Quasi-Gas Dynamic Equations

1. Auflage 2009
ISBN: 978-3-642-00292-2
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 286 Seiten

The monograph is devoted to modern mathematical models and numerical methods for solving gas- and ?uid-dynamic problems based on them. Two interconnected mathematical models generalizing the Navier–Stokes system are presented; they differ from the Navier–Stokes system by additional dissipative terms with a small parameter as a coef?cient. The new models are called the quasi-gas-dynamic and quasi-hydrodynamic equations. Based on these equations, effective ?nite-difference algorithms for calculating viscous nonstationary ?ows are constructed and examples of numerical computations are presented. The universality, the ef?ciency, and the exactness of the algorithms constructed are ensured by the ful?llment of integral conservation laws and the theorem on entropy balance for them. The book is a course of lectures and is intended for scientists and engineers who deal with constructing numerical algorithms and performing practical calculations of gas and ?uid ?ows and also for students and postgraduate students who specialize in numerical gas and ?uid dynamics.

Autoren/Hrsg.

Inhaltsverzeichnis

1;Preface;5
2;Contents;6
3;Introduction;10
4;Chapter 1 Construction of Gas-Dynamic Equations by Using Conservation Laws ;13
4.1; Averaging Procedure;13
4.1.1; Spatial Averages;14
4.1.2; Spatial-Time Averages;15
4.1.3; Galileo Transform;16
4.1.4; Continuity Equation;17
4.2; Conservation Laws in the Integral Form;19
4.3; Conservation Laws in the Differential Form;21
4.4; Euler and Navier--Stokes Equations;23
4.5; Quasi-gas-dynamic and Quasi-hydrodynamic Equations;26
4.5.1; Quasi-gas-dynamic System;26
4.5.2; Quasi-hydrodynamic System;28
4.5.3; Vector of the Mass Flux Density and the Parameter ;29
4.5.4; Comparison of the Models and the Barometric Formula;32
5;Chapter 2 Elements of Kinetic Gas Theory ;35
5.1; Boltzmann Equation;35
5.2; Equilibrium Distribution Function and the Euler System;37
5.2.1; Navier--Stokes Equations;38
5.2.2; Bhatnagar--Gross--Krook Equation;39
5.2.2.1; Mean Collision Quantities of the Particle Motion;40
5.2.2.1.1; Transport Coefficients in Equilibrium Gases;42
5.2.2.2; Numerical Simulation of Flows of Rarefied Gases;43
5.2.2.2.1; General Remarks;43
5.2.2.2.2; Monte Carlo Method;45
5.2.3; Difference Approximation of the Boltzmann Equation and Kinetically Consistent Difference Schemes;47
6;Chapter 3 Quasi-gas-dynamic Equations ;49
6.1; Regularized Kinetic Equation;49
6.2; Kinetic Deduction of the QGD Equations;52
6.3; QGD Equations in the Form of Conservation Laws;54
6.3.1; Equation of Continuity and the Vector of the Mass Flux Density;54
6.3.2; Momentum Equation and the Viscous Shear-Stress Tensor;55
6.3.3; Total Energy Equation and Heat Flux Vector;56
6.4; Dissipation Coefficients;60
6.4.1; Formulas for Dissipative Coefficients and Their Generalizations;60
6.4.2; Volume Viscosity Coefficient;60
6.5; Navier--Stokes System as an Asymptotic of the QGD system;63
6.5.1; QGD Addition to the Vector of Mass Flux Density;64
6.5.2; QGD Addition to the Viscous Shear-Stress Tensor;65
6.5.3; QGD Addition to the Work of Pressure and Viscous Friction Forces;66
6.5.4; QGD addition to the Heat Flux Vector;66
6.6; QGD Equations for Gas Flows Under the Existence of Exterior Forces and Heat Sources;67
6.7; Entropy Balance Equation;70
7;Chapter 4 Quasi-gas-dynamic Equations and Coordinate Systems ;75
7.1; Quasi-gas-dynamic Equations in an Arbitrary Coordinate System;75
7.2; Cartesian Coordinate System;78
7.3; Cylindrical Coordinate System;83
8;Chapter 5 Numerical Algorithms for Solving Gas-Dynamic Problems ;87
8.1; System for Planar Two-Dimensional Flows;87
8.2; System for Cylindrical Two-Dimensional Flows;89
8.3; Boundary Conditions;91
8.4; Dimensionless Form of the Equations;93
8.5; Finite-Difference Approximation;94
8.6; Introducing the Artificial Dissipation;98
8.7; Problem on the Strong Discontinuity Step Evolution;100
8.8; Flow Around a Cylindrical Obstacle;105
8.8.1; Problem Formulation and the Numerical Algorithm;105
8.8.2; Results of Computation;109
8.9; Nonviscid Flow in a Channel with a Forward-Facing Step;111
8.10; Numerical Algorithm for Computing Subsonic Flows;114
8.10.1; Dimensionless Form of Equations and the Regularization;116
8.10.2; Nonreflecting Boundary Conditions;117
8.11; Stability and Accuracy of QGD Algorithms;118
9;Chapter 6 Algorithms for Solving Quasi-gas-dynamic Equations on Nonstructured Grids ;121
9.1; Choice of the Grid and Constructing the Control Volume;121
9.2; Approximation of the System;123
9.3; Approximation of Partial Derivatives;124
9.4; Finite-Difference Schemes for Two-Dimensional Flows;129
9.5; Approximation of Boundary Conditions;135
9.6; Computation of the Flow in a Neighborhood of a Cylinder;136
10;Chapter 7 Quasi-hydrodynamic Equations and Flows of Viscous Incompressible Fluids ;141
10.1; Quasi-hydrodynamic System;141
10.2; Computational Algorithm;145
10.3; Backward-Facing Step Flows in a Channel;148
10.3.1; Formulation of the Problem;149
10.3.2; Results of Numerical Simulation;151
10.4; Heat Convection in a Square Cavity;154
10.5; Heat Convection for Low Prandtl Numbers;157
10.5.1; Results of Computations for the R-R Case;159
10.5.2; Results of Computations for the R-F Case;162
10.6; Marangoni Convection in the Zero Gravity;164
10.7; Flows in a Cubic Cavity with a Movable Lid;167
11;Chapter 8 Quasi-gas-dynamic Equations for Nonequilibrium Gas Flows ;178
11.1; Molecular Models and Distribution Functions;178
11.2; Coordinate Systems and Certain Integrals;180
11.3; Construction of Moment Equations;182
11.4; Calculation of Exchange Terms;187
11.5; QGDR Equations for a Gas with Two or Three Rotational Degrees of Freedom;189
11.6; Examples of Numerical Computations;192
11.6.1; Spatial Relaxation Problem;192
11.6.2; Problem of the Shock Wave Structure;193
12;Chapter 9 Quasi-gas-dynamic Equations for Binary Gas Mixtures ;197
12.1; Initial Kinetic Model;198
12.2; Construction of the Moment Equations;200
12.3; Calculation of Exchange Terms;203
12.4; Determination of the Collision Frequencies;204
12.5; Quasi-gas-dynamic Equations for Gas Mixtures;206
12.6; One-Fluid Approximations;207
12.6.1; QGDM Model in the One-Fluid Approximation;207
12.6.2; One-Fluid Model for the Navier--Stokes Equations;208
12.6.3; QGDM and Navier--Stokes One-Fluid Approximations;210
12.7; QGDM System for One-Dimensional Flows;211
12.7.1; Dimensionless Form;213
12.8; Structure of Shock Waves in the Mixture of Helium and Xenon;214
12.8.1; Problem Formulation;214
12.8.2; Computation on the Two-Fluid QGDM Model;215
12.8.3; Computation in the One-Fluid Approximation;222
12.9; Diffusion Problem of Argon and Helium;226
13;Appendix A Example of Constructing Quasi-gas-dynamic Equations ;230
13.1; Equation of Continuity;13
13.1.1; Equation for the Momentum;19
13.1.2; Equation for the Energy;21
13.1.3; Resulting System of Equations;23
14;Appendix B Flows of Viscous Compressible Gas in Microchannels ;235
14.1; Introduction;35
14.2; Poiseuille Flows in Planar Channels;37
14.2.1; Formulas for the Mass-Flow Rate;236
14.2.2; Minimum of the Mass-Flow Rate, or the Knudsen Effect;238
14.2.3; Dependence of the Mass-Flow Rate on the Pressure Overfall;240
14.3; Poiseuille Flows in Circular Tubes;38
14.4; Computation of the Mass-Flow Rate for Rarefied Flows;39
14.4.1; Correction of the Parameter for Rarefied Flows;245
14.4.2; Unified Formulas for Calculating the Mass-Flow Rate;248
14.5; Comparison with Experimental Data;40
14.5.1; Concluding Remarks;254
15;Appendix C Numerical Modelling of the Stationary Shock Wave Structure ;256
15.1; Introduction;49
15.2; Statement of the Problem;52
15.3; Results of Computations: Ascertainment Method;54
15.3.1; Solution of the Steady-State Navier--Stokes Equations;60
15.3.1.1; Results of Computations for Nitrogen;63
16;Appendix D Backward-Facing Step Flow in a Channel: Laminar--Turbulent Transition ;269
16.1; Introduction;75
16.2; Statement of the Problem;78
16.3; Numerical Computations and Discussion of the Results;83
16.3.1; Laminar Flows;272
16.3.2; Laminar--Turbulent Transition;274
17;References;279
18;Index;288