Volume of fluid method

In Computational fluid dynamics, Volume of fluid (VOF) method is a numerical technique for tracking and locating the free surface (or fluid-fluid interface).

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.

The method is based on the idea of so called fraction function $C$. 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 (there's no traced fluid inside) value of $C$ is zero, if cell is full, we have $C=1$, and when the interphasal interface cuts the cell, then . C is an discontinous function, its value jumps from 0 to 1 when the argument moves into interior of traced phase.

The fraction function $C$ is a scalar function, and while the fluid moves with velocity $mathbf\{V\}=(u(x,y,z),v(x,y,z),w(x,y,z))$ (in three-dimensional space $mathbf\{R^3\}$) 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 $C$ needs to be equal to zero:

$frac\{partial\; C\}\{partial\; t\}\; +\; mathbf\{V\}cdot\; abla\; C\; =0.$

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

There are various means to solve this equation. It cannot be solved directly easily, since $C$ is discontinous, nevertheless such attempts have been performed. But most popular approach to the equation is the so called geometrical reconstruction, originating in the works of Hirt and Nichols.