Accounting for scale-dependence in the dynamic Smagorinsky model

The dynamic model has been very successful in Large-Eddy Simulation of a variety of turbulent flows. The model makes the crucial assumption that the coefficient is scale-invariant, which is reasonable in the inertial range of turbulence. Surprisingly, the dynamic model’s success has occurred even in flows where the grid-scale is not in an ideal inertial range. It turns out that the most well-known successful applications (such as transitional flows, the viscous sublayer, etc..) are those where the true SGS stress is only a small contribution to the dynamics, and where the errors induced by the deviations from scale-invariance are of minor consequence. However, there are important flows in which the errors can have large effects, such as in LES of wall-bounded flows that do not resolve the viscous sublayer. Near the wall, the grid-scale is on the order of the local integral scale (the distance to the wall) and the SGS shear stress contributes significantly to the mean momentum balance. In this paper we show that the traditional dynamic model yields coefficients that are too small, which translates into high wavenumber pile-up of the longitudinal energy spectra near the wall. In order to correct for this problem in a dynamic fashion, a new scale-dependent dynamic model is proposed. It involves a secondary test-filter operation that allows one to probe from the simulated resolved field how the coefficient depends on scale. Applications to LES of a wall-bounded flow that does not resolve the viscous sublayer are presented, showing much improved longitudinal energy spectra and better mean velocity profiles.