A lattice-Boltzmann scheme of the Navier–Stokes equation on a three-dimensional cuboid lattice

Abstract

The standard lattice-Boltzmann method (LBM) for fluid flow simulation is based on a square (in 2D) or a cubic (in 3D) lattice grids. Recently, two new lattice Boltzmann schemes have been developed on a 2D rectangular grid using the MRT (multiple-relaxation-time) collision model, either by adding a free parameter in the definition of moments or by extending the equilibrium moments. These models satisfy all isotropy conditions and are fully consistent to the Navier–Stokes equations. Here we developed a lattice Boltzmann model on a 3D cuboid lattice, namely, a lattice grid with different grid lengths in different spatial directions. We designed the moment equations, derived from our MRT–LBM model through the Chapman–Enskog analysis, to be fully consistent with the Navier–Stokes equations. A second-order term is added to the equilibrium moments in order to not only satisfy all isotropy conditions but also to better accommodate different values of shear and bulk viscosities. The form of the second-order term and the coefficients of the extended equilibrium moments are determined through an inverse design process. An additional benefit of the model is that the shear viscosity can be adjusted, independent of the stress-moment relaxation parameter, thus potentially improving the numerical stability of the model. The resulting cuboid MRT–LBM model is then validated through benchmark simulations using the laminar channel flow, the turbulent channel flow, and the 3D time-dependent, energy-cascading Taylor–Green vortex flow. The second-order accuracy of the proposed model is also demonstrated. The aspect ratios for numerical stability appear to be constrained at high flow Reynolds numbers, especially for turbulent flow simulations, an aspect requiring further investigation.