Quasi-Gas Dynamic Equations (eBook)

Preface 5
Contents 6
Introduction 10
Chapter 1 Construction of Gas-Dynamic Equations by Using Conservation Laws 13
Averaging Procedure 13
Spatial Averages 14
Spatial-Time Averages 15
Galileo Transform 16
Continuity Equation 17
Conservation Laws in the Integral Form 19
Conservation Laws in the Differential Form 21
Euler and Navier--Stokes Equations 23
Quasi-gas-dynamic and Quasi-hydrodynamic Equations 26
Quasi-gas-dynamic System 26
Quasi-hydrodynamic System 28
Vector of the Mass Flux Density and the Parameter 29
Comparison of the Models and the Barometric Formula 32
Chapter 2 Elements of Kinetic Gas Theory 35
Boltzmann Equation 35
Equilibrium Distribution Function and the Euler System 37
Navier--Stokes Equations 38
Bhatnagar--Gross--Krook Equation 39
Mean Collision Quantities of the Particle Motion 40
Transport Coefficients in Equilibrium Gases 42
Numerical Simulation of Flows of Rarefied Gases 43
General Remarks 43
Monte Carlo Method 45
Difference Approximation of the Boltzmann Equation and Kinetically Consistent Difference Schemes 47
Chapter 3 Quasi-gas-dynamic Equations 49
Regularized Kinetic Equation 49
Kinetic Deduction of the QGD Equations 52
QGD Equations in the Form of Conservation Laws 54
Equation of Continuity and the Vector of the Mass Flux Density 54
Momentum Equation and the Viscous Shear-Stress Tensor 55
Total Energy Equation and Heat Flux Vector 56
Dissipation Coefficients 60
Formulas for Dissipative Coefficients and Their Generalizations 60
Volume Viscosity Coefficient 60
Navier--Stokes System as an Asymptotic of the QGD system 63
QGD Addition to the Vector of Mass Flux Density 64
QGD Addition to the Viscous Shear-Stress Tensor 65
QGD Addition to the Work of Pressure and Viscous Friction Forces 66
QGD addition to the Heat Flux Vector 66
QGD Equations for Gas Flows Under the Existence of Exterior Forces and Heat Sources 67
Entropy Balance Equation 70
Chapter 4 Quasi-gas-dynamic Equations and Coordinate Systems 75
Quasi-gas-dynamic Equations in an Arbitrary Coordinate System 75
Cartesian Coordinate System 78
Cylindrical Coordinate System 83
Chapter 5 Numerical Algorithms for Solving Gas-Dynamic Problems 87
System for Planar Two-Dimensional Flows 87
System for Cylindrical Two-Dimensional Flows 89
Boundary Conditions 91
Dimensionless Form of the Equations 93
Finite-Difference Approximation 94
Introducing the Artificial Dissipation 98
Problem on the Strong Discontinuity Step Evolution 100
Flow Around a Cylindrical Obstacle 105
Problem Formulation and the Numerical Algorithm 105
Results of Computation 109
Nonviscid Flow in a Channel with a Forward-Facing Step 111
Numerical Algorithm for Computing Subsonic Flows 114
Dimensionless Form of Equations and the Regularization 116
Nonreflecting Boundary Conditions 117
Stability and Accuracy of QGD Algorithms 118
Chapter 6 Algorithms for Solving Quasi-gas-dynamic Equations on Nonstructured Grids 121
Choice of the Grid and Constructing the Control Volume 121
Approximation of the System 123
Approximation of Partial Derivatives 124
Finite-Difference Schemes for Two-Dimensional Flows 129
Approximation of Boundary Conditions 135
Computation of the Flow in a Neighborhood of a Cylinder 136
Chapter 7 Quasi-hydrodynamic Equations and Flows of Viscous Incompressible Fluids 141
Quasi-hydrodynamic System 141
Computational Algorithm 145
Backward-Facing Step Flows in a Channel 148
Formulation of the Problem 149
Results of Numerical Simulation 151
Heat Convection in a Square Cavity 154
Heat Convection for Low Prandtl Numbers 157
Results of Computations for the R-R Case 159
Results of Computations for the R-F Case 162
Marangoni Convection in the Zero Gravity 164
Flows in a Cubic Cavity with a Movable Lid 167
Chapter 8 Quasi-gas-dynamic Equations for Nonequilibrium Gas Flows 178
Molecular Models and Distribution Functions 178
Coordinate Systems and Certain Integrals 180
Construction of Moment Equations 182
Calculation of Exchange Terms 187
QGDR Equations for a Gas with Two or Three Rotational Degrees of Freedom 189
Examples of Numerical Computations 192
Spatial Relaxation Problem 192
Problem of the Shock Wave Structure 193
Chapter 9 Quasi-gas-dynamic Equations for Binary Gas Mixtures 197
Initial Kinetic Model 198
Construction of the Moment Equations 200
Calculation of Exchange Terms 203
Determination of the Collision Frequencies 204
Quasi-gas-dynamic Equations for Gas Mixtures 206
One-Fluid Approximations 207
QGDM Model in the One-Fluid Approximation 207
One-Fluid Model for the Navier--Stokes Equations 208
QGDM and Navier--Stokes One-Fluid Approximations 210
QGDM System for One-Dimensional Flows 211
Dimensionless Form 213
Structure of Shock Waves in the Mixture of Helium and Xenon 214
Problem Formulation 214
Computation on the Two-Fluid QGDM Model 215
Computation in the One-Fluid Approximation 222
Diffusion Problem of Argon and Helium 226
Appendix A Example of Constructing Quasi-gas-dynamic Equations 230
Equation of Continuity 13
Equation for the Momentum 19
Equation for the Energy 21
Resulting System of Equations 23
Appendix B Flows of Viscous Compressible Gas in Microchannels 235
Introduction 35
Poiseuille Flows in Planar Channels 37
Formulas for the Mass-Flow Rate 236
Minimum of the Mass-Flow Rate, or the Knudsen Effect 238
Dependence of the Mass-Flow Rate on the Pressure Overfall 240
Poiseuille Flows in Circular Tubes 38
Computation of the Mass-Flow Rate for Rarefied Flows 39
Correction of the Parameter for Rarefied Flows 245
Unified Formulas for Calculating the Mass-Flow Rate 248
Comparison with Experimental Data 40
Concluding Remarks 254
Appendix C Numerical Modelling of the Stationary Shock Wave Structure 256
Introduction 49
Statement of the Problem 52
Results of Computations: Ascertainment Method 54
Solution of the Steady-State Navier--Stokes Equations 60
Results of Computations for Nitrogen 63
Appendix D Backward-Facing Step Flow in a Channel: Laminar--Turbulent Transition 269
Introduction 75
Statement of the Problem 78
Numerical Computations and Discussion of the Results 83
Laminar Flows 272
Laminar--Turbulent Transition 274
References 279
Index 288