The fluid velocity is shown in figures 7-31 - 7-38 at for the four waves shown in table 7-1.
Table: The four waves in figures 7-31 - 7-38
Figure 7-31: Horizontal velocity contour plot for wave (1)
with , ,
and f = 1.4 at .
Figure 7-32: Vertical velocity contour plot for wave (1)
with , ,
and f = 1.4 at .
Figure 7-33: Horizontal velocity contour plot for wave (2)
with , ,
and f = 1.05 at .
Figure 7-34: Vertical velocity contour plot for wave (2)
with , ,
and f = 1.05 at .
Figure 7-35: Horizontal velocity contour plot for wave (3)
with , ,
and f = 1.86 at .
Figure 7-36: Vertical velocity contour plot for wave (3)
with , ,
and f = 1.86 at .
Figure 7-37: Horizontal velocity contour plot for wave (4)
with , ,
and f = 1.4 at .
Figure 7-38: Vertical velocity contour plot for wave (4)
with , ,
and f = 1.4 at .
All four waves have g in the range . The value of g affects both the magnitude and
the shape of the velocity profile since is a
function of g. Wave (1) has a relatively low viscosity and the lower
fluid is significantly denser than the top fluid. Wave (2) also has
a relatively low viscosity and the two fluids are of similar densities varying
by only 5%. Wave (3) has the same viscosity as the first two waves but
the density difference is considerable, the ratio of the densities is
1.86. Wave (4) has the same density distribution as wave (1) but the
viscosity is five times larger. For each wave the velocity is symmetric
about and . The velocities are not symmetric about x = 0,
although the contour plots are similar in both fluids. The difference is
greatest when there is a large density difference corresponding
to a large value of f.
The magnitudes of the velocities are
different for each wave. Wave (2) has velocities considerable smaller than
the other waves.
These lower velocities are shown in figures 7-33 and
7-34 where the contour lines are distorted slightly near
the interface.
This is due to small, spurious interface velocities
which have been observed for the lattice Boltzmann model and which are
due to the finite space and time steps [38].
This is not observed for the other waves where the velocities are higher.
For each wave the magnitudes of
u and w are similar. The vertical velocity peaks at z = 0 to
a slightly higher value than the peak horizontal velocity, which occurs
slightly above and below the interface. In wave (1) and wave (3) u peaks
close to the interface, the contours for the higher magnitudes have
an elliptical appearance, the semi-major axis parallel to the interface.
The contours for the lower magnitudes appear more triangular in shape with
the base near the interface and the opposite angle considerably
rounded. Waves (2) and (4), on the other hand,
have u peaking further from the
interface. The high magnitude contours are much more circular and the
lower magnitude contours are almost rectangular with curved corners.
The lower magnitude contours for the vertical velocity are elliptical
for all the waves, the semi-major axis is perpendicular to the interface.
The higher magnitude contours for waves (2) and (4) are approximately
circular while for waves (1) and (3) they are more elliptical in the same sense
as the horizontal velocity.