SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Long wave models and pressure evaluation for surface waves on shear flows

H. Kalisch and A. Ali

Abstract
The effect of background vorticity on the pressure beneath steady long gravity waves at the surface of a fluid is investigated. In particular, we derive a model equation and a formula for the pressure in a flow with constant vorticity. The model equation was previously found by Benjamin [2], and is given in terms of the vorticity $ømega_0$, and three parameters $Q,R$ and $S$ representing the mass flux, total head and momentum flux, respectively. The focus of this work is on the reconstruction of the pressure from solutions of the model equation and the behavior of the surface wave profiles and the pressure distribution as the strength of the vorticity changes. In particular, it is shown that strong enough vorticity can lead to non-monotone pressure profiles under the wave. As already indicated in [4], it is also possible for the fluid pressure near the wavecrest to be below atmospheric pressure.

Bibliography
[1] A. Ali and H. Kalisch, Reconstruction of the Pressure in Long-Wave Models with Constant Vorticity, European J. Mech. - B/Fluids 37 (2013), 187-194.
[2] T. B. Benjamin, The solitary wave on a stream with an arbitrary distribution of vorticity, J. Fluid Mech. 12 (1962), 97-116.
[3] W. Choi, Strongly nonlinear long gravity waves in uniform shear flows, Phys. Rev. E 68 (2003), 026305.
[4] A.F. Teles da Silva and D.H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid Mech. 195 (1988), 281-302.

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