Thats depends on the problem itself, for example if you work with two immiscible fluids the volume of each fluid is the mean parameter, such as in VOF method. However, I think the most fundamental methods are those that are based on the Phase field methods which are totally based on thermodynamic concepts of interfaces.

Briefly it can be said that for track volume of each phase or enforcement of boundary condition. But it really depends on the formulation and physic of problem at hand. for example in diffuse interface methods usually mass fraction (or volume fraction) of one phase (in two phase flows) is explicitly appear in formulations. On the other hand in sharp interface methods the interface is used to enforce proper boundary condition which again heavily depends on the method. for example you can separate each phase and exchange forces between phases or use some methods like ghost fluid. Additionally shape of interface is very important when surface tension is assumed to be present in the flow.

Cheers

The volume fraction approach captures the interface because only its presence inside a given computational cell is known, so that if the value is say 0.5 then we know an interface is present in that particular cell otherwise the volume fraction would be either 1 or 0. An alternative approach using the same technique is the marker particle approach where each cell contains a number of particles each of which is assigned a given fluid identity, 1 for fluid 1, 0 if fluid 1 is not present. Again, the exact interface is captured as the volume fraction is returned by the particles as is the density and viscosity. This is only valid for immiscible, incompressible fluids.

Frank is right.

Please note that those techniques are highly dissipative and the interface may be very smeared, this is he reason that you must choose your F.D.scheme carefully (sometimes use downwind rather than upwind)