12.5 Free-Surface Flows
383
where AT is the bulk temperature difference across the domain, L is a characteristic length of the surface and K is the thermal diffusivity.
Fig. 12.4. On the description of
boundary conditions at the free surface
In many applications the shear stress at the free surface can be neglected
(e.g. ocean waves with no appreciable wind). The normal stress and the effect of the surface tension are also often neglected; in this case the dynamic
boundary condition reduces to pl = p,.
The implementation of these boundary conditions is not as trivial as it
would appear. If the position of the free surface were known, there would be
little problem. The mass flux can be set to zero on cell faces lying on the free
surface, and the forces acting on the cell face from outside can be calculated;
if surface tension is neglected, only the pressure force remains. The problem is
that the location of the free surface must be computed as part of the solution
and is not usually known in advance. One can therefore directly implement
only one of the boundary conditions at the free surface; the other must be
used to locate the surface. Location of the surface must be done iteratively,
greatly increasing the complexity of the task.
Many methods have been used to find the shape of the free surface. They
can be classified into two major groups:
Methods which treat the free surface as a sharp interface whose motion is
followed (interface-tracking methods). In this type of method, boundaryfitted grids are used and advanced each time the free surface is moved. In
explicit methods, which must use small time steps, the problems associated
with grid movement that were discussed earlier are often ignored.
Methods which do not define the interface as a sharp boundary (interfacecapturing methods). The computation is performed on a fixed grid, which
extends beyond the free surface. The shape of the free surface is determined
by computing the fraction of each near-interface cell that is partially filled.
This can be achieved by introducing massless particles at the free surface
a t the initial time and following their motion; this is called the Markerand-Cell or MAC scheme that was proposed by Harlow and Welch (1965).
Alternatively, one can solve a transport equation for the fraction of the cell
383
where AT is the bulk temperature difference across the domain, L is a characteristic length of the surface and K is the thermal diffusivity.
Fig. 12.4. On the description of
boundary conditions at the free surface
In many applications the shear stress at the free surface can be neglected
(e.g. ocean waves with no appreciable wind). The normal stress and the effect of the surface tension are also often neglected; in this case the dynamic
boundary condition reduces to pl = p,.
The implementation of these boundary conditions is not as trivial as it
would appear. If the position of the free surface were known, there would be
little problem. The mass flux can be set to zero on cell faces lying on the free
surface, and the forces acting on the cell face from outside can be calculated;
if surface tension is neglected, only the pressure force remains. The problem is
that the location of the free surface must be computed as part of the solution
and is not usually known in advance. One can therefore directly implement
only one of the boundary conditions at the free surface; the other must be
used to locate the surface. Location of the surface must be done iteratively,
greatly increasing the complexity of the task.
Many methods have been used to find the shape of the free surface. They
can be classified into two major groups:
Methods which treat the free surface as a sharp interface whose motion is
followed (interface-tracking methods). In this type of method, boundaryfitted grids are used and advanced each time the free surface is moved. In
explicit methods, which must use small time steps, the problems associated
with grid movement that were discussed earlier are often ignored.
Methods which do not define the interface as a sharp boundary (interfacecapturing methods). The computation is performed on a fixed grid, which
extends beyond the free surface. The shape of the free surface is determined
by computing the fraction of each near-interface cell that is partially filled.
This can be achieved by introducing massless particles at the free surface
a t the initial time and following their motion; this is called the Markerand-Cell or MAC scheme that was proposed by Harlow and Welch (1965).
Alternatively, one can solve a transport equation for the fraction of the cell