Fundamentals of Computational Fluid Dynamics

by Lomax, Harvard; Pulliam, Thomas H.; Zingg, David W.

• ISBN13: 9783540416074
• ISBN10: 3540416072
• Format: Hardcover
• Copyright: 2001-07-01
• Publisher: Springer Verlag

Summary

This book is intended as a textbook for a first course in computational fluid dynamics and will be of interest to researchers and practitioners as well. It emphasizes fundamental concepts in developing, analyzing, and understanding numerical methods for the partial differential equations governing the physics of fluid flow. The linear convection and diffusion equations are used to illustrate concepts throughout. The chosen approach, in which the partial differential equations are reduced to ordinary differential equations, and finally to difference equations, gives the book its distinctiveness and provides a sound basis for a deep understanding of the fundamental concepts in computational fluid dynamics.

Table of Contents

Introduction	p. 1
Motivation	p. 1
	p. 2
Problem Specification and Geometry Preparation	p. 2
Selection of Governing Equations and Boundary Conditions	p. 3
Selection of Gridding Strategy and Numerical Method	p. 3
Assessment and Interpretation of Results	p. 4
Overview	p. 4
Notation	p. 4
Conservation Laws and the Model Equations	p. 7
Conservation Laws	p. 7
The Navier-Stokes and Euler Equations	p. 8
The Linear Convection Equation	p. 11
Differential Form	p. 11
Solution in Wave Space	p. 12
The Diffusion Equation	p. 13
Differential Form	p. 13
SolutioninWaveSpace	p. 14
Linear Hyperbolic Systems	p. 15
Exercises	p. 17
Finite-Difference Approximations	p. 19
Meshes and Finite-Difference Notation	p. 19
Space DerivativeApproximations	p. 21
Finite-Difference Operators	p. 22
Point Difference Operators	p. 22
Matrix Difference Operators	p. 23
Periodic Matrices	p. 26
CirculantMatrices	p. 27
Constructing Differencing Schemes of Any Order	p. 28
TaylorTables	p. 28
Generalization of Difference Formulas	p. 31
Lagrange and Hermite Interpolation Polynomials	p. 33
Practical Application of Padé Formulas	p. 35
OtherHigher-OrderSchemes	p. 36
FourierErrorAnalysis	p. 37
ApplicationtoaSpatialOperator	p. 37
Difference Operators at Boundaries	p. 41
TheLinearConvectionEquation	p. 41
The Diffusion Equation	p. 44
Exercises	p. 46
The Semi-Discrete Approach	p. 49
Reduction of PDE's to ODE's	p. 50
The Model ODE's	p. 50
TheGenericMatrixForm	p. 51
ExactSolutionsofLinearODE's	p. 51
EigensystemsofSemi-discreteLinearForms	p. 52
Single ODE's of First and Second Order	p. 53
CoupledFirst-OrderODE's	p. 54
General Solution of Coupled ODE's with Complete Eigensystems	p. 56
RealSpaceandEigenspace	p. 58
Definition	p. 58
EigenvalueSpectrumsforModelODE's	p. 59
Eigenvectors of the Model Equations	p. 60
Solutions of the Model ODE's	p. 62
TheRepresentative Equation	p. 64
Exercises	p. 65
Finite-Volume Methods	p. 67
Basic Concepts	p. 67
ModelEquations in Integral Form	p. 69
TheLinearConvectionEquation	p. 69
The Diffusion Equation	p. 70
One-DimensionalExamples	p. 70
A Second-Order Approximation to the Convection Equation	p. 71
A Fourth-Order Approximation to the Convection Equation	p. 72
A Second-Order Approximation to the Diffusion Equation	p. 74
ATwo-Dimensional Example	p. 76
Exercises	p. 79
Time-Marching Methods for ODE'S	p. 81
Notation	p. 82
Converting Time-Marching Methods to O¿E's	p. 83
Solution of Linear O¿E's with Constant Coefficients	p. 84
First- and Second-Order Difference Equations	p. 84
Special Cases of Coupled First-Order Equations	p. 86
Solution of the Representative O¿E's	p. 87
The Operational Form and its Solution	p. 87
Examples of Solutions to Time-Marching O¿E's	p. 88
The ¿-¿ Relation	p. 89
Establishing the Relation	p. 89
The Principal ¿-Root	p. 90
Spurious ¿-Roots	p. 91
One-Root Time-Marching Methods	p. 92
Accuracy Measures of Time-Marching Methods	p. 92
Local and Global Error Measures	p. 92
Local Accuracy of the Transient Solution (er¿ ¿ , er&omgea;)	p. 93
Local Accuracy of the Particular Solution (er¿)	p. 94
Time Accuracy for Nonlinear Applications	p. 95
Global Accuracy	p. 96
Linear Multistep Methods	p. 96
The General Formulation	p. 97
Examples	p. 97
Two-Step Linear Multistep Methods	p. 100
Predictor-Corrector Methods	p. 101
Runge-Kutta Methods	p. 103
Implementation of Implicit Methods	p. 105
Application to Systems of Equations	p. 105
Application to Nonlinear Equations	p. 106
Local Linearization for Scalar Equations	p. 107
Local Linearization for Coupled Sets of Nonlinear Equations	p. 110
Exercises	p. 112
Stability of Linear Systems	p. 115
Dependence on the Eigensystem	p. 115
Inherent Stability of ODE's	p. 116
The Criterion	p. 116
Complete Eigensystems	p. 117
Defective Eigensystems	p. 117
Numerical Stability of O¿E's	p. 118
The Criterion	p. 118
Complete Eigensystems	p. 118
Defective Eigensystems	p. 119
Time-Space Stability and Convergence of O¿E's	p. 119
Numerical Stability Concepts in the Complex ¿-Plane	p. 121
¿-Root Traces Relative to the Unit Circle	p. 121
Stability for Small ¿t	p. 126
Numerical Stability Concepts in the Complex ¿h Plane	p. 127
Stability for Large h	p. 127
Unconditional Stability, A-Stable Methods	p. 128
Stability Contours in the Complex ¿h Plane	p. 130
Fourier Stability Analysis	p. 133
The Basic Procedure	p. 133
Some Examples	p. 134
Relation to Circulant Matrices	p. 135
Consistency	p. 135
Exercises	p. 138
Choosing a Time-Marching Method	p. 141
Stiffness Definition for ODE's	p. 141
Relation to ¿-Eigenvalues	p. 141
Driving and Parasitic Eigenvalues	p. 142
Stiffness Classifications	p. 143
Relation of Stiffness to Space Mesh Size	p. 143
Practical Considerations for Comparing Methods	p. 144
Comparing the Efficiency of Explicit Methods	p. 145
Imposed Constraints	p. 145
An Example Involving Diffusion	p. 146
An Example Involving Periodic Convection	p. 147
Coping with Stiffness	p. 149
Explicit Methods	p. 149
Implicit Methods	p. 150
A Perspective	p. 151
Steady Problems	p. 151
Exercises	p. 152
Relaxation Methods	p. 153
Formulation of the Model Problem	p. 154
Preconditioning the Basic Matrix	p. 154
The Model Equations	p. 156
Classical Relaxation	p. 157
The Delta Form of an Iterative Scheme	p. 157
The Converged Solution, the Residual, and the Error	p. 158
The Classical Methods	p. 158
The ODE Approach to Classical Relaxation	p. 159
The Ordinary Differential Equation Formulation	p. 159
ODE Form of the Classical Methods	p. 161
Eigensystems of the Classical Methods	p. 162
The Point-Jacobi System	p. 163
The Gauss-Seidel System	p. 166
The SOR System	p. 169
Nonstationary Processes	p. 171
Exercises	p. 176
Multigrid	p. 177
Motivation	p. 177
Eigenvector and Eigenvalue Identification with Space Frequencies	p. 177
Properties of the Iterative Method	p. 178
The Basic Process	p. 178
A Two-Grid Process	p. 185
Exercises	p. 187
Numerical Dissipation	p. 189
One-Sided First-Derivative Space Differencing	p. 189
The Modified Partial Differential Equation	p. 190
The Lax-Wendroff Method	p. 192
Upwind Schemes	p. 195
Flux-Vector Splitting	p. 196
Flux-Difference Splitting	p. 198
Artificial Dissipation	p. 199
Exercises	p. 200
Split and Factored Forms	p. 203
The Concept	p. 203
Factoring Physical Representations -Time Splitting	p. 204
Factoring Space MatrixOperators in 2D	p. 206
Mesh Indexing Convention	p. 206
Data-Bases and Space Vectors	p. 206
Data-Base Permutations	p. 207
Space Splitting and Factoring	p. 207
Second-Order Factored Implicit Methods	p. 211
Importance of Factored Forms in Two and Three Dimensions	p. 212
The Delta Form	p. 213
Exercises	p. 214
Analysis of Split and Factored Forms	p. 217
The Representative Equation for Circulant Operators	p. 217
Example Analysis of Circulant Systems	p. 218
Stability Comparisons of Time-Split Methods	p. 218
Analysisofa Second-Order Time-Split Method	p. 220
The Representative Equation for Space-Split Operators	p. 222
Example Analysis of the 2D Model Equation	p. 225
The Unfactored Implicit Euler Method	p. 225
The Factored Nondelta Form of the Implicit Euler Method	p. 226
The Factored Delta Form of the Implicit Euler Method	p. 227
The Factored Delta Form of the Trapezoidal Method	p. 227
Example Analysis of the 3D Model Equation	p. 228
Exercises	p. 230
Appendices	p. 231
Useful Relations from Linear Algebra	p. 231
Notation	p. 231
Definitions	p. 232
Algebra	p. 232
Eigensystems	p. 233
Vector and Matrix Norms	p. 235
Some Properties of Tridiagonal Matrices	p. 237
Standard Eigensystem for Simple Tridiagonal Matrices	p. 237
Generalized Eigensystem for Simple Tridiagonal Matrices	p. 238
The Inverse of a Simple Tridiagonal Matrix	p. 239
Eigensystems of Circulant Matrices	p. 240
Standard Tridiagonal Matrices	p. 240
General Circulant Systems	p. 241
Special Cases Found from Symmetries	p. 241
Special Cases Involving Boundary Conditions	p. 242
The Homogeneous Property of the Euler Equations	p. 245
Index	p. 247
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