An extended finite element-level set method for simulating two-phase and free-surface two-dimensional flows

Abstract

The present work discusses the application of the extended finite element method (XFEM) to model two-phase flows. The XFEM method is naturally coupled with the level set method to provide an efficient and flexible treatment of problems involving moving discontinuities. The Navier-Stokes equations are discretized using a stable finite element pair (Taylor-Hood) on triangular or quadrangular meshes. In order to account for the discontinuities in the field variables across the interface, kink or jump enrichments may be used for the velocity and/or pressure fields. However, these enrichments may lead to an unstable velocity-pressure pair. In this work, different enrichment schemes are considered, and their accuracy and stability are numerically investigated. In cases with a surface tension force, a jump in the pressure field exists across the moving interface. Because we employ the XFEM to capture this jump, the accuracy mainly relies on the precise computation of the normal vectors and the curvature. A novel method of computing the vectors normal to the interface is proposed. This method computes the vectors normal by employing the successive refinement of the mesh inside the cut elements. This provides a high resolution of the interface position. The normal vectors thus constructed are naturally perpendicular to the interface. Comparisons with analytical and numerical solutions demonstrate that the method is effective. The quadrature of the Galerkin weak form for the intersected elements is improved by employing a mesh refinement. The reinitialization of the level set field is realized by a direct approach in order to recover the signed-distance property. The proposed methods are tested and validated for various numerical examples of increasing complexity: Poiseuille two-phase flow, extensional flow problem, a rectangular tank in a horizontal acceleration, sloshing flow in a tank, a collapsing water column with and without obstacle, and a rising bubble. For all flow regimes, excellent agreement with either analytical solutions or experimental and numerical reference data is shown.

Date	9 Sept 2016
Original language	American English
Awarding Institution	École de technologie supérieure
Supervisor	Azzeddine Soulaïmani (Supervisor)