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, φ = g δ*φ, the resulting system is not closed due to the filtered nonlinear term uu. An approximate deconvolution operator D is a bounded linear operator which is an approximate filter inverse D(u) = approximation of u. Using this general deconvolution operator yields the closure approximation to the filtered nonlinear term in the NSE uu ∼ D(u)D(u). Averaging the Navier-Stokes equations using the above closure, possible including a time relaxation term to damp unresolved scales, yields the approximate deconvolution model (ADM) wt + ∇ D(w) D(w) - νw + ∇q + χw* = f and ∇ ̇ w = 0. Here ≃ u, 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. © 2008 Università degli Studi di Ferrara.