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

Abstract

We present a comprehensive analysis of the cumulant lattice Boltzmann model with the three-dimensional Taylor–Green vortex benchmark at Reynolds number 1600. The cumulant model is investigated in several different variants, using regularization, fourth-order convergent diffusion and fourth-order convergent advection with and without limiters. In addition, a cumulant model combined with a WALE sub-grid scale model is being evaluated. The turbulence model is found to filter out the high wave number contributions from the energy spectrum and the enstrophy, while the non-filtered cumulant methods show good correspondence to spectral simulations even for the high wave numbers. The application of the WALE turbulence model appears to be counter productive for the Taylor–Green vortex at a Reynolds number of 1600. At much higher Reynolds numbers ($${\hbox {Re}}=160{,}000$$ Re = 160 , 000 ) a deviation from the ideal Kolmogorov theory can be observed in the absence of an explicit turbulence model. Cumulant models with fourth-order convergent diffusion show much better results than single relaxation time methods.

Highly efficient lattice Boltzmann multiphase simulations of immiscible fluids at high-density ratios on CPUs and GPUs through code generation

“…While more advanced LB collision operators like the cumulant LBM are available in the lbmpy framework, this has not yet been employed in the phase-field algorithm. As described by Geier et al (2021), a careful numeric design becomes crucial when using the LBM, especially with more sophisticated collision operators. The annotated symbolic derivation of the algorithm can be made aware of roundoff and stability concerns.…”

Lattice Boltzmann Solver for Multiphase Flows: Application to High Weber and Reynolds Numbers

“…In the context of the present study, the low-Mach model was wrapped in a moment-based formulation where postcollision populations to be streamed as are computed as The postcollision preconditioned population is where is the moment transform matrix from preconditioned populations to the target momentum space, is the identity matrix, and is the diagonal relaxation frequency matrix. Following [ 34 ], prior to transformation to momentum space, populations are preconditioned as This preconditioning accomplishes two tasks, namely, normalizing the populations with density and thus eliminating the density dependence of the moments, and introducing the first half of the source term. As such, moments are computed as The transformation from distribution function (DF)s to cumulants is carried out using the steps suggested in [ 35 ], which allows for a more efficient algorithm.…”

Lattice Boltzmann solver for multi-phase flows: Application to high Weber and Reynolds numbers

“…where C is the moments transform matrix -from pre-conditioned populations to the target momentum space, I the identity matrix and W the diagonal relaxation frequency matrix. Following [34] prior to transformation to momentum space the populations are pre-conditioned as:…”