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

We present a D3Q19 lattice scheme based in MRT with central moments (MRT-CM), where the free parameters of the model are optimized to dissipate under-resolved flow structures with high wavenumbers. In Chávez-Modena et al. in Computers & Fluids 172:397-409, 2018 [1], we compared the BGK, MRT-RM and MRT-CM for the D2Q9 lattice scheme using von Neumann analyses and quantified their numerical properties. Based on this study, we proposed an optimized 2D MRT-CM scheme with enhanced stability for under-resolved flows. Here, we extend this idea to the D3Q19 MRT-CM scheme. As before, we base our optimization for the free parameters, on the k-1% dispersion-error rule, that states that waves with dispersive errors above 1% should be dissipated since they pollute the solution and may cause instabilities. To this aim we increase dissipation in the scheme for waves with dispersive errors above 1%. The resulting optimized scheme is verified through a von Newmann analysis and validated for the three-dimensional Taylor-Green isotropic turbulent flow. We show how the original D3Q19 MRT-CM (d’Humières version) leads to unrealistic kinetic energy accumulation at high wave numbers, whilst our optimized MRT-CM provides the correct energy dissipated rate, avoiding energy build up at high wavenumbers. These results suggest that our optimization strategy enhances stability and allows for accurate energy spectra in under-resolved flow simulations such as typically found in Large Eddy Simulations.