## D

Discussions in Chapter 2 may be referred to for explanations of the various symbols. It is straightforward to apply such conservation equations to single-phase flows. In the case of multiphase flows also, in principle, it is possible to use these equations with appropriate boundary conditions at the interface between different phases. In such cases, however, density, viscosity and all the other relevant properties will have to change abruptly at the location of the interface. These methods, which describe and track the time-dependent behavior of the interface itself, are called front tracking methods. Numerical solution of such a set of equations is extremely difficult and enormously computation intensive. The main difficulty arises from the interaction between the moving interface and the Eulerian grid employed to solve the flow field (more discussion about numerical solutions is given in Chapters 6 and 7).

The volume of fluid (VOF) approach simulates the motion of all the phases rather than tracking the motion of the interface itself. The motion of the interface is inferred indirectly through the motion of different phases separated by an interface. Motion of the different phases is tracked by solving an advection equation of a marker function or of a phase volume fraction. Thus, when a control volume is not entirely occupied by one phase, mixture properties are used while solving governing Eqs (4.1) and (4.2). This avoids abrupt changes in properties across a very thin interface. The properties appearing in Eqs (4.1) and (4.2) are related to the volume fraction of the kth phase as follows:

Y,ak PkCpk J2ak Pk

The average of any other variable <p can also be written

J2ak Pk

The volume fraction of each fluid, ak , is calculated by tracking the interface between different phases throughout the solution domain. Tracking of the interfaces between N different phases present in the system is accomplished by solving continuity equations for N - 1 phases. For the kth phase, this equation has the following form (similar to scalar advection):

It must, however, be noted that the marker function or the volume fraction does not uniquely identify the interface. Several different interface configurations may correspond to the same value of volume fraction (Fig. 4.3). Several specialized techniques have been proposed to track the interface accurately (Rider and Kothe, 1995; Rudman, 1997). Some of these techniques are discussed in Chapter 7.

The VOF approach allows one to model various interfacial phenomena; for example, wall adhesion and surface (or interfacial) tension can be modeled rigorously using this approach. Brackbill et al. (1992) developed a continuous surface force (CSF) model to describe interfacial surface tension. CSF model replaces surface force by a smoothly varying volumetric force acting on all the fluid elements in the interface transition region. For two-phase flows (dispersed or secondary phase is denoted by subscript 2), surface force, Fsf can be written (Brackbill et al., 1992):

FIGURE 4.3 Possible interface configurations for the same value of volume fraction in the control volume (volume fraction = 0.5).

where a is surface tension, k is local curvature and n is the surface normal. The surface normal is defined as the gradient of the secondary phase volume fraction:

The local surface curvature is defined as

0 0