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On the Stolz–Adams deconvolution model for the large-eddy simulation of turbulent flows. (English) Zbl 1128.76029
Summary: We consider a family of large-eddy simulation (LES) models with an arbitrarily high consistency error $O\left({\delta }^{2N+2}\right)$ for $N=1,2,3,...$ that are based on the van Cittert deconvolution procedure. This family of models has been proposed and tested for LES with success by Adams and Stolz in a series of papers, e.g., [Deconvolution methods for subgrid-scale approximation in large-eddy simulation, in: Modern Simulation Strategies for Turbulent Flow, R. T. Edwards, Philadelphia 2001, 21–41 (2001), An approximate deconvolution procedure for large eddy simulation, Phys. Fluids A 11, 1699–1701 (1999)]. We show that these models have an interesting and quite strong stability property. Using this property we prove an energy equality, existence, uniqueness, and regularity of strong solutions and give a rigorous bound on the modeling error $∥\overline{u}-w∥$, where $𝐰$ is the model’s solution and $\overline{u}$ is the true flow averages.
##### MSC:
 76F65 Direct numerical and large eddy simulation of turbulence 76D03 Existence, uniqueness, and regularity theory 76D05 Navier-Stokes equations (fluid dynamics)