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In computational fluid dynamics, the 'volume of fluid (VOF) method' is a free-surface modelling technique, i.e. a numerical technique for tracking and locating the free surface (or fluid-fluid interface). It belongs 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 is an advection scheme—a numerical recipe that allows the programmer to track the shape and position of the interface, but it is not a standalone flow solving algorithm. The Navier–Stokes equations describing the motion of the flow have to be solved separately. The same applies for all other advection algorithms.
Contents
History
The volume of fluid method is based on earlier Marker-and-cell (MAC) methods. First accounts of what is now known as VOF have been given by Noh & Woodward in 1976, where fraction function
Overview
The method is based on the idea of a so-called fraction function
The evolution of the
with the following constraint
i.e., the volume of the fluids is constant. For each cell, properties such as density
These properties are then used to solve a single momentum equation through the domain, and the attained velocity field is shared among the fluids.
The VOF method is computationally friendly, as it introduces only one additional equation and thus requires minimal storage. The method is also characterized by its capability of dealing with highly non-linear problems in which the free-surface experiences sharp topological changes. By using the VOF method, one also evades the use of complicated mesh deformation algorithms used by surface-tracking methods. The major difficulty associated with the method is the smearing of the free-surface. This problem originates from excessive diffusion of the transport equation.
Discretization
To avoid smearing of the free-surface, the transport equation has to be solved without excessive diffusion. Thus, the success of a VOF method depends heavily on the scheme used for the advection of the C field. Whereas a first order upwind scheme smears the interface, a downwind scheme of the same order will cause a false distribution problem which will cause erratic behavior in case of the flow is not oriented along a grid line. As these lower order schemes are inaccurate, and higher order schemes are unstable and induce oscillations, it has been necessary to develop schemes which keep the free-surface sharp while also producing monotonic profiles for C. Over the years, a multitude of different methods for treating the advection have been developed. In the original VOF-article by Hirt, a donor-acceptor scheme was employed. This scheme formed a basis for the compressive differencing schemes.
The different methods for treating VOF can be roughly divided into three categories, namely the donor-acceptor formulation, higher order differencing schemes and line techniques. The donor-acceptor scheme is based on two fundamental criteria, namely the boundedness criterion and the availability criterion. The first one states that the value of
In the higher order differencing schemes, as the name suggests, the convective transport equation is discretized with higher order or blended differencing schemes. Such methods include the Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM) and High Resolution Interface Capturing (HRIC) scheme, which are both based on the Normalized Variable Diagram (NVD) by Leonard.
Line techniques circumvent the problems associated with the discretization of the transport equation by not tracking the interface in a cell explicitly. Instead, the fluid distribution in a cell an interface is obtained by using the volume fraction distribution of neighbouring cells. The Simple Line Interface Calculation (SLIC) by Noh and Woodward from 1976 uses a simple geometry to reconstruct the interface. In each cell the interface is approximated as a line parallel to one of the coordinate axes and assumes different fluid configurations for the horizontal and vertical movements respectively. A widely used technique today is the Piecewise Linear Interface Calculation by Youngs. PLIC is based on the idea that the interface can be represented as a line in R2 or a plane in R3; in the latter case we may describe the interface by:
where