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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: jerome.mougel@imft.fr
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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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References

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, vol. 55. Courier Corporation.Google Scholar
Acheson, D. J. 1976 On over-reflexion. J. Fluid Mech. 77 (03), 433–472.CrossRefGoogle Scholar
Bach, B., Linnartz, E. C., Vested, M. H., Andersen, A. & Bohr, T. 2014 From Newton’s bucket to rotating polygons: experiments on surface instabilities in swirling flows. J. Fluid Mech. 759, 386–403.Google Scholar
Bender, C. M. & Orszag, S. A. 1999 Advanced Mathematical Methods for Scientists and Engineers I. Springer.Google Scholar
Bergmann, R., Tophøj, L., Homan, T. A. M., Hersen, P., Andersen, A. & Bohr, T. 2011 Polygon formation and surface flow on a rotating fluid surface. J. Fluid Mech. 679, 415–431.Google Scholar
Billant, P. & Le Dizès, S. 2009 Waves on a columnar vortex in a strongly stratified fluid. Phys. Fluids 21 (10), 106602.Google Scholar
Cairns, R. A. 1979 The role of negative energy waves in some instabilities of parallel flows. J. Fluid Mech. 92, 1–14.Google Scholar
Fabre, D. & Mougel, J. 2014 Generation of three-dimensional patterns through wave interaction in a model of free surface swirling flow. Fluid Dyn. Res. 46 (6), 061415.Google Scholar
Ford, R. 1994 The instability of an axisymmetric vortex with monotonic potential vorticity in rotating shallow water. J. Fluid Mech. 280, 303–334.Google Scholar
Funada, T. & Joseph, D. D. 2001 Viscous potential flow analysis of Kelvin–Helmholtz instability in a channel. J. Fluid Mech. 445, 263–283.CrossRefGoogle Scholar
Hecht, F. 2012 New development in freefem++. J. Numer. Maths 20 (3–4), 251–265.Google Scholar
Iga, K., Yokota, S., Watanabe, S., Ikeda, T., Niino, H. & Misawa, N. 2014 Various phenomena on a water vortex in a cylindrical tank over a rotating bottom. Fluid Dyn. Res. 46 (3), 031409.Google Scholar
Iima, M. & Tasaka, Y. 2016 Dynamics of flow structures and surface shapes in the surface switching of rotating fluid. J. Fluid Mech. 789, 402–424.Google Scholar
Jansson, T. R. N., Haspang, M. P., Jensen, K. H., Hersen, P. & Bohr, T. 2006 Polygons on a rotating fluid surface. Phys. Rev. Lett. 96, 174502.Google Scholar
Kahouadji, L. & Witkowski, L. M. 2014 Free surface due to a flow driven by a rotating disk inside a vertical cylindrical tank: axisymmetric configuration. Phys. Fluids 26 (7), 072105.Google Scholar
Le Dizès, S. & Lacaze, L. 2005 An asymptotic description of vortex Kelvin modes. J. Fluid Mech. 542, 69–96.Google Scholar
Le Dizès, S. & Riedinger, X. 2010 The strato-rotational instability of Taylor-Couette and Keplerian flows. J. Fluid Mech. 660, 147–161.CrossRefGoogle Scholar
Lewis, B. M. & Hawkins, H. F. 1982 Polygonal eye walls and rainbands in hurricanes. Bull. Am. Meteorol. Soc. 63 (11), 1294–1301.Google Scholar
Lindzen, R. S. & Barker, J. W. 1985 Instability and wave over-reflection in stably stratified shear flow. J. Fluid Mech. 151, 189–217.Google Scholar
Mougel, J., Fabre, D. & Lacaze, L. 2014 Waves and instabilities in rotating free surface flows. Mech. Ind. 15, 107–112.Google Scholar
Mougel, J., Fabre, D. & Lacaze, L. 2015 Waves in Newton’s bucket. J. Fluid Mech. 783, 211–250.CrossRefGoogle Scholar
Park, J. & Billant, P. 2013 The stably stratified Taylor-Couette flow is always unstable except for solid-body rotation. J. Fluid Mech. 725, 262–280.Google Scholar
Schecter, D. A. & Montgomery, M. T. 2004 Damping and pumping of a vortex Rossby wave in a monotonic cyclone: critical layer stirring versus inertia-buoyancy wave emission. Phys. Fluids 16 (5), 1334–1348.Google Scholar
Schecter, D. A. & Montgomery, M. T. 2006 Conditions that inhibit the spontaneous radiation of spiral inertia-gravity waves from an intense mesoscale cyclone. J. Atmos. Sci. 63 (2), 435–456.Google Scholar
Suzuki, T., Iima, M. & Hayase, Y. 2006 Surface switching of rotating fluid in a cylinder. Phys. Fluids 18 (10), 101701.Google Scholar
Takehiro, S. & Hayashi, Y. 1992 Over-reflection and shear instability in a shallow-water model. J. Fluid Mech. 236, 259–279.Google Scholar
Tasaka, Y. & Iima, M. 2009 Flow transitions in the surface switching of rotating fluid. J. Fluid Mech. 636, 475–484.CrossRefGoogle Scholar
Tophøj, L. & Bohr, T. 2013 Stationary ideal flow on a free surface of a given shape. J. Fluid Mech. 721, 28–45.Google Scholar
Tophøj, L., Mougel, J., Bohr, T. & Fabre, D. 2013 Rotating polygon instability of a swirling free surface flow. Phys. Rev. Lett. 110 (19), 194502.Google Scholar
Ursell, F. 1952 Edge waves on a sloping beach. Proc. R. Soc. Lond. A 214 (1116), 79–97.Google Scholar
Vatistas, G. H. 1990 A note on liquid vortex sloshing and Kelvin’s equilibria. J. Fluid Mech. 217, 241–248.Google Scholar
Vatistas, G. H., Wang, J. & Lin, S. 1994 Recent findings on Kelvin’s equilibria. Acta Mechanica 103, 89–102.Google Scholar