A simplified thermal lattice Boltzmann method without evolution of distribution functions

In this paper, a simplified thermal lattice Boltzmann method (STLBM) without evolution of the distribution functions is developed for simulating incompressible thermal flows. With the assistance of the fractional step technique, the macroscopic governing equations recovered from Chapman-Enskog (C-E) expansion analysis are resolved through a predictor-corrector scheme. Then in both the predictor and corrector steps, using the isentropic properties of lattice tensors and relationships of C-E analysis, the macroscopic flow variables are explicitly calculated from the equilibrium and non-equilibrium distribution functions. In STLBM, the equilibrium distribution functions are calculated from the macroscopic variables, while the non-equilibrium distribution functions are evaluated from the differences between two equilibrium distribution functions at different locations and time levels. Therefore, STLBM directly updates the macroscopic variables during the computational process, which lowers the virtual memory cost and facilitates the implementation of physical boundary conditions. Through von Neumann stability analysis, the present method is proven to be unconditionally stable, which is further validated by numerical tests. Three representative examples are presented to demonstrate the robustness of STLBM in practical simulations and its flexibility on different types of meshes and boundaries.