DIRECT METHODS FOR SOLVING THE BOLTZMANN EQUATION AND STUDY OF NONEQUILIBRIUM FLOWS

Transcription

1 DIRECT METHODS FOR SOLVING THE BOLTZMANN EQUATION AND STUDY OF NONEQUILIBRIUM FLOWS

2 FLUID MECHANICS AND ITS APPLICATIONS Volume 60 Series Editor: R. MOREAU MADYIAM Ecole Nationale Superieure d'hydraulique de Grenoble Bofte Postale Saint Martin d'heres Cedex, France Aims and Scope of the Series The purpose of this series is to focus on subjects in which fluid mechanics plays a fundamental role. As well as the more traditional applications of aeronautics, hydraulics, heat and mass transfer etc., books will be published dealing with topics which are currently in a state of rapid development, such as turbulence, suspensions and multiphase fluids, super and hypersonic flows and numerical modelling techniques. It is a widely held view that it is the interdisciplinary subjects that will receive intense scientific attention, bringing them to the forefront of technological advancement. Fluids have the ability to transport matter and its properties as well as transmit force, therefore fluid mechanics is a subject that is particulary open to cross fertilisation with other sciences and disciplines of engineering. The subject of fluid mechanics will be highly relevant in domains such as chemical, metallurgical, biological and ecological engineering. This series is particularly open to such new multidisciplinary domains. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of a field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. For a list of related mechanics titles, see final pages.

3 Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows by v.v. ARISTOV Computing Center 0/ the Russian Academy 0/ Sciences, Moscow, Russia... " SPRINGER SCIENCE+BUSINESS MEDIA, B.V.

4 A C.LP. Catalogue record for this book is available from the Library of Congress. ISBN ISBN (ebook) DOI / Printed on acid-free paper All Rights Reserved 2001 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2001 Softcover reprint of the hardcover 1 st edition 2001 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

5 Table of Contents PREFACE INTRODUCTION References IX Xill xvii 1 THE BOLTZMANN EQUATION AS A PHYSICAL AND MATHEMATICAL MODEL Different mathematical forms of the kinetic equation Peculiarities of kinetic approach for describing physical properties Formulation of problems and boundary conditions The forms of the Boltzmann equations in some physical cases 13 References 21 2 SURVEY OF MATHEMATICAL APPROACHES TO SOLV ING THE BOLTZMANN EQUATION General notes on classification of methods Methods combining analytical and numerical features. Some partial solutions Approaches based on kinetic models Numerical simulation methods Direct simulation Monte Carlo methods Methods of direct integration Comparison of direct integration and direct simulation 33 References 39 3 MAIN FEATURES OF THE DIRECT NUMERICAL AP PROACHES ~ 3.1 Discrete velocities and approximation in velocity space Approximation in physical space. Finite-difference schemes and iterations Splitting method v

6 vi 3.4 Finite volume scheme Evaluation of the collision integrals by Monte Carlo technique Quasi Monte Carlo technique References 67 4 DETERMINISTIC (REGULAR) METHOD FOR SOLV ING THE BOLTZMANN EQUATION General features of the method Approach to approximation of the collision integrals. Integration over velocity space Exact evaluation of integrals over impact parameters Approximation of the collision integrals by quadratic form with constant coefficients Simplification for velocity space in the case of isotropic symmetry References 83 5 CONSTRUCTION OF CONSERVATIVE SCHEME FOR THE KINETIC EQUATION Different definitions of conservativity Conservative splitting method Characteristics and advantages of the conservative schemes Practical verification of the method Conservative method for gas mixtures 103 References PARALLEL ALGORITHMS FOR THE KINETIC EQUA- TION Parallel implementation for the direct methods.. i Several parallel algorithms Examples of parallel applications of the algorithms 113 References APPLICATION OF THE CONSERVATIVE SPLITTING METHOD FOR INVESTIGATING NEAR CONTINUUM GAS FLOWS Some approaches to solving the Boltzmann equation for weakly rarefied gas

7 7.2 Asymptotic kinetic schemes approximating the Euler and Navier-Stokes equations Schemes for flows at low Knudsen numbers References STUDY OF UNIFORM RELAXATION IN KINETIC GAS THEORY Spatially uniform (homogeneous) relaxation problem Obtaining the test solutions for isotropic relaxation Some examples of the relaxation problem solutions Uniform relaxation for gas mixtures References NONUNIFORM RELAXATION PROBLEM AS A BASIC MODEL FOR DESCRIPTION OF OPEN SYSTEMS Formulation of the problem and solution methods Nonclassical behavior of macroscopic parameters Behavior of the distribution function and macroscopic parameters Possible entropy decrease 9.5 Some generalizations vii References 10 ONE-DIMENSIONAL KINETIC PROBLEMS 10.1 The problem of heat transfer Shock wave structure Flow in the field of an external force Recondensation of a mixture in a force field References MULTI-DIMENSIONAL PROBLEMS. STUDY OF FREE JET FLOWS Possibilities of direct integration approaches for studying multi-dimensional problems Formulation of the problem and numerical scheme Free plane jet Axisymmetric and three-dimensional free jet flows 215 References 225

8 viii 12 THE BOLTZMANN EQUATION AND THE DESCRIP- TION OF UNSTABLE FLOWS Main notions Boltzmann and Navier-Stokes description Mathematical apparatus Some results of numerical modelling 231 References SOLUTIONS OF SOME MULTI-DIMENSIONAL PROB LEMS Unsteady problem of a shock wave reflection from a wedge Solution for focusing of a shock wave Study of flows in elements of cryovacuum devices Flows in the vacuum cryomodulus Two-component mixture flows with cryocondensation. 263 References SPECIAL HYPERSONIC FLOWS AND FLOWS WITH VERY HIGH TEMPERATURES Special hypersonic flows Unsteady flows caused by a powerful point discharge of a finite gaseous mass Asymptotic solution at t Numerical analysis. Asymptotic solution at t Scattering of impulsive molecular beam References 293

9 PREFACE This book is concerned with the methods of solving the nonlinear Boltzmann equation and of investigating its possibilities for describing some aerodynamic and physical problems. This monograph is a sequel to the book 'Numerical direct solutions of the kinetic Boltzmann equation' (in Russian) which was written with F.G.Tcheremissine and published by the Computing Center of the Russian Academy of Sciences some years ago. The main purposes of these two books are almost similar, namely, the study of nonequilibrium gas flows on the basis of direct integration of the kinetic equations. Nevertheless, there are some new aspects in the way this topic is treated in the present monograph. In particular, attention is paid to the advantages of the Boltzmann equation as a tool for considering nonequilibrium, nonlinear processes. New fields of application of the Boltzmann equation are also described. Solutions of some problems are obtained with higher accuracy. Numerical procedures, such as parallel computing, are investigated for the first time. The structure and the contents of the present book have some common features with the monograph mentioned above, although there are new issues concerning the mathematical apparatus developed so that the Boltzmann equation can be applied for new physical problems. Because of this some chapters have been rewritten and checked again and some new chapters have been added. These new points correspond to new numerical algorithms, some new test results, solutions for multi-dimensional problems, and new understanding based on the methods of the Boltzmann equation theory. The method of directly solving the Boltzmann equation is a natural and simple way to study nonequilibrium flows of a rarefied gas. However, there were some obvious difficulties connected with the multi-dimensionality of the distribution function, nonlinearity of the equation, and complexity of five-fold collision integrals. Therefore, for many investigators the simulation of gas flows was supposed to be preferable. But in recent years, progress in direct approaches and the development of multiprocessor parallel computers have provided new arguments in favour of the direct integration. In fact, this approach is an alternative to the direct simulation methods, which now can be considered (under certain conditions) as the scheme for ix

10 x PREFACE solving the Boltzmann equation itself. From this point of view, there are advantages and disadvantages in each of the mentioned approaches, and they are considered and compared in this book. The first direct numerical approach was proposed by Nordsieck and co-authors in the 1960s. Theremissine then developed his own variant of the direct integration method. In the 70s the conservative splitting method was constructed (by Tcheremissine and the author of this book). Numerous improvements to this conservative method have been made and the direct numerical approach can now be applied for solving three-dimensional problems, at least for simple gases. In fact, one can consider the direct method (or some of the direct numerical schemes) as a theoretical instrument for studying complex nonequilibrium flows of a gas on the basis of the Boltzmann equation. Moreover, there are comparisons with available experimental data and predictions for desirable future experiments to verify the new theoretical results. We now describe the main features of the monograph. The book can be divided approximately into two parts: the first one (Chapters 1-7) is devoted mainly to a description of the methods, and the second one (Chapters 8-14) is devoted to a study of different problems on the basis of the schemes developed in Part 1. Chapter 1 considers the mathematical apparatus of the Boltzmann equation (without derivation). Well-known formulae contained in standard monographs and handbooks are not reproduced. The main facts concerned with the peculiarities of the physical and mathematical model of the Boltzmann kinetic equation are presented. The necessary mathematical formulae for solving the Boltzmann equation are described briefly in this chapter. The different forms of the collision integrals are presented. The peculiarities of the kinetic equations are emphasized. Chapter 2 gives a survey of the mathematical approaches (mainly concerning the numerical methods) for solving the Boltzmann equation. Of special interest is a comparison of the direct numerical and the direct simulation methods which are treated as schemes for approximating the Boltzmann equation itself. Chapter 3 presents the main features of all direct numerical approaches. These methods include discretization in velocity and physical spaces, the evaluation of the collision integrals by means of different procedures, finitedifference or iterative schemes, and the choice of a conservative algorithm. Chapter 4 is concerned with the deterministic (regular) method of integration developed by the author. In this approach the discrete velocity technique intrinsic to all direct methods is naturally combined with regular integration using the properties of the collision integrals. Finally, this procedure (in fact for a piecewise constant approximation in velocity space) results in a simple numerical scheme.

11 PREFACE xi The important mathematical technique is presented in Chapter 5, where a construction of the conservative splitting method is described. In contrast to most of the other conservative schemes providing conservation laws for the collision integral, this conservative approach obtains the exact approximation of the first five moment equations on a step of time t::..t. Peculiarities and advantages of this approach are discussed here. Note that almost all solutions for the Boltzmann equation described in the book are constructed on the basis of the conservative splitting scheme. Chapter 6 describes the development of direct numerical schemes for parallel computing. Chapter 7 deals with the possible application of the kinetic numerical schemes to study flows at low Knudsen numbers (for weakly rarefied gas). The asymptotic case of approximation of the Euler and the Navier-Stokes equations is also considered. Simple spatially uniform relaxation problems are test cases; on the other hand, three-dimensional relaxation is the main part of the splitting method. Solutions of such problems are covered in Chapter 8. The so-called nonuniform relaxation problem is the generalization of the known uniform relaxation problem for spatial processes. But this new problem for the Boltzmann equation considered in Chapter 9 is a model of an open system, where the methods of the kinetic theory can demonstrate their peculiarities and effectiveness for describing the nonequilibrium states. Interesting physical relationships are observed; in particular, the analytical and numerical solutions in some cases provide non monotonous spatial profiles of entropy in a such structure. Chapter 10 deals with simple, classical, one-dimensional problems which can be treated as tests; furthermore, problems of gas flows in the gravitational field are considered. In Chapters 11, 12 and 13 complex two- and three-dimensional flows are studied on the basis of the conservative splitting method. The nonstationary process of reflection of the shock wave from a wedge is investigated. Some inner flows with boundary conditions corresponding to cryogenic panels are considered. Particular attention focuses on solving several problems for supersonic underexpanded free jet flows. Chapter 12 discusses notions related to the description of instabilities (and turbulence) in gas flows in terms of the kinetic theory is discussed. The first results on this issue are presented, where unstable solutions for free jet flows were obtained by means of the conservative splitting method. Chapter 14 presents special problems for high Mach number and their solutions. Flows for very high temperatures and special relaxation models treated as the approximation of the kinetic equation are studied. Scattering the impulse beams and related questions are considered as well, and interest is demonstrated in the study of a simple model of two-component scattering where a travelling wave solution is constructed.

12 xii PREFACE Acknowledgements My great acknowledgements are to my colleagues in the Computing Center of the Russian Academy of Sciences: Academician V.V.Rumyantsev (a head of the department of mechanics), Prof. F.G.Tcheremissine (with whom the main part of the work was discussed and made), Prof. E.M.Schakhov (his influence and discussions were also considerable). I also thank Prof. V.A.Rykov, Dr. E.M.Limar, Dr. A.M.Bishaev and Dr. I.N.Larina for some fruitful discussions as well. Especially I thank Dr.A.I.Derzhavina for a great help in translation of the manuscript. I acknowledge some foreign colleagues for their criticism of the work in numerous symposia and seminars; I especially thank Prof. A.Frezzotti (Politecnico, Milano) and Prof. A.Palzcewskii and Prof. W.Walus (Warsaw University). The author also thank the editors at Kluwer Academic Publishers for their cooperation during the preparation of the monograph.

13 INTRODUCTION There are two main issues which arise when we try to study a mathematical and physical subject as complex as the Boltzmann equation. The first question is: can we solve it? The second one is: need we solve it? The first issue is more of a mathematical problem than the second. The second issue is concerned essentially with the physical field. Nevertheless, mathematical aspects stimulate an interest in the second question because this nonlinear equation is a very intriguing object for investigation. On the other hand, attempts to obtain solutions of this equation are stimulated by the problems originating in actual nonequilibrium nonlinear physical problems. Methods of search for particular solutions and general approaches for solving the Boltzmann equation have been developed slowly in course of a long time. The situation changed in the fifties due to aerodynamic requirements for high altitude vehicles and vacuum technology requirements. Furthermore, aspects of 'pure science' related to computational physics and mechanics resulted in the construction of effective methods for solving the Boltzmann equation. Mathematical methods for this very complicated equation were progressing concurrently with the development of computer power. A method for the description of gas states was found by Maxwell and Boltzmann more than one hundred years ago, which was thus when the foundation of the Boltzmann equation was made. The importance of solving the Boltzmann equation has slowly became evident to investigators owing to both the complexity of this equation and the fact that fields for its possible applications were unknown. In nineteenth century the entropy principle has been proved on the basis of the Boltzmann equation, and, in fact, the first and the second postulates of thermodynamics were combined to a single theoretical scheme (taking into account the statistical character of the theory). The further development of solving the Boltzmann equation has been concerned with the establishment of a connection of this equation with a solution by means of the equation for continuum media. The wellknown approach of Hilbert, Chapman and Enskog led to a way to obtain transfer properties, in particular transport coefficients, on the basis of the Boltzmann equation. In so doing, specific kinetic effects as thermodiffusion were discovered. However, solving the kinetic equation for a general non lin- Xlll

14 xiv INTRODUCTION ear case was a very heavy problem. The situation changed only in the 50s and 60s when computers and numerical methods advanced sufficiently to consider interesting problems. These attempts, of course, were stimulated by technical and technological requirements. Needs related, in particular, to space and aircraft research and to the development of a vacuum technology influenced the scientists mind concerning the import;tnce of new methods in kinetic theory. The Chapman-Enskog method proved the validity of the Navier-Stokes equations as a limit at small Knudsen numbers. But there were some restrictions on macroscopic equations in considering rarefied gas regimes. The attempts to construct rarefied gas dynamics equations similar to the macroscopic equations of hydrodynamics without using the multidimensional phase space were unsuccessful. The necessity of studying the Boltzmann kinetic equation itself (or the other kinetic equations) was obvious. It is well known that the Boltzmann equation was the starting point for constructing numerous kinetic equations in many fields of physics. Examples of such equations can easily be found: electron transfer in plasma, transport neutrons in nuclear reactors, and so on. The idea of describing processes on a scale of the order of the relaxation scales of time and space have been realized, in particular, in formulations of the relaxation equation of electron transfer in solids. On the other hand, physicists very often emphasize the restricted role of the classical Boltzmann equation because of the unknown character of the intermolecular counteractions, the validity of the kinetic equations only in the gaseous media, etc. However, the nonlinear model of the Boltzmann equation possesses the important essence of the original and physically realistic equation, so it is possible not only to consider the flows for simple media but to formulate new problems due to the ability of these equations to describe nonequilibrium states. One can expect that the relationships in simple solutions of the Boltzmann equations will also be valid for more complex physical situations. This book presents the results of development and sophistication of numerical methods for solving the classical nonlinear Boltzmann equation. The potentials of the methods are demonstrated through different examples of solutions of relaxation processes, of one-, two- and three-dimensional rarefied gas-dynamics problems with variations of flow parameters and internal parameters, of the algorithms as well. So, to date, the answer to the first question is positive (although great efforts are need to obtain solutions for physically realistic situations). This statement is supported by the solutions for different problems. The answer to the second question is also positive because the Boltzmann equation gives us a more accurate description of physical reality than some approximations. The calculations for several problems and comparison with experimental data show that the

15 INTRODUCTION xv kinetic Boltzmann equation adequately describs the processes. The Boltzmann equation could be a basic model for the study of unknown nonlinear, nonequilibrium processes (see, e.g., Chapters 9 and 12). The kinetic equation could be the origin for obtaining simple models and solutions for the description of very complicated physical situations, such as gas motion with a very large velocity (in respect to the thermal speed) considered in Chapter 14. We tried to construct such a method that would reflect the general physical properties of the Boltzmann equation in the numerical algorithm. In fact, the physical discrete model of the kinetic equation was obtained. Priority features of the numerical scheme were the monotonicity, the explicit character of the scheme, the positive solution of the distribution function, but not an order of the accuracy of the approximation. In evaluating the collision integrals the statistical peculiarities of collisions of particles in a gas were taken into account. It was obvious that for the given level of computer power it was impossible to solve problems with detailed grids. Nevertheless, an additional simplification of the kinetic equation was not included; thus the property of approximation of the original Boltzmann equation by the numerical method was provided. One of the main properties of the Boltzmann equation is its conservativity. The intrinsic feature of the kinetic equation is that the conservativity can be treated in different forms. The conservativity in the hydrodynamic level possesses some advantages due to some factors of 'freedom' (in particular connected with the parameter of a time step), which leads to a flexible algorithm that can be realized in several numerical schemes. This conservative algorithm was constructed in terms of the splitting procedure by physical processes. On the basis of the conservative splitting method the relaxation and one-dimensional problems (with accuracy controlled by decreasing the scheme steps), and examples of solutions of two- and threedimensional problems were solved. In so doing all the solutions were obtained for computers with an approximately fixed power level, i.e. progress was obtained by the improvement of the approach itself. On the other hand, there are powerful computers with parallel processors, and the property of the direct integration algorithms, which tend themselves well to parallel computing, is very important. The fruitful idea of direct statistical simulation was expressed in a study of a computer of the trajectories of mathematical points (in phase space) named particles. Now, the direct simulation approaches have obvious advantages in comparison with the other Monte Carlo imitation methods. Recently, direct simulation Monte Carlo methods have been developed up to the engineering level and can be applied for studying gas flows with complex physical-chemical processes. But there are some approximations of models

16 xvi INTRODUCTION in the description of reactions where a requirement of high accuracy cannot be given, and simulation methods are adequate for such problems (being in fact a modeling approach for the description of chemical processes). For simple gases the direct integration methods can be an adequate mathematical tool apparatus because, after the appearance of supercomputers, the test solutions for the Boltzmann equation can be obtained and can be applied for evaluating the accuracy of the simulation computations. Obtaining solutions to the Boltzmann equation provides the basis of mathematical studies in rarefied gas dynamics. However, the direct integration methods are now not only an approach to the exact solution of simple problems but as an alternative to direct simulation methods for studying complicated nonequilibrium flows. The Boltzmann equation as a physical model has important properties. Grad noted in [1]: "... the basic length scale in the Boltzmann equation is the mean-free-path and the the basic time scale is the molecular collision time. This contrasts sharply with almost all other equations of mathematical physics which exhibit the irreversibility of fluid behavior over distances which must be large compared to the mean-free-path and times which must be large com pared to the mean collision time. Another way of stating this difference is that the usual thermodynamics of irreversible processes, for example, is concerned with small (linear) deviations from equilibrium while the Boltzmann equation permits large (nonlinear) deviations. We must be careful to distinguish the nonlinearity of the equations of fluid dynamics from the linear irreversibility mechanism (e.g., heat flow proportional to temperature gradient)." It is possible that the methods of the direct numerical analysis of the Boltzmann equation will be found to be an appropriate tool to study these processes.

17 References 1. Grad H. (1958) Principles of the kinetic theory of gases, Handbtlch der Physics, Springer, Berlin, no. XII, pp xvii