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On the instabilities of a potential vortex with a free surface

Published online by Cambridge University Press:  05 July 2017

J. Mougel Open the ORCID record for J. Mougel [Opens in a new window] ,
D. Fabre ,
L. Lacaze  and
T. Bohr
J. Mougel*
Affiliation:
Institut de Mécanique des Fluides de Toulouse, Université de Toulouse, Allée du Pr. Camille Soula, 31400 Toulouse, France
D. Fabre
Affiliation:
Institut de Mécanique des Fluides de Toulouse, Université de Toulouse, Allée du Pr. Camille Soula, 31400 Toulouse, France
L. Lacaze
Affiliation:
Institut de Mécanique des Fluides de Toulouse, Université de Toulouse, Allée du Pr. Camille Soula, 31400 Toulouse, France CNRS, IMFT, 31400 Toulouse, France
T. Bohr
Affiliation:
Physics Department and Center for Fluid Dynamics, Technical University of Denmark, 2800 Kongens Lyngby, Denmark
*
†Email address for correspondence: [email protected]
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Abstract

In this paper, we address the linear stability analysis of a confined potential vortex with a free surface. This particular flow has been recently used by Tophøj et al. (Phys. Rev. Lett., vol. 110(19), 2013, article 194502) as a model for the swirling flow of fluid in an open cylindrical container, driven by rotating the bottom plate (the rotating bottom experiment) to explain the so-called rotating polygons instability (Vatistas J. Fluid Mech., vol. 217, 1990, pp. 241–248; Jansson et al., Phys. Rev. Lett., vol. 96, 2006, article 174502) in terms of surface wave interactions leading to resonance. Global linear stability results are complemented by a Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) analysis in the shallow-water limit as well as new experimental observations. It is found that global stability results predict additional resonances that cannot be captured by the simple wave coupling model presented in Tophøj et al. (2013). Both the main resonances (thought to be at the root of the rotating polygons) and these secondary resonances are interpreted in terms of over-reflection phenomena by the WKBJ analysis. Finally, we provide experimental evidence for a secondary resonance supporting the numerical and theoretical analysis presented. These different methods and observations allow to support the unstable wave coupling mechanism as the physical process at the origin of the polygonal patterns observed in free-surface rotating flows.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Vortex Flows: Vortex instability Geophysical and Geological Flows: Waves in rotating fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 824 , 10 August 2017 , pp. 230 - 264
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Copyright
© 2017 Cambridge University Press 

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