Existence theory of abstract approximate deconvolution models of turbulence

Abstract

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, $$\overline{\bf \phi} = g_{\delta}\,*\,{\bf \phi}$$ , the resulting system is not closed due to the filtered nonlinear term $$\overline{\bf uu}$$ . An approximate deconvolution operator D is a bounded linear operator which is an approximate filter inverse $$D(\overline{\bf u}) = {\rm approximation\,of}\, {\bf u}.$$ Using this general deconvolution operator yields the closure approximation to the filtered nonlinear term in the NSE $$\overline{\bf uu} \simeq \overline{D(\overline{\bf u})D(\overline{\bf u})}.$$ 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) $${\bf w}_{t} + \nabla \cdot \overline{D({\bf w})\,D({\bf w})} - \nu \triangle{\bf w}+\nabla q + \chi {\bf w}^* = \overline{\bf f} \quad {\rm and} \quad \nabla \cdot {\bf w} = 0.$$ Here $${\bf w} \simeq \overline{\bf u}$$ , 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.