Initialization of Lattice Boltzmann Models with the Help of the Numerical Chapman–Enskog Expansion

Abstract

We extend the applicability of the numerical Chapman-Enskog expansion as a lifting operator for lattice Boltzmann models to map density and momentum to distribution functions. In earlier work [Vanderhoydonc et al. Multiscale Model. Simul. 10(3): 766-791, 2012] such an expansion was constructed in the context of lifting only the zeroth order velocity moment, namely the density. A lifting operator is necessary to convert information from the macroscopic to the mesoscopic scale. This operator is used for the initialization of lattice Boltzmann models. Given only density and momentum, the goal is to initialize the distribution functions of lattice Boltzmann models. For this initialization, the numerical Chapman-Enskog expansion is used in this paper.

Constrained Runs Algorithm as a Lifting Operator for the One-Dimensional in Space Boltzmann Equation with BGK Collision Term

“…A review of these lifting operators for LBMs with an initial density as macroscopic variable is given in [30]. A generalization of the numerical Chapman-Enskog expansion is made in [29] where both density and momentum are considered to be given at initialization.…”

“…We first applied the generalization of the CR algorithm in [29] for a small number of velocities to deal with the conservation of density and momentum in lattice Boltzmann problems.…”

“…. In [28] and [29] we have applied, for LBMs, the CR on distribution functions represented as the first few terms of the Chapman-Enskog expansion. The iteration then determines the coefficients of the expansion rather than the distribution function itself.…”