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

Abstract

This paper presents the construction of a correct-energy stabilized finite element method for the incompressible Navier-Stokes equations. The framework of the methodology and the correct-energy concept have been developed in the convective-diffusive context in the preceding paper [M.F.P. ten Eikelder, I. Akkerman, Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. I: The convective-diffusive context, Comput. Methods Appl. Mech. Engrg. 331 (2018) 259-280]. The current work extends ideas of the preceding paper to build a stabilized method within the variational multiscale (VMS) setting which displays correct-energy behavior. Similar to the convection-diffusion case, a key ingredient is the proper dynamic and orthogonal behavior of the small-scales. This is demanded for correct energy behavior 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 dynamic divergence-free small-scales (GLSDD). It is locally mass-conservative for both the large-and small-scales separately. In addition, it locally conserves linear and angular momentum. The computations require and employ NURBS-based isogeometric analysis for the spatial discretization. The resulting formulation numerically shows improved energy behavior for turbulent flows comparing with the original VMS method.

Variational multiscale modeling with discretely divergence-free subscales

“…Remark 3. The Stokes projector employed here is similar to the Stokes projector presented in [22]. However, the projector here has an additional grad-div term.…”

“…Remark 5. The semi-discrete formulation given by Problem (V h ) is similar to the Galerkin/leastsquares formulation with dynamic divergence-free small-scales (GLSDD) presented in [22]. However, there are a few key differences.…”

Isogeometric analysis of linear free-surface potential flow

“…The Lagrange multiplier from equation (20) can be reconstructed by selecting q = 0, which results in…”

“…For the incompressible Navier-Stokes equations this can be exploited to create velocity-pressure approximations that result in exactly solenoidal solutions [19]. In [20,21] this exact divergence is essential for getting correct energy behavior of the single and two-fluid Navier-Stokes problem, respectively. In the current context the improved approximation and spectral properties are of paramount importance.…”

Variation entropy: a continuous local generalization of the TVD property using entropy principles

“…The well-known methods are the Streamline upwind-Petrov Galerkin (SUPG) method [16], the Galerkin/least-squares method [17] and the variational multiscale method [18][19][20]. The latter method offers a rich prospect for design new stabilized methods and has gained a lot of attention recently [21][22][23][24][25]. In the direction of TVD schemes and maximum principles, several VMS methods have been proposed.…”