Volume of Fluid Method (VOF)

When using an Euler-Lagrange coupling, the fluid is modeled with a fixed mesh. The location of the free surface cannot be calculated by looking at the displacement of the nodes. Therefore, a different approach is needed to determine the free surface; several methods exist:

In the level-set free surface tracking method, the distance function ( ) is introduced to know the distance of the surface x from its initial position at t=0. The interface corresponds to ( ) . The problem with this model is that mass is not conserved.

The most popular model is the Volume of fluid model (VOF). The volume fraction F is the fractional volume of a cell occupied by the fluid, and has a value between zero and one. Every cell has its own VOF. Cells with F = 0 or F = 1 are called pure cells;

when , the cell is called a mixed cell. The VOF-method was first introduced by Hirt and Nichols. It has two important characterizing properties: The way the interface is reconstructed and the way it is propagated. As explained later on in this section, several methods exist to reconstruct the interface. Based on the obtained interface and the velocity fluxes at the different surfaces of each cell, fluid is moved from a donor cell to an acceptor cell. Only the value of the VOF in surrounding cells is needed to calculate the VOF in cell. This makes it easy to divide the calculations in parallel processes.

Rider identifies 4 steps to calculate the VOF, as shown in Fig. 2.12. In the first step, the volume is divided in discrete parts using a mesh. Next, the free surface is discretized.

Now the material fluxes can be calculated. In the final step, the volumes are integrated to a new time level.

Fig. 2.4. Steps taken to calculate Volume Fraction F

Two ways of reconstructing the surface are much used nowadays. First, there is the Simple Linear Interface Calculation method (SLIC), where the interface is always parallel to one of the coordinate axes. As can be seen in Fig. 2.13(a), this method is not very accurate.

A more sophisticated method is the Piecewise Linear Interface Reconstruction method (PLIC), as can be seen in Fig. 2.13 (b). To reconstruct the interface, it is divided by a line in a certain number of discrete partitions equal to the number of phases present in the cell.

The line is a linear approximation of the curved interface. Also, a discontinuity between the different lines is allowed. This method is very accurate when the curvature of the

interface is small, but even when the fluid surface has a large curvature it remains robust.

This is important, especially when modelling events which can contain droplets, because infinite curvatures may appear when these droplets reconnect.

Fig. 2.5. Comparison between the SLIC and PLIC surface reconstruction algorithms The advection equation that governs F for an incompressible fluid can be written as follows:

( ) (2.20) where u is the flow field. This equation shows that volume is conserved along a streamline.

The fluxes through a cell face can be calculated using:

(2.21) with A the cell surface and the time step.

When these fluxes are calculated, the VOF is updated from level n to n + 1:

(2.22) The indices n, e, s and w stand for north, east, south and west, representing the fluxes through the different faces of a cell in two dimensions. Away from the free surface, the net flux becomes zero. Around the free surface the calculation of the fluxes becomes more complicated, a method to do so can be found in Hurt and Nichols. There are however some major drawbacks to this method. First of all, conservation of mass may be violated when rounding F when or . This is especially the case for the

lower-order methods like SLIC. To address these problems, a local height function was introduced. A detailed description is given by Gerrits. The local height function h is defined in each surface cell, and gives the height of the surface in a column of three cells each. The direction of the local height function is defined as the direction most normal to the free surface.

Fig. 2.6. Local height function h

Normally, when the fluxes at the boundaries are calculated, the individual values of the VOF are updated. Here the local height function h is updated instead. Afterwards, the VOF is calculated from the height of the fluid in each column. No underflow or overflow of the column can appear, so this method is strictly mass conserving. As a conclusion, it can be stated that the introduction of the local height function definitely improves the performance of the VOF algorithm.

In the initial time step, the initial volume occupied by the water must be determined. This can be done using a predefined field for the material assignment. The Eulerian domain must be divided in different sections to be able to do this. A volume fraction between 0 and 1 can be assigned to a chosen section of the domain. It is important to include a certain region filled with void. When the Langrangian part moves through the Eulerian domain, the Eulerian elements occupied by the lagrangian part become filled with void.

Assigning at least one layer of void, results in the formation of a free surface. This is especially important when the lagrangian part is positioned outside of the Eulerian domain in the initial state of the model. But even when this is not the case, including void makes sure that water can be pushed out of the Eulerian element by the impacting body.