A Splitting Method for Numerical Simulation of Free Surface Flows of Incompressible Fluids with Surface Tension

Abstract

The paper studies a splitting method for the numerical time-integration of the system of partial di erential equations describing the motion of viscous incompressible uid with free boundary subject to surface tension forces. The method splits one time step into a semi-Lagrangian treatment of the surface advection and uid inertia, an implicit update of viscous terms and the projection of velocity into the subspace of divergence-free functions. We derive several conservation properties of the method and a suitable energy estimate for numerical solutions. Under certain assumptions on the smoothness of the free surface and its evolution, this leads to a stability result for the numerical method. E cient computations of free surface ows of incompressible viscous uids need several other ingredients, such as dynamically adapted meshes, surface reconstruction and level set function re-initialization. These enabling techniques are discussed in the paper as well. The properties of the method are illustrated with a few numerical examples. These examples include analytical tests and the oscillating droplet benchmark problem.

Finite stopping times for freely oscillating drop of a yield stress fluid

“…then 1 2 of the viscous term kills the negative term on the left hand side of (38). Further, for the viscous term in (36), the following holds trivially…”

“…These experiments also study the dependence of the finite stopping time for the 3D droplet problem on various parameters. For the computer simulations we use the numerical approach developed in [35][36][37] for free-surface incompressible viscous flows. The numerical method is built on a staggered grid finite difference octree discretization of momentum, mass conservation and level set equations.…”

An adaptive numerical method for free surface flows passing rigidly mounted obstacles

“…After the completion of the semi-Lagrangian step, we perform the re-initialization of the level set function to satisfy equation (8). For this purpose, we use an algorithm from [31] based on the marching cubes method for free surface triangulation and a higher order closest point method. The numerical integration of (9) may also cause a divergence (loss or gain) of the fluid volume.…”

“…The numerical integration of (9) may also cause a divergence (loss or gain) of the fluid volume. So we perform the volume correction with the help of the procedure described in [31]. We note that the use of the BFECC method makes the re-initialization and volume correction steps less critical compared to the standard linear semi-Lagrangian method, but still they are necessary for long-time simulations.…”

“…Remark 1 A rigorous stability analysis of the hybrid method is an open question. We note that stability of the semi-discrete scheme from section 3.1 (only discretization in time) for free-surface flows was studied in [31]. The scheme was shown to conserve global momentum and angular momentum, and based on that an energy inequality was shown to hold.…”