Optimality of the dynamic procedure for large-eddy simulations

We present a database analysis to obtain a precise evaluation of the accuracy limitations associated with the popular dynamic eddy-viscosity model in large-eddy simulation. We consider decaying homogeneous isotropic turbulence at two different Reynolds numbers, i.e., Relambda;=50 and 100. The large-eddy simulation errors associated with the dynamic model are compared with those arising in the "static" Smagorinsky model. A large number of systematically varied simulations using the Smagorinsky model provides a detailed impression of the dependence of the total simulation error on (i) the spatial resolution and (ii) the resolution of the subgrid dissipation length. This error behavior also induces an "optimal refinement trajectory" which specifies the particular Smagorinsky parameter, in terms of the spatial resolution, for which the total error is minimal. In contrast, the dynamic model gives rise to a self-consistently determined "dynamic trajectory" that represents the dependence of the dynamic coefficient on the spatial resolution. This dynamic trajectory is compared with the optimal refinement trajectory as obtained from the full database analysis of the Smagorinsky fluid. It is shown that the dynamic procedure in which the top-hat test filter is adopted, predicts values for the eddy viscosity as function of resolution and Reynolds number, which quite closely follow the main trends established in the optimal refinement trajectory. Furthermore, a sensitivity analysis, including dependency on test-filter width and filter shape, is discussed. Total simulation errors, due to interacting discretization, and modeling errors associated with the dynamic procedure may be a factor 2 higher compared to the optimum; still the dynamic procedure represents one of the very few self-contained and efficient error-reduction strategies when increasing the spatial resolution