Monte Carlo Methods In Mechanics Of Fluid And Gas

by Oleg Mikhailovich Belotserkovskii, Yury Iv Khlopkov

Overview

This book is devoted to analysis of Monte Carlo methods developed in rarefied gas dynamics. Presented is the short history of the development of such methods, described are their main properties, their advantages and deficiencies. It is shown that the contemporary stage in the progress of computational methods cannot be regarded without a complex approach to the preparation of algorithms taking into account all the peculiarities of the problem under consideration, that is, of the physical nature of a process, the mathematical model and the theoretical aspects of computational mathematics and stochastic processes. Thoroughly investigated is the possibility of application of Monte Carlo methods in some kindred areas of science which are non-traditional for the use of statistical modeling (continuous media, turbulence). Considered are the possible directions of development of statistical modeling.

Product Details

ISBN-13:	9789814282352
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	05/10/2010
Pages:	280
Product dimensions:	5.70(w) x 9.10(h) x 0.80(d)

Table of Contents

Preface v

0 Introduction 1

1 The Main Equations and Approaches to Solutions of the Problems in Rarefied Gas Dynamics 23

1.1 The Main Equations in Rarefied Gas Dynamics 23

1.2 The Main Approaches to the Construction of Statistical Algorithms 25

1.3 Connection of the Stationary Modeling with the Solution of Equation 26

1.4 Construction of the Method of Direct Statistical Modeling 28

2 Development of the Numerical Methods of Solution of the Linear Kinetic Equations 30

2.1 The Perfection of VGK Method (Vlasov, Gorelov, Kogan) 30

2.2 Modification of the Vlasov's Method for the Solution of Linear Problems 35

2.3 Method of Solution of the Linearized Boltzmann's Equation 38

3 Methods of Solution of the Nonlinear Problems in Rarefied Gas Dynamics 43

3.1 Method of Solution of the Model Equation Based on a Stationary Modeling 43

3.2 The Possibilities of the Scheme of Splitting for the Solution of Kinetic Equations 46

3.3 Increase of the Method's Rate of Convergence 52

3.4 Method by Belotserkovskii and Yanitskii 54

4 Modeling of the Flow of Continuous Media 58

4.1 Procedure of the Monte Carlo Methods for Modeling the Flows of Rarefied Gas and Continuous Medium 58

4.2 Method "Relaxation-Transfer" for a Solution of the Problems of Gas Dynamics in the Wide Range of the Degree of Rarefaction of a Medium (see Kogan et al.⁸³) 62

4.3 Modeling of the Flows of Nonviscous Perfect Gas 66

5 Solution of the Navier-Stokes Equations (Petrov^133-139) 72

5.1 Formulation of the Problem, Initial and Boundary Conditions for the Navier-Stokes Equations in the Form by Helmholtz 72

5.2 The General Properties of the Vertical Flow Arising by the Instantaneous Start of a Body from the State of Rest 74

5.3 Initial Conditions for the Problem of the Instantaneous Start of a Body in a Viscous Fluid 78

5.4 The General Algorithm of the Numerical Solution of an Initial-Boundary Problem for the Navier-Stokes Equations in the form by Helmholtz 80

5.5 Solution of the Cauchy Problem for the Fokker-Plank Equation at Small Interval of Time 88

5.6 The Numerical Solution of the Fokker-Plank Equation by the Method of Direct Statistical Modeling 95

6 Studies of the Weakly Perturbed Flows of Rarefied Gas 103

6.1 Determination of the Velocity of Slip 103

6.2 Solution of the Problem of the Feeble Evaporation (Condensation) from the Plane Surface (see Korovkin, Khlopko¹⁰⁴) 106

6.3 The Slow Motion of a Sphere in Rarefied Gas (Brownian Motion) 108

6.4 The Coefficient of Diffusion and the Mean Shifting of a Brownian Particle in the Rarefied Gas (see Khlopkov¹⁰⁶) 110

7 Study of the Flows About Different Bodies in Transitional Regime 114

7.1 Flows About the Planar Bodies 115

7.2 Flows About Axisymmetrical Bodies 119

7.3 Influence of the Evaporation (Condensation) on the Aerodynamical Resistance of a Sphere by the Supersonic Flow About It 125

7.4 Computation of the Steady Regime of a Flow About a Body and of the Profile Resistance in a Viscous Gas (See A.S. Petrov) 128

8 Determination of the Aerodynamical Characteristics of the Returnable Space Systems (RSS) 138

8.1 Methodics of the Description of a Surface 138

8.2 Methodics of Calculation of the Aerodynamical Characteristics of the Flying Apparatus in the Conditions of a Free-Molecular Flow 142

8.3 The Engineering Methodics of the Computation of Aerodynamical Characteristics of the Bodies of Complicated Form in a Transitional Regime (see Galkin, Eropheev, Tolstykh⁸⁵) 143

8.4 The Results of the Flow About a Hypersonic Flying Apparatus "Clipper" (see Voronich, Zey Yar²²⁵) 145

9 The Flow About Blunted Bodies with the Addition of Heat (see Vorovich, Moiseev) 165

9.1 The Main Features of a Method 165

9.2 Description of the Algorithm 167

9.3 The Approximational Properties 170

9.4 The Algorithm and the Nets 172

9.5 Direct Statistical Modeling of the Inviscid Flows About Blunted Bodies by the Presence of Energy Addition 175

10 The General Models of Description of the Turbulent Flows 187

10.1 Theoretical Methods of the Description of Turbulence 187

10.2 Coherent Structures in the Turbulent Boundary Layer (see Khlopkov, Zharov, Gorelov²⁰⁵) 194

10.3 The Description of Turbulence with the Help of a Model of the Three-Wave Resonance 204

10.4 The Fluidical Model of the Description of Turbulence (Belotserkovskii, Yanitskii) 208

11 Studies of the Turbulent Flow of Fluid and Gas 211

11.1 Modeling of a Turbulent Transition within the Boundary Layer Using Monte Carlo Method (see Zharov, Tun Tun, Khlopkov²²³) 211

11.2 Study of the Dissipation of Turbulent Spots (see Belotserkovskii, Yanitskii, Bukin^12,221) 218

11.3 Evolution of the Vertical System in the Rarefied Gas (see Rovenskaya, Voronich, Zharov²²²) 219

12 The Possible Directions of Development of the Methods of Statistical Study 228

12.1 Development of the Methods of Solution of Linear Problems 228

12.2 Use of the Possibilities of the Model Equations 232

12.3 Modeling of the Flows of Continuous Medium 235

12.4 Modeling of the Turbulent Flows of Fluid and Gas 240

12.5 Parallelization of the Statistical Algorithms (Bukin, Voronich, Shtarkin) 245

Conclusions 253

References 257