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Velocity Variation Across a Horizontal Cross-Section

The velocity variation along horizontal cross-sections through the wave were also examined. These are shown in figures 7-49 - 7-52

   figure6561
Figure 7-49: The horizontal velocity u as a function of x at tex2html_wrap_inline12457 . 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.

   figure6570
Figure 7-50: The vertical velocity w as a function of x at tex2html_wrap_inline12457 . 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.

   figure6579
Figure 7-51: The horizontal velocity u as a function of x at tex2html_wrap_inline12457 . 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.

   figure6588
Figure 7-52: The horizontal velocity u as a function of x at tex2html_wrap_inline12457 . 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 tex2html_wrap_inline12457 . 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 tex2html_wrap_inline12305 than they are at x = 0 and tex2html_wrap_inline16951 . Conversely when z is negative the results for x = 0 and tex2html_wrap_inline16951 are closer to the interface than the results for tex2html_wrap_inline12305 . This distorts the results from the plotted sine curves.


next up previous contents
Next: Boundary Layer at the Up: Velocities Previous: Velocity Variation Across a

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