Dissipative Structure and Long Term Behavior of a Finite Element Approximation of Incompressible Flows with Numerical Subgrid Scale Modeling

In this chapter we summarize a finite element formulation for incompressible flows based on a two-scale decomposition of the velocity field, where the sub-grid scales are modeled numerically. The main features of the formulation are the choice for the space of the sub-grid scales, their time dependency and the fact that they are accounted for in all the terms where they appear. We present the main results obtained for this formulation, with emphasis on its dissipative structure and stability behavior in the long term, which give arguments to support the claim that it is able to model turbulent flows without any additional turbulence model.