Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows / Edition 1

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More About This Textbook

Overview

The outstanding points of our book consist of investigations into the possibility of the numerical schemes of the direct method for solving the Boltzmann equation. Both deterministic and Monte Carlo procedures are considered to evaluate the collision integrals. The main mathematical tool is the conservative splitting method on the basis of which, a set of classical and new problems are solved to study nonequilibrium gas flows. This monograph differs from other books in the same field, because, for example the book by G.A. Bird is concerned with the approach of simulation of rarefied gas flows and the book by C. Cercignani deals with the classical kinetic theory issues and describes mainly the analytical and engineering methods for solving the Boltzmann equation. Our book is the first (as we know) monograph which is devoted to the numerical direct solving of the Boltzmann equation. The intended level of readership are graduate and postgraduate students and researches. This book can be used by the target groups as the mathematical apparatus to numerical study of complex problems of nonequilibrium gas flows.

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Editorial Reviews

Booknews
Concerned with the methods of solving the nonlinear Baltzmann equation and investigating its possibilities for describing some aerodynamical and physical problems, the main purpose of this work is the study of nonequilibrium gas flows on the basis of the direct integration of the kinetic equations. The monograph is a revision of Aristov's and Thcherenessine's (in Russian), with the main difference being more attention to the advantages of the Boltzmann equation as a tool for considering nonlinear, nonequilibrium processes. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780792368311
  • Publisher: Springer-Verlag New York, LLC
  • Publication date: 1/31/2001
  • Series: Fluid Mechanics and Its Applications Series , #60
  • Edition description: 2001
  • Edition number: 1
  • Pages: 319
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.75 (d)

Table of Contents

Preface ix
Introduction xiii
References xvii
1 The Boltzmann Equation as a Physical and Mathematical Model 1
1.1 Different mathematical forms of the kinetic equation 1
1.2 Peculiarities of kinetic approach for describing physical properties 6
1.3 Formulation of problems and boundary conditions 10
1.4 The forms of the Boltzmann equations in some physical cases 13
References 21
2 Survey of Mathematical Approaches to Solving the Boltzmann Equation 23
2.1 General notes on classification of methods 23
2.2 Methods combining analytical and numerical features. Some partial solutions 25
2.3 Approaches based on kinetic models 27
2.4 Numerical simulation methods 29
2.5 Direct simulation Monte Carlo methods 30
2.6 Methods of direct integration 31
2.7 Comparison of direct integration and direct simulation 33
References 39
3 Main Features of the Direct Numerical Approaches 45
3.1 Discrete velocities and approximation in velocity space 45
3.2 Approximation in physical space. Finite-difference schemes and iterations 49
3.3 Splitting method 51
3.4 Finite volume scheme 56
3.5 Evaluation of the collision integrals by Monte Carlo technique 58
3.6 Quasi Monte Carlo technique 61
References 67
4 Deterministic (Regular) Method for Solving the Boltzmann Equation 69
4.1 General features of the method 69
4.2 Approach to approximation of the collision integrals. Integration over velocity space 69
4.3 Exact evaluation of integrals over impact parameters 70
4.4 Approximation of the collision integrals by quadratic form with constant coefficients 75
4.5 Simplification for velocity space in the case of isotropic symmetry 77
References 83
5 Construction of Conservative Scheme for the Kinetic Equation 85
5.1 Different definitions of conservativity 85
5.2 Conservative splitting method 87
5.3 Characteristics and advantages of the conservative schemes 93
5.4 Practical verification of the method 98
5.5 Conservative method for gas mixtures 103
References 107
6 Parallel Algorithms for the Kinetic Equation 109
6.1 Parallel implementation for the direct methods 109
6.2 Several parallel algorithms 111
6.3 Examples of parallel applications of the algorithms 113
References 119
7 Application of the Conservative Splitting Method for Investigating Near Continuum Gas Flows 121
7.1 Some approaches to solving the Boltzmann equation for weakly rarefied gas 121
7.2 Asymptotic kinetic schemes approximating the Euler and Navier-Stokes equations 124
7.3 Schemes for flows at low Knudsen numbers 131
References 137
8 Study of Uniform Relaxation in Kinetic Gas Theory 139
8.1 Spatially uniform (homogeneous) relaxation problem 139
8.2 Obtaining the test solutions for isotropic relaxation 140
8.3 Some examples of the relaxation problem solutions 146
8.4 Uniform relaxation for gas mixtures 148
References 153
9 Nonuniform Relaxation Problem as a Basic Model for Description of Open Systems 155
9.1 Formulation of the problem and solution methods 155
9.2 Nonclassical behavior of macroscopic parameters 159
9.3 Behavior of the distribution function and macroscopic parameters 164
9.4 Possible entropy decrease 167
9.5 Some generalizations 171
References 179
10 One-Dimensional Kinetic Problems 181
10.1 The problem of heat transfer 181
10.2 Shock wave structure 188
10.3 Flow in the field of an external force 197
10.4 Recondensation of a mixture in a force field 203
References 207
11 Multi-Dimensional Problems. Study of Free Jet Flows 211
11.1 Possibilities of direct integration approaches for studying multi-dimensional problems 211
11.2 Formulation of the problem and numerical scheme 212
11.3 Free plane jet 214
11.4 Axisymmetric and three-dimensional free jet flows 215
References 225
12 The Boltzmann Equation and the Description of Unstable Flows 227
12.1 Main notions 227
12.2 Boltzmann and Navier-Stokes description 228
12.3 Mathematical apparatus 230
12.4 Some results of numerical modelling 231
References 239
13 Solutions of Some Multi-Dimensional Problems 241
13.1 Unsteady problem of a shock wave reflection from a wedge 241
13.2 Solution for focusing of a shock wave 250
13.3 Study of flows in elements of cryovacuum devices 254
13.4 Flows in the vacuum cryomodulus 260
13.5 Two-component mixture flows with cryocondensation 263
References 269
14 Special Hypersonic Flows and Flows with Very High Temperatures 271
14.1 Special hypersonic flows 271
14.2 Unsteady flows caused by a powerful point discharge of a finite gaseous mass 274
14.3 Asymptotic solution at t [right arrow] 0 277
14.4 Numerical analysis. Asymptotic solution at t [right arrow] [infinity] 281
14.5 Scattering of impulsive molecular beam 286
References 293
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