Improving Accuracy in α-Models of Turbulence through Approximate Deconvolution

Abstract

In this report, we present several results in the theory of α -models of turbulence with improved accuracy that have been developed in recent years. The α -models considered herein are the Leray- α model, the zeroth Approximate Deconvolution Model (ADM) turbulence model, the modified Leray- α and the Navier–Stokes- α model. For all of the models from above, the accuracy is limited to α 2 in smooth flow regions. Better accuracy requires decreasing the filter radius α , which, in turn, requires a smaller mesh width that will lead in the end to a higher computational cost. Instead, one can use approximate deconvolution (without decreasing the mesh size) to attain better accuracy. Such deconvolution methods have been considered recently in many studies that show the efficiency of this approach. For smooth flows, periodic boundary conditions and van Cittert deconvolution operator of order N, the expected accuracy is α 2 N + 2 . In a bounded domain, such results are valid only in case special conditions are satisfied. In more general conditions, the author has recently proved that, in the case of the ADM, the expected accuracy of the finite element method with Taylor–Hood elements and Crank–Nicolson time stepping method is Δ t 2 + h 2 + K N α 2 , where the constant K < 1 depends on the ratio α / h , which is assumed constant. In this study, we present the extension of the result to the rest of the models.