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.

Energy transfers and spectral eddy viscosity in large-eddy simulations of homogeneous isotropic turbulence: Comparison of dynamic Smagorinsky and multiscale models over a range of discretizations Large-eddy simulation ͑LES͒ with regular explicit filtering is investigated. The filtered-scale stress due to the explicit filtering is here partially reconstructed using the tensor-diffusivity model: It provides for backscatter along the stretching direction͑s͒, and for global dissipation, both also attributes of the exact filtered-scale stress. The necessary LES truncations ͑grid and numerical method͒ are responsible for an additional subgrid-scale stress. A natural mixed model is then the tensor-diffusivity model supplemented by a dynamic Smagorinsky term. This model is reviewed, together with useful connections to other models, and is tested against direct numerical simulation ͑DNS͒ of turbulent isotropic decay starting with Re ϭ90 ͑thus moderate Reynolds number͒: LES started from a 256 3 DNS truncated to 64 3 and Gaussian filtered. The tensor-diffusivity part is first tested alone; the mixed model is tested next. Diagnostics include energy decay, enstrophy decay, and energy spectra. After an initial transient of the dynamic procedure ͑observed with all models͒, the mixed model is found to produce good results. However, despite expectations based on favorable a priori tests, the results are similar to those obtained when using the dynamic Smagorinsky model alone in LES without explicit filtering. Nevertheless, the dynamic mixed model appears as a good compromise between partial reconstruction of the filtered-scale stress and modeling of the truncations effects ͑incomplete reconstruction and subgrid-scale effects͒. More challenging 48 3 LES are also done: Again, the results of both approaches are found to be similar. The dynamic mixed model is also tested on the turbulent channel flow at Re ϭ395. The tensor-diffusivity part must be damped close to the wall in order to avoid instabilities. Diagnostics are mean profiles of velocity, stress, dissipation, and reconstructed Reynolds stresses. The velocity profile obtained using the damped dynamic mixed model is slightly better than that obtained using the dynamic Smagorinsky model without explicit filtering. The damping used so far is however crude, and this calls for further work.

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