Role of higher-order Hermite polynomials in the central-moments-based lattice Boltzmann framework

Abstract

The cascaded lattice Boltzmann method decomposes the collision stage on a basis of central moments on which the equilibrium state is assumed equal to that of the continuous Maxwellian distribution. Such a relaxation process is usually considered as an assumption, which is then justified a posteriori by showing the enhanced Galilean invariance of the resultant algorithm. An alternative method is to relax central moments to the equilibrium state of the discrete second-order truncated distribution. In this paper, we demonstrate that relaxation to the continuous Maxwellian distribution is equivalent to the discrete counterpart if higher-order (up to sixth) Hermite polynomials are used to construct the equilibrium when the D3Q27 lattice velocity space is considered. Therefore, a theoretical a priori justification of the choice of the continuous distribution is formally provided for the first time.

Under-resolved and large eddy simulations of a decaying Taylor–Green vortex with the cumulant lattice Boltzmann method

“…In addition to the basis transformation, recent research has focused on the role of the equilibrium distribution [46,47]. The cumulant method intrinsically uses the same number of monomials in the equilibrium function as the number of discrete velocities.…”

“…One goal of the current study is to investigate differences of these models through the numerical benchmark of a decaying three-dimensional Taylor-Green vortex. Even though classical MRT models [14][15][16]70] based on raw moments and central moment-based models [17,18,46,[71][72][73][74][75] including factorized central moment methods [76,77] are part of the heritage of the cumulant method, we abstain from including them into the current analysis. This is mostly because the deficiencies of these models have already been widely discussed (see in particular our discussion in [21]).…”

Stability of the lattice kinetic scheme and choice of the free relaxation parameter

“…The off-diagonal component of the stress tensor recovered by the classical lattice Boltzmann formulation with the SRT collision operator and a second-order EDF are known to deviate from the Newtonian fluid stress tensor for large values of the Mach number following O(Ma 3 ). An asymptotic analysis shows that introducing the third-order Hermite polynomial in the EDF can eliminate the leading-order error term in the stress tensor 9,38,47,49,[112][113][114] . This effect can also be observed in the spectral behavior of the scheme, namely the spectral speed and dissipation 50,101 .…”

“…The majority of these modified LB schemes were developed and proposed with the aim of extending the stability domain of the solver in order to access higher Reynolds and/or Mach numbers at reasonable costs. They include (but are not limited to) approaches such as the entropic lattice Boltzmann method (ELBM) [10][11][12][13][14][15][16][17][18] , the multiple relaxation (MRT) [19][20][21][22][23][24][25][26][27][28][29][30][31] , the central moments (CM) or cascaded LBM [32][33][34][35][36][37][38][39] , the regularized (RLBM) [40][41][42][43][44][45][46][47][48][49][50] and the Cumulant method [51][52][53][54] . In addition to these collision models, several numerical discretizations have been used to numerically solve the lattice Boltzamnn equation, see for example [55]…”

Central-Moments-Based Lattice Boltzmann for Associating Fluids: A New Integrated Approach

“…and d,c,h,i,j,ž ‚N = 0. Indeed, as stated in De Rosis and Luo 86 , the discrete equilibrium CMs are equal to those of the continuous Maxwellian distribution when the full set of Hermite polynomials is considered. Let us denote the 9 * 9 unit tensor as and the 9 * 9 relaxation matrix as…”