Download or read book A Numerical Solution of the Boltzmann Equation for a Two dimensional Gas written by Frank Robben and published by . This book was released on 1967 with total page 42 pages. Available in PDF, EPUB and Kindle. Book excerpt: An attempt to solve the Boltzmann equation numerically was carried out by means of a more direct method than the Monte-Carlo technique. The initial value problem of the energy relaxation of a two-dimensional gas of hard spheres was studied, using direct numerical integration techniques. A finite difference approximation to the Boltzmann equation was used. While the results are qualitatively correct, conservation of energy and number were violated to the extent of about 4% per collision. On comparing the detailed results to that predicted by a Krook model, quantitative differences are noted. It is believed that the inaccuracies of the solutions obtained are due to a singularity in the collision integrals. (Author).
Download or read book Numerical Solutions to the Two Dimensional Boltzmann Equation written by Christopher G. Smithtro and published by . This book was released on 1999-03-01 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: A strict kinetic, two-dimensional model of the electron kinetics within a glow discharge positive column is developed. The problem is solved in cylindrical geometry using the standard two-term Legendre expansion of the electron velocity distribution function. The model establishes a steady state solution, such that the net ionization rate is exactly balanced by the wall loss. In addition to a thorough analytic development, we present the numerical techniques used to solve the resulting elliptic partial differential equation, including an efficient method to treat sparse banded matrices. The model is validated against published results, local and nonlocal kinetic approximations, and a previous Monte Carlo treatment. Having created a working model, we conduct an investigation into current flow within the solution area of a neon column, made possible by this 2-D treatment. Furthermore, we investigate the range of applicability of the earlier local and nonlocal kinetic approximations and finally present a short discussion on the effect different forms of wall loss have on the overall distribution function.
Download or read book Solutions of the Boltzmann Equation in Two Space Dimensions written by E. L. Secrest and published by . This book was released on 1959 with total page 74 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Numerical Solution of Boltzmann s Equation written by and published by . This book was released on 1976 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The numerical solution of Boltzmann's equation is considered for a gas model consisting of rigid spheres by means of Hilbert's expansion. If only the first two terms of the expansion are retained, Boltzmann's equation reduces to the Boltzmann-Hilbert integral equation. Successive terms in the Hilbert expansion are obtained by solving the same integral equation with a different source term. The Boltzmann-Hilbert integral equation is solved by a new very fast numerical method. The success of the method rests upon the simultaneous use of four judiciously chosen expansions; Hilbert's expansion for the distribution function, another expansion of the distribution function in terms of Hermite polynomials, the expansion of the kernel in terms of the eigenvalues and eigenfunctions of the Hilbert operator, and an expansion involved in solving a system of linear equations through a singular value decomposition. The numerical method is applied to the study of the shock structure in one space dimension. Numerical results are presented for Mach numbers of 1.1 and 1.6. 94 refs, 7 tables, 1 fig.
Download or read book NUMERICAL SOLUTION OF THE BOLTZMANN EQUATION written by ALEXANDRE JOEL. CHORIN and published by . This book was released on 2018 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Numerical Solutions of the Two Dimensional Confined Non linear Poisson Boltzmann Equation written by Mark Ospeck and published by . This book was released on 1996 with total page 68 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book A Numerical Solution of Boltzmann s Equation written by Gary Andrew Sod and published by . This book was released on 1976 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows written by V.V. Aristov and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is concerned with the methods of solving the nonlinear Boltz mann 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 nonequi librium, 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 in vestigated for the first time. The structure and the contents of the present book have some com mon 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.
Download or read book Direct Numerical Solution Of The Boltzmann Equation written by and published by . This book was released on 2005 with total page 10 pages. Available in PDF, EPUB and Kindle. Book excerpt: Progress in computer hardware and improvement of numerical methods made solution of the Boltzmann equation for rather complex gas dynamic problems real. The method developed by the author is based on a projection technique for evaluation of the collision operator. The computed collision integral is conservative by density, impulse, and energy, and became equal to zero when the solution has a form of the Maxwellian distribution. The later feature sharply increases its efficiency, especially for the near equilibrium flows. The method is extended on a mixture of gases and the gases with internal degrees of freedom, where it can incorporate real physical parameters of molecular potential and of internal energy spectrum. Examples of computations for a range of Mach and Knudsen numbers are presented.
Download or read book Lecture Notes on the Discretization of the Boltzmann Equation written by N. Bellomo and published by World Scientific. This book was released on 2003 with total page 317 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents contributions on the following topics: discretization methods in the velocity and space, analysis of the conservation properties, asymptotic convergence to the continuous equation when the number of velocities tends to infinity, and application of discrete models. It consists of ten chapters. Each chapter is written by applied mathematicians who have been active in the field, and whose scientific contributions are well recognized by the scientific community.
Download or read book Lattice Gas Methods For Partial Differential Equations written by Gary Doolen and published by CRC Press. This book was released on 2019-03-01 with total page 584 pages. Available in PDF, EPUB and Kindle. Book excerpt: Although the idea of using discrete methods for modeling partial differential equations occurred very early, the actual statement that cellular automata techniques can approximate the solutions of hydrodynamic partial differential equations was first discovered by Frisch, Hasslacher, and Pomeau. Their description of the derivation, which assumes the validity of the Boltzmann equation, appeared in the Physical Review Letters in April 1986. It is the intent of this book to provide some overview of the directions that lattice gas research has taken from 1986 to early 1989.
Download or read book An Introduction to the Boltzmann Equation and Transport Processes in Gases written by Gilberto M. Kremer and published by Springer Science & Business Media. This book was released on 2010-08-18 with total page 313 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book covers classical kinetic theory of gases, presenting basic principles in a self-contained framework and from a more rigorous approach based on the Boltzmann equation. Uses methods in kinetic theory for determining the transport coefficients of gases.
Download or read book Molecular Gas Dynamics and the Direct Simulation of Gas Flows written by G. A. Bird and published by Oxford University Press on Demand. This book was released on 1994 with total page 458 pages. Available in PDF, EPUB and Kindle. Book excerpt: This second edition of a highly regarded text covers all the recent research developments in gas dynamics including the direct simulation Monte Carlo method (DSMC).
Download or read book Scientific and Technical Aerospace Reports written by and published by . This book was released on 1991 with total page 1460 pages. Available in PDF, EPUB and Kindle. Book excerpt: Lists citations with abstracts for aerospace related reports obtained from world wide sources and announces documents that have recently been entered into the NASA Scientific and Technical Information Database.
Download or read book Numerical Solution of the Boltzmann Equation written by Alexandre Joel Chorin and published by . This book was released on 1971 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Direct Numerical Solution of the Three dimensional Generalized Boltzmann Equation for Hypersonic Non equilibrium Flows written by Christopher Daniel Wilson and published by . This book was released on 2010 with total page 374 pages. Available in PDF, EPUB and Kindle. Book excerpt: The development and applications of a computer code for solving the three-dimensional generalized Boltzmann equation (GBE) using a direct numerical method are presented. The Boltzmann solver of Professor Felix G. Tcheremissine of the Russian Academy of Science serves as the foundation for the development effort. This jet code includes only the translational and rotational energy states of a diatomic gas and has been applied to simulate the flow field of a jet issuing into a vacuum. In this dissertation, this code is employed to accomplish three distinct developmental steps. First, the code is extended for calculating hypersonic shock waves in an inert mixture of gases. For this purpose, the GBE is formulated in an impulse space (instead of the conventional velocity space). The computational methodology is then applied to a binary mixture of gases, which requires the simultaneous solution of four GBE's. Simulations are performed using a gas mixture including both diatomic and monatomic gases in proportions similar to that in air. The solutions are validated against existing hypersonic shock wave experimental data for a single specie gas (nitrogen) in rotational-translational non-equilibrium and available computational data for a binary mixture of monatomic gases. Simulations are then performed for an inert binary mixture of monatomic and diatomic gases in translational non-equilibrium for various concentrations. The effect of mass ratio and molecular diameter ratio of the gases on the structure of the shock is also investigated. Second, boundary conditions necessary for accurately simulating the flows around immersed bodies are developed and evaluated. This research on boundary conditions constitutes a significant advancement beyond the adsorptive boundary condition used in the original Boltzmann solver of Tcheremissine. Five types of boundary conditions at the solid boundary are investigated: (a) the standard adsorptive boundary condition, (b) the specular reflection boundary condition, (c) the diffuse reflection boundary condition, (d) the Maxwellian boundary condition, and (e) the adsorptive Maxwellian boundary condition with different values for the accommodation coefficient. These boundary conditions are tested for hypersonic flow past a flat plate to evaluate their accuracy. Third, the original Boltzmann code, hard-coded for solving the flow field of a jet issuing into a vacuum, is modified to enable simulations of rarefied flows around immersed bodies. The computations are performed for three benchmark geometries, extensively used in the literature for Navier-Stokes simulations, at various hypersonic inflow conditions for flow of a diatomic gas (N2) in rotational-translational non-equilibrium. The three geometries used in the simulations are an axisymmetric blunt body, an axisymmetric bicone, and an axisymmetric hollow-flared-cylinder. Initially, a relatively coarse Cartesian grid was employed in the three-dimensional simulations because of the limitations of physical memory on the available computers. As a result, a shared memory parallel computing platform was developed and built for the sole purpose of being able to perform the fine grid solutions. Consequently, refined grid solutions were generated on the parallel computing platform. For this purpose, the code was parallelized and the parallelization issues for a Boltzmann type solver were addressed. A comparison between the coarse and refined grid solutions is presented to show the influence of grid density on solution accuracy. In light of these results, the issues of accuracy and efficiency of the three-dimensional Boltzmann solver are addressed.
Download or read book An Introduction to the Theory of the Boltzmann Equation written by Stewart Harris and published by Courier Corporation. This book was released on 2012-12-27 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introductory graduate-level text emphasizes physical aspects of the theory of Boltzmann's equation in a detailed presentation that doubles as a practical resource for professionals. 1971 edition.
