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Numerical analysis and computational testing of a high accuracy Leray-deconvolution model of turbulence. (English) Zbl 1191.76061
Summary: We study a computationally attractive algorithm (based on an extrapolated Crank-Nicolson method) for a recently proposed family of high accuracy turbulence models, the Leray-deconvolution family. First we prove convergence of the algorithm to the solution of the Navier-Stokes equations and delineate its (optimal) accuracy. Numerical experiments are presented which confirm the convergence theory. Our 3d experiments also give a careful comparison of various related approaches. They show the combination of the Leray-deconvolution regularization with the extrapolated Crank-Nicolson method can be more accurate at higher Reynolds number that the classical extrapolated trapezoidal method of Baker (Report, Harvard University, 1976). We also show the higher order Leray-deconvolution models (e.g. $N=1,2,3$) have greater accuracy than the $N=0$ case of the Leray-$\alpha$ model. Numerical experiments for the $2d$ step problem are also successfully investigated. Around the critical Reynolds number, the low order models inhibit vortex shedding behind the step. The higher order models, correctly, do not. To estimate the complexity of using Leray-deconvolution models for turbulent flow simulations we estimate the models’ microscale.

##### MSC:
 76F65 Direct numerical and large eddy simulation of turbulence 76F02 Fundamentals of turbulence 35Q35 PDEs in connection with fluid mechanics 65M06 Finite difference methods (IVP of PDE) 76D05 Navier-Stokes equations (fluid dynamics) 76M25 Other numerical methods (fluid mechanics)