BGK Equation Research Articles

A hybrid kinetic–fluid model for solving the Vlasov–BGK equation

Our purpose is to derive a model for charged particles which combines a kinetic description of the fast particles with a fluid description of the slow ones. In a previous paper, a similar model was derived from a kinetic BGK equation that uses a constant relaxation time and does not include the effect of an electric field. In this paper, we consider a more general kinetic equation including an electric field and a varying relaxation time. Fast particles will be described through a collisional kinetic equation of Vlasov–BGK type while thermal particles will be modeled by a hydrodynamic model. Then, we construct a numerical scheme for this model and validate the approach by presenting various numerical tests.

The kinetic scheme for the full-Burnett equations

The present paper concerns two aspects for the Burnett equations. First, we are going to theoretically show the consistency between the traditional Chapman–Enskog expansion and the successive approximation for the BGK equation up to the super-Burnett order. Second, we will design a numerical scheme to efficiently solve the Burnett equations. The current approach is an improvement of the BGK-Burnett scheme [K. Xu, Phys. Fluids 15 (2003) 2077], where the locally constant collision frequency is considered. Based on the Burnett distribution function, a high order time accurate numerical flux is derived by different approaches. The resulting scheme is tested in the problem of normal shock wave and that of force-driven Poiseuille flow.

ON THE ASYMPTOTIC ANALYSIS OF KINETIC MODELS TOWARDS THE COMPRESSIBLE EULER AND ACOUSTIC EQUATIONS

This paper deals with the analysis of the asymptotic limit for models of the mathematical kinetic theory to the nonlinearized compressible Euler equations or to the acoustic equations when the Knudsen number ε tends to zero. Existence and uniqueness theorems are proven for analytic initial fluctuations on the time interval independent of the small parameter ε. As ε tends to zero, the solution of kinetics models converges strongly to the Maxwellian whose fluid-dynamics parameters solve the Euler and the acoustic systems. The general results are specifically applied to the analysis of the Boltzmann and BGK equations.

A hybrid kinetic/fluid model for solving the gas dynamics Boltzmann–BGK equation

Our purpose is to derive a hybrid model for particle systems which combines a kinetic description of the fast particles with a fluid description of the thermal ones. In the present work, fast particles will be described through a collisional kinetic equation of Boltzmann–BGK type while thermal particles will be modeled by means of a system of Euler type equations. Then, we construct a numerical scheme for this model. This scheme satisfies exact conservation properties. We validate the approach by presenting various numerical tests.

Numerical analysis of metal melting and evaporating with liquid surface depression

A metal molten-pool is described by the Navier-Stokes and energy equations and the depressed liquid surface is described by the Young-Laplace equations,and the metal evaporation is described by the BGK equation. The velocity and temperature distributions of the pool and the vapor heated by an electron gun are solved by the finite difference method. The numerical results show the evaporation rate of the metal increases with the power of the electron gun, so the vapor density increases and the vapor temperature decreases and the vapor velocity increases. The surface temperature of the pool with the depressed surface is less than the one of the pool with the plane surface, so is the metal evaporation rate. The difference of the evaporation rate between them increases with the power of the electron gun, so the liquid surface depression should be taken into consideration for a high power electron gun.

Solution of a Model Boltzmann Equation via Steepest Descent in the 2-Wasserstein Metric

We study a model Boltzmann equation closely related to the BGK equation using a steepest-descent method in the Wasserstein metric, and prove global existence of energy-and momentum-conserving solutions. We also show that the solutions converge to the manifold of local Maxwellians in the large-time limit, and obtain other information on the behavior of the solutions. We show how the Wasserstein metric is natural for this problem because it is adapted to the study of both the free streaming and the ‘‘collisions’’.

Study on gas kinetic unified algorithm for flows from rarefied transition to continuum

The modified BGK equation adapted to various flow regimes can be presented by the aid of the basic characteristics on molecular movement and collision approaching to equilibrium. The discrete velocity ordinate method is developed and applied to the velocity distribution function to remove its continuous dependency on the velocity space, and then the velocity distribution function equation is cast into hyperbolic conservation law form with nonlinear source terms. Based on the unsteady time-splitting method and the non-oscillatory, containing no free parameters, and dissipative (NND) scheme, the gas kinetic finite difference second-order scheme is constructed for the computation of the discrete velocity distribution functions. The mathematical model on the interaction of molecules with solid surface is studied and used in the numerical method. Four types of numerical quadrature rules, such as the modified Gauss–Hermite formula, the composite Newton–Cotes integration method, the Gauss–Legendre numerical quadrature rule, and the Golden Section number-theoretic integral method, are developed and applied to the discretized velocity space to evaluate the macroscopic flow parameters at each point in the physical space. As a result, a unified simplified gas kinetic algorithm is established for the flows from rarefied transition to continuum regime. Based on analyzing the inner parallel degree of the unified algorithm, the parallel strategy adapted to the gas kinetic numerical algorithm is studied, and then the HPF parallel processing software for the unified algorithm is developed. To test the present method, the one-dimensional shock-tube problems, the flows past two-dimensional circular cylinder, and the flows around three-dimensional sphere and spacecraft shape with various Knudsen numbers are simulated. The computational results are found in high resolution of the flow fields and good agreement with the theoretical, DSMC, N–S, and experimental results.

HYBRID FINITE-DIFFERENCE THERMAL LATTICE BOLTZMANN EQUATION

We analyze the acoustic and thermal properties of athermal and thermal lattice Boltzmann equation (LBE) in 2D and show that the numerical instability in the thermal lattice Boltzmann equation (TLBE) is related to the algebraic coupling among different modes of the linearized evolution operator. We propose a hybrid finite-difference (FD) thermal lattice Boltzmann equation (TLBE). The hybrid FD-TLBE scheme is far superior over the existing thermal LBE schemes in terms of numerical stability. We point out that the lattice BGK equation is incompatible with the multiple relaxation time model.

Global Existence and Large-Time Behavior for BGK Model for a Gas with Non-constant Cross Section

This article deals with the analysis of Cauchy problem for BGK equation with cross section depending on the temperature and density of the gas. It is proven that the solutions of BGK equation exist globally in time, and converge to equilibrium, when time becomes large. The analysis can be regarded as a substantial improvement of previous results related to the equation with constant cross section.

Analytic Solution of an Inhomogeneous Kinetic Equation in the Problem of Second-Order Thermal Creep

An analytic solution of the problem of second-order thermal creep is obtained. A method for solving the half-space boundary value problem for an inhomogeneous linearized kinetic BGK equation forms the basis of the solution. The general solution of the input equation is constructed in the form of an expansion of the corresponding characteristic equation in terms of the eigenfunctions. Substitution of the solution in the boundary conditions leads to a Riemann boundary value problem. The unknown thermal creep velocity is found from the condition of solvability of the boundary value problem. The numerical analysis performed confirms the existence of negative thermophoresis (in the direction of the temperature gradient) for high-conductivity aerosol particles at low Knudsen numbers.

DISCRETE TIME NAVIER-STOKES LIMIT FOR THE BGK BOLTZMANN EQUATION

ABSTRACT The discrete time Navier-Stokes equations in whole space are obtained as a fluid limit for the properly scaled discrete time BGK equation by the method developed by Bardos, Golse and Levermore to study hydrodynamic limits of the Boltzmann equation (Bardos, C.; Golse, F.; Levermore, C.D. Fluid Dynamic Limits of Kinetic Equations II: Convergence Proofs for the Boltzmann Equation. Comm. Pure Appl. Math. 1993, 46(5), 667–753.). Two new ideas lead to a completely rigorous result in the BGK case. First we get some control on large velocities by remarking that the microscopic density can be decomposed as , where Mf is the Maxwellian distribution associated with f and is controlled by means of the entropy dissipation. Secondly some equiintegrability is obtained from a new velocity averaging result for the advection operator . Both estimates allow to fulfill the program in (Bardos, C.; Golse, F.; Levermore, C.D. Fluid Dynamic Limits of Kinetic Equations II: Convergence Proofs for the Boltzmann Equation. Comm. Pure Appl. Math. 1993, 46(5), 667–753.).

Lattice Kinetic Schemes for Magnetohydrodynamics

Lattice kinetic equations for simulating incompressible magnetohydrodynamics in two or three dimensions are constructed. The fluid is simulated via a conventional low Mach number lattice Boltzmann scheme, modified to include the Lorentz force due to the magnetic field. The magnetic field is represented by a separate vector-valued magnetic distribution function which obeys a vector Boltzmann–BGK equation. The two distribution functions are only coupled via the macroscopic density, momentum, and magnetic field evaluated at lattice points. This allows a reduced lattice to be used for the magnetic distribution function, with a corresponding saving in storage, which becomes comparable to that for the scalar hydrodynamic distribution function. The magnetic diffusivity may be adjusted independently of the fluid viscosity, unlike an earlier formulation. Numerical experiments with Hartmann flow, the Orszag–Tang vortex, and the doubly periodic coalescence instability compare favorably with results obtained using a spectral method, and with previously published results. The scheme preserved a consistent approximation to the divergence-free condition ∇· B=0 to round-off error.

A coupled lattice BGK model for the Boussinesq equations

AbstractIn this paper, a thermal lattice BGK model is developed for the Boussinesq incompressible fluids. The basic idea is to solve the velocity field and the temperature field using two independent lattice BGK equations, respectively, and then combine them into one coupled model for the whole system. The porous plate problem and the two‐dimensional natural convection flow in a square cavity with Pr=0.71 and various of Rayleigh numbers are simulated using the model. The numerical results are found to be in good agreement with the analytical solutions or those of previous studies. Copyright © 2002 John Wiley & Sons, Ltd.

Du modèle BGK de l'équation de Boltzmann aux équations d'Euler des fluides incompressibles

Dissipative solutions [12] of the Euler equations of incompressible fluids are obtained as the hydrodynamic limit of a properly scaled BGK equation. This stability result comes from refined entropy and entropy dissipation bounds. It uses in a crucial way the local conservation laws which are known to hold for weak solutions of this simplified model of the Boltzmann equation.

Velocity averaging in [formula omitted] for the transport equation

A new result of L 1-compactness for velocity averages of solutions to the transport equation is stated and proved in this Note. This result, proved by a new interpolation argument, extends to the case of any space dimension Lemma 8 of Golse–Lions–Perthame–Sentis [J. Funct. Anal. 76 (1988) 110–125], proved there in space dimension 1 only. This is a key argument in the proof of the hydrodynamic limits of the Boltzmann or BGK equations to the incompressible Euler or Navier–Stokes equations. To cite this article: F. Golse, L. Saint-Raymond, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 557–562.

ν-DSMC: A Fast Simulation Method for Rarefied Flow

A new approximate simulation method for rarefied flows, ν-DSMC, is described and tested by comparison with results from Bird's direct simulation Monte–Carlo (DSMC) method, and with a previously proposed ‘relaxation time’ simulation method (RTSM) which solves the BGK equation. Both ν-DSMC and RTSM execute about twice (or more) as fast as DSMC. In ν-DSMC all collision pairs (amongst near neighbours) are equally likely so the collision loops are suitable for parallel execution. In order to reproduce any desired viscosity law μ=μ(T) the collision rate must be altered by making the total collision cross-section a function of the local kinetic temperature. The approximate methods have been compared to DSMC in three test cases: (1) simple relaxation calculations, (2) high speed Couette flow, and (3) the plane normal shock problem. Both approximate methods produce the same relaxation rate as DSMC, but only the new method produces a nonequilibrium distribution function similar to that for DSMC. For the Couette flow, ν-DSMC produces profiles of flow velocity, density, and temperature which agree with DSMC results to within 3–5%, while the BGK solution produces temperatures which are up to 18% greater. Similar results were found for the normal shock wave. ν-DSMC predicts the location and size of the peak Tx temperature more accurately than does RTSM. The deviation of the temperature profiles from the DSMC results are 2–3 times greater for RTSM than for ν-DSMC. The conclusion from all test cases is clear: the incorrect Prandtl number predicted by the BGK equation detracts from the usefulness of the BGK equation as a model for rarefied flow; the new approximate simulation method is just as fast as solving the BGK equation and gives results which are in much better agreement with DSMC.

BGK-equation simulation of the exhaust plume formed by a manoeuvre thruster in outer space

The general principle of utilizing the BGK equation to simulate a macroscopic gas flow is illustrated. Two typical examples, i.e., a low-speed axisymmetric submerged jet and the Prandtl-Meyer expansion to a vacuum, are presented for validating the feasibility and accuracy of the BGK-equation simulation in continuum and non-continuum flow regimes. This approach is then used to simulate the exhaust plume formed by a small manoeuvre thruster of an artificial satellite in the outer space. The plume impingement on a flat surface perpendicular to the nozzle axis is also simulated by the same method. In the latter case the impingement force acting on the flat surface is calculated. When the flow reaches to the steady state the calculated impingement force is reasonably compared with the theoretical value of the nozzle thrust.

Entropy Inequalities for Evaporation Condensation Problem in Rarefied Gas Dynamics

The present paper is devoted mainly to the half space problem for stationary Boltzmann-type equations. Using only conservation laws and the Boltzmann H-theorem we derive an inequality for unknown constant fluxes of mass, energy, and momentum. This inequality is expressed in terms of three parameters (pressure p, temperature T and the Mach number M) of the asymptotic Maxwellian at infinity. Geometrically the inequality describes a “physical” domain with positive entropy production in the 3-d space of the parameters. The domain appears to be qualitatively different for evaporation and condensation problems. We show that for given M, the curve p=p(M), T=T(M) of maximal entropy production practically coincides with the experimental evaporation curve obtained by Sone et al. on the basis of numerical solutions of BGK equation. Similar consideration for the condensation problem is also in qualitative agreement with known numerical results.

DISCRETE VELOCITY MODEL AND IMPLICIT SCHEME FOR THE BGK EQUATION OF RAREFIED GAS DYNAMICS

We present a numerical method for computing transitional flows as described by the BGK equation of gas kinetic theory. Using the minimum entropy principle to define a discrete equilibrium function, a discrete velocity model of this equation is proposed. This model, like the continuous one, ensures positivity of solutions, conservation of moments, and dissipation of entropy. The discrete velocity model is then discretized in space and time by an explicit finite volume scheme which is proved to satisfy the previous properties. A linearized implicit scheme is then derived to efficiently compute steady-states; this method is then verified with several test cases.

Long-jump probabilities in a BGK model for surface diffusion

The jump-length probability distribution (JLPD) in a periodic system is studied using the solution of the linearized Boltzmann equation with BGK collision kernel. The equation is solved both by the matrix-continued-fraction method and by direct numerical simulations. The JLPDs obtained by the BGK equation are compared to those derived from other kinetic models, such as the Langevin (Fokker–Planck) model.