Comprehensive comparison of collision models in the lattice Boltzmann framework: Theoretical investigations

Abstract

Over the last decades, several types of collision models have been proposed to extend the validity domain of the lattice Boltzmann method (LBM), each of them being introduced in its own formalism. The present article proposes a formalism that describes all these methods within a common mathematical framework, and in this way allows us to draw direct links between them. Here, the focus is put on single and multirelaxation time collision models in either their raw moment, central moment, cumulant or regularized form. In parallel with that, several bases (non orthogonal, orthogonal, Hermite) are considered for the polynomial expansion of populations. General relationships between moments are first derived to understand how moment spaces are related to each other. In addition, a review of collision models further sheds light on collision models that can be rewritten in a linear matrix form. More quantitative mathematical studies are then carried out by comparing explicit expressions for the post collision populations. Thanks to this, it is possible to deduce the impact of both the polynomial basis (raw, Hermite, central, central Hermite, cumulant) and the inclusion of regularization steps on isothermal LBMs. Extensive results are provided for the D1Q3, D2Q9, and D3Q27 lattices, the latter being further extended to the D3Q19 velocity discretization. Links with the most common two and multirelaxation time collision models are also provided for the sake of completeness. The present work ends by emphasizing the importance of an accurate representation of the equilibrium state, independently of the choice of moment space. As an addition to the theoretical purpose of the present article, general instructions are provided to help the reader with the implementation of the most complicated collision models.

Extensive analysis of the lattice Boltzmann method on shifted stencils

“…since its derivation does not account for the conservation of the isothermal energy as compared to its polynomial counterpart [22].…”

“…The concept of optimal equilibration paths in the context of the LBM has been an area of active research and resulted in a variety of modified collision models [22]. The first class of modified collision models is based on d'Humière's generalized lattice Boltzmann equations [23].…”

“…Attempts at further reducing velocity-dependent higher-order error terms later resulted in approaches such as the cascaded method based on relaxation of central moments (evaluated in the local fluid velocity frame) instead of raw moments, and the cumulants LBM [32,33]. It is also worth noting that similar asymptotic behaviors could also be obtained by adopting extended equilibria, even with the very simple Bhatnagar-Gross-Krook (BGK) collision operator [22].…”

“…While most EDFs recover the same moments appearing at the Navier-Stokes (NS) level, they differ in their higher-order moments. Although a wide variety of EDFs have been proposed over the years, two systematic approaches to constructing discrete EDFs have become widely popular [22]: (a) discretization of the continuous EDF via projection onto Hermite polynomials space [52,53] and (b) construction of a discrete equilibrium state based on extremization of a discrete entropy functional [46,48,54].…”

“…These errors come from inconsistencies in the third-order moments tensor of the EDF. While diagonal components of this tensor cannot be directly corrected due to insufficient symmetry of the first-neighbor stencils, the off-diagonal components can be correctly recovered using higher-order Hermite expansions of the EDF that are naturally accounted for in the most sophisticated collision models [22]. For the D2Q9 stencil, for example, keeping up to third-order terms of the expansion is sufficient to correctly recover the off-diagonal components.…”