Optimizing free parameters in the D3Q19 Multiple-Relaxation lattice Boltzmann methods to simulate under-resolved turbulent flows

Linear and brute force stability of orthogonal moment multiple-relaxation-time lattice Boltzmann methods applied to homogeneous isotropic turbulence

“…[31]). Further, in [36] s^q and s^μ (table 1) are implicitly dependent on the lattice Mach number italicMa and ν. Whereas this dependency is not further discussed therein, here it is consciously emphasized.…”

“…Here and in the following i,j∈falsefalse{0,1,2,…,q−1falsefalse} are reserved as discrete velocity counters. The MRT collision operator can be generically formulated as [36] Ji=−Kifalse[bold-italicf−feqfalse], where feq=false(fieqfalse)i∈double-struckRq is the typical second-order truncated Maxwellian [4,5] to obtain (2.1) in the diffusive limit, and Ki=false(Ki,jfalse)j∈double-struckRq denotes the i-th row vector of the matrix K=M−1SM∈double-struckRq×q. The relaxation matrix S=diagfalse(bold-italicsfalse)∈double-struckRq×q contains the relaxation frequencies ...…”

“…Since the ordering of rows and columns in M is non-unique, it is convenient to use the tensor specification alongside population indexing. Table 1 summarizes the orthogonal moment MRT construction for D3Q19 and additionally lists the relaxation frequency configurations proposed in [33,36]. Note that the present relaxation times contained in S are kept variable on purpose and characterized via the introduction of stability sets given below.…”

“…The orthogonal polynomial basis which spans the moment space constructed in [33] can be reduced to obtain the basis given in [36] (see e.g. [31]).…”

“…Chávez-Modena et al [35] proposed optimized relaxation times to ensure the dissipation of under-resolved flow structures. The stability of the thus obtained MRT schemes in two [35] and three dimensions [36] for orthogonal and central moments was studied with von Neumann (VN) analysis and successfully applied to the three-dimensional Taylor–Green vortex (TGV) flow in a specific range of resolutions, Mach and Reynolds numbers. Nevertheless, the aforementioned optimization leads to specific sets of constant relaxation times, and has been carried out exclusively in acoustic scaling (AS).…”