3,433 research outputs found
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
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.Comment: Update to postprint versio
The dissipative structure of variational multiscale methods for incompressible flows
In this paper, we present a precise definition of the numerical dissipation for the orthogonal projection version of the variational multiscale method for incompressible flows. We show that, only if the space of subscales is taken orthogonal to the finite element space, this definition is physically reasonable as the coarse and fine scales are properly separated. Then we compare the diffusion introduced by the numerical discretization of the problem with the diffusion introduced by a large eddy simulation model. Results for the flow around a surface-mounted obstacle problem show that numerical dissipation is of the same order as the subgrid dissipation introduced by the Smagorinsky model. Finally, when transient subscales are considered, the model is able to predict backscatter, something that is only possible when dynamic LES closures are used. Numerical evidence supporting this point is also presented
On the stability of bubble functions and a stabilized mixed finite element formulation for the Stokes problem
In this paper we investigate the relationship between stabilized and enriched
finite element formulations for the Stokes problem. We also present a new
stabilized mixed formulation for which the stability parameter is derived
purely by the method of weighted residuals. This new formulation allows equal
order interpolation for the velocity and pressure fields. Finally, we show by
counterexample that a direct equivalence between subgrid-based stabilized
finite element methods and Galerkin methods enriched by bubble functions cannot
be constructed for quadrilateral and hexahedral elements using standard bubble
functions.Comment: 25 pages, 13 figures (The previous version was compiled by mistake
with the wrong style file, the current one uses amsart, and there is no
difference in the text or the figures
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