Incorporating forcing terms in cascaded lattice Boltzmann approach by method of central moments

Abstract

Cascaded lattice-Boltzmann method (Cascaded-LBM) employs a new class of collision operators aiming to stabilize computations and remove certain modeling artifacts for simulation of fluid flow on lattice grids with sizes arbitrarily larger than the smallest physical dissipation length scale (Geier et al., Phys. Rev. E 63, 066705 (2006)). It achieves this and distinguishes from other collision operators, such as in the standard single or multiple relaxation time approaches, by performing relaxation process due to collisions in terms of moments shifted by the local hydrodynamic fluid velocity, i.e. central moments, in an ascending order-by-order at different relaxation rates. In this paper, we propose and derive source terms in the Cascaded-LBM to represent the effect of external or internal forces on the dynamics of fluid motion. This is essentially achieved by matching the continuous form of the central moments of the source or forcing terms with its discrete version.Different forms of continuous central moments of sources, including one that is obtained from a local Maxwellian, are considered in this regard. As a result, the forcing terms obtained in this new formulation are Galilean invariant by construction. To alleviate lattice artifacts due to forcing terms in the emergent macroscopic fluid equations, they are proposed as temporally semi-implicit and second-order, and the implicitness is subsequently effectively removed by means of a transformation to facilitate computation. It is shown that the impressed force field influences the cascaded collision process in the evolution of the transformed distribution function. The method of central moments along with the associated orthogonal properties of the moment basis completely determines the analytical expressions for the source terms as a function of the force and macroscopic velocity fields. In contrast to the existing forcing schemes, it is found that they involve higher order terms in velocity space. It is shown that the proposed approach implies "generalization" of both local equilibrium and source terms in the usual lattice frame of reference, which depend on the ratio of the relaxation times of moments of different orders. An analysis by means of the Chapman-Enskog multiscale expansion shows that the Cascaded-LBM with forcing terms is consistent with the Navier-Stokes equations. Computational experiments with canonical problems involving different types of forces demonstrate its accuracy.

Three-dimensional cascaded lattice Boltzmann method: Improved implementation and consistent forcing scheme

“…According to the binomial theorem, a central moment can be expressed by raw moments from the lowest order to the same order [23,28], and thus the shift matrix N is a lower triangular matrix. As seen in Eq.…”

“…Compared with the forcing schemes in Refs. [23,28,41], the consistent forcing scheme has better numerical performances in terms of accuracy, isotropy, and faithful implementation of the nonslip boundary condition. Therefore, the present paper aims to simplify the implementation and reduce the computational cost for 3D CLBM.…”

Disproportionate entrance length in superfluid flows and the puzzle of counterflow instabilities

“…Finally, let us mention that the standard BGK collision term has been eventually replaced by an equivalent central-moment collision operator, as detailed in Refs. [78,79], in order to enhance the numerical stability of the scheme.…”

PROTEUS: A coupled iterative force-correction immersed-boundary cascaded lattice Boltzmann solver for moving and deformable boundary applications

“…, 8 are the discrete forcing terms. Using the transformation matrix K [42,12] the collision operator takes the form Ω a = (K ·k) a , wherek =k a (x, t) : α = 0, 1, . .…”

“…Chosing the nominal moment basis and the orthogonalised matrix K [42], the equlibrium distribution contains higher order velocity terms as compared to the standard LBM [6]. …”