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Existence theory of abstract approximate deconvolution models of turbulence. (English) Zbl 1191.76058

By using approximate deconvolution, the author develops an abstract approach to modeling the motion of large eddies in a turbulent flow. The first part represents approximate deconvolution models (ADM) and an approximate deconvolution or approximate/asymptotic inverse of the filtering operator D. For large eddy simulation (LES), the author examines only two operators which have been studied earlier: the van Cittert deconvolution operator, and the Geurt’s approximate inverse filter. The second part reviews the averaging/filtering in LES and defines the function spaces and norms needed for variational formulation of the scale similarity model. In the third part the author finds conditions on the approximate deconvolution operator D that guarantee that ADM has a weak solution. Thus, the operator D:L 2 (Q)L 2 (Q), Q=(0,L) 3 , L being the period, has to be a self-adjoint positive definite bounded linear operator which commutes with differentiation. The fourth part demonstrates that the weak solution is really a unique strong solution, and that the model satisfies an energy equality rather than inequality and correctly captures the global energy balance of large scales.

A few examples of deconvolution operators and their properties are reviewed in the fifth part: the van Cittert deconvolution operator, the accelerated van Cittert deconvolution operator, Tikhonov regularization of a deconvolution operator, the Geurt’s approximate filter inverse and its variation. The main conclusions are given in the sixth part.

MSC:
76F02Fundamentals of turbulence
76F65Direct numerical and large eddy simulation of turbulence
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