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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.