A stabilized finite element method using a discontinuous level set approach for the computation of bubble dynamics

Abstract

A novel numerical method for solving three-dimensional two phase flow problems is presented. This method combines a quadrature free discontinuous Galerkin method for the level set equation with a pressure stabilized finite element method for the Navier Stokes equations.The main challenge in the computation of such flows is the accurate evaluation of surface tension forces. This involves the computation of the curvature of the fluid interface. In the context of the discontinuous Galerkin method, we show that the use of a curvature computed by means of a direct derivation of the level set function leads to inaccurate and oscillatory results. A more robust, second-order, least squares computation of the curvature that filters out the high frequencies and produces converged results is presented.This whole numerical technology allows to simulate a wide range of flow regimes with large density ratios, to accurately capture the shape of the deforming interface of the bubble and to maintain good mass conservation.

On stability condition for bifluid flows with surface tension: Application to microfluidics

“…This formulation of the surface tension has been used by Unverdi 5 and Tryggvason [55] and Brackbill, Kothe and Zemach [5]. This complete Level Set formulation with the Navier-Stokes equations for two-fluids flows was derived by Chang, Hou, Merriman and Osher [6] and later used in many other works (Sussman et al [48,47], Olsson and Kreiss [29], Marchandise et al [26], to new a few).…”

“…see [34,42] for VOF method or [49,26] for Level Set method. The idea is to treat the surface tension as a body force in the momentum equation.…”

“…First observed in Boltzmann interfacial methods, parasitic currents are also presented by Lafaurie et al in [24] where they suggested the alternative Continuum Surface Stress (CSS) method. Then follow several approaches to tackle this problem [35,32,53,34,41,52,26]. Their key ideas in suppressing parasitic currents, usually mentioned in this literature, are (i) improvement of curvature computation, (ii) achievement of discrete balance between surface tension and pressure gradient (iii) adaptive time integration scheme to tackle the stiffness induced by surface tension [25].…”

An accurate adaptive solver for surface-tension-driven interfacial flows

“…Numerical methods using an implicit representation of the interface such as Volume-OfFluid (vof) [1,2,3,4] or Levelset (ls) [5,6,7,8,9,10,11,12,13] can robustly and efficiently represent evolving, topologically complex interfaces. The accurate representation of surface tension within these methods is typically more delicate than for schemes using an explicit representation of the interface [14,15,16,17] and care must be taken to avoid any imbalance between discrete surface tension and pressure gradient terms [18,19,15,20,21,22,23].…”

Fast and robust solvers for pressure-correction in bubbly flow problems

“…Computation of two-phase flows and, in particular, bubbly flows is a very active research topic in computational fluid dynamics (CFD), see, for instance, [7,12,37,[44][45][46][47] and, more recently, [17,23,35,38]. Understanding the dynamics and interaction of bubbles and droplets in a large variety of industrial processes is crucial for economically and ecologically optimized design.…”