Existence theory of abstract approximate deconvolution models of turbulence

This report studies an abstract approach to modeling the motion of large eddies in a turbulent flow. If the Navier-Stokes equations (NSE) are averaged with a local, spatial convolution type filter, , the resulting system is not closed due to the filtered nonlinear term . An approximate deconvolution operator D is a bounded linear operator which is an approximate filter inverseUsing this general deconvolution operator yields the closure approximation to the filtered nonlinear term in the NSEAveraging the Navier-Stokes equations using the above closure, possible including a time relaxation term to damp unresolved scales, yields the approximate deconvolution model (ADM)Here , χ ≥ 0, and w ^* is a generalized fluctuation, defined by a positive semi-definite operator. We derive conditions on the general deconvolution operator D that guarantee the existence and uniqueness of strong solutions of the model. We also derive the model’s energy balance. The author is partially supported by NSF grant DMS 0508260.