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Existence theory of abstract approximate deconvolution models of turbulence

Authors:

Iuliana Stanculescu

Institution:

(1) Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

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. The author is partially supported by NSF grant DMS 0508260.

Keywords:

Large eddy simulation Turbulence Deconvolution

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