Analysis and reconstruction of the simplified thermal lattice Boltzmann method

Abstract

The previous thermal lattice Boltzmann models are generally based on the lattice framework, which leads to some intrinsic drawbacks including the limitation of uniform mesh, coupled time step with mesh spacing, extra memory size, and complex boundary treatment. Aiming to decrease memory size and achieve good numerical stability for high Rayleigh number thermal flows, the simplified thermal lattice Boltzmann method (STLBM) was proposed. However, STLBM still has two fundamental problems. Firstly, as a weakly compressible model, its reason for the good numerical stability at high Rayleigh number thermal flows has not been revealed in the perspective of macroscopic scale yet. Secondly, it still depends on the uniform mesh, which makes the implementation of non-uniform meshes time-consuming and complicated. To tackle the two problems, the macroscopic equations of STLBM (MEs-STLBM) with actual numerical dissipative terms are derived firstly by approximating its actual computational process. By solving MEs-STLBM with the least-squares-based finite difference (LSFD) method, the limitation of the uniform mesh of STLBM can be easily overcome. Numerical investigation proves that the numerical dissipative terms in MEs-STLBM are the reason for the good numerical stability at high Rayleigh number thermal flows. By retaining these numerical dissipative terms, the discretized MEs-STLBM can recover the accuracy and numerical stability of STLBM well. Besides, they have significantly higher computational efficiency than STLBM.