12.5 Free-Surface Flows
389
CV, but the prescribed pressure at the free surface produces a velocity
correction there, so that mass fluxes through the free surface are non-zero.
0 Correct the position of the free surface to enforce the kinematic boundary
condition. Each free-surface cell face is moved so that the volume flux due
to its movement compensates the flux obtained in the previous step.
0 Iterate until no further adjustment is necessary and both the continuity
equation and momentum equations are satisfied.
0 Advance to the next time step.
The critical issue for the efficiency and stability of the method is the
algorithm for the movement of the free surface. The problem is that there
is only one discrete equation per free-surface cell face but a larger number
of grid nodes that have to be moved. Correct treatment of the intersections
of the free surface with other boundaries (inlet, outlet, symmetry, walls) is
essential if wave reflection and/or instability is to be avoided. We shall briefly
describe one such method. Only two-time-level schemes are considered, but
the approach can be extended.
The mass flux through a moving free-surface cell face is (see Eqs. (12.18)
and (12.22)):
The superscript T denotes the time (t, < t, < tn+l) at which the quantity
is calculated; for the implicit Euler scheme, t, = tn+l, while for the CrankNicolson scheme, t, = f racl2(t, + t,+l).
The mass fluxes obtained from the pressure-correction equation with prescribed pressure a t the free surface are non-zero; we compensate by displacing
the free surface, i.e.:
From this equation we obtain the volume of fluid his which has to flow into or
out of the CV due to free-surface motion. We need to obtain the coordinates
of the CV vertices that lie on the free surface from this equation. This has
to be done carefully and thus requires special attention. Because there is a
single volumetric flow rate for each cell but a greater number of CV vertices,
there are more unknowns than equations.
Thk et al. (1994) suggested using staggered CVs in the layer adjacent to
free surface, but only in the continuity (pressure- correction) equation. The
method was applied to several problems in 2D and showed good performance.
However, it requires substantial adaptation of the solution method, especially
in 3D; see Thk et al. (1994) for more details.
Another possibility is to define the CVs under the free surface not by
vertices but by cell-face centers; the vertices are then defined by interpolating
cell-face center locations, as shown in Fig. 12.6 for a 2D structured grid.
389
CV, but the prescribed pressure at the free surface produces a velocity
correction there, so that mass fluxes through the free surface are non-zero.
0 Correct the position of the free surface to enforce the kinematic boundary
condition. Each free-surface cell face is moved so that the volume flux due
to its movement compensates the flux obtained in the previous step.
0 Iterate until no further adjustment is necessary and both the continuity
equation and momentum equations are satisfied.
0 Advance to the next time step.
The critical issue for the efficiency and stability of the method is the
algorithm for the movement of the free surface. The problem is that there
is only one discrete equation per free-surface cell face but a larger number
of grid nodes that have to be moved. Correct treatment of the intersections
of the free surface with other boundaries (inlet, outlet, symmetry, walls) is
essential if wave reflection and/or instability is to be avoided. We shall briefly
describe one such method. Only two-time-level schemes are considered, but
the approach can be extended.
The mass flux through a moving free-surface cell face is (see Eqs. (12.18)
and (12.22)):
The superscript T denotes the time (t, < t, < tn+l) at which the quantity
is calculated; for the implicit Euler scheme, t, = tn+l, while for the CrankNicolson scheme, t, = f racl2(t, + t,+l).
The mass fluxes obtained from the pressure-correction equation with prescribed pressure a t the free surface are non-zero; we compensate by displacing
the free surface, i.e.:
From this equation we obtain the volume of fluid his which has to flow into or
out of the CV due to free-surface motion. We need to obtain the coordinates
of the CV vertices that lie on the free surface from this equation. This has
to be done carefully and thus requires special attention. Because there is a
single volumetric flow rate for each cell but a greater number of CV vertices,
there are more unknowns than equations.
Thk et al. (1994) suggested using staggered CVs in the layer adjacent to
free surface, but only in the continuity (pressure- correction) equation. The
method was applied to several problems in 2D and showed good performance.
However, it requires substantial adaptation of the solution method, especially
in 3D; see Thk et al. (1994) for more details.
Another possibility is to define the CVs under the free surface not by
vertices but by cell-face centers; the vertices are then defined by interpolating
cell-face center locations, as shown in Fig. 12.6 for a 2D structured grid.