Lattice BGK kinetic model for high speed compressible flows: hydrodynamic and nonequilibrium behaviors

TL;DR

This paper introduces a flexible lattice BGK kinetic model for high-speed compressible flows that accurately captures hydrodynamic and nonequilibrium behaviors, enabling simulations at Mach numbers up to 30.

Contribution

It presents a novel approach to formulate the lattice BGK model with a flexible discrete velocity model and equilibrium distribution, applicable in multiple dimensions and capable of high Mach number simulations.

Findings

Model recovers Navier-Stokes equations in the continuum limit.

Simulation Mach number can be increased up to 30 or higher.

System behavior near shock waves shows deviations from thermodynamic equilibrium.

Abstract

We present a simple and general approach to formulate the lattice BGK model for high speed compressible flows. The main point consists of two parts: an appropriate discrete equilibrium distribution function (DEDF) $\mathbf{f}^{eq}$ and a discrete velocity model with flexible velocity size. The DEDF is obtained by $\mathbf{f}^{eq}=\mathbf{C}^{-1}\mathbf{M}$, where $\mathbf{M}$ is a set of moment of the Maxwellian distribution function, and $\mathbf{C}$ is the matrix connecting the DEDF and the moments. The numerical components of $\mathbf{C}$ are determined by the discrete velocity model. The calculation of $\mathbf{C}^{-1}$ is based on the analytic solution which is a function of the parameter controlling the sizes of discrete velocity. The choosing of discrete velocity model has a high flexibility. The specific heat ratio of the system can be flexible. The approach works for the one-, two- and three-dimensional model constructions. As an example, we compose a new lattice BGK kinetic model which works not only for recovering the Navier-Stokes equations in the continuum limit but also for measuring the departure of system from its thermodynamic equilibrium. Via adjusting the sizes of the discrete velocities the stably simulated Mach number can be significantly increased up to 30 or even higher. The model is verified and validated by well-known benchmark tests. Some macroscopic behaviors of the system due to deviating from thermodynamic equilibrium around the shock wave interfaces are shown.