A Two-Level LES Constructed with Unfiltered Equations
and SGS Models of Physical Variables
We describe a two-level LES formalism based on three key notions: i) filter solution variables rather than equations; ii) model physical variables on sub-grid scales, and iii) directly use SGS model results to enhance fidelity of under-resolved large-scale solutions. The underlying mathematics associated with each of these is mollification, use of symbols of differential operators (as well as a generalization of KolmogorovŐs K41 theory), and multi-level construction, respectively. Mollification of solutions (in lieu of filtering equations) widens the range of possibilities for SGS model formulation, in particular, permitting straightforward use of models of physical variables. This, in turn, allows at least approximate account of interactions of turbulence with other physics on sub-grid scales via symbols of the differential operators of governing equations. The multi-level formalism then leads to propagation of these effects through the resolved-scale solution. We will present details of each of these steps, along with a priori tests associated with the first two, and analysis of a complete solution, to demonstrate the last step. Finally, we will note (numerous) remaining open questions arising from this approach.