The Leray-$\alpha \beta$-deconvolution model: Energy analysis and numerical algorithms

Abstract

We study a fluid–flow regularization based on the Leray-$\alpha$ model that uses deconvolution in the nonlinear term and dissipation scale modeling in the viscous term. In particular, we establish that this ‘Leray-$\alpha \beta$-deconvolution model’ has an energy cascade with an enhanced energy dissipation that enlarges the microscale of the model relative to the Kolmogorov microscale, but captures more of the small scales than does the Leray-$\alpha$ model. These theoretical results are confirmed via numerically determined energy spectra. We also propose and analyze an efficient finite-element algorithm method for the proposed model. In addition to establishing stability of the method, the essential ingredient for any numerical study, we demonstrate convergence to a Navier–Stokes solution. A numerical experiment for the two-dimensional flow around an obstacle is also discussed. Results show that enhancing the Leray-$\alpha$ model with deconvolution and dissipation scale modeling can significantly increase accuracy.