Free Surface Boundary Condition

The geometry of free surface boundary is not known a priori. This shape is part of the solution, which means we have a very difficult boundary condition to cope with. In the case of surface wave of negligible surface tension, we call them gravity wave. The free surface of a wave can be described as:

where p is the displacement of the free surface above the horizontal plane, say z = 0,. If the surface varies with time, as would the water surface, then the total derivative of the surface with respect to time would be zero on the surface. In other word, if we move with the surface, it does not changes.

where p is the displacement of the free surface above the horizontal plane,

Figure 2.17: Free surface wave

Where the unit vector normal to the surface has been introduced as n = TVF. Rearranging the above equation, we get:

The boundary condition at the free surface using equation 2.35 is:

Carrying out the dot product:

dt dx dz Neglecting the convection terms and approximating pressure we get:

We also know:

Differentiating equation 2.36 with respect to t and cancelling p with equation 2.37 gives:

1 dp

A second differentiation with respect to t and elimination of vz with equation 2.38 then substitution of ptotai = P + P9^, in which p represent the excess pressure due to motion and pg^stands for hydrostatic pressure, we get:

1 d2p dp

g dt2 dz

This is an approximate to the surface boundary condition.

The above boundary condition is usually replaced with the boundary condition that:

This assumption of no surface wave is common assumption in concrete dams and is valid. It was shown that the surface waves are negligible.

The more complicated free surface boundary condition can be established in which we can assume pressure distribution due to interaction of wind and surface. These cases are above the scope of the book and can be found in appropriate references of water wave mechanics.

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