Page 308 - Process Modelling and Simulation With Finite Element Methods

Modeling of Multi-Phase Flow Using the Level Set Method 295

monitoring the free surface position with mesh re-gridding or by solving an
auxiliary transport equation for a field variable (VOF, diffuse interface, or level
set methods) is arguable. The latter auxiliary equation methods have ease of
coding in their favor, which will be illustrated in this chapter with the level set
method.
The level set method is used in this chapter to illustrate the coalescence of
two axisymmetric and non-axisymetric drops. Computations are performed using
FEMLAB. This FEM approach simplifies the level set method by eliminating all
the complexities in grid discretization required for free surfacehnterface tracking
methods. The governing equations for the level set method are described in
following section.

8.2 Governing Equations of the Level Set Method

In the level set method, a smooth function called a level set function is used to
represent the interface between two phases. The level set function is always
positive in the continuous phase and is always negative in the dispersed phase.
The free surface is implicitly represented by the set of points in which level set
function is always zero. Hence we have,
for the continuous phase @(X,Y,t)>O (8.la)

for the interface @(x, Y,t> = 0 (8.1 b)

for the dispersed phase @(X, Y 3 t> < 0 (8.lc)
From such a representation of the free surface, the unit normal on the interface
pointing from dispersed phase to continuous phase and curvature of the interface
can be expressed in terms of level set function as,

The motion of the interface can be captured by advection of the level set
function,

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-+u.V@=O (8.4)
at