Comparison of dynamic Smagorinsky and anisotropic subgrid-scale models

LES models using dynamically computed coefficient values have been extensively used since the pioneering work of Germano (Germano et al. 1991). Starting from the Smagorinsky subgrid-scale model, the common idea to all these models is to extrapolate the information on the resolved fields at two scale levels to compute optimal coefficient values. A simple solution to the integral equations which follow from this approach can be obtained by restricting the test filter action to certain directions and assuming that the model coefficient only varies along the orthogonal directions. The coefficient values are then derived from simple least-square formulas. The resulting global dynamic model has been successfully used for a number of flows having at least one direction of homogeneity. For more general flows, the brute force solution of the integral equations may lead to persistent local negative values, which have a destabilizing effect on the numerical solver. Clipping these values lead to discard up to 50% of the coefficients. Moreover, local large values of the coefficient require an implicit treatment of the eddy viscosity term to avoid prohibitively small time-steps. Ghosal et al. (1995) have proposed two models to address this issue. In the first one the positivity of the coefficient is rigorously constrained in the integral equation. In the second one, negative coefficient values are allowed, but the model is supplemented by a transport equation for the subgrid-scale kinetic energy. These techniques do alleviate the restrictions of the global dynamic model, but at the expanse of a significant computational overhead (Cabot, 1994). Our work here has been motivated by practical considerations. It seems clear that the difficulties associated with the solution of integral equations in the dynamic Smagorinsky model would be avoided if the variations of the coefficient over a scale of the order of the grid-size could be assumed to be small. This, in turn, requires that the underlying subgrid-scale model is well-conditioned for the dynamic procedure, or, in other words, has good correlation properties. It is well known that this is not the case for the Smagorinsky model. Leonard’s expansion of the subgridscale residual stress, by contrast, is known to have very good correlation properties. Its drawback in actual implementations is that it contains backscatter as well as dissipation. The backscatter control strategies which have been proposed so far do