On the interaction of deep water waves and exponential shear currents |
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Authors: | Jun Cheng Jie Cang and Shi-Jun Liao |
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Institution: | (2) Department of Mathematics, Firat University, Elazig, Turkey;(3) Dean of Engineering Faculty, Ardahan University, Ardahan, Turkey; |
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Abstract: | A train of periodic deep-water waves propagating on a steady shear current with a vertical distribution of vorticity is investigated
by an analytic method, namely the homotopy analysis method (HAM). The magnitude of the vorticity varies exponentially with
the magnitude of the stream function, while remaining constant on a particular streamline. The so-called Dubreil–Jacotin transformation
is used to transfer the original exponentially nonlinear boundary-value problem in an unknown domain into an algebraically
nonlinear boundary-value problem in a known domain. Convergent series solutions are obtained not only for small amplitude
water waves on a weak current but also for large amplitude waves on a strong current. The nonlinear wave-current interaction
is studied in detail. It is found that an aiding shear current tends to enlarge the wave phase speed, sharpen the wave crest,
but shorten the maximum wave height, while an opposing shear current has the opposite effect. Besides, the amplitude of waves
and fluid velocity decay over the depth more quickly on an aiding shear current but more slowly on an opposing shear current
than that of waves on still water. Furthermore, it is found that Stokes criteria of wave breaking is still valid for waves
on a shear current: a train of propagating waves on a shear current breaks as the fiuid velocity at crest equals the wave
phase speed. Especially, it is found that the highest waves on an opposing shear current are even higher and steeper than
that of waves on still water. Mathematically, this analytic method is rather general in principle and can be employed to solve
many types of nonlinear partial differential equations with variable coefficients in science, finance and engineering. |
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