On the modelling of the subgrid-scale and filtered-scale stress tensors in large-eddy simulation

The large-eddy simulation (LES) equations are obtained from the application of two operators to the Navier-Stokes equations: a smooth filter and a discretization operator. The introduction ab initio of the discretization influences the structure of the unknown stress in the LES equations, which now contain a subgrid-scale stress tensor mainly due to discretization, and a filtered-scale stress tensor mainly due to filtering. Theoretical arguments are proposed supporting eddy viscosity models for the subgrid-scale stress tensor. However, no exact result can be derived for this term because the discretization is responsible for a loss of information and because its exact nature is usually unknown. The situation is different for the filtered-scale stress tensor for which an exact expansion in terms of the large-scale velocity and its derivatives is derived for a wide class of filters including the Gaussian, the tophat and all discrete filters. As a consequence of this generalized result, the filtered-scale stress tensor is shown to be invariant under the change of sign of the large-scale velocity. This implies that the filtered-scale stress tensor should lead to reversible dynamics in the limit of zero molecular viscosity when the discretization effects are neglected. Numerical results that illustrate this effect are presented together with a discussion on other approaches leading to reversible dynamics like the scale similarity based models and, surprisingly, the dynamic procedure.