Development of a coupled simplified lattice Boltzmann method for thermal flows

•A more general evolution model is derived by considering the source term at the distribution function level which can achieve better accuracy.•An analytical approach to interpret physical boundary conditions is given for temperature fields so our method physically more robust.•Present method is superior to the previous simplified thermal lattice Boltzmann method in terms of accuracy without using high-order interpolation. The simplified lattice Boltzmann method (SLBM) is relatively new in the LBM community, which lowers the cost in virtual memories significantly and has better numerical stability compared with the single-relaxation-time (SRT) LBM. Recently, SLBM has been extended to simulate thermal flows based on the simplified thermal energy distribution function model. However, the existing thermal models developed for SLBM are not strict in theory. In this work, a coupled simplified lattice Boltzmann method (CSLBM) for thermal flows and its boundary treatment are proposed, where the Navier-Stokes equations for the hydrodynamic field and the convection-diffusion equation for the temperature field are solved independently by two sets of SLBM equations. The consistent forcing scheme is adopted to couple the contribution of the temperature field to the hydrodynamic field. The boundary treatment for temperature field proposed in this work offers an analytical interpretation of the no-slip boundary condition. To validate the accuracy, efficiency, and stability of the present CSLBM, several canonical test cases, including the porous plate problem, the Rayleigh-Bénard convection, and the natural convection in a square cavity are conducted. The numerical results agree well with the analytical solutions or numerical results in the literatures, which shows the present algorithm is of second-order accuracy in space and demonstrates the robustness of CSLBM in practical simulations.