Volume of fluid method

In computational fluid dynamics, the volume of fluid (VOF) method is a family of free-surface modelling techniques, i.e. numerical techniques for tracking and locating the free surface (or fluid–fluid interface). They belong to the class of Eulerian methods which are characterized by a mesh that is either stationary or is moving in a certain prescribed manner to accommodate the evolving shape of the interface. As such, VOF methods are advection schemes capturing the shape and position of the interface, but are not standalone flow solving algorithms. The Navier–Stokes equations describing the motion of the flow have to be solved separately.

History The volume of fluid method is based on earlier Marker-and-cell (MAC) methods developed at Los Alamos National Laboratory. MAC used Lagrangian marker particles to track the distribution of fluid in a fixed Eulerian grid. The use of marker particles was computationally expensive because it required many marker particles per grid cell, to reduce numerical noise when discrete marker particles move across grid cells. The original idea of the VOF method was to replace marker particles with a single scalar variable per grid cell representing the volume fraction of fluid in it. Thereby, the volume of fluid is governed by an advection equation. This idea arose from studies of two-phase mixture (water and steam) problems where it was customary to use a volume of steam variable. The VOF approach was first demonstrated in a 1975 publication “Methods for Calculating Multi-Dimensional, Transient Free Surface Flows Past Bodies” by Nichols and Hirt. This publication described how to advect the fluid fraction with a Donor-Acceptor scheme, how to estimate the orientation and position of the free surface inside surface cells, and how to prescribe appropriate boundary conditions (continuity and zero shear stress) at the free surface. This approach was much simpler than other techniques tracking the surface of fluid, yet more versatile as it could model the coalescence and breakup of fluid regions. In 1976, Noh & Woodward presented the Simple Line Interface Calculation (SLIC), a technique to approximate fluid interfaces based on volume fractions, designed for directional-split advection scheme of volume fractions. SLIC could also handle an arbitrary number of immiscible fluid phases per grid cells. Thereby, SLIC was well suited to the VOF approach, although the two methods were initially independent and remained separate till the 90s. The term “Volume of Fluid method” and it acronym “VOF” method were coined in the 1980 Los Alamos Scientific Laboratory report, “SOLA-VOF: A Solution Algorithm for Transient Fluid Flow with Multiple Free Boundaries,” by Nichols, Hirt and Hotchkiss and in the journal publication “Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries” by Hirt and Nichols in 1981. These two publications provided more details about the specific procedures used to approximate the position of the free surface (locally represented by an inclined line in surface cells) and apply the free surface boundary conditions on it. Since VOF method surpassed MAC by lowering computer storage requirements, it quickly became popular. Early applications of the SOLA-VOF program developed at Los Alamos include light-water-reactor safety studies. A variant of the SOLA-VOF code was also adopted by NASA. In 1982, Youngs developed the Piecewise-Linear Interface Calculation (PLIC) scheme, which improved accuracy of interface reconstruction upon the SLIC and early VOF methods.

Overview The method is based on the idea of a fraction function C {\displaystyle C} . It is a scalar function, defined as the integral of a fluid's characteristic function in the control volume, namely the volume of a computational grid cell. The volume fraction of each fluid is tracked through every cell in the computational grid, while all fluids share a single set of momentum equations, i.e. one for each spatial direction. From a cell-volume averaged perspective, when a cell is empty of the tracked phase, the value of C {\displaystyle C} is zero; when the cell is full of tracked phase, C = 1 {\displaystyle C=1} ; and when the cell contains an interface between the tracked and non-tracked volumes, 0 < C < 1 {\displaystyle 0<C<1} . From a perspective of a local point that contains no volume, C {\displaystyle C} is a discontinuous function insofar as its value jumps from 0 to 1 when the local point moves from the non-tracked to the tracked phase. The normal direction of the fluid interface is found where the value of C {\displaystyle C} changes most rapidly. With this method, the free-surface is not defined sharply, instead it is distributed over the height of a cell. Thus, in order to attain accurate results, local grid refinements have to be done. The refinement criterion is simple, cells with 0 < C < 1 {\displaystyle 0<C<1} have to be refined. A method for this, known as the marker and micro-cell method, has been developed by Raad and his colleagues in 1997. The evolution of the m {\displaystyle m} -th fluid in a system on n {\displaystyle n} fluids is governed by the transport equation (actually the same equation that has to be fulfilled by the level-set method distance function ϕ {\displaystyle \phi } ):

∂ C m ∂ t + v ⋅ ∇ C m = 0 , {\displaystyle {\frac {\partial C_{m}}{\partial t}}+\mathbf {v} \cdot \nabla C_{m}=0,}

with the following constraint

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