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
Using this general deconvolution operator yields the closure approximation to the filtered nonlinear term in the NSE
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)
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.
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The author is partially supported by NSF grant DMS 0508260.
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Stanculescu, I. Existence theory of abstract approximate deconvolution models of turbulence. Ann. Univ. Ferrara 54, 145–168 (2008). https://doi.org/10.1007/s11565-008-0039-z
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DOI: https://doi.org/10.1007/s11565-008-0039-z