An exact solution to the two-particle Boltzmann equation system for Maxwell gases is obtained with use of Bobylev approach.The relationship between the exact solution and the self-similar solution of the boltzmann equ...An exact solution to the two-particle Boltzmann equation system for Maxwell gases is obtained with use of Bobylev approach.The relationship between the exact solution and the self-similar solution of the boltzmann equation is also given.展开更多
In this paper, a novel model is proposed to investigate the neutron transport in scattering and absorbing medium. This solution to the linear Boltzmann equation is expanded from the idea of lattice Boltzmann method(LB...In this paper, a novel model is proposed to investigate the neutron transport in scattering and absorbing medium. This solution to the linear Boltzmann equation is expanded from the idea of lattice Boltzmann method(LBM) with the collision and streaming process. The theoretical derivation of lattice Boltzmann model for transient neutron transport problem is proposed for the first time.The fully implicit backward difference scheme is used to ensure the numerical stability, and relaxation time and equilibrium particle distribution function are obtained. To validate the new lattice Boltzmann model, the LBM formulation is tested for a homogenous media with different sources, and both transient and steady-state LBM results get a good agreement with the benchmark solutions.展开更多
Many macroscopic equations are proposed to describe the rarefied gas dynamics beyond the Navier-Stokes level,either from the mesoscopic Boltzmann equation or some physical arguments,including(i)Burnett,Woods,super-Bur...Many macroscopic equations are proposed to describe the rarefied gas dynamics beyond the Navier-Stokes level,either from the mesoscopic Boltzmann equation or some physical arguments,including(i)Burnett,Woods,super-Burnett,augmented Burnett equations derived from the Chapman-Enskog expansion of the Boltzmann equation,(ii)Grad 13,regularized 13/26 moment equations,rational extended thermodynamics equations,and generalized hydrodynamic equations,where the velocity distribution function is expressed in terms of low-order moments and Hermite polynomials,and(iii)bi-velocity equations and“thermo-mechanically consistent"Burnett equations based on the argument of“volume diffusion”.This paper is dedicated to assess the accuracy of these macroscopic equations.We first consider the RayleighBrillouin scattering,where light is scattered by the density fluctuation in gas.In this specific problem macroscopic equations can be linearized and solutions can always be obtained,no matter whether they are stable or not.Moreover,the accuracy assessment is not contaminated by the gas-wall boundary condition in this periodic problem.Rayleigh-Brillouin spectra of the scattered light are calculated by solving the linearized macroscopic equations and compared to those from the linearized Boltzmann equation.We find that(i)the accuracy of Chapman-Enskog expansion does not always increase with the order of expansion,(ii)for the moment method,the more moments are included,the more accurate the results are,and(iii)macroscopic equations based on“volume diffusion"do not work well even when the Knudsen number is very small.Therefore,among about a dozen tested equations,the regularized 26 moment equations are the most accurate.However,for moderate and highly rarefied gas flows,huge number of moments should be included,as the convergence to true solutions is rather slow.The same conclusion is drawn from the problem of sound propagation between the transducer and receiver.This slow convergence of moment equations is due to the incapability of Hermite polynomials in the capturing of large discontinuities and rapid variations of the velocity distribution function.This study sheds some light on how to choose/develop macroscopic equations for rarefied gas dynamics.展开更多
As is known,the numerical stiffness arising from the small mean free path is one of the main difficulties in the kinetic equations.In this paper,we derive both the split and the unsplit schemes for the linear semicond...As is known,the numerical stiffness arising from the small mean free path is one of the main difficulties in the kinetic equations.In this paper,we derive both the split and the unsplit schemes for the linear semiconductor Boltzmann equation with a diffusive scaling.In the two schemes,the anisotropic collision operator is realized by the“BGK”-penalty method,which is proposed by Filbet and Jin[F.Filbet and S.Jin,J.Comp.Phys.229(20),7625-7648,2010]for the kinetic equations and the related problems having stiff sources.According to the numerical results,both of the schemes are shown to be uniformly convergent and asymptotic-preserving.Besides,numerical evidences suggest that the unsplit scheme has a better numerical stability than the split scheme.展开更多
The derivation of the quantum lattice Boltzmann model is reviewed with special emphasis on recent developments of the model,namely,the extension to a multi-dimensional formulation and the application to the computatio...The derivation of the quantum lattice Boltzmann model is reviewed with special emphasis on recent developments of the model,namely,the extension to a multi-dimensional formulation and the application to the computation of the ground state of the Gross-Pitaevskii equation(GPE).Numerical results for the linear and nonlinear Schr odinger equation and for the ground state solution of the GPE are also presented and validated against analytical results or other classical schemes such as Crank-Nicholson.展开更多
The linear barotropic vorticity equation describing wind-driven oceancirculation is considered as a convection-diffusion equation that can be numerically solved bylattice Boltzmann method. Numerical experiments are ca...The linear barotropic vorticity equation describing wind-driven oceancirculation is considered as a convection-diffusion equation that can be numerically solved bylattice Boltzmann method. Numerical experiments are carried out to examine the validity of the modelfor the wind-driven circulation. When horizontal viscosity is constant and spatially uniform, allnumerical solutions for different parameters approach analytical solutions well. The spatiallyvarying horizontal viscosity is also included in this model. It is shown that the variant horizontalviscosity increases the meridional transport significantly in west boundary current. By theinvestigation of numerical results, it was concluded that this model is competent for simulatingwestern boundary current.展开更多
Sound propagation and the initial value problems in gas mixtures of two components are investigated. By using the eigen theory of linearized Boltzmann equations, a model equations is formed, with the use of the Fourie...Sound propagation and the initial value problems in gas mixtures of two components are investigated. By using the eigen theory of linearized Boltzmann equations, a model equations is formed, with the use of the Fourier-Laplace transform for model equations derived, the dispersion relations for both components are obtained.展开更多
文摘An exact solution to the two-particle Boltzmann equation system for Maxwell gases is obtained with use of Bobylev approach.The relationship between the exact solution and the self-similar solution of the boltzmann equation is also given.
基金supported by the Foundation of National Key Laboratory of Reactor System Design Technology(No.HT-LW-02-2014003)the State Key Program of National Natural Science of China(No.51436009)
文摘In this paper, a novel model is proposed to investigate the neutron transport in scattering and absorbing medium. This solution to the linear Boltzmann equation is expanded from the idea of lattice Boltzmann method(LBM) with the collision and streaming process. The theoretical derivation of lattice Boltzmann model for transient neutron transport problem is proposed for the first time.The fully implicit backward difference scheme is used to ensure the numerical stability, and relaxation time and equilibrium particle distribution function are obtained. To validate the new lattice Boltzmann model, the LBM formulation is tested for a homogenous media with different sources, and both transient and steady-state LBM results get a good agreement with the benchmark solutions.
文摘Many macroscopic equations are proposed to describe the rarefied gas dynamics beyond the Navier-Stokes level,either from the mesoscopic Boltzmann equation or some physical arguments,including(i)Burnett,Woods,super-Burnett,augmented Burnett equations derived from the Chapman-Enskog expansion of the Boltzmann equation,(ii)Grad 13,regularized 13/26 moment equations,rational extended thermodynamics equations,and generalized hydrodynamic equations,where the velocity distribution function is expressed in terms of low-order moments and Hermite polynomials,and(iii)bi-velocity equations and“thermo-mechanically consistent"Burnett equations based on the argument of“volume diffusion”.This paper is dedicated to assess the accuracy of these macroscopic equations.We first consider the RayleighBrillouin scattering,where light is scattered by the density fluctuation in gas.In this specific problem macroscopic equations can be linearized and solutions can always be obtained,no matter whether they are stable or not.Moreover,the accuracy assessment is not contaminated by the gas-wall boundary condition in this periodic problem.Rayleigh-Brillouin spectra of the scattered light are calculated by solving the linearized macroscopic equations and compared to those from the linearized Boltzmann equation.We find that(i)the accuracy of Chapman-Enskog expansion does not always increase with the order of expansion,(ii)for the moment method,the more moments are included,the more accurate the results are,and(iii)macroscopic equations based on“volume diffusion"do not work well even when the Knudsen number is very small.Therefore,among about a dozen tested equations,the regularized 26 moment equations are the most accurate.However,for moderate and highly rarefied gas flows,huge number of moments should be included,as the convergence to true solutions is rather slow.The same conclusion is drawn from the problem of sound propagation between the transducer and receiver.This slow convergence of moment equations is due to the incapability of Hermite polynomials in the capturing of large discontinuities and rapid variations of the velocity distribution function.This study sheds some light on how to choose/develop macroscopic equations for rarefied gas dynamics.
文摘As is known,the numerical stiffness arising from the small mean free path is one of the main difficulties in the kinetic equations.In this paper,we derive both the split and the unsplit schemes for the linear semiconductor Boltzmann equation with a diffusive scaling.In the two schemes,the anisotropic collision operator is realized by the“BGK”-penalty method,which is proposed by Filbet and Jin[F.Filbet and S.Jin,J.Comp.Phys.229(20),7625-7648,2010]for the kinetic equations and the related problems having stiff sources.According to the numerical results,both of the schemes are shown to be uniformly convergent and asymptotic-preserving.Besides,numerical evidences suggest that the unsplit scheme has a better numerical stability than the split scheme.
文摘The derivation of the quantum lattice Boltzmann model is reviewed with special emphasis on recent developments of the model,namely,the extension to a multi-dimensional formulation and the application to the computation of the ground state of the Gross-Pitaevskii equation(GPE).Numerical results for the linear and nonlinear Schr odinger equation and for the ground state solution of the GPE are also presented and validated against analytical results or other classical schemes such as Crank-Nicholson.
文摘The linear barotropic vorticity equation describing wind-driven oceancirculation is considered as a convection-diffusion equation that can be numerically solved bylattice Boltzmann method. Numerical experiments are carried out to examine the validity of the modelfor the wind-driven circulation. When horizontal viscosity is constant and spatially uniform, allnumerical solutions for different parameters approach analytical solutions well. The spatiallyvarying horizontal viscosity is also included in this model. It is shown that the variant horizontalviscosity increases the meridional transport significantly in west boundary current. By theinvestigation of numerical results, it was concluded that this model is competent for simulatingwestern boundary current.
基金Supported by National Natural Science Foundation of China under Grant No.10861008the "211 Project" Innovative Talents Training Program of Inner Mongolia University and Grant-in-Aid for Scientific Research from Inner Mongolia University of Technology under Grant No.ZS201032
文摘Sound propagation and the initial value problems in gas mixtures of two components are investigated. By using the eigen theory of linearized Boltzmann equations, a model equations is formed, with the use of the Fourier-Laplace transform for model equations derived, the dispersion relations for both components are obtained.
