A gas-kinetic BGK scheme for the finite volume lattice Boltzmann method for nearly incompressible flows

Abstract

This work presents a gas kinetic BGK scheme for the finite volume lattice Boltzmann method (FVLBM). In the present method, the formal analytical solution of the lattice Boltzmann BGK equation is used to determine the fluxes on cell interfaces. With such formal analytical solution, there are two remarkable features in the present FVLBM. One is that the effects of the collision term in the lattice Boltzmann equation (LBE) is included in the reconstructed fluxes on cell interfaces, which reduces numerical dissipation, and another is that, not as conventional flux schemes of FVLBM, the present flux scheme is a function of both space and time, which gives second order accuracy of both time and space in one step for unsteady flows. Furthermore, thanks to the collision invariant, by introducing a prediction step for the macro variables, the implicit feature of the equilibrium distribution function is removed and thus, the time-step in the present scheme is only limited by the Courant–Friedrichs–Lewy (CFL) condition. Additionally, some numerical implementations of boundary conditions are discussed and, to validate accuracy of the present scheme, comparison studies of several two dimensional incompressible unsteady and steady benchmark flows are also provided. The accuracy and effectiveness of the present scheme are clearly verified.