When the governing equations are solved on a fixed grid, using one set of equations for the whole flow field, the different fluids must be identified in some way. This is generally done by using a marker function that takes different values in the different fluids. Sometimes a material property, such as the fluid density for incompressible fluids, can serve as a marker function, but here we shall assume that the rôle of the marker function is only to identify the different fluids. As the fluids move, and the boundary between the different fluids changes location, the marker function must be updated. Updating the marker function accurately is both critical for the success of simulations of multiphase flows and also surprisingly difficult. In this chapter we discuss the difficulties with advecting the marker function directly and the various methods that have been developed to overcome these difficulties.

The VOF method is the oldest method to advect a marker function and – after many improvements and innovations – continues to be widely used. Other marker function methods include the level-set method, the phase-field method, and the CIP method. Instead of advecting the marker function directly, the boundary between the different fluids can also be tracked using marker points, and the marker function then reconstructed from the location of the interface. Methods using marker points are generally referred to as “front-tracking” methods to distinguish them from “front-capturing” methods, where the marker function is advected directly.