An improved gas kinetic BGK scheme for finite volume lattice Boltzmann method for nearly incompressible flows

Abstract

The recently gas kinetic Bhatnagar–Gross–Krook (BGK) scheme proposed by Li et al. for finite volume lattice Boltzmann method (FVLBM) is improved in this work to enhance the numerical stability. Compared with the original scheme, there are two striking features in the present improved method. One is that two auxiliary distribution functions, i.e., f̄α and f̃α, with the inclusion of collision effect are introduced in the present scheme, which eliminates the implicit feature for the update of distribution function. Meanwhile, the update of the distribution function relies on f̄α instead of the original one. Similarly with the original scheme, the fluxes on the cell interfaces are determined by the formal analytical solution of the lattice Boltzmann BGK equation, that is, the collision effects are included in the reconstructed fluxes, which can reduce excessive numerical dissipation. Another different feature is that an auxiliary distribution function f̃α is applied to the reconstructed fluxes on the cell interfaces. These two different features make the distribution function related to the reconstructed fluxes has better positivity-preserving properties, thus Δt/τ in the present method can be much higher than Li et al. scheme, and the numerical stability is enhanced considerably. Additionally, various benchmark studies, including the Kovasznay flow, the lid-driven cavity flow, the laminar boundary layer flow and the unsteady Taylor–Green vortex flow, are conducted to assess the numerical accuracy and stability of the proposed method. The numerical results demonstrate that the present scheme has second-order accuracy and better numerical stability for simulating nearly incompressible flows.