Volume of Fluid Method - Specification

Specification

The method is based on the idea of so-called fraction function . It is defined as the integral of fluid's characteristic function in the control volume (namely, volume of a computational grid cell). Basically, when the cell is empty, with no traced fluid inside. the value of is zero; when the cell is full, ; and when the interphasal interface cuts the cell, then . is a discontinuous function, its value jumps from 0 to 1 when the argument moves into interior of traced phase.

The fraction function is a scalar function, and while the fluid moves with velocity (in three-dimensional space ) every fluid particle retains its identity, i.e. when a particle is a given phase, it doesn't change the phase – like a particle of air, that is a part of air bubble in water remains air particle, regardless of the bubble movement (actually, for this to hold, we have to disregard processes such as dissolving of air in water). If that is so, then the substantial derivative of fraction function needs to be equal to zero:

This is actually the same equation that has to be fulfilled by the level set distance function .

This equation cannot be easily solved directly, since is discontinuous, but such attempts have been performed. But the most popular approach to the equation is the so called geometrical reconstruction, originating in the works of Hirt and B. D. Nichols. The most popular approach to the interface reconstruction, the PLIC (Piecewise Linear Interface Calculation), bases on the idea, that the interface can be represented as a line in or a plane in, in the latter case wa may describe the interface by:

,

where is a vector normal to the interface. Compontents of the normal are found i.e. by using the Finite Difference method or its combination with least squares optimization. The free term is then found (analytically or by approximation) by enforcing mass conservation within computational cell.

Once the description of the interface is established, the advection equation of is solved using geometrical techniques such as finding the flux of between grid cells, or advecting the endpoints of interface using discrete values of fluid velocity.

The VOF method is known for its ability to conserve the "mass" of the traced fluid, also, when fluid interface changes its topology, this change is traced easily, so the interfaces can for example join, or break apart.