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Peak Horizontal Velocity

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 tex2html_wrap_inline16861 . Here we look more closely at the dependence of d, the vertical distance between the two peeks, on the simulation parameters, tex2html_wrap_inline17031 and tex2html_wrap_inline12375 . The values of d was found from the simulations when t = T/4 at tex2html_wrap_inline12645 and are correct to within one lu. The theoretical values were obtained by finding the zeros of tex2html_wrap_inline17041 and tex2html_wrap_inline17043 [84] where tex2html_wrap_inline17045 and tex2html_wrap_inline17047 are given in equations (6.55) and (6.56). The results are shown in figures 7-56 - 7-58.

  figure6668

   figure6678
Figure 7-57: The vertical separation d of the horizontal velocity peaks as a function of the the viscosity tex2html_wrap_inline12375 when tex2html_wrap_inline12721 and tex2html_wrap_inline12323 .

   figure6688
Figure 7-58: The vertical separation d of the horizontal velocity peaks as a function of the wavelength tex2html_wrap_inline12727 when tex2html_wrap_inline12721 and tex2html_wrap_inline12731 .

The results in figure 7-56 are for three cases. Case (1) has tex2html_wrap_inline17095 and tex2html_wrap_inline12161 varied. Case (2) has tex2html_wrap_inline17099 and tex2html_wrap_inline12323 . The values of tex2html_wrap_inline15539 and tex2html_wrap_inline15541 are varied to give different values of f while keeping g fixed. Case (3) has tex2html_wrap_inline17111 and tex2html_wrap_inline12323 . 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the boundary layer is large compared to the wavelength the assumptions made in deriving the velocity expressions are no longer valid.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 tex2html_wrap_inline12189 , tex2html_wrap_inline12315 , f = 1.4 and tex2html_wrap_inline12323 . 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 tex2html_wrap_inline17131 , 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 tex2html_wrap_inline12721 and tex2html_wrap_inline12731 . A square grid was used here so tex2html_wrap_inline17139 and tex2html_wrap_inline17141 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 tex2html_wrap_inline17143 , which is shown in figure 7-59, decreases with increasing tex2html_wrap_inline12727 .

   figure6711
Figure 7-59: The horizontal velocity profile, at tex2html_wrap_inline12645 , tex2html_wrap_inline12457 , as a function of the dimensionless parameter tex2html_wrap_inline12737 . This is for waves with tex2html_wrap_inline12739 and tex2html_wrap_inline12731 .

This is as expected since as tex2html_wrap_inline12727 , 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 tex2html_wrap_inline12457 , tex2html_wrap_inline12305 is also shown in figure 7-60

   figure6722
Figure 7-60: The vertical velocity profile, at tex2html_wrap_inline12305 , tex2html_wrap_inline12457 , as a function of the the dimensionless parameter tex2html_wrap_inline12737 . This is for waves with tex2html_wrap_inline12739 and tex2html_wrap_inline12731 .

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.


next up previous contents
Next: Summary Up: Velocities Previous: Boundary Layer at the

James Buick
Tue Mar 17 17:29:36 GMT 1998