Preconditioned Central Moment Lattice Boltzmann Method on a Rectangular Lattice Grid for Accelerated Computations of Inhomogeneous Flows

Convergence acceleration of flow simulations to their steady states at lower Mach numbers can be achieved via preconditioning the lattice Boltzmann (LB) schemes that alleviate the associated numerical stiffness, which have so far been constructed on square lattices. We present a new central moment LB method on rectangular lattice grids for efficient computations of inhomogeneous and anisotropic flows by solving the preconditioned Navier-Stokes (PNS) equations. Moment equilibria corrections are derived via a Chapman-Enskog analysis for eliminating the truncation errors due to grid-anisotropy arising from the use of the rectangular lattice and the non-Galilean invariant cubic velocity errors resulting from an aliasing effect on the standard D2Q9 lattice for consistently recovering the PNS equations. Such corrections depend on the diagonal components of the velocity gradients, which are locally obtained from the second order non-equilibrium moments and parameterized by an associated grid aspect ratio $r$ and a preconditioning parameter $\gamma$, and the speed of sound in the collision model is naturally adapted to $r$ via a physically consistent strategy. We develop our approach by using a robust non-orthogonal moment basis and the central moment equilibria are based on a matching principle, leading to simpler expressions for the corrections for using the rectangular grids and for representing the viscosities as functions of the relaxation parameters, $r$ and $\gamma$, and its implementation is modular allowing a ready extension of the existing LB schemes based on the square lattice. Numerical simulations of inhomogeneous and anisotropic shear-driven bounded flows using the preconditioned rectangular central moment LB method demonstrate the accuracy and significant reductions in the numbers of steps to reach the steady states for various sets of characteristic parameters.