Multi-level lattice Boltzmann model on square lattice for compressible flows

Abstract

A compressible lattice Boltzmann model is established on a square lattice. The model allows large variations in the mean velocity by introducing a large particle-velocity set. To maintain tractability, the support set of the equilibrium distribution is chosen to include only four directions and three particle-velocity levels in which the third level is introduced to improve the stability of the model. This simple structure of the equilibrium distribution makes the model efficient for the simulation of flows over a wide range of Mach numbers and gives it the capability of capturing shock jumps. Unlike the standard lattice Boltzmann model, the formulation eliminated the fourth-order velocity tensors, which were the source of concerns over the homogeneity of square lattices. A modified collision invariant eliminates the second-order discretization error of the fluctuation velocity in the macroscopic conservation equation from which the Navier–Stokes equation and energy equation are recovered. The model is suitable for both viscous and inviscid compressible flows with or without shocks. Two-dimensional shock-wave propagations and boundary layer flows were successfully simulated. The model can be easily extended to three-dimensional cubic lattices.