12.5 Free-Surface Flows
391
the end of each time step, when the outer iterations converge, all corrections
are zero. For a detailed description of this approach and its implementation
on arbitrary unstructured 3D-grids, see Muzaferija and PeriC (1997,1999).
H
Fig. 12.7. Geometry and boundary conditions for the
sloshing problem
Flows with free surfaces, like open channel flows, flows around ships etc.
are characterized by the Froude number:
where g is the gravity acceleration, v is the reference velocity, L is the reference length; fl is the velocity of a wave of length L in deep water. When
Fr > 1, the fluid velocity is greater than the wave speed and the flow is
said to be supercritical and waves cannot travel upstream (as in supersonic
compressible flow). When Fr < 1, waves can travel in all directions. If the
method of calculating the free surface shape is not properly implemented,
disturbances in the form of small waves may be generated and it may not
be possible to obtain a steady solution. A method which does not generate
waves where they physically should not be (e.g. in front of ships) is said to
satisfy the radiation condition.
We now present three examples of free-surface flow problems. The first is
small-amplitude sloshing in a 2D tank. The initial free surface is a sine wave
with amplitude 1% of the water depth; see Fig. refsloshgeo. If the liquid is
inviscid, sloshing continues forever, i.e. the wave is not damped.
Figure 12.8 shows results of computations on a 40 x 40 CV uniform grid.
The free-surface heights at the two side walls are shown as functions of time.
In the inviscid case, (left figure) the amplitude is nearly unchanged after
several oscillation periods. When viscosity is introduced (right figure), the
wave amplitude decays with time and the period is changed. Linear theory,
which is accurate for small amplitude waves, predicts the period to be T =
3.55. Lilek (1995) obtained T = 3.55 for the inviscid case and T = 3.65 for
the viscous case.
The second example is the flow over a hydrofoil under a free surface.
The geometry of the problem and the grid used are shown in Fig. 12.9.
The grid is block-structured with non-matching interfaces and the steadystate shape of the free surface is shown. The hydrofoil of length L = 0.203
391
the end of each time step, when the outer iterations converge, all corrections
are zero. For a detailed description of this approach and its implementation
on arbitrary unstructured 3D-grids, see Muzaferija and PeriC (1997,1999).
H
Fig. 12.7. Geometry and boundary conditions for the
sloshing problem
Flows with free surfaces, like open channel flows, flows around ships etc.
are characterized by the Froude number:
where g is the gravity acceleration, v is the reference velocity, L is the reference length; fl is the velocity of a wave of length L in deep water. When
Fr > 1, the fluid velocity is greater than the wave speed and the flow is
said to be supercritical and waves cannot travel upstream (as in supersonic
compressible flow). When Fr < 1, waves can travel in all directions. If the
method of calculating the free surface shape is not properly implemented,
disturbances in the form of small waves may be generated and it may not
be possible to obtain a steady solution. A method which does not generate
waves where they physically should not be (e.g. in front of ships) is said to
satisfy the radiation condition.
We now present three examples of free-surface flow problems. The first is
small-amplitude sloshing in a 2D tank. The initial free surface is a sine wave
with amplitude 1% of the water depth; see Fig. refsloshgeo. If the liquid is
inviscid, sloshing continues forever, i.e. the wave is not damped.
Figure 12.8 shows results of computations on a 40 x 40 CV uniform grid.
The free-surface heights at the two side walls are shown as functions of time.
In the inviscid case, (left figure) the amplitude is nearly unchanged after
several oscillation periods. When viscosity is introduced (right figure), the
wave amplitude decays with time and the period is changed. Linear theory,
which is accurate for small amplitude waves, predicts the period to be T =
3.55. Lilek (1995) obtained T = 3.55 for the inviscid case and T = 3.65 for
the viscous case.
The second example is the flow over a hydrofoil under a free surface.
The geometry of the problem and the grid used are shown in Fig. 12.9.
The grid is block-structured with non-matching interfaces and the steadystate shape of the free surface is shown. The hydrofoil of length L = 0.203