**103**(2013) 24003

## Lattice BGK kinetic model for high-speed compressible flows: Hydrodynamic and nonequilibrium behaviors

^{1} North China Institute of Aerospace Engineering - Langfang 065000, PRC ^{2} National Key Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics P. O. Box 8009-26, Beijing 100088, PRC ^{3} China Petroleum Pipeline Bureau - Langfang 065000, PRC

^{(a)} Xu_Aiguo@iapcm.ac.cn (corresponding author)

Received: 17 May 2013

Accepted: 19 July 2013

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) **f**^{eq} and a discrete velocity model with flexible velocity size. The DEDF is obtained by **f**^{eq} = **C**^{−1}**M**, where **M** is a set of moments of the Maxwellian distribution function, and **C** is the matrix connecting the DEDF and the moments. The numerical components of **C** are determined by the discrete velocity model. The calculation of **C**^{−1} is based on the analytic solution which is a function of the parameter controlling the sizes of discrete velocity. The choice of the 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 the 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 the deviation from thermodynamic equilibrium around the shock wave interfaces are shown.

PACS: 47.11.-j – Computational methods in fluid dynamics / 51.10.+y – Kinetic and transport theory of gases / 05.20.Dd – Kinetic theory

*© EPLA, 2013*