Table Of ContentJohn F. Wendt (Editor)
Computational
Fluid Dynamics
An Introduction
With Contributions by
ID. Anderson, G. Degrez, E. Dick,
and R. Grundmann
-.-=--" A =-
~~
-~O~-
~VW
A von Karman Institute Book
Springer-Verlag Berlin Heidelberg GmbH
Prof. Dr. John F. Wendt
Director von Kannan Institute
for Fluid Dynamics
72 Chaussee de Waterloo
B-1640 Rhode-Saint-Genese, Belgium
ISBN 978-3-662-11352-3
Libmry of Congress Cataloging-in-Publication Data
Introduction to computational fluid dynamics / Iohn F. Wendt (editor);
authors, I. Anderson ... (et a!.).
An outgrowth oflecture series given at the von Karman Institute for
Fluid Dynamics. "A von Karman Institute book".
Includes bibliographical references and index.
ISBN 978-3-662-11352-3 ISBN 978-3-662-11350-9 (eBook)
DOI 10.1007/978-3-662-11350-9
1. Fluid dynamics--Mathematics. 2. Numerical analysis.
1. Wendt, Iohn F., 1936- . II. Anderson, Iohn David.
lll. von Karman Institute for Fluid Dynamcis.
TA357.I67 1992
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CI Springer-Verlag Berlin Heidelberg 1992
Originally published by Springer-Verlag Berlin Heidelberg New York in 1992
The use of general descriptive names, registered names, trademarks, etc. in this
publication does not imply, even in the absence of a specific statement, that such
names are exempt from tbe relevant protective laws and regulations and therefore
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Typesetting: Asco Trade Typesetting Ud., Hong Kong
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Preface
This book is an outgrowth of a von Kannan Institute Lecture Series by the same title
first presented in 1985 and repeated with modifications in succeeding years.
The objective, then and now, was to present the subject of computational fluid
dynamics (CFD) to an audience unfamiliar with all but the most basic aspects of
numerical techniques and to do so in such a way that the practical application ofCFD
would become clear to everyone.
Remarks from hundreds of persons who followed this course encouraged the
editor and the authors to improve the content and organization year by year and
eventually to produce the present volume.
The book is divided into two parts. In the first part, John Anderson lays out the
subject by first describing the governing equations offluid dynamics, concentration
on their mathematical properties which contain the keys to the choice of the
numerical approach. Methods of discretizing the equations are discussed next and
then transformation techniques and grids are also discussed. This section closes with
two examples of numerical methods which can be understood easily by all
concerned: source and vortex panel methods and the explicit method.
The second part of the book is devoted to four self-contained chapters on more
advanced material: Roger Grundmann treats the boundary layer equations and
methods of solution; Gerard Degrez treats implicit time-marching methods for
inviscid and viscous compressible flows, and Eric Dick treats, in two separate
articles, both finite-volume and finite-element methods.
The editor and authors will consider the book to have been successful if the
readers conclude they are well prepared to examine the literature in the field and
to begin applying CFD methods to the resolution of problems in their areas of
concern.
Thanks are due to many persons at the YKI who assisted in the organization of
the Lecture Series and the resulting book, but the editor wishes to take this
opportunity to thank specifically the authors for their contributions , their enthusiasm
for this project, and for teaching the editor the rudiments of computational fluid
dynamics.
November 1991 John F. Wendt
von Karman Institute
Biographical Sketches of the Authors
Professor J. D. Anderson Jr.,
University of Maryland, Dept. of Aerospace Engineering,
College Park, Maryland 20902, USA
John D. Anderson, Jr. is Professor of Aerospace Engineering at the University of
Maryland. He received his B. Eng. degree from the University ofF lorida and Ph.D.
degree in aeronautical and astronautical engineering from the Ohio State
University. He has held a number of positions including Chief of the Department
ofthe Hypersonic Group ofthe U. S .N aval Ordinance Laboratory, Chairman ofthe
Department of Aerospace Engineering at the University of Maryland, and the
Charles Lindbergh Chair at the National Air and Space Museum of the
Smithsonian Institute.
Dr. Anderson is the author of five textbooks and over 90 professional papers. In
1989, he was given the John Leland Atwood Award by the American Institute of
Aeronautics and Astronautics and the American Society ofE ngineering Education
award for exellence in aerospace engineering education.
Professor G. Degrez
Aerospace!A eronautics Dept., von Karman Institute,
72, Chaussee de Waterloo,
1640 Rhode-Saint-Genese, Belgium
Gerard Degrez, Associate Professor at the von Karman Institute for Fluid
Dynamics,Belgium,received his engineering degree (IngenieurCivil Mecanicien)
from the University ofBrussels.He then attended the graduate school at Princeton
University where he received a Master of Science degree in Engineering. He went
on to become an assistant atthe University offirussels while conducting research at
the von Karman Institute thatled to aPh.D. delivered by the University offirussels.
Assistant, then Associate Professor in Aeronautical Engineering at the University
ofSherbrooke (Canada), he recently joined the von Karman Institute. His current
research interests concern high speed viscous flows both from experimental and
computational standpoints.
VIII Biographical Scetches of the Authors
Dr.E.Dick
University of Ghent,
S1. Pietersnieuwstraat 25,
9000 Ghent, Belgium
Erik Dick obtained a degree in Mechanical Engineering from the State University
of Ghent, Belgium in 1973, and his Ph. D. degree in fluid mechanics from the same
university in 1980. Since 1974 he has worked at the State University of Ghent in the
Department of Machinery, Division of Turbomachines. Presently he is research
leader and lecturer. His main scientific activity is in computational fluid dynamics
and in recent years he has concentrated on upwind finite volume methods
combined with multigrid techniques. He teaches theoretical fluid mechanics and
computational fluid machanics.
Professor R. Grundmann
DLR, BunsenstraBe 10,
3400 G6ttingen, Germany
Roger Grundmann received the Dipl.-Ing. degree for aircraft turbines and the
Dr.-Ing. degree from the Technische UniversiHit ofBerlin. Since 1972 he has been a
member ofthe Deutsche Forschungsanstalt fUr Luft-und Raumfahrt (DLR) and is
presently at the Institute fUrTheoretische Str6mungsmechanik. From 1985 to 1987
he was an Associate Professor at the von Karman Institute for Fluid Dynamics
(VKI) in Rhode-St-Genese, Belgium. After returning to the D LR in G6ttingen he
was sent to the VKI as a Visiting Professor. His main field of research is viscous
hypersonic flows by means of numerical methods.
Contents
Part 1
1 Basic Philosophy of CFD .. . . . . . . . . . . . 3
1.1 Motivation: An Example .... . . . . . . . . 3
1.2 Computational Fluid Dynamics: What is it? . 5
1.3 The Role of Computational Fluid Dynamics in Modern Fluid Dynamics 6
1.4 The Role of This Course ......... 13
2 Governing Equations of Fluid Dynamics 15
2.1 Introduction........ 15
2.2 Modelling of the Flow 15
2.3 The Substantial Derivative 16
2.4 Physical Meaning of V . V 22
2.5 The Continuity Equation . 23
2.6 The Momentum Equation 27
2.7 The Energy Equation . . . 33
2.8 Summary of the Governing Equations for Fluid Dynamics: With Comments 40
2.8.1 Equations for Viscous Flow ....... 40
2.8.2 Equations for Inviscid Flow ....... 42
2.8.3 Comments on the Governing Equations 43
2.8.4 Boundary Conditions . . . . . . . . . . . 44
2.9 Forms of the Governing Equations Particularly Suited for CFD: Comments on the
Conservation Form ... . . . . . . . . . . . . . . . . . . . . . . 45
3 Incompressible Inviscid Flows: source and vortex panel methods 52
3.1 Introduction........................ 52
3.2 Some Basic Aspects of Incompressible, Inviscid Flow 52
3.2.1 Uniform Flow . 53
3.2.2 Source Flow .................... 54
3.2.3 Vortex Flow .................... 54
3.3 Non-lifting Flows Over Arbitrary Two-Dimensional Bodies: The Source
Panel Method ................................ 55
3.4 Lifting Flows over Arbitrary Two-Dimensional Bodies: The Vortex
Panel Method ................................ 64
3.5 An Application-the Aerodynamics of Drooped Leading-Edge Wings Below and
Above Stall . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4 Mathematical Properties of the Fluid Dynamic Equations 75
4.1 Introduction.......................... 75
4.2 Classification of Partial Differential Equations 75
4.3 General Behaviour of the Different Classes of Partial Differential Equations
and their Relation to Fluid Dynamics 79
4.3.1 Hyperbolic Equations 79
4.3.2 Parabolic Equations 80
4.3.3 Elliptic Equations .. 82
4.3.4 Some Comments . . . 83
4.3.5 Well-Posed Problems 84
X Contents
5 Discretization of Partial Differential Equations 85
5.1 Introduction.................... 85
5.2 Derivation of Elementary Finite Difference Quotients 86
5.3 Basic Aspects of Finite-Difference Equations .... . 92
5.4 Errors and Analysis of Stability ............ . 95
6 Transformations and Grids .............. . 101
6.1 Introduction ........................ . 101
6.2 General Transformation of the Equations ....... . 103
6.3 Metrics and lacobians .................. . 107
6.4 Coordinate Stretching ......... . 109
6.5 Boundary-Fitted Coordinate Systems . 113
6.6 Adaptive Grids ............. . 118
7 Explicit Finite Difference Methods: Some Selected Applications to Inviscid
and Viscous Flows. . . . . . . . . . . . . . . . .. ............. 123
7.1 Introduction................................... 123
7.2 The Lax-WendroffMethod . . . . . . . . . . . . . . . . . . . . . . 124
7.3 MacCormack's Method ............................ 128
7.4 Stability Criterion .............................. 131
7.5 Selected Applications of the Explicit Time-Dependent Technique . . . . 132
7.5.1 Non-equilibrium Nozzle Flows . . . . . . . . . . . . . . . . 132
7.5.2 Flow Field Over a Supersonic Blunt Body ............. 135
7.5.3 Internal Combustion Engine Flows ......................... 137
7.5.4 Supersonic Viscous Flows Over a Rearward-Facing Step With Hydrogen
Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 140
7.5.5 Supersonic Viscous Flow Over a Base . . . . . . . . . . . . . . . . . . . . . . .. 142
7.5.6 Compressible Viscous Flow Over an Airfoil .................... 143
7.6 Final Comment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 145
7.7 References (for Chapters 1-7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 145
Part 2
8 Boundary Layer Equations and Methods of Solution . . . . . . . . .. 151
8.1 Introduction............................. . . . . . . . . .. 151
8.2 Description ofPrandtl's Boundary Layer Equations ......... , 152
8.3 Hierarchy of the Boundary Layer Equations ........................ 156
8.4 Transformation of the Boundary Layer Equations. . . . . . . . . . . . . . . . . . . . 160
8.5 Numerical Solution Method .............. . . . . . . . . . . . . . 162
8.5.1 Choice of Discretization Model. . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.5.2 Generalized Crank-Nicholson Scheme ...................... 164
8.5.3 Discretization of the Boundary Layer Equation ................. 166
8.5.4 Solution of a Tridiagonal System of Linear Algebraic Equations ....... 170
8.6 Sample Calculations ..................................... 172
8.6.1 Three-dimensional Boundary Layer Calculation Along Lines
of Symmetry ...................................... 172
8.6.2 Geometrical Conditions .... ........ 173
8.6.3 Fluid Mechanical Equations ................... 174
8.6.4 Boundary Conditions ...... . . . . . . . . . 175
8.6.5 Solution Scheme .......................... 176
8.6.6 Numerical Results ......................... ......... 177
8.7 References............................................ 178
9 Implicit Time-Dependent Methods for Inviscid and Viscous Compressible Flows,
With a Discussion of the Concept of Numerical Dissipation . . . . . . . . . . 180
9.1 Introduction........................................... 180
9.2 Solution Techniques for Simpler Flows and Reason of Their Failure for
EulerfNavier-Stokes Equations ............................... 181
9.2.1 Solution Technique for Steady Subsonic Potential Flow ............. 181
Contents XI
9.2.2 Solution Technique for Steady Supersonic Potential Flow ........... . 184
9.2.3 More Complicated Problems ............................ . . 184
9.2.4 A Solution: The Time-dependent Approach ................... . 186
9.3 Respective Stability Properties of Explicit and Implicit Methods ........... . 187
9.3.1 Stability of the Numerical Methods for the Integration of Ordinary Differential
Equations ....................................... . 187
9.3.1.1 Definition-Examples ............................... . 188
9.3.1.2 Weak Instability .................................. . 189
9.3.1.3 Region of(absolute) Stability ........................... . 191
9.3.1.4 Stiff Problems .................................... . 192
9.3.1.5 Absolute Stability .................................. . 192
9.3.2 Stability of Numerical Methods for the Integration of Partial Differential
Equations ........................................ . 194
9.4 Construction of Implicit Methods for Time-dependent Problems ........... . 196
9.4.1 Systems of Ordinary Differential Equations: Local Linearization ....... . 196
9.4.2 Application of a Linearized Implicit Method to a Nonlinear
One-Dimensional PDE ............................... . 197
9.4.3 Application of a Linearized Implicit Method to a Two-Dimensional
PDE-Approximate Factorization ......................... . 198
9.4.3.1 PDE without Cross Derivative-The Approximate Factorization
Technique ...................................... . 198
9.4.3.2 PDE with Cross Derivatives ........................... . 201
9.4.3.3 Systems ofPDEs in Several Space Dimensions-Euler and Navier-Stokes
Equations ...................................... . 202
9.4.3.4 Summary of Stability Properties of Linearized Implicit Schemes
for PDE ....................................... . 203
9.5 Numerical Dissipation .................................... . 203
9.5.1 Definition-Need for Dissipation ......... . .......... . 203
9.5.1.1 Definition ...................................... . 203
9.5.1.2 The Need for Dissipation ............................. . 205
9.5.1.2.1 The Inherent Limitation of Numerical Methods ............... . 205
9.5.1.2.2 Instances in which Dissipation is Needed: Inherent Inadequacy of
the Mesh ..................................... . 206
9.5.1.2.3 Time-wise Dissipation and Steady-state Dissipation ............. . 209
9.5.2 The Various Ways of Generating Dissipation .................. . 211
9.5.2.1 Artificial Viscosity ........................... . 211
9.5.2.2 Dissipation Through Time-Differencing ............... . 212
9.5.2.2.1 Dissipation of the Time Integrator Itself ............. . 212
9.5.2.2.2 Simultaneous Time and Space Discretization ........... . 212
9.5.2.3 Upwind Differencing .......................... . 214
9.6 Conservative Upwind Discretization for Hyperbolic Systems-Further Advantages
of Upwind Discretization ............................ . 216
9.6.1 Conservative Upwind Discretization for Systems .......... . 216
9.6.1.1 Roe's Flux Difference Splitting Scheme ............... . 216
9.6.1.2 Steger and Warming's Flux Vector Splitting Scheme ....... . 219
9.6.2 Further Advantages of Upwind Differencing ................... . 220
9.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
9.8 References ........................................... . 221
10 Introduction to Finite Element Techniques in Computational Fluid Dynamics 223
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
10.2 Strong and Weak Formulations of a Boundary Value Problem ............ . 225
10.2.1 Strong Formulation ................................. . 225
10.2.2 Weighted Residual Formulation ......................... . 228
10.2.3 Galerkin Formulation ............................... . 229
10.2.4 Weak Formulation ................................. . 230
10.2.5 Variational Formulation ............................. . 232
10.2.6 Conclusion ..................................... . 232
10.3 Piecewise Defined Shape Functions ........................... . 233
10.3.1 The Finite Element Interpolation ........................ . 233
XII Contents
10.3.2 Finite Elements with Co Continuity in Two-dimensions 237
10.3.2.1 Triangular Elements ............. . 237
10.3.2.2 Quadrilateral Lagrange Elements ...... . 241
10.3.2.3 Quadrilateral Serendipity Elements ....................... . 241
10.3.2.4 Isoparametric Elements ............................. . 242
10.3.3 Finite Elements with C Continuity ............. . 243
1
10.4 Implementation of the Finite Element Method .............. . 244
10.4.1 The Assembly ............................ . 244
10.4.2 Numerical Integration ................. . 245
10.4.3 Solution Procedure .. . . . . . . . . . . . . . . . . . . . . . . . 246
10.5 Examples ................................... . 246
10.5.1 Steady Incompressible Potential Flow ............. . 247
10.5.2 Incompressible Navier-Stokes Equations in w - '" Formulation ... 249
10.5.3 Incompressible Steady Navier-Stokes Equations in u, v, p Formulation .. . 254
10.5.4 Compressible Euler Equations ..................... . 257
10.6 References ........................................... . 259
11 Introduction to Finite Volume Techniques in Computational Fluid Dynamics . . . 261
ILl Introduction...................... . . . . . . . . . . . . . . 261
11.2 FEM-like Finite Volume Techniques ......................... 265
11.2.1 Cell-centred Formulation ........................... 266
11.2.Ll Lax-WendroffTime Stepping. . . . . . . . . . . . . . . . . . . . . . . . 267
11.2.1.2 Runge-Kutta Time Stepping-Multi-stage Time Stepping . . . . . . . . .. 272
11.2.1.3 Accuracy ....................................... 277
11.2.2 Cell-vertex Formulation ....................... 277
11.2.2.1 Multi-stage Time Stepping-Overlapping Control Volumes . 277
11.2.2.2 Lax-WendroffTime Stepping-Non-overlapping Control Volumes .. 279
11.3 FDM-like Finite Volume Tec:miques ....... . . . . . . ... . . . . . 280
11.3.1 Central Type Discretizations ......................... 281
11.3.2 Upwind Type Discretizations ......................... 281
11.4 Other Formulations ................................... 286
11.5 Treatment of Derivatives ................................ 286
11.6 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 287
Subject Index . . 289
