Central moment lattice Boltzmann method on a rectangular lattice

Abstract

Simulating inhomogeneous flows with different characteristic scales in different coordinate directions using the collide-and-stream based lattice Boltzmann methods (LBMs) can be accomplished efficiently using rectangular lattice grids. We develop and investigate a new rectangular central moment LBM based on a non-orthogonal moment basis and involving multiple relaxation times. The equilibria to which the central moments relax under collision in this approach are obtained from matching with those corresponding to the continuous Maxwell distribution. A Chapman–Enskog analysis is performed to derive the correction terms to the second order moment equilibria involving the grid aspect ratio and velocity gradients that restore the isotropy of the viscous stress tensor and eliminate the non-Galilean invariant cubic velocity terms in the resulting hydrodynamical equations. A special case of this rectangular formulation involving the raw moments is also constructed. The resulting schemes represent a considerable simplification, especially for the transformation matrices and isotropy corrections, and improvement over the existing lattice Boltzmann schemes based on raw moments on rectangular lattice grids that use orthogonal moment basis. Numerical validation study of both the proposed rectangular LBMs for a variety of benchmark flows is performed, which shows good accuracy at various grid aspect ratios. The ability of our proposed schemes to simulate flows at relatively low grid aspect ratios and higher Reynolds numbers than considered in prior approaches is demonstrated. Furthermore, simulations reveal the superior stability characteristics of the rectangular central moment LBM over that based on raw moments in handling shear flows at lower viscosities and/or higher characteristic velocities. In addition, computational advantages of using our rectangular LB formulation in lieu of that based on the square lattice are shown.