Three-dimensional non-orthogonal multiple-relaxation-time lattice Boltzmann model for multiphase flows

In the classical multiple-relaxation-time (MRT) lattice Boltzmann (LB) method, the transformation matrix is formed by constructing a set of orthogonal basis vectors. In this paper, a theoretical and numerical study is performed to investigate the capability and efficiency of a non-orthogonal MRT-LB model for simulating multiphase flows. First, a three-dimensional non-orthogonal MRT-LB is proposed. A non-orthogonal MRT collision operator is devised based on a set of non-orthogonal basis vectors, through which the transformation matrix and its inverse matrix are considerably simplified as compared with those of an orthogonal MRT collision operator. Furthermore, through the Chapman-Enskog analysis, it is theoretically demonstrated that the three-dimensional non-orthogonal MRT-LB model can correctly recover the macroscopic equations at the Navier-Stokes level in the low Mach number limit. Numerical comparisons between the non-orthogonal MRT-LB model and the usual orthogonal MRT-LB model are made by simulating multiphase flows on the basis of the pseudopotential multiphase LB approach. The numerical results show that, in comparison with the usual orthogonal MRT-LB model, the non-orthogonal MRT-LB model can retain the numerical accuracy while simplifying the implementation.