A multiple-relaxation-time collision model by Hermite expansion

TL;DR

This paper introduces a multiple-relaxation-time collision model based on Hermite expansion for the Boltzmann equation, allowing independent relaxation of tensor components to improve fluid simulation accuracy.

Contribution

It proposes a novel collision model with independent relaxations of Hermite coefficients, enhancing the traditional BGK approach with multiple relaxation rates.

Findings

Model captures shear, bulk viscosity, and thermal diffusivity.

Avoids numerical dissipation via binomial transform.

Numerical verification confirms improved accuracy.

Abstract

The Bhatnagar-Gross-Krook (BGK) single-relaxation-time collision model for the Boltzmann equation serves as the foundation of the lattice BGK (LBGK) method developed in recent years. The description of the collision as a uniform relaxation process of the distribution function towards its equilibrium is, in many scenarios, simplistic. Based on a previous series of papers, we present a collision model formulated as independent relaxations of the irreducible components of the Hermit coefficients in the reference frame moving with the fluid. These components, corresponding to the irreducible representation of the rotation group, are the minimum tensor components that can be separately relaxed without violating rotation symmetry. For the 2nd, 3rd and 4th moments respectively, two, two and three independent relaxation rates can exist, giving rise to the shear and bulk viscosity, thermal diffusivity and some high-order relaxation process not explicitly manifested in the Navier-Stokes-Fourier equations. Using the binomial transform, the Hermite coefficients are evaluated in the absolute frame to avoid the numerical dissipation introduced by interpolation. Extensive numerical verification is also provided.