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Oscillating line source in a shear flow with a free surface: critical layer-like contributions

Published online by Cambridge University Press:  31 May 2016

Simen Å. Ellingsen Open the ORCID record for Simen Å. Ellingsen [Opens in a new window]  and
Peder A. Tyvand
Simen Å. Ellingsen*
Affiliation:
Department of Energy and Process Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
Peder A. Tyvand
Affiliation:
Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, N-1432 Ås, Norway
*
†Email address for correspondence: [email protected]
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Abstract

The linearized water wave radiation problem for an oscillating submerged line source in an inviscid shear flow with a free surface is investigated analytically at finite, constant depth in the presence of a shear flow varying linearly with depth. The surface velocity is taken to be zero relative to the oscillating source, so that Doppler effects are absent. The radiated wave out from the source is calculated based on Euler’s equation of motion with the appropriate boundary and radiation conditions, and differs substantially from the solution obtained by assuming potential flow. To wit, an additional wave is found in the downstream direction in addition to the previously known dispersive wave solutions; this wave is non-dispersive and we show how it is the surface manifestation of a critical layer-like flow generated by the combination of shear and mass flux at the source, passively advected with the flow. As seen from a system moving at the fluid velocity at the source’s depth, streamlines form closed curves in a manner similar to Kelvin’s cat’s eye vortices. A resonant frequency exists at which the critical wave resonates with the downstream propagating wave, resulting in a downstream wave pattern diverging linearly in amplitude away from the source.

JFM classification

Waves/Free-surface Flows: Critical layers Waves/Free-surface Flows: Shear waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 798 , 10 July 2016 , pp. 201 - 231
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Copyright
© 2016 Cambridge University Press 

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Ellingsen et al. supplementary movie

Movie of Figure 3 from article. Multiple examples of waves patterns for different values of the Froude number. See Fig. 3 of article for further information.

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Ellingsen et al. supplementary movie

Movie animation of Fig. 11a showing cat\'s eye-like vortex structures advected downstream in a critical layer. See Fig. 11 of article for further information.

Download Ellingsen et al. supplementary movie(Video)
Video 2 MB

Ellingsen et al. supplementary movie

Movie animation of Fig. 11b showing cat\'s eye-like vortex structures advected downstream in a critical layer. See Fig. 11 of article for further information.

Download Ellingsen et al. supplementary movie(Video)
Video 2.6 MB

Ellingsen et al. supplementary movie

Movie animation of Fig. 11c showing cat\'s eye-like vortex structures advected downstream in a critical layer. See Fig. 11 of article for further information.

Download Ellingsen et al. supplementary movie(Video)
Video 3.2 MB