Numerical simulation of bubble and droplet deformation by a level set approach with surface tension in three dimensions

Abstract

SUMMARYIn this paper we present a three-dimensional Navier-Stokes solver for incompressible two-phase flow problems with surface tension and apply the proposed scheme to the simulation of bubble and droplet deformation. One of the main concerns of this study is the impact of surface tension and its discretization on the overall convergence behavior and conservation properties.Our approach employs a standard finite difference/finite volume discretization on uniform Cartesian staggered grids and uses Chorin's projection approach. The free surface between the two fluid phases is tracked with a level set (LS) technique. Here, the interface conditions are implicitly incorporated into the momentum equations by the continuum surface force method. Surface tension is evaluated using a smoothed delta function and a third-order interpolation. The problem of mass conservation for the two phases is treated by a reinitialization of the LS function employing a regularized signum function and a global fixed point iteration. All convective terms are discretized by a WENO scheme of fifth order. Altogether, our approach exhibits a second-order convergence away from the free surface. The discretization of surface tension requires a smoothing scheme near the free surface, which leads to a first-order convergence in the smoothing region.We discuss the details of the proposed numerical scheme and present the results of several numerical experiments concerning mass conservation, convergence of curvature, and the application of our solver to the simulation of two rising bubble problems, one with small and one with large jumps in material parameters, and the simulation of a droplet deformation due to a shear flow in three space dimensions. Furthermore, we compare our three-dimensional results with those of quasi-two-dimensional and twodimensional simulations. This comparison clearly shows the need for full three-dimensional simulations of droplet and bubble deformation to capture the correct physical behavior.

An SPH modeling of bubble rising and coalescing in three dimensions

“…Volume of Fluid (VOF) method [4], Level Set (LS) method [5,6], Front Tracking Method (FTM) [7][8][9], Lattice Boltzmann Method (LBM) [10][11][12][13], etc. The advantage and disadvantage of these mesh-based methods are analyzed (see e.g.…”

A multi-GPU accelerated solver for the three-dimensional two-phase incompressible Navier-Stokes equations

“…The simulation of incompressible two-phase flows utilizes the level set function and is based on the two-phase NavierStokes equations with surface tension [5]. They consist of the momentum equation…”

“…In our CPU code [5], we apply Chorin's projection method and the level set method to solve the two-phase unsteady Navier-Stokes equations. This results in Algorithm 1 which describes the time-discrete computation of the velocity, the pressure and the level set function for a new time step t n+1 with time step size δt in our simulation.…”

“…Engineers and researchers need a parallel high order double precision fluid solver that scales on large clusters. Our in-house CPU Navier-Stokes solver NaSt3DGPF [4,5] meets these requirements. It solves the full two-phase non-stationary incompressible NavierStokes equations in three dimensions.…”

“…Studies of droplet behavior [5], gas bubble dynamics or the physics of rivers crossing a dam construction [20] make the simultaneous simulation of two fluid phases (e.g. air and water) necessary.…”