The results above all show that the magnitude of the horizontal velocity |u| peaks above and below the interface within the boundary layer region. The height at which the peaks occur has been seen to depend on the . Here we look more closely at the dependence of d, the vertical distance between the two peeks, on the simulation parameters, and . The values of d was found from the simulations when t = T/4 at and are correct to within one lu. The theoretical values were obtained by finding the zeros of and [84] where and are given in equations (6.55) and (6.56). The results are shown in figures 7-56 - 7-58.
Figure 7-57: The vertical separation d of the horizontal velocity peaks
as a function of the the viscosity when and .
Figure 7-58: The vertical separation d of the horizontal velocity peaks
as a function of the wavelength
when and .
The results in figure 7-56 are for three cases. Case (1) has and varied. Case (2) has and . The values of and are varied to give different values of f while keeping g fixed. Case (3) has and . There is good agreement with the theory except for the lowest values of f, the density ratio, where the theory predicts a value for d which is significantly larger than the one measured in the simulations. When the boundary layer is large compared to the wavelength the assumptions made in deriving the velocity expressions are no longer valid. The results in figure 7-57 show the variation in d for different values of the viscosity when , , f = 1.4 and . Here the simulation results also show good agreement with the theory except at the highest values of the viscosity where, again, the size of the boundary layer is becoming significant compared to the wavelength. The results here suggest that, provided the size of the boundary layer (taken here to be d/2) is less than , there is good agreement between the expressions found from the theory and the simulation results. Figure 7-58 shows the value of d for different wavelengths when and . A square grid was used here so and for each wavelength. Again a good fit is observed between the simulation points and the theory. The size of the boundary layer about the interface is seen, in figure 7-58, to increase with the wavelength when all other parameters are unchanged. Despite this increase in the size of the boundary layer the dimensionless size of the boundary layer , which is shown in figure 7-59, decreases with increasing .
Figure 7-59: The horizontal velocity profile, at , ,
as a function of the dimensionless parameter . This is
for waves with and .
This is as expected since as , and hence the Reynolds number of the wave, is increased it is expected that the viscous effects become less significant. The profile of the vertical velocity at , is also shown in figure 7-60
Figure 7-60: The vertical velocity profile, at , ,
as a function of the the dimensionless parameter . This is
for waves with and .
for the same waves. The viscosity has less affect on the vertical velocity and all the profiles have the same shape with a different amplitude. There is some small variations in w which can be observed near the solid boundaries of the waves with the larger wavelengths. These are similar to those observed in figure 7-55 and are also due to the initial separation of the interface. If it also possible that there may be a small residual oscillation from the initial settling of the density gradient. However, every care was taken to ensure this had damped out before the simulations were performed.