A mixed Smagorinsky and Langevin-equation-based subgrid-scale model for large-eddy simulation of turbulence including stochastic backscatter

Traditional subgrid-scale (SGS) models in large-eddy simulation (LES), such as the dynamic Smagorinsky model, typically predict deterministic relationships between the subgrid-scales and the resolved scales, as well as non-negative SGS energy dissipation everywhere. However, for flows in which unknowable effects from unresolved scales can play a role, traditional SGS models remain problematic due to the purely deterministic and dissipative nature of eddy-viscosity. Effective SGS closures must therefore incorporate some effects of the small-scale dynamics, including energy backscatter, i.e. local inverse cascade of energy. Here, we introduce a mixed Smagorinsky Langevin-equation-based SGS (MSLS) model, combining a traditional dynamic Smagorinsky closure with a stochastic component modeled by a Langevin equation. The stochastic term is constructed to ensure momentum conservation, Galilean invariance, and numerical stability. Stochastic SGS velocity fluctuations are advanced from a Lagrangian perspective on the LES grid and are used to construct the stochastic part of the SGS model. We first verify by means of one-dimensional scenarios that the stochastic portion replicates Kolmogorov-scale scaling laws also identifying suitable coefficient ranges. Next, the full SGS model is tested in LES of decaying homogeneous isotropic turbulence at two different Reynolds numbers, using a versatile finite difference code. Results yield good agreement with data, with the added stochastic part showing minimal impact on the energy spectrum which, as is well-known, can be modeled successfully with traditional eddy viscosity models. However, unlike traditional SGS models, the MSLS approach predicts significant amounts of energy backscatter without causing numerical instabilities, as demonstrated by measuring probability densities of SGS energy dissipation and comparing with DNS. We also illustrate how the embedded Wiener process causes variability across realizations to grow exponentially in time initially, transitioning to linear growth, and ultimately saturating after several eddy turnover times.