*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.