On why dynamic subgrid-scale models work

Dynamic subgrid models have proved to be remarkably successful in predicting the behavior of turbulent flows. Part of the reasons for their success are well understood. Since they are constructed to generate an effective viscosity which is proportional to some measure of the turbulent energy at the high wavenumber end of the spectrum, their eddy viscosity vanishes as the flow becomes laminar. This alone would justify their use over simpler models. But beyond this obvious advantage, which is confined to inhomogeneous and evolving flows, the reason why they also work better in simpler homogeneous cases, and how they do it without any obvious adjustable parameter, is not clear. This lack of understanding of the internal mechanisms of a useful tool is disturbing, not only as an intellectual challenge, but because it raises the doubt of whether it will work in all cases. This note is an attempt to clarify those mechanisms. We will see why dynamic models are robust and how they can get away with even comparatively gross errors in their formulations. This will suggest that they are only particular cases of a larger family of robust models, all of which would be relatively insensitive to large simplifications in the physics of the flow. We will also construct some such models, although mostly as research tools. It will turn out, however, that the standard dynamic formulation is not only robust to errors, but also behaves as if it were substantially well formulated. The details of why this is so will still not be clear at the end of this note, specially since it will be shown that the 'a priori' testing of the stresses gives, as is usual in most subgrid models, very poor results. But it will be argued that the basic reason is that the dynamic formulation mimics the condition that the total dissipation is approximately equal to the production measured at the test filter level.