Scale-separation models for the larger eddies in turbulent flow

Large-eddy simulation (LES) seeks to predict the dynamics of spatially filtered turbulent flows. The very essence of LES is that the LES-field contains only scales of size $\geq \delta$, where $\delta$ denotes the (user-chosen) length of the spatial filter. In the present approach we continue the work that was conducted during the 2010 CTR Summer Program by addressing the following two basic questions: (a) when does a LES-model stop the production of smaller scales of motion from continuing at the filter scale; and (b) when does it dissipate any disturbances having a length scale smaller than $\delta$ initially. In this way we find two scale separation conditions that ensure that all subfilter scales are dynamically insignificant. These conditions can be applied to any type of LES-model. In case of a mixed model, for instance, they imply that the eddy viscosity $\nu_t$ has to depend on the invariants $q = \mbox{$\frac{1}{2}$}{\rm tr}(S2)$ and $r= -\mbox{$\frac{1}{3}$}{\rm tr}(S3)$ of the (filtered) strain rate tensor $S$. The simplest model is then given by $\nu_t = \mbox{$\frac{1}{96}$}\,\delta2 |r|/q$. This model is successfully tested for a turbulent channel flow (Re$_\tau$=590 and 1,000).