Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. II: The incompressible Navier-Stokes equations

摘要

This paper presents the construction of a correct-energy stabilized finiteelement method for the incompressible Navier-Stokes equations. The framework ofthe methodology and the correct-energy concept have been developed in theconvective-diffusive context in the preceding paper [M.F.P. ten Eikelder, I.Akkerman, Correct energy evolution of stabilized formulations: The relationbetween VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometricanalysis. The convective-diffusive context, CMAME, Accepted 2018]. This workextends ideas of this paper to build a stabilized method within the variationalmultiscale (VMS) setting which displays correct-energy behavior. Similar to theconvection-diffusion case, a key ingredient is the proper dynamic andorthogonal behavior of the small-scales. This is demanded for correct energybehavior and links the VMS framework to the streamline-upwind Petrov-Galerkin(SUPG) and the Galerkin/least-squares method (GLS). The presented method is a Galerkin/least-squares formulation with dynamicdivergence-free small-scales (GLSDD). It is locally mass-conservative for boththe large- and small-scales separately. In addition, it locally conserveslinear and angular momentum. The computations require and employ NURBS-basedisogeometric analysis for the spatial discretization. The resulting formulationnumerically shows improved energy behavior for turbulent flows comparing withthe original VMS method.