  Simulating inhomogeneous flows with different characteristic scales in
different coordinate directions using the collide-and-stream based lattice
Boltzmann methods (LBM) can be accomplished efficiently using rectangular
lattice grids. We develop and investigate a new rectangular central moment LBM
based on non-orthogonal moment basis (referred to as RC-LBM). 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 restores the isotropy of the viscous stress tensor and
eliminates the non-Galilean invariant cubic velocity terms of the resulting
hydrodynamical equations. A special case of this rectangular formulation
involving the raw moments (referred to as the RNR-LBM) is also constructed. The
resulting schemes represent a considerable simplification, especially for the
transformation matrices and isotropy corrections, and improvement over the
existing MRT-LB schemes on rectangular lattice grids that use orthogonal moment
basis. Numerical validation study of both the RC-LBM and RNR-LBM for a variety
of benchmark flow problems are performed that show good accuracy at various
grid aspect ratios. The ability of our proposed schemes to simulate flows using
relatively lower grid aspect ratios than considered in prior rectangular LB
approaches is demonstrated. Furthermore, simulations reveal the superior
stability characteristics of the RC-LBM over RNR-LBM 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 is shown.
