Impact of collision models on the physical properties and the stability of lattice Boltzmann methods

Abstract

The lattice Boltzmann method (LBM) is known to suffer from stability issues when the collision model relies on the BGK approximation, especially in the zero viscosity limit and for non-vanishing Mach numbers. To tackle this problem, two kinds of solutions were proposed in the literature. They consist in changing either the numerical discretization (finite-volume, finite-difference, spectral-element, etc.) of the discrete velocity Boltzmann equation (DVBE), or the collision model. In this work, the latter solution is investigated in detail. More precisely, we propose a comprehensive comparison of (static relaxation time based) collision models, in terms of stability, and with preliminary results on their accuracy, for the simulation of isothermal high-Reynolds number flows in the (weakly) compressible regime. It starts by investigating the possible impact of collision models on the macroscopic behaviour of stream-and-collide based D2Q9-LBMs, which clarifies the exact physical properties of collision models on LBMs. It is followed by extensive linear and numerical stability analyses, supplemented with an accuracy study based on the transport of vortical structures over long distances. In order to draw conclusions as generally as possible, the most common moment spaces (raw, central, Hermite, central Hermite and cumulant), as well as regularized approaches, are considered for the comparative studies. LBMs based on dynamic collision mechanisms (entropic collision, subgrid-scale models, explicit filtering, etc.) are also briefly discussed. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.

Efficient supersonic flow simulations using lattice Boltzmann methods based on numerical equilibria

“…A crucial importance is therefore attributed to the explicit calculation of the equilibrium term, as a function of the macroscopic variables density, velocity and temperature. As a matter of fact, the expression of the equilibrium distribution can be considered to entirely determine the physics expressed by the model [94,95], if and only if the different non-equilibrium contributions are relaxed to recover the correct transport coefficients of the macroscopic equations of interest (see [96] for a systematic approach in the context of fluid mechanics).…”

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

“…We will call Eq. (4) the cumulant SRT equation or KSRT (where we use 'K' for cumulants [20,53] due to the fact that 'C' has been established for central moments in the lattice Boltzmann context and because cumulant is written with 'K' in many European languages). Further, we denote the BGK operator with the same equilibrium as the cumulant operator as BGK+ in order to distinguish it from the standard lattice BGK model which uses a Taylor expanded Maxwellian as equilibrium.…”

“…The idea is always that the collision operator is not applied to the distributions directly but to some sort of moments such that different relaxation parameters can be attached to different observable quantities. While the original multiple relaxation time models used unweighted orthogonal raw moments [14][15][16], a wide range of other options have now been promoted, including central moments [17][18][19], Hermite moments [20] and cumulants [21]. Besides these complete transformations there exist methods separating hydrodynamic relevant information from so-called ghost modes using only a small set of categories (two or three).…”

“…This is not only true for the incompressible Navier-Stokes equation, but it is intrinsic to the fact that a static velocity set is used in connection with explicit time marching algorithm that allows information only to travel a finite distance per time step. Despite this theoretical obstacle it has become quite popular to apply the lattice Boltzmann method to aero-acoustic problems and to dismiss analysis based on diffusive scaling accordingly [20]. It is therefor of considerable practical interest to investigate the fourth-order convergent cumulant method in both diffusive and acoustic scaling, which is one of the main aims of the current study.…”

Extensive analysis of the lattice Boltzmann method on shifted stencils

“…For the Hermite polynomial formulations, one obtains the upper limit Ma max = √ 3 − 1 ≈ 0.73, with Ma max obtained by considering all possible orientations of the mean flow and keeping the minimal value. Interestingly, one cannot increase this critical Mach number by changing the collision model [43,79] or the numerical scheme [80]. In fact, this can only be achieved by modifying the velocity discretization and the equilibrium state [9,16,43].…”