The velocity variation along horizontal cross-sections through the wave were also examined. These are shown in figures 7-49 - 7-52
Figure 7-49: The horizontal velocity u as a function of x at
.
The results are for
wave (1) at different heights z within the inviscid body of the wave.
The solid lines are sine curves with an appropriate amplitude.
Figure 7-50: The vertical velocity w as a function of x at .
The results are for
wave (1) at different heights z within the inviscid body of the wave.
The solid lines are cosine curves with an appropriate amplitude.
Figure 7-51: The horizontal velocity u as a function of x at .
The results are for
wave (1) at different heights z within the viscous boundary layer at the
solid boundaries.
The solid lines are sine curves with an appropriate amplitude.
Figure 7-52: The horizontal velocity u as a function of x at .
The results are for
wave (1) at different heights z within the viscous boundary layer at the
interface.
The solid lines are sine curves with an appropriate amplitude.
for wave (1) at .
Figures 7-49 and 7-50 show the horizontal and vertical
velocities along cross-sections through the inviscid body of the fluid.
Figures 7-51 and 7-52 show the horizontal velocity
along cross-sections through the viscous boundary layers near the
solid boundaries and the interfacial region respectively. Also plotted
in the figures are sine and cosine curves with
selected amplitudes, these are represented by the solid lines. The vertical
velocity is very small within the boundary layer at the solid boundary. In the interfacial boundary layer the variation in w is the same as that
shown in figure 7-50.
The results in figures 7-49 and 7-50 show very good agreement
with the sine and cosine curves. Thus the wave velocities are seen to be
following the expected variation with horizontal distance. Note that
the amplitude of the sine and cosine curves in figures 7-49 and
7-50 (and in figures 7-51 and 7-52) have
been picked arbitrarily to give a good fit to the results and are not the
amplitudes predicted by theory. This was done because any small deviation
of the velocity profile from a sinusoidal variation would not be
obvious if the simulation results were being compared to a sine curve
with a different amplitude. The difference between the amplitude of
the results and the amplitude predicted by equations (6.55)
- (6.58)
is typically
small and can be seen in figures 7-39 and 7-40.
The results in figures 7-51 show the comparison
inside the boundary layer at the fixed boundaries.
The fit here is not quite as good, the results vary slightly
from the sine curves.
Any small phase shift introduced
by the irrotational part of the velocity will be the same at all points
and so should not affect the shape of the curves.
The results in figure 7-52 show a poorer fit close to the
interface. This is because the interface is not completely flat,
see figures 7-31 and 7-32.
For z positive the results are closer to the
interface, and hence smaller, at than they are at
x = 0 and . Conversely when z is negative the results
for x = 0 and are closer to the interface than the results
for . This distorts the results from the plotted sine curves.