An improved discrete unified gas kinetic scheme for simulating compressible natural convection flows

Discrete unified gas-kinetic scheme (DUGKS) has been developed recently as a general method for simulating flows at all Knudsen numbers. In this study, we extend DUGKS to simulate fully compressible thermal flows. We introduce a source term to the Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) collision model [1] to adjust heat flux and thus the Prandtl number. The fully compressible Navier-Stokes equations can be recovered by the current model. As a mesoscopic CFD approach, it requires an accurate mesoscopic implementation of the boundary conditions. Using the Chapman-Enskog approximation, we derive the “bounce-back” expressions for both temperature and velocity distribution functions, which reveal the need to consider coupling terms between the velocity and thermal fields. To validate our scheme, we first reproduce the Boussinesq flow results by simulating natural convection in a square cavity with a small temperature difference $(ϵ=0.01)$ and a low Mach number. Then we perform simulations of steady natural convection $(Ra=1.0\times {10}^6)$ in a square cavity with differentially heated side walls and a large temperature difference $(ϵ=0.6)$, where the Boussinesq approximation becomes invalid. Temperature, velocity profiles, and Nusselt number distribution are obtained and compared with the benchmark results from the literature. Finally, the unsteady compressible natural convection with $Ra=5.0\times {10}^9,ϵ=0.6$ is studied and the turbulent fluctuation statistics are computed and analyzed.