Download or read book Direct Numerical Solution of Nonlinear Model Boltzmann Equations written by Ryan Hulguin and published by . This book was released on 2008 with total page 108 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 Computational Fluid Dynamics written by Dieter Leutloff and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computational methods and modelling is of growing importance in fundamental science as well as in applications in industry and in environmental research. In this topical volume the readers find important contributions in the field of turbulent boundary layers, the Tsunami problem, group invariant solution of hydrodynamic equations, non-linear waves, modelling of the problem of evaporation-condensation, the exact solution of discrete models of the Boltzmann equation etc. The book addresses researchers and engineers both in the mechanical sciences and in scientific computing.
Download or read book Asymptotic Equivalence of the Navier Stokes and Nonlinear Boltzmann Equations written by Harold Grad and published by . This book was released on 1964 with total page 66 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Modern Solution Methods for Nonlinear Multidimensional Problems written by O.M. Belotserkovskii and published by . This book was released on 2000 with total page 446 pages. Available in PDF, EPUB and Kindle. Book excerpt: Preparing the English translation the author has completely revised the Russian edition of 1994. The new versions and generalizations of computational algorithms which were intensively developed by the method of splitting with respect to physical processes are described.
Download or read book Numerical Solution for Nonlinear Poisson Boltzmann Equations and Numerical Simulations for Spike Dynamics written by Zhonghua Qiao and published by . This book was released on 2006 with total page 101 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the second part of this thesis, numerical schemes based on adaptive grid methods are designed for simulating the spike dynamics governed by the two-dimensional Gierer-Meinhardt Model. The moving mesh finite element method is employed in the numerical simulations. Numerically, we verify some stability properties of the spike dynamics following some recent asymptotic analysis results. Moreover, some spike oscillation and splitting phenomenon observed in one-dimensional analysis and computations are discussed using our two-dimensional computational results. On the computational side, it is observed that we can obtain accurate numerical results with much less grid resolution if the moving mesh methods are employed.
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 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 Lecture Notes on the Mathematical Theory of the Boltzmann Equation written by N. Bellomo and published by World Scientific. This book was released on 1995 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a collection of four lectures on some mathematical aspects related to the nonlinear Boltzmann equation. The following topics are dealt with: derivation of kinetic equations, qualitative analysis of the initial value problem, singular perturbation analysis towards the hydrodynamic limit and computational methods towards the solution of problems in fluid dynamics.
Download or read book Deterministic Numerical Simulation of the Boltzmann and Kinetic Model Equations for Classical and Quantum Dilute Gases written by Lei Wu and published by . This book was released on 2013 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the areas of low-density aerodynamics, vacuum industry, and micro-electromechanical systems, the Navier-Stokes-Fourier equations fail to describe the gas dynamics when the molecular mean free path is not negligible compared to the characteristic flow length. Instead, the Boltzmann equation is used to account for the non-continuum nature of the rarefied gas. Although many efforts have been made to derive the macroscopic equations from the Boltzmann equation, the numerical simulation of the Boltzmann equation is indispensable in the study of moderately and highly rarefied gas. We aim to develop an accurate and efficient deterministic numerical method to solve the Boltzmann equation. The fast spectral method [1], originally developed by Mouhot and Pareschi for the numerical approximation of the collision operator, is extended to deal with other collision kernels, such as those corresponding to the soft, Lennard-Jones, and rigid attracting potentials. The accuracy of the fast spectral method is checked by comparing our numerical results with the exact Bobylev-Krook-Wu solutions of the space-homogeneous Boltzmann equation for a gas of Maxwell molecules. It is found that the accuracy is improved by replacing the trapezoidal rule with Gauss-Legendre quadrature in the calculation of the kernel mode, and the conservation of momentum and energy are ensured by the Lagrangian multiplier method without loss of spectral accuracy. The relax-to-equilibrium processes of different collision kernels with the same value of shear viscosity are then compared and the use of special collision kernels is justified. An iteration scheme, where the numerical errors decay exponentially, is employed to obtain stationary solutions of the space-inhomogeneous Boltzmann equation. Sever classical benchmarking problems (the normal shock wave, and the planar Fourier/Couette/force-driven Poiseuille flows) are investigated. For normal shock waves, our numerical results are compared with the finite-difference solution of the Boltzmann equation for hard sphere molecules, the experimental data, and the molecular dynamics simulation of argon using the realistic Lennard-Jones potential. For the planar Fourier/Couette/force-driven Poiseuille flows, our results are compared with the Direct Simulation Monte Carlo method. Excellent agreements are observed in all test cases. The fast spectral method is then applied to the linearised Boltzmann equation. With appropriate velocity discretization, the classical Poiseuille and thermal creep flows are solved up to Kn 106, where the accuracy in the mass and heat flow rates is comparable to those from the finite-difference method and the efficiency is much better than the low-noise Direct Simulation Monte Carlo method. The fast spectral method is also extended to solve the Boltzmann equation for binary gas mixtures, both in the framework of classical and quantum mechanics. With the accurate numerical solution provided by the fast spectral method, we check the accuracy of kinetic model equations to find out at what flow regime can the complicated Boltzmann collision kernel be replaced by the simple kinetic ones. We also solve the collective oscillation of quantum gas confined in external trap and compare the numerical solutions with the experimental data, indicating the applicability of quantum kinetic model.
Download or read book The Boltzmann Equation and Its Applications written by Carlo Cercignani and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 467 pages. Available in PDF, EPUB and Kindle. Book excerpt: Statistical mechanics may be naturally divided into two branches, one dealing with equilibrium systems, the other with nonequilibrium systems. The equilibrium properties of macroscopic systems are defined in principle by suitable averages in well-defined Gibbs's ensembles. This provides a frame work for both qualitative understanding and quantitative approximations to equilibrium behaviour. Nonequilibrium phenomena are much less understood at the present time. A notable exception is offered by the case of dilute gases. Here a basic equation was established by Ludwig Boltzmann in 1872. The Boltzmann equation still forms the basis for the kinetic theory of gases and has proved fruitful not only for a study of the classical gases Boltzmann had in mind but also, properly generalized, for studying electron transport in solids and plasmas, neutron transport in nuclear reactors, phonon transport in superfluids, and radiative transfer in planetary and stellar atmospheres. Research in both the new fields and the old one has undergone a considerable advance in the last thirty years.
Download or read book Rarefied Gas Dynamics written by Lei Wu and published by Springer Nature. This book was released on 2022-09-09 with total page 293 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book highlights a comprehensive description of the numerical methods in rarefied gas dynamics, which has strong applications ranging from space vehicle re-entry, micro-electromechanical systems, to shale gas extraction. The book consists of five major parts: The fast spectral method to solve the Boltzmann collision operator for dilute monatomic gas and the Enskog collision operator for dense granular gas; The general synthetic iterative scheme to solve the kinetic equations with the properties of fast convergence and asymptotic preserving; The kinetic modeling of monatomic and molecular gases, and the extraction of critical gas parameters from the experiment of Rayleigh-Brillouin scattering; The assessment of the fluid-dynamics equations derived from the Boltzmann equation and typical kinetic gas-surface boundary conditions; The applications of the fast spectral method and general synthetic iterative scheme to reveal the dynamics in some canonical rarefied gas flows. The book is suitable for postgraduates and researchers interested in rarefied gas dynamics and provides many numerical codes for them to begin with.
Download or read book Lecture Notes on the Discretization of the Boltzmann Equation written by Nicola Bellomo and published by World Scientific. This book was released on 2003 with total page 320 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. Contents: From the Boltzmann Equation to Discretized Kinetic Models (N Bellomo & R Gatignol); Discrete Velocity Models for Gas Mixtures (C Cercignani); Discrete Velocity Models with Multiple Collisions (R Gatignol); Discretization of the Boltzmann Equation and the Semicontinuous Model (L Preziosi & L Rondoni); Semi-continuous Extended Kinetic Theory (W Koller); Steady Kinetic Boundary Value Problems (H Babovsky et al.); Computational Methods and Fast Algorithms for the Boltzmann Equation (L Pareschi); Discrete Velocity Models and Dynamical Systems (A Bobylev & N Bernhoff); Numerical Method for the Compton Scattering Operator (C Buet & S Cordier); Discrete Models of the Boltzmann Equation in Quantum Optics and Arbitrary Partition of the Velocity Space (F Schrrer). Readership: Higher level postgraduates in applied mathematics.
Download or read book Spectral Methods in Chemistry and Physics written by Bernard Shizgal and published by Springer. This book was released on 2015-01-07 with total page 431 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a pedagogical presentation of the application of spectral and pseudospectral methods to kinetic theory and quantum mechanics. There are additional applications to astrophysics, engineering, biology and many other fields. The main objective of this book is to provide the basic concepts to enable the use of spectral and pseudospectral methods to solve problems in diverse fields of interest and to a wide audience. While spectral methods are generally based on Fourier Series or Chebychev polynomials, non-classical polynomials and associated quadratures are used for many of the applications presented in the book. Fourier series methods are summarized with a discussion of the resolution of the Gibbs phenomenon. Classical and non-classical quadratures are used for the evaluation of integrals in reaction dynamics including nuclear fusion, radial integrals in density functional theory, in elastic scattering theory and other applications. The subject matter includes the calculation of transport coefficients in gases and other gas dynamical problems based on spectral and pseudospectral solutions of the Boltzmann equation. Radiative transfer in astrophysics and atmospheric science, and applications to space physics are discussed. The relaxation of initial non-equilibrium distributions to equilibrium for several different systems is studied with the Boltzmann and Fokker-Planck equations. The eigenvalue spectra of the linear operators in the Boltzmann, Fokker-Planck and Schrödinger equations are studied with spectral and pseudospectral methods based on non-classical orthogonal polynomials. The numerical methods referred to as the Discrete Ordinate Method, Differential Quadrature, the Quadrature Discretization Method, the Discrete Variable Representation, the Lagrange Mesh Method, and others are discussed and compared. MATLAB codes are provided for most of the numerical results reported in the book - see Link under 'Additional Information' on the the right-hand column.
Download or read book Scientific and Technical Aerospace Reports written by and published by . This book was released on 1977 with total page 1006 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Modeling and Computational Methods for Kinetic Equations written by Pierre Degond and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt: In recent years kinetic theory has developed in many areas of the physical sciences and engineering, and has extended the borders of its traditional fields of application. This monograph is a self-contained presentation of such recently developed aspects of kinetic theory, as well as a comprehensive account of the fundamentals of the theory. Emphasizing modeling techniques and numerical methods, the book provides a unified treatment of kinetic equations not found in more focused works. Specific applications presented include plasma kinetic models, traffic flow models, granular media models, and coagulation-fragmentation problems. The work may be used for self-study, as a reference text, or in graduate-level courses in kinetic theory and its applications.
Download or read book Applied mechanics reviews written by and published by . This book was released on 1948 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt:
