Springer computational methods for fluid dynamics

Joel H. Ferziger 1 Milovan PeriC
Computational Methods for Fluid Dynamics


Springer
Berlin
Heidelberg
New York
Barcelona
Hong Kong
London
Milan
Paris
Tokyo


Joel H. Ferziger 1 Milovan PeriC

Computarional 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 Peril
Computational Dynamics
DiirrenhofstraBe 4
D-90402 Niirnberg

ISBN 3-540-42074-6 Springer-Verlag Berlin Heidelberg NewYork

Library of Congress Cataloging-in-Publication Data
Ferziger, Joel H.:
Computational Methods for Fluid Dynamics / Joel H. Ferziger / Milovan Perit. - 3., rev. ed. Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Tokyo: Springer, 2002
ISBN 3-540-42074-6
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
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of September 9, 1965, in its current version, and permission for use must always be obtained from
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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 a t 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 (eg. t o 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 t o 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


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 t o 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 t o thank all our present and former students, colleagues, and
friends, who helped us in one way or another t o finish this work; the complete
list of names is too long t o list here. Names that we cannot avoid mentioning
include Drs. Ismet DemirdZiC, Samir Muzaferija, ~ e l j k oLilek, 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 TEX,@TEX, 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 t o 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

Joel H. Ferziger



Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V

1. Basic Concepts of Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Conservation Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 MomentumConservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Conservation of Scalar Quantities . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Dimensionless Form of Equations . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Simplified Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.1 Incompressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.2 Inviscid (Euler) Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.3 Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.4 Creeping (Stokes) Flow . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.7.5 Boussinesq Approximation . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.6 Boundary Layer Approximation . . . . . . . . . . . . . . . . . . . .
1.7.7 Modeling of Complex Flow Phenomena . . . . . . . . . . . . .
1.8 Mathematical Classification of Flows . . . . . . . . . . . . . . . . . . . . . .
1.8.1 Hyperbolic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.2 Parabolic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.3 Elliptic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.4 Mixed Flow Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9 Plan of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
3
4
5
9
11
12
12
13
13
14
14
15
16
16
17
17
17
18

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.7 Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Properties of Numerical Solution Methods . . . . . . . . . . . . . . . . .
2.5.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.4 Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.5 Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5.6 Realizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.7 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Discretization Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.2 Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.3 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . .

30
31
31
31
32
32
33
33
33
34
35
35
36
36

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 AD1 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 Multipoinf 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


XI1

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 Staggered Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .188
7.5.2 Treatment of Pressure for Colocated Variables . . . . . . . 196
7.5.3 SIMPLE Algorithm for a Colocated Variable Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

8.7
8.8
8.9
8.10

8.11
9

.

XI11

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
Control-Volume-Based Finite Element Methods . . . . . . . . . . . . 245
Pressure-Correction Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
Axi-Symmetric Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .252
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
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

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

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 . l List of Computer Codes and How t o Access Them . . . . . . . . . . 405
A.2 List of Frequently Used Abbreviations . . . . . . . . . . . . . . . . . . . . . 407

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


1. Basic Concepts of Fluid Flow

1.1 Introduction
Fluids are substances whose molecular structure offers no resistance t o external shear forces: even the smallest force causes deformation of a fluid particle.
Although a significant distinction exists between liquids and gases, both types
of fluids obey the same laws of motion. In most cases of interest, a fluid can
be regarded as continuum, i.e. a continuous substance.
Fluid flow is caused by the action of externally applied forces. Common
driving forces include pressure differences, gravity, shear, rotation, and surface tension. They can be classified as surface forces (e.g. the shear force due
to wind blowing above the ocean or pressure and shear forces created by a
movement of a rigid wall relative t o the fluid) and body forces (e.g. gravity
and forces induced by rotation).
While all fluids behave similarly under action of forces, their macroscopic
properties differ considerably. These properties must be known if one is t o
study fluid motion; the most important properties of simple fluids are the
density and viscosity. Others, such as Prandtl number, specific h,eat, and surface tension affect fluid flows only under certain conditions, e.g. when there
are large temperature differences. Fluid properties are functions of other thermodynamic variables (e.g. temperature and pressure); although it is possible
to estimate some of them from statistical mechanics or kinetic theory, they
are usually obtained by laboratory measurement.
Fluid mechanics is a very broad field. A small library of books would be
required t o cover all of the topics that could be included in it. In this book
we shall be interested mainly in flows of interest to mechanical engineers but
even that is a very broad area so we shall try to classify the types of problems
that may be encountered. A more mathematical, but less complete, version
of this scheme will be found in Sect. 1.8.
The speed of a flow affects its properties in a number of ways. At low
enough speeds, the inertia of the fluid may be ignored and we have creeping flow. This regime is of importance in flows containing small particles
(suspensions), in flows through porous media or in narrow passages (coating
techniques, micro-devices). As the speed is increased, inertia becomes important but each fluid particle follows a smooth trajectory; the flow is then
said t o be laminar. Further increases in speed may lead t o instability that

2

1. Basic Concepts of Fluid Flow

eventually produces a more random type of flow that is called turbulent; the
process of laminar-turbulent transition is an important area in its own right.
Finally, the ratio of the flow speed to the speed of sound in the fluid (the
Mach number) determines whether exchange between kinetic energy of the
motion and internal degrees of freedom needs to be considered. For small
Mach numbers, Ma < 0.3, the flow may be considered incompressible; otherwise, it is compressible. If Ma < 1, the flow is called subsonic; when Ma > 1,
the flow is supersonic and shock waves are possible. Finally, for Ma > 5 , the
compression may create high enough temperatures to change the chemical
nature of the fluid; such flows are called hypersonic. These distinctions affect
the mathematical nature of the problem and therefore the solution method.
Note that we call the flow compressible or incompressible depending on the
Mach number, even though compressibility is a property of the fluid. This
is common terminology since the flow of a compressible fluid at low Mach
number is essentially incompressible.
In many flows, the effects of viscosity are important only near walls, so
that the flow in the largest part of the domain can be considered as inviscid.
In the fluids we treat in this book, Newton's law of viscosity is a good approximation and it will be used exclusively. Fluids obeying Newton's law are
called Newtonian; non-Newtonian fluids are important for some engineering
applications but are not treated here.
Many other phenomena affect fluid flow. These include temperature differences which lead to heat transfer and density differences which give rise to
buoyancy. They, and differences in concentration of solutes, may affect flows
significantly or, even be the sole cause of the flow. Phase changes (boiling,
condensation, melting and freezing), when they occur, always lead to important modifications of the flow and give rise to multi-phase flow. Variation of
other properties such as viscosity, surface tension etc. may also play important role in determining the nature of the flow. With only a few exceptions,
these effects will not be considered in this book.
In this chapter the basic equations governing fluid flow and associated

phenomena will be presented in several forms: (i) a coordinate-free form,
which can be specialized to various coordinate systems, (ii) an integral form
for a finite control volume, which serves as starting point for an important
class of numerical methods, and (iii) a differential (tensor) form in a Cartesian
reference frame, which is the basis for another important approach. The basic
conservation principles and laws used to derive these equations will only
be briefly summarized here; more detailed derivations can be found in a
number of standard texts on fluid mechanics (e.g. Bird et al., 1962; Slattery,
1972; White, 1986). It is assumed that the reader is somewhat familiar with
the physics of fluid flow and related phenomena, so we shall concentrate on
techniques for the numerical solution of the governing equations.


1.2 Conservation Principles

3

1.2 Conservation Principles
Conservation laws can be derived by considering a given quantity of matter or
control mass (CM) and its extensive properties, such as mass, momentum and
energy. This approach is used to study the dynamics of solid bodies, where the
CM (sometimes called the system) is easily identified. In fluid flows, however,
it is difficult to follow a parcel of matter. It is more convenient to deal with
the flow within a certain spatial region we call a control volume (CV), rather
than in a parcel of matter which quickly passes through the region of interest.
This method of analysis is called the control volume approach.
We shall be concerned primarily with two extensive properties, mass and
momentum. The conservation equations for these and other properties have
common terms which will be considered first.
The conservation law for an extensive property relates the rate of change

of the amount of that property in a given control mass to externally determined effects. For mass, which is neither created nor destroyed in the flows
of engineering interest, the conservation equation can be written:

On the other hand, momentum can be changed by the action of forces and
its conservation equation is Newton's second law of motion:

where t stands for time, m for mass, v for the velocity, and f for forces acting
on the control mass.
We shall transform these laws into a control volume form that will be used
throughout this book. The fundamental variables will be intensive rather than
extensive properties; the former are properties which are independent of the
amount of matter considered. Examples are density p (mass per unit volume)
and velocity v (momentum per unit mass).
If 4 is any conserved intensive property (for mass conservation, 4 = 1; for
momentum conservation, 4 = v ; for conservation of a scalar, 4 represents the
conserved property per unit mass), then the corresponding extensive property
@ can be expressed as:

where OcM stands for volume occupied by the CM. Using this definition,
the left hand side of each conservation equation for a control volume can be
written:'
This equation is often called control volume equation or the Reynolds' transport
theorem.


4

1. Basic Concepts of Fluid Flow

where flcv is the CV volume, Scv is the surface enclosing CV, n is the unit

vector orthogonal to Scv and directed outwards, v is the fluid velocity and vb
is the velocity with which the CV surface is moving. For a fixed CV, which
we shall be considering most of the time, vb = 0 and the first derivative
on the right hand side becomes a local (partial) derivative. This equation
states that the rate of change of the amount of the property in the control
mass, @, is the rate of change of the property within the control volume plus
the net flux of it through the CV boundary due to fluid motion relative to
CV boundary. The last term is usually called the convective (or sometimes,
advective) flux of q5 through the CV boundary. If the CV moves so that its
boundary coincides with the boundary of a control mass, then v = vb and
this term will be zero as required.
A detailed derivation of this equation is given in in many textbooks on
fluid dynamics (e.g. in Bird et al., 1962; Fox and McDonald, 1982) and will not
be repeated here. The mass, momentum and scalar conservation equations
will be presented in the next three sections. For convenience, a fixed CV will
be considered; fl represents the CV volume and S its surface.

1.3 Mass Conservation
The integral form of the mass conservation (continuity) equation follows directly from the control volume equation, by setting 4 = 1:

By applying the Gauss' divergence theorem to the convection term, we can
transform the surface integral into a volume integral. Allowing the control
volume to become infinitesimally small leads to a differential coordinate-free
form of the continuity equation:
ap
at

-

+ div (pv) = 0

This form can be transformed into a form specific to a given coordinate
system by providing the expression for the divergence operator in that system.
Expressions for common coordinate systems such as the Cartesian, cylindrical
and spherical systems can be found in many textbooks (e.g. Bird et al., 1962);
expressions applicable to general non-orthogonal coordinate systems are given
e.g. in Truesdell (1977), Aris (1989), Sedov (1971). We present below the
Cartesian form in both tensor and expanded notation. Here and throughout
this book we shall adopt the Einstein convention that whenever the same


1.4 Momentum Conservation

5

index appears twice in any term, summation over the range of that index is
implied:

where xi (i=1,2,3) or (x, y , z ) are the Cartesian coordinates and ui or
(ux,u,, u,) are the Cartesian components of the velocity vector v. The conservation equations in Cartesian form are often used and this will be the case
in this work. Differential conservation equations in non-orthogonal coordinates will be presented in Chap. 8.

1.4 Moment um Conservation
There are several ways of deriving the momentum conservation equation. One
approach is to use the control volume method described in Sect. 1.2; in this
method, one uses Eqs. (1.2) and (1.4) and replaces 4 by v , e.g. for a fixed
fluid-containing volume of space:

To express the right hand side in terms of intensive properties, one has to
consider the forces which may act on the fluid in a CV:

0
0

surface forces (pressure, normal and shear stresses, surface tension etc.);
body forces (gravity, centrifugal and Coriolis forces, electromagnetic forces,
etc.).

The surface forces due to pressure and stresses are, from the molecular point
of view, the microscopic momentum fluxes across a surface. If these fluxes
cannot be written in terms of the properties whose conservation the equations govern (density and velocity), the system of equations is not closed;
that is there are fewer equations than dependent variables and solution is
not possible. This possibility can be avoided by making certain assumptions.
The simplest assumption is that the fluid is Newtonian; fortunately, the Newtonian model applies to many actual fluids.
For Newtonian fluids, the stress tensor T, which is the molecular rate of
transport of momentum, can be written:

where ,LL is the dynamic viscosity, I is the unit tensor, p is the static pressure
and D is the rate of strain (deformation) tensor:


6

1. Basic Concepts of Fluid Flow

These two equations may be written, in index notation in Cartesian coordinates, as follows:

where Sij is Kronecker symbol ( S i j = 1 if i = j and Sij = 0 otherwise).
For incompressible flows, the second term in brackets in Eq. (1.11) is zero
by virtue of the continuity equation. The following notation is often used in
literature to describe the viscous part of the stress tensor:

rij

2
= 2pDij - -pSij div v .
3

(1.13)

For non-Newtonian fluids, the relation between the stress tensor and the
velocity is defined by a set of partial differential equations and the total
problem is far more complicated. In fact, different types of non-Newtonian
fluids require different constitutive equations which may, in turn, require
different solution methods. This subject is complex and is just beginning to
be explored. For these reasons, it will not be considered further in this book.
With the body forces (per unit mass) being represented by b, the integral
form of the momentum conservation equation becomes:

A coordinate-free vector form of the momentum conservation equation (1.14)
is readily obtained by applying Gauss' divergence theorem to the convective
and diffusive flux terms:

at

+ div ( p v v ) = div T + pb

The corresponding equation for the ith Cartesian component is:
a(~ui)
div ( p u i v ) = div ti + phi .
at

+

Since momentum is a vector quantity, the convective and diffusive fluxes
of it through a CV boundary are the scalar products of second rank tensors
( p v v and T) with the surface vector n d S . The integral form of the above
equations is:


1.4 Momentum Conservation

7

where (see Eqs. (1.9) and (1.10)):

Here bi stands for the ith component of the body force, superscript
means
transpose and ii is the Cartesian unit vector in the direction of the coordinate
xi. In Cartesian coordinates one can write the above expression as:

A vector field can be represented in a number of different ways. The basis
vectors in terms of which the vector is defined may be local or global. In curvilinear coordinate systems, which are often required when the boundaries are
complex (see Chap. 8) one may choose either a covariant or a contravariant
basis, see Fig. 1.1. The former expresses a vector in terms of its components
along the local coordinates; the latter uses the projections normal to coordinate surfaces. In a Cartesian system, the two become identical. Also, the basis
vectors may be dimensionless or dimensional. Including all of these options,
over 70 different forms of the momentum equations are possible. Mathematically, all are equivalent; from the numerical point of view, some are more
difficult to deal with than others.

Fig. 1.1. Representation of a vector through different components: u i - W t e s i a n
components; vi - contravariant components; v i - covariant components [ V A = V B ,

( % ) A = ( u ~ ) B ,( V i ) A # ( V i ) B , ( v i ) #
~ ( v i ) ~ ]

The momentum equations are said to be in "strong conservation form" if
all terms have the form of the divergence of a vector or tensor. This is possi-


8

1. Basic Concepts of Fluid Flow

ble for the component form of the equations only when components in fixed
directions are used. A coordinate-oriented vector component turns with the
coordinate direction and an "apparent force" is required to produce the turning; these forces are non-conservative in the sense defined above. For example,
in cylindrical coordinates the radial and circumferential directions change so
the components of a spatially constant vector (e.g. a uniform velocity field)
vary with r and 8 and are singular at the coordinate origin. To account
for this, the equations in terms of these components contain centrifugal and
Coriolis force terms.
Figure 1.1 shows a vector v and its contravariant, covariant and Cartesian
components. Obviously, the contravariant and covariant components change
as the base vectors change even though the vector v remains constant. We
shall discuss the effect of the choice of velocity components on numerical
solution methods in Chap. 8.
The strong conservation form of the equations, when used together with a
finite volume method, automatically insures global momentum conservation
in the calculation. This is an important property of the conservation equations
and its preservation in the numerical solution is equally important. Retention
of this property can help to insure that the numerical method will not diverge
during the solution and may be regarded as a kind of "realizability".

For some flows it is advantageous to resolve the momentum in spatially
variable directions. For example, the velocity in a line vortex has only one
component us in cylindrical coordinates but two components in Cartesian
coordinates. Axisymmetric flow without swirl is two-dimensional (2D) when
analyzed in a polar-cylindrical coordinate frame, but three-dimensional (3D)
when a Cartesian frame is used. Some numerical techniques that use nonorthogonal coordinates require use of contravariant velocity components. The
equations then contain so-called "curvature terms", which are hard to compute accurately because they contain second derivatives of the coordinate
transformations that are difficult to approximate.
Throughout this book we shall work with velocity vectors and stress tensors in terms of their Cartesian components, and we shall use conservative
form of the Cartesian momentum equations.
Equation (1.16)is in strong conservation form. A non-conservative form
of this equation can be obtained by employing the continuity equation; since
div (pvui)= ui div (pv)+ pv . gradui ,
it follows that:

The pressure term contained in ti can also be written as
div ( p i i ) = gradp . ii .


1.5 Conservation of Scalar Quantities

9

The pressure gradient is then regarded as a body force; this amounts to
non-conservative treatment of the pressure term. The non-conservative form
of equations is often used in finite difference methods, since it is somewhat
simpler. In the limit of a very fine grid, all equation forms and numerical
solution methods give the same solution; however, on coarse grids the nonconservative form introduces additional errors which may become important.
If the expression for the viscous part of the stress tensor, Eq. (1.13),
is substituted into Eq. (1.16) written in index notation and for Cartesian

coordinates, and if gravity is the only body force, one has:

where g, is the component of the gravitational acceleration g in the direction
of the Cartesian coordinate xi. For the case of constant density and gravity,
the term pg can be written as grad (pg . r ) , where r is the position vector,
r = x,ii (usually, gravity is assumed to act in the negative t-direction, i.e.
g = g t k , g, being negative; in this case g . r = g,z). Then -pg,t is the
hydrostatic pressure, and it is convenient - and for numerical solution more
efficient - to define 5 = p-pg,z as the head and use it in place of the pressure.
The term pgi then disappears from the above equation. If the actual pressure
is needed, one has only t o add pg,z to @.
Since only the gradient of the pressure appears in the equation, the absolute value of the pressure is not important except in compressible flows.
In variable density flows (the variation of gravity can be neglected in all
flows considered in this book), one can split the pgi term into two parts:
pogi (p - po)gi, where po is a reference density. The first part can then be
included with pressure and if the density variation is retained only in the
gravitational term, we have the Boussinesq approximation, see Sect. 1.7.

+

1.5 Conservation of Scalar Quantities
The integral form of the equation describing conservation of a scalar quantity,
4, is analogous to the previous equations and reads:

where fm represents transport of 4 by mechanisms other than convection and
any sources or sinks of the scalar. Diffusive transport is always present (even
in stagnant fluids), and it is usually described by a gradient approximation,
e.g. Fourier's law for heat diffusion and Fick's law for mass diffusion:

10

1. Basic Concepts of Fluid Flow

where r is the diffusivity for the quantity 4. An example is the energy equation which, for most engineering flows, can be written:

where h is the enthalpy, T is the temperature, k is the thermal conductivity,
k = pc,/Pr, and S is the viscous part of the stress tensor, S = T+pl. P r is the
Prandtl number and c, is the specific heat at constant pressure. The source
term represents work done by pressure and viscous forces; it may be neglected
in incompressible flows. Further simplification is achieved by considering a
fluid with constant specific heat, in which case a convection/diffusion equation for the temperature results:

Species concentration equations have the same form, with T replaced by
the concentration c and Pr replaced by Sc, the Schmidt number.
It is useful to write the conservation equations in a general form, as all
of the above equations have common terms. The discretization and analysis
can then be carried out in a general manner; when necessary, terms peculiar
to an equation can be handled separately.
The integral form of the generic conservation equation follows directly
from Eqs. (1.22) and (1.23):

where q$ is the source or sink of 4. The coordinate-free vector form of this
equation is:
a(p4)
at

+ div (pdv) = div ( T grad 4) + q$ .

In Cartesian coordinates and tensor notation, the differential form of the

generic conservation equation is:

Numerical methods will first be described for this generic conservation equation. Special features of the continuity and momentum equations (which are
usually called Navier-Stokes equations) will be described afterwards as an
extension of the methods for the generic equation.


1.6 Dimensionless Form of Equations

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1.6 Dimensionless Form of Equations
Experimental studies of flows are often carried out on models, and the results
are displayed in dimensionless form, thus allowing scaling to real flow conditions. The same approach can be undertaken in numerical studies as well.
The governing equations can be transformed to dimensionless form by using
appropriate normalization. For example, velocities can be normalized by a
reference velocity vo, spatial coordinates by a reference length Lo, time by
some reference time t o , pressure by pvi, and temperature by some reference
temperature difference TI - To. The dimensionless variables are then:

If the fluid properties are constant, the continuity, momentum and temperature equations are, in dimensionless form:

dT*
at*

St-+-----

a(ujT*)-

ax;

1 d2T*
Re P r dxj2

- --

The following dimensionless numbers appear in the equations:

which are called Strouhal, Reynolds, and Froude numbers, respectively. yi is
the component of the normalized gravitational acceleration vector in the xi
direction.
For natural convection flows, the Boussinesq approximation is often used,
in which case the last term in the momentum equations becomes:

where Ra is the Rayleigh number, defined as:

and

p is the coefficient of thermal

expansion.