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Next: Standing Waves Up: Waves in a Viscous Previous: Frequency and Damping Parameter

Wave Velocities

  The linearised Navier-Stokes equation permits the velocity to be split into a potential part, tex2html_wrap_inline15875 , and a rotational part, tex2html_wrap_inline15877 [78, 79, 77, 80, 81]. The solution for tex2html_wrap_inline13033 is simply the inviscid solution which has already been given in equation (6.17). To find the rotational velocity we note [81, 80] that provided

  equation5632

is small in both fluids; the motion is essentially irrotational except near viscous boundary layers which are of thickness tex2html_wrap_inline15881 and which occur at solid boundaries and at the interface. Thus we look for solutions for tex2html_wrap_inline15883 which satisfy the Navier-Stokes equation for the rotational part of the velocity,

  equation5642

and the appropriate boundary conditions. We only consider solutions which have a significant value in the boundary layer and a negligible value outside the boundary layer. Since we expect [79, 80] that tex2html_wrap_inline15889 will be an order of magnitude smaller then tex2html_wrap_inline15891 we will solve for tex2html_wrap_inline15891 , the component in the direction of wave propagation. The solution for tex2html_wrap_inline15889 can then be found by integrating tex2html_wrap_inline15891 [80]. Looking first at the area of the interface we consider solutions of the form [78, 80]

  equation5651

Now consider the boundary conditions at the interface

  1. The conservation of horizontal velocity at the interface requires

      equation5666


  2. Conservation of the stress tex2html_wrap_inline15825 across the interface requires

      equation5675

Substituting the velocities, equation (6.48), into the boundary conditions, equations (6.49) and (6.50) gives

  equation5687

where tex2html_wrap_inline15901 and tex2html_wrap_inline15903 .

Next consider the fluid adjacent to the solid boundaries. We look for solutions, for the rotational velocity close to the boundary layer, of the form [78, 80]

  equation5700

Substituting these into the no-slip boundary condition

equation5714

gives the constants

equation5725

The final solution for tex2html_wrap_inline12875 can be found by summing all the horizontal velocity terms. In doing this we neglect tex2html_wrap_inline12303 in terms ( tex2html_wrap_inline15909 ) except where it appears in an exponential multiplied by t, which can be large. This gives

  equation5730

and

  equation5757

where we have set tex2html_wrap_inline15913 and tex2html_wrap_inline15915 . Note that the total velocity can be taken as the sum of the three velocity terms, equations (6.17), (6.48) and (6.52) since the rotational velocities must be negligible far from their respective boundary layers. The first term in both equation (6.55) and equation (6.56) is the inviscid solution and the second and third terms are the rotational corrections at the interface and the solid boundaries respectively. The vertical velocity can be found from the continuity equation and is

  equation5789

and

  equation5802

It can easily be seen that integrating the first term in equations (6.55) and (6.56) gives the inviscid solution tex2html_wrap_inline12185 . The integral of the second and third terms are smaller than the original terms by a factor tex2html_wrap_inline15919 and so give only a very small correction term.

  figure5813

  figure5824

Figures 6-2 and 6-3 show the different components of the horizontal and vertical velocities for a wave with amplitude tex2html_wrap_inline15945 , tex2html_wrap_inline15947 , f = 1.4, tex2html_wrap_inline12323 and tex2html_wrap_inline15953 . The solid line in figure 6-2 shows the irrotational velocity which has a discontinuity at z = 0 and is symmetric under a rotation of tex2html_wrap_inline15957 about (z=0, u=0). The magnitude of the irrotational velocity is maximum at z=0. The dashed line is the sum of the irrotational velocity and the rotational component at the interface. This velocity distribution changes continuously from its smallest value slightly below z=0 to its maximum value slightly above z=0. In most of the boundary layer the rotational velocity has the opposite sign to the irrotational velocity and a smaller magnitude. Thus it has the effect of reducing the velocity from its inviscid value. At the outer region of the boundary layer, where the rotational component is small but not negligible, the sign of the rotational component is the same as the irrotational component and so the combined velocity is larger than the inviscid velocity. Further from the interface the rotational component becomes negligible and the solid and dashed lines appear to merge. The velocity represented by the dashed line is not symmetric about (z=0, u=0). This is due to the ratio tex2html_wrap_inline15969 being f and not unity. The dotted line represents the final solution for the velocity incorporating the rotational components at the interface and the solid boundaries. It is coincidental with the dashed line except close to the solid boundaries at the top and bottom of the wave. Here, as with the rotational velocity at the interface, the sign of U' is opposite to the sign of u close to the boundary, where its magnitude is largest. In this region the rotational component acts to reduce the magnitude of the total velocity, reducing it to zero on the boundary where the two components have the same magnitude. Further from the boundary there is a region where the signs are the same and the rotational component adds to the magnitude of the horizontal velocity. The magnitude of the rotational component is small here and the effect is not as obvious in figure 6-2 as it is near the interface. This is because tex2html_wrap_inline15977 is increasing away from the solid boundary but it is decreasing away from the interface. There is much less difference between the inviscid vertical velocity and the viscous vertical velocity shown in figure 6-3. The main difference is near the interface where the magnitude of the velocity is maximum. The magnitude of the viscous solution is slightly smaller and slightly more rounded at z = 0.


next up previous contents
Next: Standing Waves Up: Waves in a Viscous Previous: Frequency and Damping Parameter

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