Statistics of turbulence subgrid-scale stresses: Necessary conditions and experimental tests

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Abstract
Some thoughts are presented regarding the question: when can a subgrid‐scale model yield correct statistics of resolved fields in a large‐eddy simulation (LES) of turbulent flow. The filtered Navier–Stokes equations are used to find necessary conditions on the statistical properties of the modeled subgrid‐scale stress tensor, for statistical equivalence between a ‘‘real’’ and a modeled (via LES) turbulent velocity field. When trying to formulate sufficient conditions, an unclosed hierarchy of expressions is obtained, essentially due to the ‘‘turbulence problem’’ of the resolved scales of motion. Experimental (statistical a priori) testing of subgrid‐scale models is performed, based on single‐probe measurements in grid turbulence and on several key assumptions. Three versions of the eddy‐viscosity model are considered: constant eddy viscosity, subgrid kinetic energy, and the usual Smagorinsky eddy viscosity. Measured joint moments between filtered velocity and real or modeled subgrid scale stresses show that both energy and enstrophy dissipation can be properly captured, with a single value of the model constants over a significant range of filter widths. These results are used to examine a new subgrid model based on enstrophy equilibrium. The cross‐correlation function of filtered velocity with the subgrid stress tensor is measured, which is of special importance for large‐scale energy spectra. No significant differences are observed between the different models, and it is found that they predict trends in the stress‐velocity cross‐correlation quite well. The results show that, in nearly isotropic turbulence, the eddy‐viscosity subgrid models correctly reproduce statistical trends necessary for the accurate LES prediction of energy spectra and enstrophy evolution.

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