On the instabilities of a potential vortex with a free surface

J. Mougel, D. Fabre, L. Lacaze, Tomas Bohr

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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 Tophoj 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 Tophoj 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.
Original languageEnglish
JournalJournal of Fluid Mechanics
Volume824
Pages (from-to)230-264
Number of pages35
ISSN0022-1120
DOIs
Publication statusPublished - 2017

Keywords

  • Surface gravity waves
  • Vortex instability
  • Waves in rotating fluids

Cite this

Mougel, J. ; Fabre, D. ; Lacaze, L. ; Bohr, Tomas. / On the instabilities of a potential vortex with a free surface. In: Journal of Fluid Mechanics. 2017 ; Vol. 824. pp. 230-264.
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title = "On the instabilities of a potential vortex with a free surface",
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 Tophoj 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 Tophoj 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.",
keywords = "Surface gravity waves, Vortex instability, Waves in rotating fluids",
author = "J. Mougel and D. Fabre and L. Lacaze and Tomas Bohr",
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On the instabilities of a potential vortex with a free surface. / Mougel, J.; Fabre, D.; Lacaze, L.; Bohr, Tomas.

In: Journal of Fluid Mechanics, Vol. 824, 2017, p. 230-264.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - On the instabilities of a potential vortex with a free surface

AU - Mougel, J.

AU - Fabre, D.

AU - Lacaze, L.

AU - Bohr, Tomas

PY - 2017

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N2 - 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 Tophoj 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 Tophoj 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.

AB - 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 Tophoj 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 Tophoj 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.

KW - Surface gravity waves

KW - Vortex instability

KW - Waves in rotating fluids

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DO - 10.1017/jfm.2017.341

M3 - Journal article

VL - 824

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JO - Journal of Fluid Mechanics

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