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

Abstract

Lattice Boltzmann (LB) methods are usually developed on cubic lattices that discretize the configuration space using uniform grids. For efficient computations of anisotropic and inhomogeneous flows, it would be beneficial to develop LB algorithms involving the collision-and-stream steps based on orthorhombic cuboid lattices. We present a new 3D central moment LB scheme based on a cuboid D3Q27 lattice. This scheme involves two free parameters representing the ratios of the characteristic particle speeds along the two directions with respect to those in the remaining direction, and these parameters are referred to as the grid aspect ratios. Unlike the existing LB schemes for cuboid lattices, which are based on orthogonalized raw moments, we construct the collision step based on the relaxation of central moments and avoid the orthogonalization of moment basis, which leads to a more robust formulation. Moreover, prior cuboid LB algorithms prescribe the mappings between the distribution functions and raw moments before and after collision by using a moment basis designed to separate the trace of the second order moments (related to bulk viscosity) from its other components (related to shear viscosity), which lead to cumbersome relations for the transformations. By contrast, in our approach, the bulk and shear viscosity effects associated with the viscous stress tensor are naturally segregated only within the collision step and not for such mappings, while the grid aspect ratios are introduced via simpler pre- and post-collision diagonal scaling matrices in the above mappings. These lead to a compact approach, which can be interpreted based on special matrices. It also results in a modular 3D LB scheme on the cuboid lattice, which allows the existing cubic lattice implementations to be readily extended to those based on the more general cuboid lattices. To maintain the isotropy of the viscous stress tensor of the 3D Navier–Stokes equations using the cuboid lattice, corrections for eliminating the truncation errors resulting from the grid anisotropy as well as those from the aliasing effects are derived using a Chapman–Enskog analysis. Such local corrections, which involve the diagonal components of the velocity gradient tensor and are parameterized by two grid aspect ratios, augment the second order moment equilibria in the collision step. We present a numerical study validating the accuracy of our approach for various benchmark problems at different grid aspect ratios. In addition, we show that our 3D cuboid central moment LB method is numerically more robust than its corresponding raw moment formulation. Finally, we demonstrate the effectiveness of the 3D cuboid central moment LB scheme for the simulations of anisotropic and inhomogeneous flows and show significant savings in memory storage and computational cost when used in lieu of that based on the cubic lattice.

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

“…Also, in Ref. [35], we also explicitly demonstrated the computational advantages of using a rectangular lattice in lieu of a square lattice in solving inhomogeneous and anisotropic flows. Moreover, the central moment LB method on a cuboid lattice presented in Ref.…”

“…However, many of these methods involved cumbersome implementations, complicated expressions for the corrections, and numerical stability issues when the grid aspect ratio of the rectangular lattice (defined later) is significantly far off from unity (i.e., characterizing strong grid stretching in one of the directions relative to the other) or for simulating flows with relatively low viscosities or high Reynolds numbers. On the other hand, recognizing that the use of central moments, which naturally preserves the Galilean invariance of those moments independently supported by the lattice, can significantly improve the stability and accuracy when compared to the use of raw moments [6,18,19,20,21,22,23,24,25,26,27,28,29,30,31,31,32,33], we recently constructed a rectangular central moment LB method (RC-LBM) [34], which was then further extended to three-dimensions with an improved implementation strategy [35]. While the original central moment LB scheme was constructed using an orthogonal moment basis [6], Geier et al [7] in 2015 provided a detailed discussion on the role of the moment basis in their development of a cumulant LB method and also constructed a variety of collision models, including those based on raw moments, central moments and cumulants using non-orthogonal moment basis and presented them in the various appendices of [7].…”

“…Moreover, the central moment LB method on a cuboid lattice presented in Ref. [35] is modular in construction thereby allowing ready extension of the existing algorithms on a cubic lattice to cuboid lattices, and provides a unified formulation with corrections that are applicable for a wide variety of all the standard collision models.…”

“…By contrast, in this paper, following our recent work in Ref. [35], the PRC-LBM will segregate the bulk viscosity from the shear viscosity only within the step involving relaxation of moments to preconditioned equilibria under collision, and the pre-and post-collision mapping matrices involve a simpler moment basis and account for the grid aspect ratios only via certain diagonal scaling matrices. This results in a modular implementation of the PRC-LBM, and its formulation involves simpler expressions for the necessary corrections and the transport coefficients.…”

Rectangular multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations: General equilibrium and some important issues

“…However, these two models require a relatively complex quasiequilibrium collision step. In addition, Yahia et al [26] developed a rectangular central-moment MRT-LBM based on a non-orthogonal moment basis, and then extended it to three-dimensional central-moment LBM on a cuboid lattice [27]. The equilibrium to which the central moments relax under collision in this approach is obtained from those corresponding to the continuous Maxwell distribution.…”