Self-consistent Force Scheme in the Discrete Boltzmann Equation

Physical Review E, 1998

A representation of the forcing term in the Boltzmann equation based on a Hermite expansion of the Boltzmann distribution function in velocity phase space is derived. Based on this description of the forcing term, a systematic comparison of previous lattice Boltzmann models describing a nonideal gas behavior is given. ͓S1063-651X͑98͒12010-X͔

SIAM Journal on Numerical Analysis, 1997

Physical Review Letters, 1998

We point out an equivalence between the discrete velocity method of solving the Boltzmann equation, of which the lattice Boltzmann equation method is a special example, and the approximations to the Boltzmann equation by a Hermite polynomial expansion. Discretizing the Boltzmann equation with a BGK collision term at the velocities that correspond to the nodes of a Hermite quadrature is shown to be equivalent to truncating the Hermite expansion of the distribution function to the corresponding order. The truncated part of the distribution has no contribution to the moments of low orders and is negligible at small Mach numbers. Higher order approximations to the Boltzmann equation can be achieved by using more velocities in the quadrature.

Physical Review E, 2006

The velocity discretization is a critical step in deriving the lattice Boltzmann ͑LBE͒ from the continuous Boltzmann equation. This problem is considered in the present paper, following an alternative approach and giving the minimal discrete velocity sets in accordance with the order of approximation that is required for the LBE with respect to the continuous Boltzmann equation and with the lattice structure. Considering N to be the order of the polynomial approximation to the Maxwell-Boltzmann equilibrium distribution, it is shown that solving the discretization problem is equivalent to finding the inner product in the discrete space induced by the inner product in the continuous space that preserves the norm and the orthogonality of the Hermite polynomial tensors in the Hilbert space generated by the functions that map the velocity space onto the real numbers space. As a consequence, it is shown that, for each order N of approximation, the even-parity velocity tensors are isotropic up to rank 2N in the discrete space. The norm and the orthogonality restrictions lead to space-filling lattices with increased dimensionality when compared with presently known lattices. This problem is discussed in relation with a discretization approach based on a finite set of orthogonal functions in the discrete space. Two-dimensional square lattices intended to be used in thermal problems and their respective equilibrium distributions are presented and discussed.

AIP Conference Proceedings, 2005

Physica A: Statistical Mechanics and its Applications, 1992

This investigation presents a dynamical formulation of Boltzmann-like discrete kinetic equations. The central concept of such a formulation is based on the idea that a dissipative system is equipped with a functional Poisson bracket and a dissipative bracket, and with two functionals generating the dynamical evolution. We present non-canonical brackets and the generating functionals. We also discuss their basic properties and investigate dynamical invariants as well as the production of entropy generated by the entropy functional via the dissipative part of the bracket. When collisions are the only sources of dissipation we are coming up with a consistent up to now unknown version of the discrete Boltzmann-Vlasov kinetic equation.

arXiv: Fluid Dynamics, 2017

In the frame of the Boltzmann equation, wall-bounded flows of rarefied gases require the implementation of boundary conditions at the kinetic level. Such boundary conditions induce a discontinuity in the distribution function with respect to the component of the momentum which is normal to the boundary. Expanding the distribution function with respect to half-range polynomials allows this discontinuity to be captured. The implementation of this concept has been reported in the literature only for force-free flows. In the case of general forces which can have non-zero components in the direction perpendicular to the walls, the implementation of the force term requires taking the momentum space gradient of a discontinuous function. Our proposed method deals with this difficulty by employing the theory of distributions. We validate our procedure by considering the simple one-dimensional flow between diffuse-reflective walls of equal or different temperatures driven by the constant grav...

Transactions of the American Mathematical Society, 1972

In this paper we compare the nonlinear Boltzmann equation appearing in the kinetic theory of gases, with its linearized version. We exhibit an intertwining operator for the two semigroups involved. We do not assume from the reader any familiarity with Boltzmann's equation but rather start from scratch. 0. Introduction. Consider a dilute gas composed of a very large number of molecules moving in space according to the laws of classical mechanics, and colliding in pairs from time to time. Assume that we can disregard all external effects, such as gravity, so that the motion is completely specified by giving the intermolecular forces. One is interested in the number of molecules which at time t have position r and velocity v, within dr dv. This is given by n(t,r,v) = Nf(t,r,v)drdv(2) where/is called the density function. It is clear that this quantity is going to change in time due to the motion of the molecules and to the effect of the collisions. Boltzmann derived an equation for the rate of change off with time. It has the form of a nonlinear integro-differential equation: if' \f(vV)f(^-f(vi)f(v2)]\v,-v2\I(\v,-v2\, 8) sin 0dBd</>dv2.

2005

The particles model, the collision model, the polynomial development used for the equilibrium distribution, the time discretization and the velocity discretization are factors that let the lattice Boltzmann framework (LBM) far away from its conceptual support: the continuous Boltzmann equation (BE). Most collision models are based on the BGK, single parameter, relaxation-term leading to constant Prandtl numbers. The polynomial expansion used for the equilibrium distribution introduces an upper-bound in the local macroscopic speed. Most widely used time discretization procedures give an explicit numerical scheme with second-order time step errors. In thermal problems, quadrature did not succeed in giving discrete velocity sets able to generate multi-speed regular lattices. All these problems, greatly, difficult the numerical simulation of LBM based algorithms. In present work, the systematic derivation of lattice-Boltzmann models from the continuous Boltzmann equation is discussed. The collision term in the linearized Boltzmann equation is modeled by expanding the distribution function in Hermite tensors. Thermohydrodynamic macroscopic equations are correctly retrieved with a second-order model. Velocity discretization is the most critical step in establishing regular-lattices framework. In the quadrature process, it is shown that the integrating variable has an important role in defining the equilibrium distribution and the lattice-Boltzmann model, leading, alternatively, to temperature dependent velocities (TDV) and to temperature dependent weights (TDW) lattice-Boltzmann models.

Physical Review E, 1997

The lattice Boltzmann equation ͑LBE͒ is directly derived from the Boltzmann equation by discretization in both time and phase space. A procedure to systematically derive discrete velocity models is presented. A LBE algorithm with arbitrary mesh grids is proposed and a numerical simulation of the backward-facing step is conducted. The numerical result agrees well with experimental and previous numerical results. Various improvements on the LBE models are discussed, and an explanation of the instability of the existing LBE thermal models is also provided. ͓S1063-651X͑97͒51106-8͔