كتاب Fundamentals of Computational Fluid Dynamics

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Fundamentals of Computational Fluid Dynamics
Harvard Lomax and Thomas H. Pulliam
NASA Ames Research Center
David W. Zingg
University of Toronto Institute for Aerospace Studies

ويتناول الموضوعات الأتية :

1 INTRODUCTION 1
1.1 Motivation 1
1.2 Background 2
1.2.1 Problem Specifcation and Geometry Preparation 2
1.2.2 Selection of Governing Equations and Boundary Conditions 3
1.2.3 Selection of Gridding Strategy and Numerical Method 3
1.2.4 Assessment and Interpretation of Results 4
1.3 Overview 4
1.4 Notation 5
2 CONSERVATION LAWS AND THE MODEL EQUATIONS 7
2.1 Conservation Laws 7
2.2 The Navier-Stokes and Euler Equations 8
2.3 The Linear Convection Equation 12
2.3.1 Diferential Form 12
2.3.2 Solution in Wave Space 13
2.4 The Difusion Equation 14
2.4.1 Diferential Form 14
2.4.2 Solution in Wave Space 15
2.5 Linear Hyperbolic Systems 16
2.6 Problems 17
3 FINITE-DIFFERENCE APPROXIMATIONS 21
3.1 Meshes and Finite-Diference Notation 21
3.2 Space Derivative Approximations 24
3.3 Finite-Diference Operators 25
3.3.1 Point Diference Operators 25
3.3.2 Matrix Diference Operators 25
3.3.3 Periodic Matrices 29
3.3.4 Circulant Matrices 30
3.4 Constructing Diferencing Schemes of Any Order 31
3.4.1 Taylor Tables 31
3.4.2 Generalization of Diference Formulas 34
3.4.3 Lagrange and Hermite Interpolation Polynomials 35
3.4.4 Practical Application of Pade Formulas 37
3.4.5 Other Higher-Order Schemes 38
3.5 Fourier Error Analysis 39
3.5.1 Application to a Spatial Operator 39
3.6 Diference Operators at Boundaries 43
3.6.1 The Linear Convection Equation 44
3.6.2 The Difusion Equation 46
3.7 Problems 47
4 THE SEMI-DISCRETE APPROACH 51
4.1 Reduction of PDE's to ODE's 52
4.1.1 The Model ODE's 52
4.1.2 The Generic Matrix Form 53
4.2 Exact Solutions of Linear ODE's 54
4.2.1 Eigensystems of Semi-Discrete Linear Forms 54
4.2.2 Single ODE's of First- and Second-Order 55
4.2.3 Coupled First-Order ODE's 57
4.2.4 General Solution of Coupled ODE's with Complete Eigensystems 59
4.3 Real Space and Eigenspace 61
4.3.1 Defnition 61
4.3.2 Eigenvalue Spectrums for Model ODE's 62
4.3.3 Eigenvectors of the Model Equations 63
4.3.4 Solutions of the Model ODE's 65
4.4 The Representative Equation 67
4.5 Problems 68
5 FINITE-VOLUME METHODS 71
5.1 Basic Concepts 72
5.2 Model Equations in Integral Form 73
5.2.1 The Linear Convection Equation 73
5.2.2 The Difusion Equation 74
5.3 One-Dimensional Examples 74
5.3.1 A Second-Order Approximation to the Convection Equation 75
5.3.2 A Fourth-Order Approximation to the Convection Equation 77
5.3.3 A Second-Order Approximation to the Difusion Equation 78
5.4 A Two-Dimensional Example 80
5.5 Problems 83
6 TIME-MARCHING METHODS FOR ODE'S 85
6.1 Notation 86
6.2 Converting Time-Marching Methods to OE's 87
6.3 Solution of Linear OE's With Constant Coecients 88
6.3.1 First- and Second-Order Diference Equations 89
6.3.2 Special Cases of Coupled First-Order Equations 90
6.4 Solution of the Representative OE's 91
6.4.1 The Operational Form and its Solution 91
6.4.2 Examples of Solutions to Time-Marching OE's 92
6.5 The Relation 93
6.5.1 Establishing the Relation 93
6.5.2 The Principal Root 95
6.5.3 Spurious Roots 95
6.5.4 One-Root Time-Marching Methods 96
6.6 Accuracy Measures of Time-Marching Methods 97
6.6.1 Local and Global Error Measures 97
6.6.2 Local Accuracy of the Transient Solution (er; jj ; er!) 98
6.6.3 Local Accuracy of the Particular Solution (er) 99
6.6.4 Time Accuracy For Nonlinear Applications 100
6.6.5 Global Accuracy 101
6.7 Linear Multistep Methods 102
6.7.1 The General Formulation 102
6.7.2 Examples 103
6.7.3 Two-Step Linear Multistep Methods 105
6.8 Predictor-Corrector Methods 106
6.9 Runge-Kutta Methods 107
6.10 Implementation of Implicit Methods 110
6.10.1 Application to Systems of Equations 110
6.10.2 Application to Nonlinear Equations 111
6.10.3 Local Linearization for Scalar Equations 112
6.10.4 Local Linearization for Coupled Sets of Nonlinear Equations 115
6.11 Problems 117
7 STABILITY OF LINEAR SYSTEMS 121
7.1 Dependence on the Eigensystem 122
7.2 Inherent Stability of ODE's 123
7.2.1 The Criterion 123
7.2.2 Complete Eigensystems 123
7.2.3 Defective Eigensystems 123
7.3 Numerical Stability of OE 's 124
7.3.1 The Criterion 124
7.3.2 Complete Eigensystems 125
7.3.3 Defective Eigensystems 125
7.4 Time-Space Stability and Convergence of OE's 125
7.5 Numerical Stability Concepts in the Complex Plane 128
7.5.1 Root Traces Relative to the Unit Circle 128
7.5.2 Stability for Small t 132
7.6 Numerical Stability Concepts in the Complex h Plane 135
7.6.1 Stability for Large h. 135
7.6.2 Unconditional Stability, A-Stable Methods 136
7.6.3 Stability Contours in the Complex h Plane. 137
7.7 Fourier Stability Analysis 141
7.7.1 The Basic Procedure 141
7.7.2 Some Examples 142
7.7.3 Relation to Circulant Matrices 143
7.8 Consistency 143
7.9 Problems 146
8 CHOICE OF TIME-MARCHING METHODS 149
8.1 Stifness Defnition for ODE's 149
8.1.1 Relation to -Eigenvalues 149
8.1.2 Driving and Parasitic Eigenvalues 151
8.1.3 Stifness Classifcations 151
8.2 Relation of Stifness to Space Mesh Size 152
8.3 Practical Considerations for Comparing Methods 153
8.4 Comparing the Eciency of Explicit Methods 154
8.4.1 Imposed Constraints 154
8.4.2 An Example Involving Difusion 154
8.4.3 An Example Involving Periodic Convection 155
8.5 Coping With Stifness 158
8.5.1 Explicit Methods 158
8.5.2 Implicit Methods 159
8.5.3 A Perspective 160
8.6 Steady Problems 160
8.7 Problems 161
9 RELAXATION METHODS 163
9.1 Formulation of the Model Problem 164
9.1.1 Preconditioning the Basic Matrix 164
9.1.2 The Model Equations 166
9.2 Classical Relaxation 168
9.2.1 The Delta Form of an Iterative Scheme 168
9.2.2 The Converged Solution, the Residual, and the Error 168
9.2.3 The Classical Methods 169
9.3 The ODE Approach to Classical Relaxation 170
9.3.1 The Ordinary Diferential Equation Formulation 170
9.3.2 ODE Form of the Classical Methods 172
9.4 Eigensystems of the Classical Methods 173
9.4.1 The Point-Jacobi System 174
9.4.2 The Gauss-Seidel System 176
9.4.3 The SOR System 180
9.5 Nonstationary Processes 182
9.6 Problems 187
10 MULTIGRID 191
10.1 Motivation 191
10.1.1 Eigenvector and Eigenvalue Identifcation with Space Frequencies191
10.1.2 Properties of the Iterative Method 192
10.2 The Basic Process 192
10.3 A Two-Grid Process 200
10.4 Problems 202
11 NUMERICAL DISSIPATION 203
11.1 One-Sided First-Derivative Space Diferencing 204
11.2 The Modifed Partial Diferential Equation 205
11.3 The Lax-Wendrof Method 207
11.4 Upwind Schemes 209
11.4.1 Flux-Vector Splitting 210
11.4.2 Flux-Diference Splitting 212
11.5 Artifcial Dissipation 213
11.6 Problems 214
12 SPLIT AND FACTORED FORMS 217
12.1 The Concept 217
12.2 Factoring Physical Representations | Time Splitting 218
12.3 Factoring Space Matrix Operators in 2{D 220
12.3.1 Mesh Indexing Convention 220
12.3.2 Data Bases and Space Vectors 221
12.3.3 Data Base Permutations 221
12.3.4 Space Splitting and Factoring 223
12.4 Second-Order Factored Implicit Methods 226
12.5 Importance of Factored Forms in 2 and 3 Dimensions 226
12.6 The Delta Form 228
12.7 Problems 229
13 LINEAR ANALYSIS OF SPLIT AND FACTORED FORMS 233
13.1 The Representative Equation for Circulant Operators 233
13.2 Example Analysis of Circulant Systems 234
13.2.1 Stability Comparisons of Time-Split Methods 234
13.2.2 Analysis of a Second-Order Time-Split Method 237
13.3 The Representative Equation for Space-Split Operators 238
13.4 Example Analysis of 2-D Model Equations 242
13.4.1 The Unfactored Implicit Euler Method 242
13.4.2 The Factored Nondelta Form of the Implicit Euler Method 243
13.4.3 The Factored Delta Form of the Implicit Euler Method 243
13.4.4 The Factored Delta Form of the Trapezoidal Method 244
13.5 Example Analysis of the 3-D Model Equation 245
13.6 Problems 247
A USEFUL RELATIONS AND DEFINITIONS FROM LINEAR AL-
GEBRA 249
A.1 Notation 249
A.2 Defnitions 250
A.3 Algebra 251
A.4 Eigensystems 251
A.5 Vector and Matrix Norms 254
B SOME PROPERTIES OF TRIDIAGONAL MATRICES 257
B.1 Standard Eigensystem for Simple Tridiagonals 257
B.2 Generalized Eigensystem for Simple Tridiagonals 258
B.3 The Inverse of a Simple Tridiagonal 259
B.4 Eigensystems of Circulant Matrices 260
B.4.1 Standard Tridiagonals 260
B.4.2 General Circulant Systems 261
B.5 Special Cases Found From Symmetries 262
B.6 Special Cases Involving Boundary Conditions 263
C THE HOMOGENEOUS PROPERTY OF THE EULER EQUATIONS265

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