Designing correct fluid hydrodynamics on a rectangular grid using MRT lattice Boltzmann approach

While the lattice Boltzmann method (LBM) has become a powerful numerical approach for solving complex flows, the standard lattice Boltzmann method typically uses a square lattice grid in two spatial dimensions and cubic lattice grid in three dimensions. For inhomogeneous and anisotropic flows, it is desirable to have a LBM model that utilizes a rectangular grid. There were two previous attempts to extend the multiple-relaxation-time (MRT) LBM to a rectangular lattice grid in 2D, however, the resulting hydrodynamic momentum equation was not fully consistent with the Navier–Stokes equation, due to anisotropy of the transport coefficients. In the present work, a new MRT model with an additional degree of freedom is developed in order to match precisely the Navier–Stokes equation when a rectangular lattice grid is used. We first revisit the previous attempts to understand the origin and nature of anisotropic transport coefficients by conducting an inverse design analysis within the Chapman–Enskog procedure. Then an additional adjustable parameter that governs the relative orientation in the energy–normal stress subspace is introduced. It is shown that this adjustable parameter can be used to fully eliminate the anisotropy of transport coefficients, thus the exact Navier–Stokes equation can be derived on a rectangular grid. Our theoretical findings are confirmed by numerical solutions using three two-dimension benchmark problems, i.e. the channel flow, the cavity flow, and the decaying Taylor–Green vortex flow. The numerical results demonstrate that the proposed model shows remarkably good performance with appropriate choice of model parameters.