Multifluid flows with weak and strong discontinuous interfaces using an elemental enriched space

Abstract

In a previous paper the authors present an elemental enriched space to be used in a finite element framework (EFEM) capable to reproduce kinks and jumps in an unknown function using a fixed mesh in which the jumps and kinks do not coincide with the inter-element boundaries. In this previous publication, only scalar transport problems where solved (thermal problems). In the present work these ideas are generalized to vectorial unknowns, in particular the incompressible Navier-Stokes equations for multi-fluid flows presenting internal moving interfaces. The advantage of the EFEM compared with the global enrichment is the important reduction of the computing time when the internal interface is moving. In the EFEM the matrix to be solved at each time-step has, not only the same amount of degrees of freedom (DOFs) but also has always the same connectivity between the DOFs. This frozen matrix-graph improves enormously the efficiency of the solver. Another characteristic of the elemental enriched space presented here is that allows a linear variation of the jump, improving the convergence rate compared with other enriched spaces that have a constant variation of the jump. Furthermore, the implementation in any existing finite element code is extremely easy with the version presented here because the new shape functions are based on the usual FEM shape functions for triangles or tetrahedrals and, once * Corresponding author: Sergio R. Idelsohn; Postal Address; Gran Capitan s/n; Edificio C1, Campus Nord UPC; 08034 Barcelona, Spain; Tel.: +34 93 401 1829; E-mail: sergio@cimne.upc.edu Preprint submitted to Journal Name April 26, 2017 statically condensed the internal DOFs, the resulting elements have exactly the same number of unknowns as the non-enriched finite elements.

A discontinuous Nitsche‐based finite element formulation for the imposition of the Navier‐slip condition over embedded volumeless geometries

“…2D elbow with internal wall. Problem geometry (source Reference 64) [Colour figure can be viewed at wileyonlinelibrary.com]…”

“…The previously described elbow geometry is solved using several mesh refinement levels and considering both a slip interface, as was done by the original authors in Reference 64, as well as a no‐slip one. The obtained solutions are compared with the ones obtained with a reference body fitted solver.…”

Solution of Navier–Stokes equations on a fixed mesh using conforming enrichment of velocity and pressure

“…In this numerical example, a static simulation has been conducted using two identical fluids in an elbow shaped domain. The case had previously been studied in [24]. Since the original problem has two different domains with slip-condition on the interface (velocity normal to the wall is zero), to simulate the same physics, the slip condition presented in the sub-section.…”

“…Eulerian based methodologies use special techniques to capture the interface such as: Level Set approaches [1,2], Volume of Fluid methods [3], etc. Since the element could be cut by the interface in any arbitrary position, further techniques, such as X-FEM and E-FEM, are needed for the representation of weak and/or strong discontinuities in the pressure and/or velocity fields [5,17,20,25,24,26]. These methodologies exhibit deficiencies in conservation of the mass.…”

“…Although this remedy was successful, it was still not capable of capturing the strong discontinuities in the pressure and/or strong and weak discontinuities in the velocity field. In [24], a new methodology has been proposed to deal with both weak and strong discontinuities in the velocity and the pressure field. However, this methodology was neither applied onto non-stationary problems, nor was employed with the PFEM-2 strategy.…”