A link between energy conserving schemes and LES models of tensor-diffusivity type

Large-eddy simulation (LES) mainly consists of a “projection”: an effective “truncation” to much fewer modes than what is required to perform a direct numerical simulation (DNS). This can be done in different ways: using a reduced set of discrete field values (formally equivalent to sampling, on the coarse grid, of the continuous DNS field), or using a reduced set of continuous functions, etc. One can also first assume a filtering of the DNS field (using an regular filter of effective width ∆; this is basically the view in the method of finite volumes, where the filter is a top hat of width ∆ = h), but this step is necessarily followed by the projection step. The main property of a projector is that applying the projector twice is like applying it once. Letting ũ be the LES projection of u (i.e., the u field as “seen” on the coarse LES grid), one indeed has that ̃̃ u = ũ: projection is idempotent. The product of the LES fields is also captured on the LES grid. Thus, in LES, one really solves a “tilde” equation (as all terms have the same status):