 Example: Stratified Homogeneous Shear Flow ........ 287

9.4 RANS Models .......................................... 292 9.4.1 Reynolds-Averaged Navier-Stokes (RANS) Equations. 292 9.4.2 Simple Turbulence Models and their Application ..... 294 9.4.3 The v2f Model ................................... 301 9.4.4 Example: Flow Around an Engine Valve ............. 302

9.5 Reynolds Stress Models ................................. 304 9.6 Very Large Eddy Simulation ............................. 306

10. Compressible Flow ....................................... 309 10.1 Introduction ........................................... 309 10.2 Pressure-Correction Methods for Arbitrary Mach Number ... 310

10.2.1 Pressure-Velocity-Density Coupling ................ 311 10.2.2 Boundary Conditions ............................. 315 10.2.3 Examples ....................................... 319

10.3 Methods Designed for Compressible Flow .................. 324 10.3.1 An Overview of Some Specific Methods ............. 326

XIV Contents

11. Efficiency and Accuracy Improvement .................... 329 11.1 Error Analysis and Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

11.1.1 Description of Errors ............................. 329 11.1.2 Estimation of Errors .............................. 332 11.1.3 Recommended Practice for CFD Uncertainty Analysis 337

11.2 Grid quality and optimization ............................ 341 11.3 Multigrid Methods for Flow Calculation ................... 344 11.4 Adaptive Grid Methods and Local Grid Refinement ......... 351 11.5 Parallel Computing in CFD .............................. 356

11.5.1 Iterative Schemes for Linear Equations .............. 357 11.5.2 Domain Decomposition in Space ................... 360 11.5.3 Domain Decomposition in Time .................... 363 11.5.4 Efficiency of Parallel Computing ................... 364

12. Special Topics ............................................ 369 12.1 Introduction ........................................... 369 12.2 Heat and Mass Transfer ................................. 370 12.3 Flows With Variable Fluid Properties ..................... 373 12.4 Moving Grids .......................................... 373 12.5 Free-Surface Flows ...................................... 381

12.5.1 Interface-Tracking Methods. . . . . . . . . . . . . . . . . . . . . . . . 388 12.5.2 Hybrid Methods .................................. 396

12.6 Meteorological and Oceanographic Applications ............ 397 12.7 Multiphase flows ....................................... 399 12.8 Combustion ........................................... 400

A. Appendices ............................................... 405 A.1 List of Computer Codes and How to Access Them .......... 405 A.2 List of Frequently Used Abbreviations ..................... 407

References ................................................. 409

Index . ................................................... 421