Computational Methods for Fluid Dynamics

375 Pages·1996·15.139 MB·English

by  Prof. Dr. Joel H. Ferziger, Prof. Dr. Milovan Perić (auth.)

Computational Methods
for Fluid Dynamics
Springer
Berlin
Heidelberg
New York
Barcelona
Budapest
Hong Kong
London
Milan
Paris
Santa Clara
Singapore
Tokyo
J. H. Ferziger · M. Peric
Computational Methods
for Fluid Dynamics
With 115 Figures
, Springer
Prof. Dr. Joel H. Ferziger
Department ofM echanical Engineering
Stanford University
Stanford, CA 94305-3030
USA
Prof. Dr. Milovan Perie
Institut ffir Schiffbau
Universitat Hamburg
Lammersieth 90
22305 Hamburg
Germany
Corrected 2nd printing 1997
All computer codes can be accessed from the publisher's server
ftp.springer.de on the internet.
ISBN-13:978-3-540-59434-5
Cip data applied for
Die Deutsche Bibliothek -CIP-Einheitsaufnahme
Ferziger, Joel H.: Computational methods for fluid dynamics I J. H. Ferziger; M. Perie . -
Corr. 2. print. -Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong;
London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer, 1997
ISBN-13: 978-3-540-59434-5 e-lsBN-13: 978-3-642-97651-3
DOl: 10.1007/978-3-642-97651-3
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Typesetting: Camera-ready by authors
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Preface
Computational fluid dynamics, commonly known under 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 the use of finite element methods is 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 and
this 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 fluid
mechanics and numerical analysis; significant errors have been made by peo
ple 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 meth
ods 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 Uni
versity of Erlangen-Niirnberg and the University of Hamburg. It reflects the
authors' experience in both writing CFD codes and using them to solve engi
neering 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, differen
tiation and integration approximations, etc.). This is important as one does
not really know a method until s/he has programmed and/or run it.
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.
VI
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 meth
ods of the type; other possible approaches and extensions are briefly de
scribed. We have tried to emphasize common elements of methods rather
than their differences.
There is a vast literature devoted to numerical methods for fluid mechan
ics. -Even if we restrict our attention to incompressible flows, it would be
impossible to cover everything in a single work. To do so, would also 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 di
rect 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 informa
tion 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
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, and Messrs. Eberhard Schreck, Volker Seidl, Kishan Shah, and
David Briggs. The help provided by those people who created arid made
VII
available 'lEX, I9\.TEYC, 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 Perie 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 Re
search Organization). Without their support, this work would never have
come into existence and we cannot express sufficient thanks to them.
Milovan Perie
[e-mail]
Joel H. Ferziger
[e-mail]
Table of 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 Table of Contents
2.4.6 Solution Method ................................. 30
2.4.7 Convergence Criteria ............................. 30
2.5 Properties of Numerical Solution Methods . . . . . . . . . . . . . . . .. 31
2.5.1 Consistency..................................... 31
2.5.2 Stability........................................ 31
2.5.3 Convergence..................................... 32
2.5.4 Conservation.................................... 32
2.5.5 Boundedness..................................... 33
2.5.6 Realizibility..................................... 33
2.5.7 Accuracy........................................ 34
2.6 Discretization Approaches. . . . . . .. . . . .. . .. . . .. .. . . . . .. . .. 35
2.6.1 Finite Difference Method. . . . . . . . . . . . . . . . . . . . . . . . .. 35
2.6.2 Finite Volume Method. . . . . . . . . . . . . . . . . . . . . . . . . . .. 35
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 ............................... 46
3.4 Approximation of the Second Derivative.. . . .. . . . . .. . . . . . .. 47
3.5 Approximation of Mixed Derivatives . . . . . . . . . . . . . . . . . . . . .. 50
3.6 Approximation of Other Terms. . . . . . . . . . . . . . . . . . . . . . . . . .. 50
3.7 Implementation of Boundary Conditions ............. '. . . . .. 50
3.8 An Introduction to Spectral Methods ..................... 52
3.8.1 Basic Concept. . .. .. .. . . .. .. . . .. . . .. . . .. . . . . .. . .. 52
3.8.2 Another View of Discretization Error .. . . . . . . . . . . . .. 54
3.9 The Algebraic Equation System ......................... , 55
3.10 Discretization Errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 59
3.11 Example .............................................. 60
4. Finite Volume Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 67
4.1 Introduction........................................... 67
4.2 Approximation of Surface Integrals ................ . . . . . .. 68
4.3 Approximation of Volume Integrals . . . . . . . . . . . . . . . . . . . . . .. 71
4.4 Interpolation Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 72
4.4.1 Upwind Interpolation (UDS) . . .. .. . . .. . . .... . . . . . .. 72
4.4.2 Linear Interpolation (CDS) .... , . . .. .... . . . . .. . . . .. 73
4.4.3 Quadratic Upwind Interpolation (QUICK).. . . . .. . . .. 74
4.4.4 Higher-Order Schemes ............................ 75
4.4.5 Other Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76
4.5 Deferred Correction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76
4.6 Implementation of Boundary Conditions. . . . . . . . . . . . . . . . . .. 77
4.7 The Algebraic Equation System . . . . . . . . . . . . . . . . . . . . . . . . .. 77
Table of Contents XI
4.8 Examples.............................................. 78
5. Solution of Linear Equation Systems. . . . . . . . . . . . . . . . . . . . .. 85
5.1 Introduction........................................... 85
5.2 Direct Methods ........................................ 86
5.2.1 Gauss Elimination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 86
5.2.2 LV Decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 88
5.2.3 Tridiagonal Systems .............................. 89
5.2.4 Cyclic Reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 90
5.3 Iterative Methods ...................................... 91
5.3.1 Basic Concept. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 91
5.3.2 Convergence..................................... 92
5.3.3 Some Basic Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 94
5.3.4 Incomplete LV Decomposition: Stone's Method ...... 95
5.3.5 ADI and Other Splitting Methods ................. , 99
5.3.6 Conjugate Gradient Methods ...................... 101
5.3.7 Biconjugate Gradients and CGSTAB ............... 104
5.3.8 Multigrid Methods ............................... 106
5.3.9 Other Iterative Solvers ............................ 110
5.4 Coupled Equations and Their Solution .................... 110
5.4.1 Simultaneous Solution ............................ 111
5.4.2 Sequential Solution ............................... 111
5.5 Non-Linear Equations and their Solution .................. 113
5.5.1 Newton-like Techniques .......................... , 113
5.5.2 Other Techniques ................................ 115
5.6 Convergence Criteria .................................... 115
5.7 Examples .............................................. 120
6. Methods for Unsteady Problems .......................... 127
6.1 Introduction ........................................... 127
6.2 Methods for Initial Value Problems in ODEs ............... 127
6.2.1 Two-Level Methods ............................... 127
6.2.2 Predictor-Corrector and Multipoint Methods ......... 130
6.2.3 Runge-Kutta Methods ............................ 132
6.2.4 Other Methods ................................... 134
6.3 Application to the Generic Transport Equation ............. 134
6.3.1 Explicit Methods ................................ , 135
6.3.2 Implicit Methods ................................. 140
6.3.3 Other Methods ................................... 143
6.4 Examples .............................................. 144