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

摘要

We present a simple and general approach to formulate the lattice BGK modelfor high speed compressible flows. The main point consists of two parts: anappropriate discrete equilibrium distribution function (DEDF) $\mathbf{f}^{eq}$and a discrete velocity model with flexible velocity size. The DEDF is obtainedby $\mathbf{f}^{eq}=\mathbf{C}^{-1}\mathbf{M}$, where $\mathbf{M}$ is a set ofmoment of the Maxwellian distribution function, and $\mathbf{C}$ is the matrixconnecting 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 theparameter controlling the sizes of discrete velocity. The choosing of discretevelocity model has a high flexibility. The specific heat ratio of the systemcan be flexible. The approach works for the one-, two- and three-dimensionalmodel constructions. As an example, we compose a new lattice BGK kinetic modelwhich works not only for recovering the Navier-Stokes equations in thecontinuum limit but also for measuring the departure of system from itsthermodynamic equilibrium. Via adjusting the sizes of the discrete velocitiesthe stably simulated Mach number can be significantly increased up to 30 oreven higher. The model is verified and validated by well-known benchmark tests.Some macroscopic behaviors of the system due to deviating from thermodynamicequilibrium around the shock wave interfaces are shown.