One-step simplified lattice Boltzmann method of thermal flows under the Boussinesq approximation.

In recent years, the simplified lattice Boltzmann method without evolution of distribution functions was developed, which adopts predictor-corrector steps to solve the constructed macroscopic equations. To directly solve the constructed macroscopic equations in a single step, we propose the present one-step simplified lattice Boltzmann method and apply it to simulate thermal flows under the Boussinesq approximation. The present method is derived by reconstructing the evolution equation of the lattice Boltzmann method and constructing nonequilibrium distribution functions. This method inherits the advantages of the simplified lattice Boltzmann method, such as low virtual memory cost, convenient boundary treatment, and good numerical stability at relaxation time close to 0.5. In addition, compared to the traditional artificial compressible method (ACM), the present method is more efficient in computation when a small time step is applied in the ACM to ensure numerical stability. Several numerical examples, including natural convection in a square cavity, the porous plate problem, and natural convection in a concentric annulus, are conducted to test the accuracy of the present method. The results show that this method can accurately simulate thermal flow problems and has good numerical stability.