On the model coefficients for the standard and the variational multi-scale Smagorinsky model

A theoretical analysis is presented on the behaviour of the model coefficients for the well-known Smagorinsky model and two variational multi-scale (VMS) variants of the Smagorinsky model. The dependency on two important parameters is addressed, i.e. the ratio of the LES-filter width Delta and the Kolmogorov scale eta on the one hand, and the ratio of the integral length scale L and the LES-filter width Delta on the other hand. First of all, it is demonstrated that the model coefficients vary strongly with Delta/eta. By evaluating the model coefficients as functions of the subgrid activity s (which expresses the relative contribution of the subgrid-scale model in the total dissipation, and corresponds to a nonlinear transformation of Delta/eta), we show that a classical Lilly-Smagorinsky model overestimates the dissipation, even in cases where the dissipation of the subgrid-scale model is dominant. Therefore, generic and easy-to-use modifications to the different models are proposed, which provide close approximations to the models employing 'exact' coefficients. For the standard Smagorinsky model, this modified model corresponds to approximating the eddy viscosity nu(1) as nu(1) = (nu(2)(Lilly) + nu(2))(1/2)-nu, with nu(Lilly) the turbulent viscosity obtained by employing Lilly's classical Smagorinsky constant and nu the laminar viscosity. Similar easy-to-use relations are presented for the variational multi-scale Smagorinsky models. Next to the Delta/eta dependence of the model coefficients, the L/Delta behaviour is also elaborated. Although a strong dependence on L/Delta is observed for low values of the ratio, we do not advocate the use of L/Delta-dependent model coefficients. Rattler, the asymptotic L/Delta independence and the speed of asymptotic convergence are used as a tool to compare the quality of subgrid-scale models (e.g. L/Delta > 10 is a minimum order of magnitude for the small-small VMS model), and differences are observed between the standard Smagorinsky model and its two VMS variants. Finally, for the VMS models, the influence of the shape of the high-pass filter, used in the variational multi-scale formulation, is investigated. We observed that smooth high-pass filters result in more robust VMS Smagorinsky models.