Variational multi-scale stabilized formulations for the stationary three-field incompressible viscoelastic flow problem

Abstract

In this paper, three-field finite element stabilized formulations are proposed for the numerical solution of incompressible viscoelastic flows. These methods allow one to use equal interpolation for the problem unknowns $\sigma -u-p$ (elastic deviatoric stress–velocity–pressure) and to stabilize dominant convective terms. Starting from residual-based stabilized formulations, the proposed method introduces a term-by-term stabilization which is shown to have a superior behavior when there are stress singularities. A general discontinuity-capturing technique for the elastic stress component is also proposed, which allows one to eliminate the local oscillations that can appear when the Weissenberg number is high and the fluid flow finds an abrupt change in the geometry. The formulations are tested in the classical 4:1 planar contraction benchmark up to $We=5$ in the inertial case, with Reynolds number $Re=1$, and up to $We=6.5$ in the quasi non-inertial case, with $Re=0.01$. The standard Oldroyd-B constitutive model is used for the rheological behavior and linear and quadratic elements for the spatial approximation.