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A note on relative equilibria in a rotating shallow water layer

Published online by Cambridge University Press:  08 May 2013

Hamid Ait Abderrahmane*
Affiliation:
Division of Mathematical and Computer Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia
Mohamed Fayed
Affiliation:
Department of Mechanical Engineering, Alexandria University, Egypt
Hoi Dick Ng
Affiliation:
Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Quebec, Canada H3G 1M8
Georgios H. Vatistas
Affiliation:
Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Quebec, Canada H3G 1M8
*
Email address for correspondence: haitabd@hotmail.com

Abstract

Relative equilibria of two and three satellite vortices in a rotating shallow water layer have been recorded via particle image velocimetry (PIV) and their autorotation speed was estimated. This study shows that these equilibria retain the fundamental characteristics of Kelvin’s equilibria, and could be adequately described by the classical idealized point vortex theory. The same conclusion can also be inferred using the experimental dataset of Bergmann et al. (J. Fluid Mech., vol. 679, 2011, pp. 415–431; J. Fluid Mech., vol. 691, 2012, pp. 605–606) if the assigned field’s contribution to pattern rotation is included.

Type
Papers
Copyright
©2013 Cambridge University Press 

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References

Ait Abderrahmane, H., Siddiqui, K., Vatistas, G. H., Fayed, M. & Ng, H. D. 2011 Symmetrization of polygonal hollow-core vortex through beat-wave resonance. Phys. Rev. E 83, 056319.Google Scholar
Ait Abderrahmane, H., Siddiqui, K. & Vatistas, G. H. 2011 Rotating waves within a hollow vortex core. Exp. Fluids 50, 677688.Google Scholar
Ait Abderrahmane, H., Siddiqui, K. & Vatistas, G. H. 2009 The transition between Kelvin’s equilibria. Phys. Rev. E 80, 066305.CrossRefGoogle Scholar
Bauer, L. & Morikawa, G. K. 1976 Stability of rectilinear geostrophic vortices in stationary equilibrium. Phys. Fluids 19, 929942.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, 415431.Google Scholar
Bergmann, R., Tophøj, L., Homan, T. A. M., Hersen, P., Andersen, A. & Bohr, T. 2012 Erratum: polygon formation and surface flow on a rotating fluid surface. J. Fluid Mech. 691, 605606.Google Scholar
Durkin, D. & Fajans, J. 2000 Experiments on two-dimensional vortex patterns. Phys. Fluids 12, 289293.CrossRefGoogle Scholar
Fine, K., Cass, A., Flynn, W. & Driscoll, C. 1995 Relaxation of 2D turbulence to vortex crystals. Phys. Rev. Lett. 75, 32773280.CrossRefGoogle ScholarPubMed
Grzybowski, B. A., Stone, H. A. & Whitesides, G. M. 2000 Dynamic self-assembly of magnetized, millimetre-sized objects rotating at a liquid–air interface. Nature 405, 10331036.Google Scholar
Havelock, T. H. 1931 The stability of motion of rectilinear vortices in ring formation. Philos. Mag. 11, 617633.CrossRefGoogle 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
Mayer, A. M. 1878 Experiments with floating magnets. Nature 17, 487488.Google Scholar
Morikawa, G. K. & Swenson, E. V. 1971 Interacting motion of rectilinear geostrophic vortices. Phys. Fluids 14, 10581071.CrossRefGoogle Scholar
Okulov, V. L. 2004 On the stability of multiple helical vortices. J. Fluid Mech. 521, 319342.Google Scholar
Poncet, S. & Chauve, M. P. 2007 Shear-layer instability in a rotating system. J. Flow Visual. Image Process. 14, 85105.Google Scholar
Rabaud, M. & Couder, Y. 1983 A shear-flow instability in a circular geometry. J. Fluid Mech. 136, 291319.Google Scholar
Sørensen, J. N., Naumov, I. V. & Okulov, V. L. 2011 Multiple helical modes of vortex breakdown. J. Fluid Mech. 683, 430441.CrossRefGoogle Scholar
Thomson, J. J. 1883 A Treatise on the Motion of Vortex Rings. Macmillan & Co.Google Scholar
Thomson, W. (Lord Kelvin) 1878 Floating magnets. Nature 18, 1314.CrossRefGoogle Scholar
Vatistas, G. H. 1990 A note on liquid vortex sloshing and Kelvin’s equilibria. J. Fluid Mech. 217, 241248.Google Scholar
Vatistas, G. H., Ait Abderrahmane, H. & Siddiqui, K. 2008 Experimental confirmation of Kelvin’s equilibria. Phys. Rev. Lett. 100, 174503.Google Scholar
Yarmuchk, E., Gordon, M. & Packard, R. 1979 Observation of stationary vortex array in rotating superfluid helium. Phys. Rev. Lett. 43, 214217.Google Scholar