A mixed multiscale model better accounting for the cross term of the subgrid-scale stress and for backscatter

Abstract

In the large-eddy simulation (LES) of turbulent flows, models are used to account for the subgrid-scale (SGS) stress. We here consider LES with “truncation filtering only” (i.e., that due to the LES grid), thus without regular explicit filtering added. The SGS stress tensor is then composed of two terms: the cross term that accounts for interactions between resolved scales and unresolved scales, and the Reynolds term that accounts for interactions between unresolved scales. Both terms provide forward- (dissipation) and backward (production, also called backscatter) energy transfer. Purely dissipative, eddy-viscosity type, SGS models are widely used: Smagorinsky-type models, or more advanced multiscale-type models. Dynamic versions have also been developed, where the model coefficient is determined using a dynamic procedure. Being dissipative by nature, those models do not provide backscatter. Even when using the dynamic version with local averaging, one typically uses clipping to forbid negative values of the model coefficient and hence ensure the stability of the simulation; hence removing the backscatter produced by the dynamic procedure. More advanced SGS model are thus desirable, and that better conform to the physics of the true SGS stress, while remaining stable. We here investigate, in decaying homogeneous isotropic turbulence, and using a de-aliased pseudo-spectral method, the behavior of the cross term and of the Reynolds term: in terms of dissipation spectra, and in terms of probability density function (pdf) of dissipation in physical space: positive and negative (backscatter). We then develop a new mixed model that better accounts for the physics of the SGS stress and for the backscatter. It has a cross term part which is built using a scale-similarity argument, further combined with a correction for Galilean invariance using a pseudo-Leonard term: this is the term that also does backscatter. It also has an eddy-viscosity multiscale model part that accounts for all the remaining phenomena (also for the incompleteness of the cross term model), that is dynamic and that adjusts the overall dissipation. The model is tested, both a priori and a posteriori, and is compared to the direct numerical simulation and to the exact SGS terms, also in time. The model is seen to provide accurate energy spectra, also in comparison to the dynamic Smagorinsky model. It also provides significant backscatter (although four times less than the real SGS stress), while remaining stable.