Computational Aerodynamics and Fluid Dynamics: An Introduction

190 Pages·2002·5.58 MB·English

by  Professor Jean-Jacques Chattot (auth.)

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J.-J. Chattot, Davis, CA, USA
P. Colella, Berkeley, CA, USA
R. Glowinski, Houston, TX, USA
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Jean -Jacques Chattot
Com putationa I
Aerodynamics
and Fluid Dynamics
An Introduction
With 77 Figures
Springer
Professor Jean-Jacques Chattot
University of California
Department of Mechanical
and Aeronautical Engineering
One Shields Avenue
Davis, CA 95616, USA
Library of Congress Cataloging-in-Publication Data. Chattot, J. J. Computational aerodynamics and fluid dynamics:
an introductionl Jean-Jacques Chattot. p. cm. -(Scientific computation, ISSN 1434-8322) Includes bibliographical ref
erences and index.
ISBN 978-3-642-07798-2 ISBN 978-3-662-05064-4 (eBook)
DOI 10.1007/978-3-662-05064-4
1. Fluid dynamics-Data processing. 2. Fluid dynamics-Mathematical
models. L Title. II. Series. QA911.C435 2002 531'.05-dc21 2002021726
ISSN 1434-8322
ISBN 978-3-642-07798-2
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Preface
The field of computational fluid dynamics (CFD) has matured since the au
thor was first introduced to electronic computation in the mid-sixties. The
progress of numerical methods has paralleled that of computer technology
and software. Simulations are used routinely in all branches of engineering as
a very powerful means for understanding complex systems and, ultimately,
improve their design for better efficiency.
Today's engineers must be capable of using the large simulation codes
available in industry, and apply them to their specific problem by implemen
ting new boundary conditions or modifying existing ones.
The objective of this book is to give the reader the basis for understanding
the way numerical schemes achieve accurate and stable simulations of phy
sical phenomena, governed by equations that are related, yet simpler, than
the equations they need to solve. The model problems presented here are
linear, in most cases, and represent the propagation of waves in a medium,
the diffusion of heat in a slab, and the equilibrium of a membrane under
distributed loads. Yet, regardless of the origin of the problem, the partial
differential equations (PDE's) reflect the physical phenomena to be modeled
and can be classified as being of hyperbolic, parabolic or elliptic type. The
numerical treatment depends on the equation type that can represent several
physical situations as diverse as heat conduction and viscous fluid flow. Non
linear model problems are also presented and solved, such as the transonic
small disturbance equation and the equations of gas dynamics. The model
problems are given a full treatment, from the exact analytical solution, the
analysis of the scheme's consistency and accuracy, the study of stability, to
the detailed implementation of the scheme and of the boundary and/or initial
conditions. It is the author's hope that this will entice the reader to write
his/her own programs, and by doing so, learn more about CFD than a book
can teach.
Davis, March 2002 Jean-Jacques Chattot
Table of Contents
1. Introduction.............................................. 1
1.1 Motivation............................................ 1
1.2 Content............................................... 1
2. Basics of the Finite-Difference Method..... . . ... . . .... ... 5
2.1 Representation of a Function by Discrete Values. . . . . . . . . . . . 5
2.2 Representation of a First Derivative ...................... 6
2.3 Representation of a Second Derivative. . . . . . . . . . . . . . . . . . . . . 7
2.4 Geometric Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Taylor Expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.6 Consistency and Accuracy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.7 Stability............................................... 12
2.8 Complements on Truncation Error. . . . . . . . . . . . . . . . . . . . . . .. 14
3. Application to the Integration
of Ordinary Differential Equations. . . . . . . . . . . . . . . . . . . . . . .. 19
3.1 Introduction........................................... 19
3.2 The Euler-Cauchy Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20
3.3 Improved Euler Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21
3.4 The Runge-Kutta Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22
3.5 Integration of Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22
3.6 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23
4. Partial Differential Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27
4.1 Introduction........................................... 27
4.2 General Classification
and Notion of Characteristic Surface. . . . . . . . . . . . . . . . . . . . .. 28
4.3 Model Equations and Types ............................. 30
4.3.1 Linear Convection Equation. . . . . . . . . . . . . . . . . . . . . .. 30
4.3.2 The Wave Equation.. . . . .... . .... . . ... . . ... . . . ... 31
4.3.3 Laplace's Equation ............................... 32
4.3.4 The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 33
4.3.5 Burgers' Equation (Inviscid) . . . . . . . . . . . . . . . . . . . . . .. 34
4.3.6 Other Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35
X Table of Contents
4.4 Conservation Laws and Jumps for a System of PDEs. . . . . . .. 35
4.4.1 Jump Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36
4.4.2 Examples....................................... 37
5. Integration of a Linear Hyperbolic Equation. . . . . . . . . . . . .. 41
5.1 Introduction........................................... 41
5.2 The Linear Convection Equation. . . . . . . . . . . . . . . . . . . . . . . .. 41
5.2.1 A Centered Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41
5.2.2 An Upwind Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42
5.2.3 The Lax Scheme ................................. 43
5.2.4 The Lax-Wendroff Scheme. . . . . . . . . . . . . . . . . . . . . . . .. 43
5.2.5 The MacCormack Scheme... .. . . ... . . . . .. . . . . . .... 44
5.3 The Wave Equation... . . ... . . . .... . . . . .... . . .... . . . ..... 45
5.3.1 Exact Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46
5.3.2 Numerical Scheme, Consistency and Accuracy ....... 47
5.3.3 Numerical Implementation. . . . . . . . . . . . . . . . . . . . . . . .. 49
5.3.4 Stability........................................ 49
5.4 Implicit Scheme for the Wave Equation ................... 52
6. Integration of a Linear Parabolic Equation ............... 53
6.1 Introduction........................................... 53
6.2 Exact Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 54
6.3 A Simple Explicit Scheme ............................... 55
6.3.1 Consistency and Accuracy. . . . . . . . . . . . . . . . . . . . . . . .. 55
6.3.2 Numerical Implementation. . . . . . . . . . . . . . . . . . . . . . . .. 56
6.3.3 Stability........................................ 57
6.4 Simple Implicit Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58
6.5 Combined Method A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 59
6.6 Solution of a Linear System with Tridiagonal Matrix. . . . . . .. 59
7. Integration of a Linear Elliptic Equation. . . . . . . . . . . . . . . . .. 63
7.1 Introduction........................................... 63
7.2 Numerical Scheme, Consistency, Accuracy. . . . . . . . . . . . . . . .. 63
7.3 Matrix Formulation; Direct Solution. . . . . . . . . . . . . . . . . . . . .. 64
7.4 Outlook of Iterative Methods. . . . . . . . . . . . . . . . . . . . . . . . . . .. 65
7.4.1 The Method of Jacobi ............................ 67
7.4.2 The Gauss-Seidel Method. . . . . . . . . . . . . . . . . . . . . . . .. 68
7.4.3 The Successive Over-Relaxation Method (SOR) ...... 69
7.5 Other Iterative Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 71
7.5.1 The SLOR Method. . . . . ... . . . ... . . . .... . . . .... . .. 72
7.5.2 AD! Methods. . ... . . . .... . ..... . . . .... . . ..... . . .. 74
Table of Contents XI
8. Finite Difference Scheme
for a Convection-Diffusion Equation ...... . . . . . . . . . . . . . . .. 75
8.1 Introduction........................................... 75
8.2 FTCS Method .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76
8.3 The Box and Modified Box Methods. . . . . . . . . . . . . . . . . . . . .. 79
8.4 A Mixed-Type Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 80
9. The Method of Murman and Cole ........................ 81
9.1 Introduction........................................... 81
9.2 The Model Problem .................................... 81
9.3 The Murman-Cole Scheme (1970) ..... . . . . . . . . . . . . . . . . . .. 85
9.4 The Four-Operator Scheme of Murman (1973) ............. 87
10. Treatment of Non-Linearities . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 93
10.1 Introduction....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 93
10.2 An Explicit Mixed-Type Scheme ......................... 93
10.3 An Implicit Mixed-Type Scheme ......................... 96
10.4 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99
11. Application to a System of Equations ..................... 101
11.1 Introduction ........................................... 101
11.2 The Equations of Gas Dynamics .......................... 101
11.3 Jump Conditions ....................................... 103
11.4 The Riemann Problem .................................. 104
11.5 A Box-Scheme for the Equations of Gas Dynamics .......... 105
11.6 Some Results .......................................... 108
11. 7 The Bigger Picture ..................................... 111
Appendix ..................................................... 113
A. Problems ................................................ 113
B. Solutions to Problems ..................................... 137
References .................................................... 181
Index ......................................................... 183
1. Introduction
1.1 Motivation
The material in this book is based on lecture notes on computational fluid
dynamics (CFD) that the author has developed over the past twenty years
in France, at Centre National d'Etudes Superieures de Mecanique and at the
Universite de Paris-Sud, and in the US at the University of California, Davis.
It is intended for senior undergraduate and first year graduate students
who will be developing or using codes in the simulation of fluid flows or other
physical phenomena governed by partial differential equations (PDEs).
It is the belief of the author, that a numerical method is not fully un
derstood until it has been coded by the user and applied in simulation; each
model and scheme in this book is presented with this goal in mind.
1.2 Content
The book is self contained and kept at a simple enough level that the reader
will not need further references in order to understand the material.
The approach is based on the finite difference method (FD), which is wi
dely employed as a method of discretization on cartesian mesh systems, in
the physical domain, or in the computational domain after coordinate trans
formation. The extension to the finite volume method on arbitrary mesh
sytems, including unstructured meshes, although feasible with a similar ap
proach, would require all analyses to be performed numerically, instead of
analytically in closed form, as is the case here.
The book is organized in chapters that build up each on material covered
in the previous chapters, particularly Chaps. 2, 3 and 4 and Chaps. 8-11.
Chapters 5, 6 and 7 can be read in any order.
The basics of the finite difference method are presented in Chap. 2. The
tools that will be used throughout the book are introduced: the Taylor expan
sion and the complex mode analysis, which requires some complex algebra.
They are the tools for the accuracy and stability analyses.
Chapter 3 is devoted to ordinary differential equations (ODEs) and their
integration. ODEs represent an important particular case of partial differen
tial equations (PDEs), when the number of independent variables reduces to
J.-J. Chattot, Computational Aerodynamics and Fluid Dynamics
© Springer-Verlag Berlin Heidelberg 2002