ANNALI DELL'UNIVERSITA' DI FERRARA

, Volume 54, Issue 1, pp 145–168 | Cite as

Existence theory of abstract approximate deconvolution models of turbulence

Article

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.

Keywords

Large eddy simulation Turbulence Deconvolution 

Mathematics Subject Classification (2000)

76F65 76D03 

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Copyright information

© Università degli Studi di Ferrara 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA

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