Elements of numerical methods for compressible flows

✍ Scribed by Knight D.D.

• Publisher: CUP
• Year: 2006
• Tongue: English
• Leaves: 266
• Series: Cambridge Aerospace Series
• Category: Library

✦ Synopsis

The purpose of this book is to present the basic elements of numerical methods for compressible flows. It is appropriate for advanced undergraduate and graduate students and specialists working in high speed flows. The focus is on the unsteady one-dimensional Euler equations which form the basis for numerical algorithms in compressible fluid mechanics. The author believes it is therefore essential to understand the development and implementation of these algorithms in their original one-dimensional context. The text is supplemented by numerous end-of-chapter exercises.

✦ Table of Contents

Cover......Page 1
About......Page 2
Cambridge Aerospace Series......Page 4
Elements of Numerical Methods for Compressible Flows......Page 6
9780521554749......Page 7
Contents......Page 10
List of Illustrations......Page 14
List of Tables......Page 18
Preface......Page 20
1.1 Introduction......Page 22
1.2 Conservation Laws......Page 23
1.5 Entropy......Page 24
1.6 Speed of Sound......Page 25
1.7 Alternate Forms......Page 26
2.2 Differential Forms of One-Dimensional Euler Equations......Page 30
2.2.1 Conservative Form......Page 31
2.2.2 Nonconservative Form......Page 32
2.2.3 Characteristic Form......Page 35
2.3 Discontinuous Waves......Page 38
2.4 Method of Characteristics......Page 41
2.5 Expansion Fan......Page 43
2.6 Domains of Dependence and Influence......Page 46
2.7 Shock Formation......Page 47
2.8 Shock Formation from Sinusoidal Disturbance......Page 51
2.9 General Riemann Problem......Page 53
2.9.1 Case 1. Two Shock Waves: p1 < p and p4 < p......Page 55
2.9.2 Case 2. Shock and Expansion: p1 < p and p4 > p......Page 56
2.9.3 Case 3. Expansion and Shock: p1 > p and p4 < p......Page 57
2.9.4 Case 4. Two Expansions: p1 > p and p4 > p......Page 58
2.10 Riemann Shock Tube......Page 60
3.2 The Problem......Page 66
3.3 Discretization......Page 67
3.4 Four Issues......Page 68
3.5 A Class of Discrete Approximations......Page 69
3.6 Accuracy......Page 70
3.7 Consistency......Page 72
3.8.1 A Simple Flux Quadrature......Page 74
3.8.2 Another Simple Flux Quadrature......Page 77
3.8.3 Numerical Domain of Dependence......Page 80
3.8.4 Shock Waves and Weak Solutions......Page 81
3.9.1 A Simple Flux Quadrature......Page 82
3.9.2 Another Simple Flux Quadrature......Page 87
3.10 Convergence......Page 89
3.11 Conclusion......Page 90
4.1 Introduction......Page 94
4.2 Reconstruction Using the Primitive Function......Page 96
4.3 No New Extrema......Page 100
4.4 Modified Upwind Scheme for Conservation Laws......Page 102
4.4.1 Case 1: ΔQ_{i+1/2} ≥ 0, ΔQ_{i−1/2} ≥ 0......Page 105
4.4.2 Case 2: ΔQ_{i+1/2} ≥ 0, ΔQ_{i−1/2} ≤ 0......Page 108
4.4.4 Case 4: ΔQ_{i+1/2} ≤ 0, ΔQ_{i−1/2} ≥ 0......Page 110
4.4.5 Summary......Page 111
4.5 Essentially Non-Oscillatory Methods......Page 113
4.5.1 Determination of the Value of a......Page 118
4.5.2 Results......Page 120
5.2 Godunov’s Method......Page 126
5.2.1 Algorithm......Page 127
5.2.2 Stability......Page 128
5.2.3 Accuracy, Consistency, and Convergence......Page 129
5.3 Roe’s Method......Page 130
5.3.1 Algorithm......Page 131
5.3.2 Stability......Page 138
5.3.3 Accuracy, Consistency, and Convergence......Page 141
5.3.4 Entropy Fix......Page 143
5.4.1 Algorithm......Page 148
5.4.2 Stability......Page 160
5.4.3 Accuracy, Consistency, and Convergence......Page 162
6.1 Introduction......Page 168
6.2.1 Algorithm......Page 169
6.2.2 Stability......Page 174
6.2.3 Accuracy, Consistency, and Convergence......Page 176
6.3.1 Algorithm......Page 178
6.3.2 Stability......Page 184
6.3.3 Accuracy, Consistency, and Convergence......Page 185
7.1 Introduction......Page 191
7.2.1 Runge-Kutta......Page 192
7.3 Implicit Methods......Page 193
7.3.1 Beam-Warming......Page 194
7.4.1 Runge-Kutta......Page 196
7.4.2 Beam-Warming......Page 201
8.1 Introduction......Page 206
8.2 Total Variation......Page 208
8.3.1 Algorithm......Page 212
8.3.2 Accuracy, Consistency, and Convergence......Page 230
8.3.3 Total Variation......Page 232
8.4 MUSCL-Hancock Method......Page 233
8.4.1 Algorithm......Page 234
8.4.3 Total Variation......Page 237
Notes......Page 245
Bibliography......Page 261
Index......Page 265