Elements of numerical methods for compressible flows

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010164664	QA911 K54 2006	Open Access Book	Book	Searching... Unknown
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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 book is restricted to the basic concepts of finite volume methods, and even in this regard is not intended to be exhaustive in its treatment. Although the practical applications of the one-dimensional Euler equations are limited, virtually all numerical algorithms for inviscid compressible flow in two and three dimensions owe their origin to techniques developed in the context of the one-dimensional Euler equations. 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.

Author Notes

Doyle D. Knight is Professor of Aerospace and Mechanical Engineering in the Department of Mechanical and Aerospace Engineering at Rutgers - The State University of New Jersey

List of Illustrations	p. xiii
List of Tables	p. xvii
Preface	p. xix
1 Governing Equations	p. 1
1.1 Introduction	p. 1
1.2 Conservation Laws	p. 2
1.3 Convective Derivative	p. 3
1.4 Vector Notation	p. 3
1.5 Entropy	p. 3
1.6 Speed of Sound	p. 4
1.7 Alternate Forms	p. 5
2 One-Dimensional Euler Equations	p. 9
2.1 Introduction	p. 9
2.2 Differential Forms of One-Dimensional Euler Equations	p. 9
2.2.1 Conservative Form	p. 10
2.2.2 Nonconservative Form	p. 11
2.2.3 Characteristic Form	p. 14
2.3 Discontinuous Waves	p. 17
2.4 Method of Characteristics	p. 20
2.5 Expansion Fan	p. 22
2.6 Domains of Dependence and Influence	p. 25
2.7 Shock Formation	p. 26
2.8 Shock Formation from Sinusoidal Disturbance	p. 30
2.9 General Riemann Problem	p. 32
2.9.1 Case 1. Two Shock Waves: p[subscript 1] [less than] p[superscript ] and p[subscript 4] [less than] p[superscript ]	p. 34
2.9.2 Case 2. Shock and Expansion: p[subscript 1] [less than] p[superscript ] and p[subscript 4] [greater than] p[superscript ]	p. 35
2.9.3 Case 3. Expansion and Shock: p[subscript 1] [greater than] p[superscript ] and p[subscript 4] [less than] p[superscript ]	p. 36
2.9.4 Case 4. Two Expansions: p[subscript 1] [greater than] p[superscript ] and p[subscript 4] [greater than] p[superscript ]	p. 37
2.10 Riemann Shock Tube	p. 39
3 Accuracy, Consistency, Convergence, and Stability	p. 45
3.1 Introduction	p. 45
3.2 The Problem	p. 45
3.3 Discretization	p. 46
3.4 Four Issues	p. 47
3.5 A Class of Discrete Approximations	p. 48
3.6 Accuracy	p. 49
3.7 Consistency	p. 51
3.8 Flux Quadrature and Stability	p. 53
3.8.1 A Simple Flux Quadrature	p. 53
3.8.2 Another Simple Flux Quadrature	p. 56
3.8.3 Numerical Domain of Dependence	p. 59
3.8.4 Shock Waves and Weak Solutions	p. 60
3.9 Stability	p. 61
3.9.1 A Simple Flux Quadrature	p. 61
3.9.2 Another Simple Flux Quadrature	p. 66
3.10 Convergence	p. 68
3.11 Conclusion	p. 69
4 Reconstruction	p. 73
4.1 Introduction	p. 73
4.2 Reconstruction Using the Primitive Function	p. 75
4.3 No New Extrema	p. 79
4.4 Modified Upwind Scheme for Conservation Laws	p. 81
4.4.1 Case 1: [Delta]Q[subscript i]+[Fraction 12] [greater than Equal] 0, [Delta]Q[subscript i]-[Fraction 12] [greater than equal] 0	p. 84
4.4.2 Case 2: [Delta]Q[subscript i]+[Fraction 12] [greater than Equal] 0, [Delta]Q[subscript i]-[Fraction 12] [less than equal] 0	p. 87
4.4.3 Case 3: [Delta]Q[subscript i]+[Fraction 12] [less than Equal] 0, [Delta]Q[subscript i]-[Fraction 12] [less than equal] 0	p. 89
4.4.4 Case 4: [Delta]Q[subscript i]+[Fraction 12] [less than Equal] 0, [Delta]Q[subscript i]-[Fraction 12] [greater than equal] 0	p. 89
4.4.5 Summary	p. 90
4.4.6 Results	p. 92
4.5 Essentially Non-Oscillatory Methods	p. 92
4.5.1 Determination of the Value of a	p. 97
4.5.2 Results	p. 99
5 Godunov Methods	p. 105
5.1 Introduction	p. 105
5.2 Godunov's Method	p. 105
5.2.1 Algorithm	p. 106
5.2.2 Stability	p. 107
5.2.3 Accuracy, Consistency, and Convergence	p. 108
5.3 Roe's Method	p. 109
5.3.1 Algorithm	p. 110
5.3.2 Stability	p. 117
5.3.3 Accuracy, Consistency, and Convergence	p. 120
5.3.4 Entropy Fix	p. 122
5.4 Osher's Method	p. 127
5.4.1 Algorithm	p. 127
5.4.2 Stability	p. 139
5.4.3 Accuracy, Consistency, and Convergence	p. 141
6 Flux Vector Splitting Methods	p. 147
6.1 Introduction	p. 147
6.2 Steger and Warming's Method	p. 148
6.2.1 Algorithm	p. 148
6.2.2 Stability	p. 153
6.2.3 Accuracy, Consistency, and Convergence	p. 155
6.3 Van Leer's Method	p. 157
6.3.1 Algorithm	p. 157
6.3.2 Stability	p. 163
6.3.3 Accuracy, Consistency, and Convergence	p. 164
7 Temporal Quadrature	p. 170
7.1 Introduction	p. 170
7.2 Explicit Methods	p. 171
7.2.1 Runge-Kutta	p. 171
7.3 Implicit Methods	p. 172
7.3.1 Beam-Warming	p. 173
7.4 Stability of Selected Methods	p. 175
7.4.1 Runge-Kutta	p. 175
7.4.2 Beam-Warming	p. 180
8 TVD Methods	p. 185
8.1 Introduction	p. 185
8.2 Total Variation	p. 187
8.3 Flux Corrected Transport	p. 191
8.3.1 Algorithm	p. 191
8.3.2 Accuracy, Consistency, and Convergence	p. 209
8.3.3 Total Variation	p. 211
8.4 MUSCL-Hancock Method	p. 212
8.4.1 Algorithm	p. 213
8.4.2 Accuracy, Consistency, and Convergence	p. 216
8.4.3 Total Variation	p. 216
Notes	p. 224
Bibliography	p. 240
Index	p. 244

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