Bubble Velocity Enrichment for Embedded Interfaces in Multi-Fluid Flows: A Crime that Does Not Pay

Abstract

We discuss ongoing work on the addition of a statically condensable bubble enrichment to capture kinks (surfaces where the function is continuous but its gradient exhibits a jump) in the velocity field at immersed interfaces not conforming with the element boundaries. Such kinks are frequent in multi-fluid flows because they arise whenever there are viscosity jumps or thermo-capillary effects. The enrichment is applied only at those elements of the finite element mesh cut by the interface, which is parameterized in this case with a level set function. The bubble function used in this work was introduced elsewhere (see Codina and Coppola-Owen, Int. J. Num. Meth. in Fluids 2005; 49:1287-1304) to capture discontinuities in the pressure gradient for two-phase flows, which arise because of density discontinuities under gravity. Its applicability as enrichment of the velocity field is not obvious due to a consistency error it creates in the variational formulation (the enrichment velocity fields are not in H1, since they are discontinuous at some inter-element boundaries). In this work we assess the accuracy, robustness and limitations of this new enrichment in some problems involving interfaces. In its current form, the non-conformity of the bubbles introduces an unphysical numerical error which is of the same order as the interpolation error the bubbles are there to alleviate. Notice that these same bubble functions, when used for the pressure, do not lead to consistency error because they do belong to L2.