General Order Virtual Element Approximation for the Smagorinsky turbulence model

In this paper, we investigate a Smagorinsky model in a virtual element framework to simulate convection-dominated Navier-Stokes equations. We conduct a two-dimensional numerical investigation to assess the performance of the general order virtual element approximation in this context. First, we examine numerically the convergence of the method with respect to the meshsize to certify the novel virtual element numerical discretization, which includes, for the first time, a discretization of the Smagorinsky term. Moreover, we present a numerical study of a lid-driven cavity for different Reynolds numbers (up to 10000) and meshes (uniform, anisotropic, and isotropic with hanging nodes). The results highlight the main advantage of using the virtual elements method in this context: the isotropic refinement with hanging nodes enhances the accuracy of the solution compared to the anisotropic mesh, uses fewer degrees of freedom with respect to the uniform mesh, and yields the most stable behavior in terms of convergence of the Newton solver.