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

Abstract

This work presents a coupled gas-kinetic Bhatnagar-Gross-Krook (BGK) scheme based finite volume lattice Boltzmann method (FVLBM) for nearly incompressible thermal flows. In the present formulation, a thermal lattice BGK model with double distribution functions coupled through the Boussinesq approximation is adopted. Compared with the conventional thermal finite volume/finite difference lattice Boltzmann schemes, the present method has two striking advantages. One is that the fluxes on cell interfaces are determined by the formal analytical solution of the lattice BGK equation rather than interpolation or solving the Riemann problem, which reduces numerical dissipation considerably. Another is that, an implicit collision of the lattice Boltzmann equations (LBE) is adopted to remove the time-step constrain from the relaxation time, which improves the efficiency for high Rayleigh number thermal flows. Additionally, the accuracy of the proposed scheme are validated by the numerical simulations of four test cases in two dimensions: (a) the thermal Couette flow with wall injection; (b) the natural convection in a square cavity with differently heated vertical walls; (c) the Rayleigh-Bénard convection in a rectangle heated from bottom; and (d) the turbulent Rayleigh-Bénard convection in a square cavity heated from bottom. The numerical results show that the present scheme has second-order accuracy and can be a reliable tool for incompressible thermal flows.