384
12. Special Topics
occupied by the liquid phase (the Volume-of-Fluid or VOF scheme, Hirt
and Nichols, 1981).
There are also hybrid methods. All of these methods can also be applied
to some kinds of two-phase flows as will be discussed in the following section.
Interface-Capturing Methods. The MAC scheme is attractive because it
can treat complex phenomena like wave breaking. However, the computing
effort is large, especially in three dimensions because, in addition to solving
the equations governing fluid flow, one has to follow the motion of a large
number of marker particles.
In the VOF method, in addition to the conservation equations for mass
and momentum, one solves an equation for the filled fraction of each control
volume, c so that c = 1 in filled CVs and c = 0 in empty CVs. From the
continuity equation, one can show that the evolution of c is governed by the
transport equation:
dc
- + div (cv) = 0 .
d t
In incompressible flows this equation is invariant with respect to interchange
of c and 1 - c; for this to be assured in the numerical method, mass conservation has to be strictly enforced.
The critical issue in this type of method is the discretization of convective
term in Eq. (12.29). Low-order schemes (like the first-order accurate upwind
method) smear the interface and introduce artificial mixing of the two fluids,
so higher-order schemes are preferred. Since c must satisfy the condition
it is important to ensure that the method does not generate overshoots or
undershoots. Fortunately, it is possible to derive schemes which both keep
the interface sharp and produce monotone profiles of c across it; see Leonard
(1997) for some examples and Ubbink (1997) or Muzaferija and Perid (1999)
for methods specifically designed for interface-capturing in free surface flows.
This approach is more efficient than the MAC scheme and can be applied
to complex free surface shapes including breaking waves. However, the free
surface profile is not sharply defined; it is usually smeared over one to three
cells (similar to shocks in compressible flows). Local grid refinement is important for accurate resolution of the free surface. The refinement criterion
is simple: cells with 0 < c < 1 need to be refined. A method of this kind,
called the marker and micro-cell method, has been developed by Raad and
his colleagues (see, for example Chen et al., 1997).
There are several variants of the above approach. In the original VOFmethod (Hirt and Nichols, 1981) Eq. (12.29) is solved in the whole domain to
find the location of the free surface; the mass and momentum conservation
equations are solved for the liquid phase only. The method can calculate flows
12. Special Topics
occupied by the liquid phase (the Volume-of-Fluid or VOF scheme, Hirt
and Nichols, 1981).
There are also hybrid methods. All of these methods can also be applied
to some kinds of two-phase flows as will be discussed in the following section.
Interface-Capturing Methods. The MAC scheme is attractive because it
can treat complex phenomena like wave breaking. However, the computing
effort is large, especially in three dimensions because, in addition to solving
the equations governing fluid flow, one has to follow the motion of a large
number of marker particles.
In the VOF method, in addition to the conservation equations for mass
and momentum, one solves an equation for the filled fraction of each control
volume, c so that c = 1 in filled CVs and c = 0 in empty CVs. From the
continuity equation, one can show that the evolution of c is governed by the
transport equation:
dc
- + div (cv) = 0 .
d t
In incompressible flows this equation is invariant with respect to interchange
of c and 1 - c; for this to be assured in the numerical method, mass conservation has to be strictly enforced.
The critical issue in this type of method is the discretization of convective
term in Eq. (12.29). Low-order schemes (like the first-order accurate upwind
method) smear the interface and introduce artificial mixing of the two fluids,
so higher-order schemes are preferred. Since c must satisfy the condition
it is important to ensure that the method does not generate overshoots or
undershoots. Fortunately, it is possible to derive schemes which both keep
the interface sharp and produce monotone profiles of c across it; see Leonard
(1997) for some examples and Ubbink (1997) or Muzaferija and Perid (1999)
for methods specifically designed for interface-capturing in free surface flows.
This approach is more efficient than the MAC scheme and can be applied
to complex free surface shapes including breaking waves. However, the free
surface profile is not sharply defined; it is usually smeared over one to three
cells (similar to shocks in compressible flows). Local grid refinement is important for accurate resolution of the free surface. The refinement criterion
is simple: cells with 0 < c < 1 need to be refined. A method of this kind,
called the marker and micro-cell method, has been developed by Raad and
his colleagues (see, for example Chen et al., 1997).
There are several variants of the above approach. In the original VOFmethod (Hirt and Nichols, 1981) Eq. (12.29) is solved in the whole domain to
find the location of the free surface; the mass and momentum conservation
equations are solved for the liquid phase only. The method can calculate flows