Table of Contents
Preface vii
1 Direct Modeling for Computational Fluid Dynamics 1
1.1 Physical Modeling and Numerical Solution of Fluid Dynamic Equations 2
1.2 Direct Modeling of Fluid Motion 4
1.3 Direct Modeling and Multiscale Coarse-graining Models 7
1.4 Necessity of Direct Modeling 8
2 Introduction to Gas Kinetic Theory 11
2.1 Macroscopic Gas Dynamic Equations 12
2.2 Gas Distribution Function of Equilibrium Flow 15
2.3 Boltzmann Equation 22
2.4 Understanding of Boltzmann Equation 29
2.5 Relation between Kinetic Theory and Hydrodynamic Equations 31
2.6 Kinetic Model Equations 35
2.6.1 Bhatnagar-Gross-Krook (BGK) Model 36
2.6.2 BGK-Shakhov Model 41
2.6.3 ES-BGK Model 43
2.6.4 Combined Model 45
2.7 Summary 47
3 Introduction to Nonequilibrium Flow Simulations 49
3.1 Nonequilibrium Flow Study 49
3.2 Numerical Methods for Non-equilibrium Flows 50
3.2.1 DSMC and Direct Boltzmann Solver 51
3.2.2 Hybrid Scheme 56
3.2.3 Extended Hydrodynamics and Moment Methods 57
3.3 Direct Modeling in Unified Gas Kinetic Scheme 60
3.3.1 General Methodology 61
3.3.2 Modeling in Discretized Space 63
3.3.3 Multiple Scale Gas Evolution Model 65
3.3.4 Discretization and Integration in Particle Velocity Space 67
3.4 Summary 70
4 Gas Kinetic Scheme 71
4.1 Introduction 71
4.2 Gas Kinetic Scheme 75
4.3 Analysis of Gas Kinetic Scheme 82
4.4 Numerical Examples 95
4.5 Comparison of GKS and Godunov Method 106
4.6 Principle of CFD 113
5 Unified Gas Kinetic Scheme 119
5.1 Introduction 120
5.2 Unified Gas Kinetic Scheme in One-dimensional Space 121
5.3 Unified Gas Kinetic Scheme in Two-dimensional Space 133
5.4 Unified Gas Kinetic Scheme in Three-dimensional Space 141
5.5 Boundary Conditions 148
5.6 Analysis of Unified Gas Kinetic Scheme 151
5.7 Summary 164
6 Low Speed Microflow Studies 165
6.1 Introduction 165
6.2 Numerical Methods for Microflow 166
6.3 Unified Gas Kinetic Scheme for Microflow 169
6.4 Microflow Studies 171
6.4.1 Couette Flow 171
6.4.2 Pressure Driven Poiseuille Flow 173
6.4.3 Slider Air Bearing Problem 175
6.4.4 Unsteady Rayleigh Flow 177
6.4.5 Thermal Transpiration 181
6.4.6 Flow Induced by Temperature Discontinuity 185
6.4.7 Thermal Creep Flow Instability 187
6.4.8 Cavity Flow 191
6.4.9 Slit Flow 200
6.4.10 Sound Wave Propagation 203
6.4.11 Crookes Radiometer 204
6.4.12 Knudsen Forces on Heated Microbcams 209
6.5 Summary 212
7 High Speed Flow Studies 215
7.1 Introduction 215
7.2 Physical Modeling and Numerical Difficulties 216
7.3 Supersonic Flow Studies 218
7.3.1 Shock Structures 219
7.3.2 Flow around Circular Cylinder 224
7.3.3 Moving Ellipse 229
7.3.4 Continuum Flow Expansion into Rarefied Environment 231
7.4 Hypersonic Flow Studies 236
7.4.1 Flow Fields 239
7.4.2 Surface Properties 243
7.5 Summary 244
8 Unified Gas Kinetic Scheme for Diatomic Gas 253
8.1 Diatomic Gas Model 253
8.2 Unified Gas Kinetic Scheme 255
8.3 Diatomic Gas Studies 259
8.3.1 Rotational Relaxation of a Homogenous Gas 259
8.3.2 Shock Structures 260
8.3.3 Flow around a Flat Plate 260
8.3.4 Flow around a Blunt Circular Cylinder 262
8.4 Summary 263
9 Conclusion 271
Appendix A Non-dimensionlizing Fluid Dynamic Variables 275
Appendix B Connection between BGK, Navier Stokes and Euler Equations 281
Appendix C Moments of Maxwellian Distribution Function and Expansion Coefficients 287
Appendix D Flux Evaluation through Stationary and Moving Cell Interfaces 293
Bibliography 297
Index 317
