In this talk we will present numerical methods for the simulation of several
two-phase flow problems requiring high accuracy at the interface. Throughout,
we will motivate and illustrate our approach using examples such as falling
liquid films, partial coalescence, bouncing droplets on a soap film, and
walking droplets on a periodically excited bath.
We identify the different domains present in such systems using a level set
function. A common challenge is how to correctly enforce boundary conditions
across the different domains. Another challenge is how to represent and track
small structures (possibly with sub-grid features) without loss of mass over
long computations.
Toward achieving these goals, we will introduce a new method to solve the
advection equation for the level set. Our approach relies on carrying both the
values and gradients of the level set function as coupled quantities via a
projection step. Using a local stencil, our method achieves third order global
accuracy and removes the need to solve a reinitialization equation. We will
then show the connection of the new method with enforcing sub-grid interfacial
jump conditions in the two-phase Navier-Stokes equations.
Finally, we will discuss some advantages of the proposed approach e.g.
locality
and higher order, and also extensions beyond two-phase liquid/gas systems.