## Nonlinear free surface flows past a semi-infinite flat plate in water of finite depth

preprint

posted on 10.07.2007 by Montri Maleewong, Roger Grimshaw#### preprint

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We consider the steady free surface two-dimensional flow past a semi-infinite flat plate in
water of a constant finite depth. The fluid is assumed to be inviscid, incompressible and
the flow is irrotational; surface tension at the free surface is neglected. Our concern is with
the periodic waves generated downstream of the plate edge. These can be characterized
by a depth-based Froude number, F, and the depth d (draft) of the depressed plate.
For small d and subcritical flows, we may use the linearized problem, combined with
conservation of momentum, to obtain some analytical results. These linear results are
valid when F is not close to 0 or 1. As F approaches 1, we use a weakly nonlinear longwave
analysis, and in particular show that the results can be extended to supercritical
flows. For larger d nonlinear effects need to be taken account, and so we solve the fully
nonlinear problem numerically using a boundary integral equation method. Here the
predicted wavelength from the linear and weakly nonlinear results is used to set the
mean depth condition for the nonlinear problem. The results by these three approaches
are in good agreement when d is relatively small. For larger d our numerical results are
compared with known results for the highest wave.We also find some wave-free solutions,
which when compared with the weakly nonlinear results are essentially just one-half of
a solitary wave solution.

### History

#### School

- Science

#### Department

- Mathematical Sciences