Filter-matrix lattice Boltzmann model for incompressible thermal flows

In this study, a new filter-matrix lattice Boltzmann (FMLB) model is proposed and extended to include incompressible thermal flows. A new equilibrium solution is found in the improved FMLB model, which is derived from the Hermite expansion. As a result, the velocity-dependent pressure is removed, which is an inherent defect of Somers's FMLB model. In addition, the improved model is extended to include incompressible thermal flows by introducing a class of temperature-distribution function for evaluating the temperature field. Two different temperature-distribution functions are discussed. The improved FMLB model and the temperature-evaluation equation are combined into one coupled model. Numerical simulations are performed on the two-dimensional (2D) lid-driven square cavity flow and the 2D natural convection flow in a square cavity using the improved FMLB model and the two coupled models, respectively. The numerical results of the 2D lid-driven square cavity flow show that the improved FMLB model is superior to the lattice Bhatnagar-Gross-Krook (LBGK) model in terms of both accuracy and stability. When compared with the multi-relaxation-time (MRT) model, the similar accuracy and slightly enhanced stability can be obtained by the improved model. The advantage of the improved model is that it no longer relies on difficult selection of the free parameters requested by the MRT model; in addition, the force term is already included in the collision operator of the improved model. In the case of 2D natural convection flow, the numerical results of the two present models are almost the same, and both exhibit good agreement with the benchmark solution.