This paper presents a self-contained description on the configuration of propagator method(PM)to calculate the electron velocity distribution function(EVDF) of electron swarms in gases under DC electric and magnetic f...This paper presents a self-contained description on the configuration of propagator method(PM)to calculate the electron velocity distribution function(EVDF) of electron swarms in gases under DC electric and magnetic fields crossed at a right angle. Velocity space is divided into cells with respect to three polar coordinates v,θ and f. The number of electrons in each cell is stored in three-dimensional arrays. The changes of electron velocity due to acceleration by the electric and magnetic fields and scattering by gas molecules are treated as intercellular electron transfers on the basis of the Boltzmann equation and are represented using operators called the propagators or Green’s functions. The collision propagator, assuming isotropic scattering, is basically unchanged from conventional PMs performed under electric fields without magnetic fields. On the other hand, the acceleration propagator is customized for rotational acceleration under the action of the Lorentz force. The acceleration propagator specific to the present cell configuration is analytically derived. The mean electron energy and average electron velocity vector in a model gas and SF6 were derived from the EVDF as a demonstration of the PM under the Hall deflection and they were in a fine agreement with those obtained by Monte Carlo simulations. A strategy for fast relaxation is discussed, and extension of the PM for the EVDF under AC electric and DC/AC magnetic fields is outlined as well.展开更多
The function and physical mechanism of heat flow and the viscous stress in the velocity distribution function expanded by Maxwellian distribution are presented. With the introduction of effective temperature Tf, incoh...The function and physical mechanism of heat flow and the viscous stress in the velocity distribution function expanded by Maxwellian distribution are presented. With the introduction of effective temperature Tf, incoherent scatter spectra from plasma for electromagnetic wave in arbitrary line of sight are given. The effect of asymmetry and anisotropy provided by heat flow and the viscous stress on power spectra is discussed. Radar spectra are calculated for different cases of electric field, direction, collision frequency and temperature. The effect of heat flow and the viscous stress on inversion results is analyzed. With a large electric field, the character of non-Maxwellian must be considered.展开更多
A Lagrangian lattice Boltzmann method for solving Euler equations is proposed. The key step in formulating this method is the introduction of the displacement distribution function. The equilibrium distribution functi...A Lagrangian lattice Boltzmann method for solving Euler equations is proposed. The key step in formulating this method is the introduction of the displacement distribution function. The equilibrium distribution function consists of macroscopic Lagrangian variables at time steps n and n + 1. It is different from the standard lattice Boltzmann method. In this method the element, instead of each particle, is required to satisfy the basic law. The element is considered as one large particle, which results in simpler version than the corresponding Eulerian one, because the advection term disappears here. Our numerical examples successfully reproduce the classical results.展开更多
This paper investigates the convergence proof of the Direct Simulation Monte Carlo(DSMC) method and the Gas-Kinetic Unified Algorithm in simulating the Boltzmann equation.It can be shown that the particle velocity dis...This paper investigates the convergence proof of the Direct Simulation Monte Carlo(DSMC) method and the Gas-Kinetic Unified Algorithm in simulating the Boltzmann equation.It can be shown that the particle velocity distribution function obtained by the DSMC method converges to a modified form of the Boltzmann equation,which is the equation of the gas-kinetic unified algorithm to directly solve the molecular velocity distribution function.Their convergence is derived through mathematical treatment.The collision frequency is presented using various molecular models and the local equilibrium distribution function is obtained by Enskog expansion using the converged equation of the DSMC method.These two expressions agree with those used in the unified algorithm.Numerical validation of the converging consistency between these two approaches is illustrated by simulating the pressure driven Poiseuille flow in the slip transition flow regime and the two-dimensional and three-dimensional flows around a circular cylinder and spherical-cone reentry body covering the whole flow regimes from low speed micro-channel flow to high speed non-equilibrium aerothermodynamics.展开更多
A multispeed lattice Boltzmann equation with nine velocity directions on a two-dimension square lattice is investigated. In the macroscopic hydrodynamic equation, the coefficient before convective term becomes 1 and t...A multispeed lattice Boltzmann equation with nine velocity directions on a two-dimension square lattice is investigated. In the macroscopic hydrodynamic equation, the coefficient before convective term becomes 1 and the viscosity can be decreased to zero. Computer simulation data are exactly coherent with theoretical results.展开更多
We present the lowest order quantum correction to the semiclassical Boltzmann distribution function,and the equation satisfied by this correction is given. Our equation for the quantum correction is obtained from the ...We present the lowest order quantum correction to the semiclassical Boltzmann distribution function,and the equation satisfied by this correction is given. Our equation for the quantum correction is obtained from the conventional quantum Boltzmann equation by explicitly expressing the Planck constant in the gradient approximation,and the quantum Wigner distribution function is expanded in pow-ers of Planck constant,too. The negative quantum correlation in the Wigner dis-tribution function which is just the quantum correction terms is naturally singled out,thus obviating the need for the Husimi’s coarse grain averaging that is usually done to remove the negative quantum part of the Wigner distribution function. We also discuss the classical limit of quantum thermodynamic entropy in the above framework.展开更多
The homotopy analysis method (HAM), as a new mathematical tool, has been employed to solve many nonlinear problems. As a fundamental equation in non-equilibrium statistical mechanics, the Boltzmann integro-differentia...The homotopy analysis method (HAM), as a new mathematical tool, has been employed to solve many nonlinear problems. As a fundamental equation in non-equilibrium statistical mechanics, the Boltzmann integro-differential equation (BE) describing the movement of particles is of strong nonlinearity. In this work, HAM is preliminarily applied to dilute granular flow which is relatively simple. By choosing the Maxwell velocity distribution function as the initial solution, the concrete expression of the first-order approximate solution to BE with collision term being the BGK model is given. Furthermore it is consistent with the solution using Chapman-Enskog method but does not rely on little parameters.展开更多
文摘This paper presents a self-contained description on the configuration of propagator method(PM)to calculate the electron velocity distribution function(EVDF) of electron swarms in gases under DC electric and magnetic fields crossed at a right angle. Velocity space is divided into cells with respect to three polar coordinates v,θ and f. The number of electrons in each cell is stored in three-dimensional arrays. The changes of electron velocity due to acceleration by the electric and magnetic fields and scattering by gas molecules are treated as intercellular electron transfers on the basis of the Boltzmann equation and are represented using operators called the propagators or Green’s functions. The collision propagator, assuming isotropic scattering, is basically unchanged from conventional PMs performed under electric fields without magnetic fields. On the other hand, the acceleration propagator is customized for rotational acceleration under the action of the Lorentz force. The acceleration propagator specific to the present cell configuration is analytically derived. The mean electron energy and average electron velocity vector in a model gas and SF6 were derived from the EVDF as a demonstration of the PM under the Hall deflection and they were in a fine agreement with those obtained by Monte Carlo simulations. A strategy for fast relaxation is discussed, and extension of the PM for the EVDF under AC electric and DC/AC magnetic fields is outlined as well.
基金the National Natural Science Foundation of China (Grant No. 40310223)
文摘The function and physical mechanism of heat flow and the viscous stress in the velocity distribution function expanded by Maxwellian distribution are presented. With the introduction of effective temperature Tf, incoherent scatter spectra from plasma for electromagnetic wave in arbitrary line of sight are given. The effect of asymmetry and anisotropy provided by heat flow and the viscous stress on power spectra is discussed. Radar spectra are calculated for different cases of electric field, direction, collision frequency and temperature. The effect of heat flow and the viscous stress on inversion results is analyzed. With a large electric field, the character of non-Maxwellian must be considered.
文摘A Lagrangian lattice Boltzmann method for solving Euler equations is proposed. The key step in formulating this method is the introduction of the displacement distribution function. The equilibrium distribution function consists of macroscopic Lagrangian variables at time steps n and n + 1. It is different from the standard lattice Boltzmann method. In this method the element, instead of each particle, is required to satisfy the basic law. The element is considered as one large particle, which results in simpler version than the corresponding Eulerian one, because the advection term disappears here. Our numerical examples successfully reproduce the classical results.
基金supported by the National Natural Science Foundation of China (Grant Nos. 91016027 and 91130018)
文摘This paper investigates the convergence proof of the Direct Simulation Monte Carlo(DSMC) method and the Gas-Kinetic Unified Algorithm in simulating the Boltzmann equation.It can be shown that the particle velocity distribution function obtained by the DSMC method converges to a modified form of the Boltzmann equation,which is the equation of the gas-kinetic unified algorithm to directly solve the molecular velocity distribution function.Their convergence is derived through mathematical treatment.The collision frequency is presented using various molecular models and the local equilibrium distribution function is obtained by Enskog expansion using the converged equation of the DSMC method.These two expressions agree with those used in the unified algorithm.Numerical validation of the converging consistency between these two approaches is illustrated by simulating the pressure driven Poiseuille flow in the slip transition flow regime and the two-dimensional and three-dimensional flows around a circular cylinder and spherical-cone reentry body covering the whole flow regimes from low speed micro-channel flow to high speed non-equilibrium aerothermodynamics.
文摘A multispeed lattice Boltzmann equation with nine velocity directions on a two-dimension square lattice is investigated. In the macroscopic hydrodynamic equation, the coefficient before convective term becomes 1 and the viscosity can be decreased to zero. Computer simulation data are exactly coherent with theoretical results.
基金the National Natural Science Foundation of China (Grant No 10404037)the Scientific Research Fund of GUCAS (Grant No 055101BM03)
文摘We present the lowest order quantum correction to the semiclassical Boltzmann distribution function,and the equation satisfied by this correction is given. Our equation for the quantum correction is obtained from the conventional quantum Boltzmann equation by explicitly expressing the Planck constant in the gradient approximation,and the quantum Wigner distribution function is expanded in pow-ers of Planck constant,too. The negative quantum correlation in the Wigner dis-tribution function which is just the quantum correction terms is naturally singled out,thus obviating the need for the Husimi’s coarse grain averaging that is usually done to remove the negative quantum part of the Wigner distribution function. We also discuss the classical limit of quantum thermodynamic entropy in the above framework.
基金Supported by the National Basic Research Program of China (Grant No. 2007CB714101)Research Fund of the State Key Laboratory for Hydroscience and Engineering in Tsinghua University (Grant No. 2008-ZY-6)
文摘The homotopy analysis method (HAM), as a new mathematical tool, has been employed to solve many nonlinear problems. As a fundamental equation in non-equilibrium statistical mechanics, the Boltzmann integro-differential equation (BE) describing the movement of particles is of strong nonlinearity. In this work, HAM is preliminarily applied to dilute granular flow which is relatively simple. By choosing the Maxwell velocity distribution function as the initial solution, the concrete expression of the first-order approximate solution to BE with collision term being the BGK model is given. Furthermore it is consistent with the solution using Chapman-Enskog method but does not rely on little parameters.
