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A high accuracy Leray-deconvolution model of turbulence and its limiting behavior. (English) Zbl 1210.76084

Summary: In 1934, J. Leray [Acta Math. 63, 193–248 (1934; JFM 60.0726.05)] proposed a regularization of the Navier-Stokes equations whose limits were weak solutions of the Navier-Stokes equations. Recently, a modification of the Leray model, called the Leray-alpha model, has attracted interest for turbulent flow simulations. One common drawback of the Leray type regularizations is their low accuracy. Increasing the accuracy of a simulation based on a Leray regularization requires cutting the averaging radius, i.e. remeshing and resolving on finer meshes. This article analyzes on a family of Leray type models of arbitrarily high orders of accuracy for a fixed averaging radius. We establish the basic theory of the entire family including limiting behavior as the averaging radius decreases to zero (a simple extension of results known for the Leray model). We also give a more technically interesting result on the limit as the order of the models increases with a fixed averaging radius. Because of this property, increasing the accuracy of the model is potentially cheaper than decreasing the averaging radius (or meshwidth) and high order models are doubly interesting.

MSC:

76F02 Fundamentals of turbulence
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
76F65 Direct numerical and large eddy simulation of turbulence

Citations:

JFM 60.0726.05
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References:

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