Volume of fluid method

In computational fluid dynamics, the Volume of fluid method is one of the most well known methods for volume tracking and locating the free surface. The motion of all phases is modelled by solving a single set of transport equations with appropriate jump boundary conditions at the interface [1]. It can model two or more immiscible fluids by solving a single set of momentum equations and tracking the volume fraction of each of the fluids throughout the domain. Typical applications include the motion of large bubbles in a liquid, the motion of liquid after a dam break, the prediction of jet breakup, and the steady or transient tracking of any liquid-gas interface.

In general, the steady or transient VOF formulation relies on the fact that two or more

fluids (or phases) are not interpenetrating. During the numerical calculation in each control volume, the sum of the volume fractions of all phases remains to unity. In ad-dition, the fields for all properties and variables are shared by the phases and represent volume-averaged values, as long as the volume fraction of each of the phases is known at each location. Thus in any given cell, the properties and variables are either purely rep-resentative of one of the phases, or reprep-resentative of a mixture of the phases, depending upon the volume fraction values. Also in other words, if p^th fluid’s volume fraction in the cell is denoted as α_p , then the following three different conditions are possible:

• α_p = 0: The computational cell is empty of the p^th fluid.

• αp = 1: The computational cell is full of thep^th fluid.

• 0< α <1: The computational cell contains interface betweenp^th fluid and one or more other fluids available.

Volume fraction equation

In VOF method, the interface(s) tracking between the phases is established by getting the solution of a continuity equation for the volume fraction of one (or more) of the phases [1] . For thep^th phase, this equation has the following form:

1 phase q and m˙qp is the mass transfer from phase q to phase p. This volume fraction equation will be solved for the secondary phase. It will not be solved for the primary phase. The primary-phase volume fraction will be calculated based on the following constraint:

n

X

p=1

α_p = 1 (17)

The volume fraction equation may be solved either through implicit or explicit time discretization scheme.

Momentum equation

In the VOF method, a single set of momentum equation is solved throughout the whole computational domain. The resulting velocity field is shared among the phases. Where the momentum equation is dependent on the volume fractions of all the phases through the densityρ and viscosityµ. The momentum equation is as follow:

∂

The VOF method can also include the effects of surface tension along the interface between each pair of phases. The model can be augmented by the additional specification of the contact angles between the phases and the walls. The continuum surface tension force (CSF) of Brackbill et al. [2] have been widely used to model surface tension in multiphase flow in volume of fluid (VOF), level-set (LS) and front tracking (FT) methods [3]. The solver will include the additional tangential stress terms that aries due to the avariation in surface tension coefficient. The effect of variable surface tension are usually important only in zero/near-zero gravity conditions.

The surface tension is a force, acting only at the surface, that is required to maintain equilibrium in such instances. Surface tension aries as a result of attractive forces between molecules in a fluid. For example, consider an air bubble in water. Within the bubble, the net force on a molecule due to its neighbors is zero. At the surface, the net force is radially inward, and the combined effect of the radial components of force across the entire spherical surface is to make the surface contract, thereby increasing the pressure on the concave side of the surface. It acts to balance the radially inward intermolecular attractive force with the radially outward pressure gradient force across the surface. In regions where two fluids are separated, but one of them is not in the form of spherical bubbles, the surface tension acts to minimize free energy by decreasing the area of the interface [4].

Reconstruction based schemes

For the special interpolation treatment to the computational cells that lie near the inter-face between two phases, there are two reconstruction based schemes as Geo-Reconstruct and Donor-Acceptor.

Figure 9 shows an actual interface shape along with the interfaces assumed during com-putation by these two methods.

Figure 9: Interface calculation

• The Geometric Reconstruction Scheme

Whenever a cell is completely filled with one phase or another, the geometric recon-struction approach, the standard interpolation schemes that are used to obtain the face fluxes. When the cell is near the interface between two phases, the geometric recon-struction scheme is used. The geometric reconrecon-struction scheme represents the interface between fluids using a piecewise-linear approach. In ANSYS FLUENT this scheme is the most accurate and is applicable for general unstructured meshes. It assumes that the interface between two fluids has a linear slope within each cell, and uses this linear shape for calculation of the advection of fluid through the cell faces. We can see in the above figure. The procedure of the geometric reconstruction approach is as follow,

• To calculate the position of the linear interface relative to the center of each partially-filled cell, based on information about the volume fraction and its deriva-tives in the cell

• To calculate the advecting amount of fluid through each face using the computed linear interface representation and information about the normal and tangential velocity distribution on the face

• To calculate the volume fraction in each cell using the balance of fluxes calculated during the previous step

• The Donor-Acceptor Scheme

When the cell is near the interface between two phases, a “the Donor-Acceptor” scheme is used to determine the amount of fluid advected through the face [4]. This scheme

identifies one cell as a donor of an amount of fluid from one phase and another (neighbor) cell as the acceptor of that same amount of fluid, and is used to prevent numerical diffusion at the interface. The amount of fluid from one phase that can be convected across a cell boundary is limited by the minimum of two values: the filled volume in the donor cell or the free volume in the acceptor cell. The orientation of the interface is also used in determining the face fluxes. It is either horizontal or vertical, depending on the direction of the volume fraction gradient of the p^th phase within the cell, and that of the neighbor cell that shares the face in question. The flux values are obtained by pure upwinding, pure downwinding, or some has a combination of the both. In addition it depends on the interface’s orientation as well as its motion.