Computational Fluid Dynamics Based on the Unified Coordinates

Cover......Page 1
Title Page......Page 3
Copyright Page......Page 4
Preface......Page 5
Table of Contents......Page 7
1.1 CFD as Numerical Solution to Nonlinear Hyperbolic PDEs......Page 10
1.2.1 Theoretical Issues......Page 11
1.2.2 Computational Issues......Page 13
1.3 Outline of the Book......Page 14
References......Page 15
2.1 Fluid as a Continuum......Page 18
2.2 Derivation of Conservation Law Equations in Fixed Coordinates......Page 19
2.4 Integral Equations versus Partial Differential Equations......Page 23
2.5 The Entropy Condition for Inviscid Flow Computation......Page 26
References......Page 27
3.1.1 Special Case: Stationary Surface of Discontinuity......Page 28
3.1.2 General Case: Moving Surface of Discontinuity......Page 29
3.2.1 Special Case: Stationary Surface of Discontinuity......Page 30
3.2.2 General Case: Moving Surface of Discontinuity......Page 32
3.3.1 Elementary Waves......Page 35
3.3.2 Existence and Uniqueness of Solution......Page 36
3.3.3 Structure of Riemann Solution for 1-D Flow......Page 37
3.3.4 Solution of the Elementary Waves......Page 38
3.3.5 Solution Algorithm......Page 42
3.4.1 The Question of Weak Solution to Partial Differential Equations......Page 43
3.5 Godunov Scheme......Page 44
3.6 High Resolution Schemes and Limiters......Page 47
3.7 Defects of Eulerian Computation......Page 48
References......Page 49
4.1 Gas Dynamics Equations Based on the Unified Coordinates......Page 51
4.2 Shock-Adaptive Godunov Scheme......Page 53
4.3 The Use of Entropy Conservation Law for Smooth Flow Computation......Page 55
4.4 The Unified Computer Code......Page 56
4.5 Cure of Defects of Eulerian and Lagrangian Computation by the UC Method......Page 60
4.5.1 Contact smearing......Page 61
4.5.2 Slow moving shock......Page 62
4.5.3 Sonic point glitch......Page 64
4.5.4 Start-up errors[12]......Page 65
4.5.5 Low pressure flow......Page 67
4.5.6 Wall-overheating......Page 68
4.5.7 Strong rarefaction waves......Page 71
References......Page 74
5.1 Eulerian Computation......Page 77
5.2 Lagrangian Computation......Page 79
5.4 Moving Mesh Methods......Page 81
5.5 Optimal Coordinates......Page 82
References......Page 83
6.1 Hui Transformation......Page 86
6.3 Derivation of Governing Equations in Conservation Form......Page 87
References......Page 92
7.2 Relation to Classical Lagrangian Coordinates......Page 93
7.3 Relation to Arbitrary-Lagrangian-Eulerian Computation......Page 94
7.5 Mesh Orthogonality......Page 95
7.6 Unified Coordinates for Steady Flow......Page 97
7.8 Relation to Other Moving Mesh Methods......Page 98
References......Page 100
8.1 Lagrangian Gas Dynamics Equations......Page 102
8.2 Weak Hyperbolicity......Page 103
8.3 Non-Equivalency of Lagrangian and Eularian Formulation......Page 104
References......Page 105
9.1 The Unified Coordinates for Steady Flow......Page 106
9.2 Euler Equations in the Unified Coordinates......Page 107
9.3 The Space-Marching Computation......Page 109
9.4 Examples......Page 110
9.5 3-D Flow......Page 116
References......Page 119
10.1 Summary of Solution to the 2-D Euler Equations Using the Unified Coordinates......Page 121
10.2 Computation Procedure......Page 123
10.3 Examples......Page 126
References......Page 129
11.1 Navier-Stokes Equations in the Unified Coordinates......Page 130
11.2 The Angle-preserving Equation......Page 133
11.3 Advantages of the g-equation Over the h-equation......Page 134
11.4 Boundary Condition and Movement of Boundary Cells......Page 136
11.5 Solution Strategies......Page 137
11.6 Test Examples: Shock/Boundary Flow Interaction and Shock/Shock Interaction......Page 141
References......Page 148
12.1 Brief Introduction of Gas-Kinetic Theory......Page 149
12.2 Gas-Kinetic BGK Model Under the Unified Coordinate Transformation......Page 154
12.3 Numerical BGK-NS Scheme in a Moving Mesh System......Page 155
12.4 Numerical Procedure......Page 159
12.5 Numerical Examples......Page 160
References......Page 170
13. Summary......Page 172
Nonlinearity of Characteristic Fields......Page 174
Smooth Solution......Page 175
Discontinuous Solutions......Page 176
Appendix B Computer Code for 1-D Flow in the Unified Coordinate......Page 178