Development of an efficient gas kinetic scheme for simulation of two-dimensional incompressible thermal flows

In this work, an efficient gas kinetic scheme is presented for simulation of two-dimensional incompressible thermal flows. In the scheme, the macroscopic governing equations for mass, momentum, and energy conservation are discretized by the finite volume method and the numerical fluxes at the cell interface are reconstructed by the local solution of the Boltzmann equation. To compute these fluxes, two distribution functions are involved. One is the circular function, which is used to calculate the numerical fluxes of mass and momentum equations. Due to the incompressible limit, the circle at the cell interface can be approximately considered to be symmetric so that the expressions for the conservative variables and numerical fluxes at the cell interface can be given explicitly and concisely. Another one is the D2Q4 model, which is utilized to compute the numerical flux of the energy equation. By following the process for derivation of numerical fluxes of mass and momentum equations, the numerical flux of the energy equation can also be given explicitly. The accuracy, efficiency, and stability of the present scheme are validated by simulating several thermal flow problems. Numerical results showed that the present scheme can provide accurate numerical results for incompressible thermal flows at a wide range of Rayleigh numbers with less computational cost than that needed by the thermal lattice Boltzmann flux solver (TLBFS), which has been proven to be more efficient than the thermal lattice Boltzmann method (TLBM).