Deconvolution-based stabilization of the incompressible Navier–Stokes equations

Abstract

The numerical simulation of the incompressible Navier–Stokes equations may suffer from instabilities due to the energy cascades activating small-scale dynamics even when high-frequency components are not in the initial conditions. The energy cascade is generally triggered by the non-linear convective term. However, the process can be triggered by particular geometries, such that the instabilities occur even with relatively low Reynolds numbers. While instabilities for high convective fields are well known and investigated (for instance, in the framework of the Variational Multiresolution formulation designed by T.J. Hughes and his collaborators), numerical stabilization of low-convection instabilities is less investigated. In this paper, we present a novel method where the backbone of classical stabilizations is merged with a localization of the potentially unstable regions by means of a deconvolution-filter indicator inspired by the Large Eddie Simulation (LES) turbulence modeling advocated by W. Layton and his collaborators. We introduce the method for steady incompressible Navier–Stokes problems, we motivate the design of the method in two different variants, the Streamline-diffusion one and the strongly consistent one. We provide some analysis of the approach and numerical results proving the improved performances in comparison with classical schemes.