In Chapter 4 we saw several families of methods for locating and advecting the interface. Here, we focus on one of them: the VOF method. Historically, it was used for free surface flows with only one “fluid,” although it is now routinely used for two-fluid flows. In the VOF method the marker function is represented by the fraction of a computational grid cell which is occupied by the fluid assumed to be the reference phase.

A very large number (probably dozens) of VOF methods have been proposed. When we choose the method we try to strike a balance between several qualities: accuracy, simplicity and volume conservation.

Basic properties

The volume fraction or color function C is the discrete version of the characteristic function H; see Equation (4.3). We will be considering only two-phase or free-surface flows, so that the C data represent the fraction of each grid cell occupied by the reference phase. Furthermore, we restrict our analysis to Cartesian grids with square cells of side h = Δx = Δy.

The function C varies between the constant value one in full cells to zero in empty cells, while mixed cells with an intermediate value of C define the transition region where the interface is localized.

Low-order VOF methods do not need to specify the location of the interface in the transition region, but a geometrical interpretation of these methods shows that in two dimensions the interface line in each mixed cell is represented by a segment parallel to one of the two coordinate axes.