Numerical Methods for Two-Phase Flow with Contact Lines

Abstract

This thesis focuses on numerical methods for two-phase ows, and especially ows with a moving contact line. Moving contact lines occur where the interface between two uids is in contact with a solid wall. At the location where both uids and the wall meet, the common continuum descriptions for uids are not longer valid, since the dynamics around such a contact line are governed by interactions at the molecular level. Therefore the standart numerical continuum models have to be adjusted to handle moving contact lines.

In the main part of the thesis a method to manipulate the position and the velocity of a contact line in a two-phase solver, is described. The Navier-Stokes equations are discretised using an explicit nite di erence method on a staggered grid. The position of the interface is tracked with the level set method and the discontinuities at the interface are treated in a sharp manner with the ghost uid method. The contact line is tracked explicitly and its dynamics can be described by an arbitrary function. The key part of the procedure is to enforce a coupling between the contact line and the Navier-Stokes equations as well as the level set method. Results for di erent contact line models are presented and it is demonstrated that they are in agreement with analytical solutions or results reported in the literature.

The presented Navier-Stokes solver is applied as a part in a multiscale method to simulate capillary driven ows. A relation between the contact angle and the contact line velocity is computed by a phase eld model resolving the micro scale dynamics in the region around the contact line. The relation of the microscale model is then used to prescribe the dynamics of the contact line in the macro scale solver. This approach allows to exploit the scale separation between the contact line dynamics and the bulk ow. Therefore coarser meshes can be applied for the macro scale ow solver compared to global phase eld simulations, reducing the overall computational coasts.

One of the major drawbacks of the level set method is that it does not conserve the mass of the uids. The application of the conservative level set method (CLSM) o ers a solution to this problem. Three of the attached articles address details concerning the implementation of the CLSM using a nite di erence method. A nite di erence discretisation of the CLSM based on stencils used in the Navier-Stokes solver is described and tested. Various methods to compute the curvature in the CLSM are assessed for the use in the ghost uid method. It is shown that the reinitialisation of the CLSM can lead to spurious deformations of the interface, a stabilised constrained reinitialisation is developed in an attempt to prevent the interface from deforming