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 (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.

Three-Dimensional Central Moment Lattice Boltzmann Method on a Cuboid Lattice for Anisotropic and Inhomogeneous Flows

“…As discussed in the introduction section, we will not orthogonalize the above set of basis vectors further to maintain simplicity and robustness and retain them in their natural forms in the following derivation based on the 3D cuboid lattice analogous to our rectangular central moment LB formulation [26].…”

“…The rationale for the limitations of the prior rectangular LB schemes were clarified in our recent work [24][25][26]. These include their choice of the orthogonal moment basis, construction of the discrete equilibria involving only the lower order velocity terms and without correcting for the non-Galilean invariant cubic velocity errors arising from aliasing effects, and the use of collision models based on raw moments.…”

“…The construction of the collision step in the LB formulations using central moments based on the peculiar velocity [29,30] naturally maintains the Galilean invariance of the moments independently supported by the lattice and its advantages have been demonstrated for a variety of fluid dynamical problems (see e.g., [31][32][33][34][35][36]). Based on these considerations, we proposed a 2D central moment rectangular LB scheme recently and demonstrated its superior numerical features for simulating flows at higher Reynolds numbers using relatively small grid aspect ratio when compared to the other existing LB methods based on the rectangular lattice [24][25][26].…”

“…Moreover, as mentioned above, the use of central moments is expected to provide significant improvements to the state-of-the-art in LBM based on non-cubic grids. However, a 3D central moment LBM based on a cuboid lattice does not currently exist in the literature, which can be constructed via an extension and significant modification of our recent work on the rectangular lattice [26] and is the main objective of this present paper.…”

“…In contrast to the prior approaches, our cuboid LB scheme will be constructed by using a natural moment basis that avoids orthogonalization, and the discrete central moment equilibria are specified by matching with those corresponding to the continuous Maxwell distribution function. Moreover, the prior LB algorithms on stretched lattice grids based on the various collision models discussed above, including our recent 2D rectangular central moment LB scheme [26], prescribe the transformations between the distribution functions and raw moments before and after collision by using a moment basis that separates the trace of the second order moments (related to the bulk viscosity) from its other components (related to shear viscosity).…”