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Evolving geometry of a vortex triangle

Vikas S. Krishnamurthy*

Hassan Aref†

Mark A. Stremler

  • Departamento de Física, Universidade Federal de Pernambuco, 50670-901 Recife, Brazil

  • Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, Virginia 24061, USA

  • Department of Biomedical Engineering and Mechanics, Virginia Tech, Blacksburg, Virginia 24061, USA

  • *vikas.krishnamurthy2@gmail.com
  • †Deceased.
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Phys. Rev. Fluids 3, 024702 – Published 23 February, 2018

DOI: https://doi.org/10.1103/PhysRevFluids.3.024702

Abstract

The motion of three interacting point vortices in the plane can be thought of as the motion of three geometrical points endowed with a dynamics. This motion can therefore be reformulated in terms of dynamically evolving geometric quantities, viz., the circle that circumscribes the vortex triangle and the angles of the vortex triangle. In this study, we develop the equations of motion for the center, Z, and radius, R, of this circumcircle; and for the angles of the vortex triangle, A,B, and C; and for the triangle orientation given by φ1. The equations of motion for R,A,B, and C form an autonomous dynamical system. A number of known results in the three-vortex problem follow readily from the equations, giving an alternate geometrical perspective on the problem.

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References (27)

  1. G. J. F. van Heijst, R. C. Kloosterziel, and C. W. M. Williams, Laboratory experiments on the tripolar vortex in a rotating fluid, J. Fluid Mech. 225, 301 (1991).
  2. W. Gröbli, Spezielle Probleme über die Bewegung geradliniger paralleler Wirbelfäden, Vierteljahrsschr. Naturforsch. Ges. Zür. 22, 37 (1877); 22, 129 (1877).
  3. J. L. Synge, On the motion of three vortices, Can. J. Math. 1, 257 (1949).
  4. E. A. Novikov, Dynamics and statistics of a system of vortices, Sov. Phys. JETP 41, 937 (1975).
  5. H. Aref, Motion of three vortices, Phys. Fluids 22, 393 (1979).
  6. H. Aref, N. Rott, and H. Thomann, Gröbli's solution of the three-vortex problem, Annu. Rev. Fluid Mech. 24, 1 (1992).
  7. H. Aref, A transformation of the point vortex equations, Phys. Fluids 14, 2395 (2002).
  8. H. Lamb, Hydrodynamics, 6th ed. (Cambridge University Press, Cambridge, 1932).
  9. G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge, UK, 1967).
  10. P. G. Saffman, Vortex Dynamics (Cambridge University Press, Cambridge, UK, 1992).
  11. P. K. Newton, The N-vortex Problem: Analytical Techniques, Applied Mathematical Sciences Vol. 145 (Springer-Verlag, New York, 2001).
  12. H. S. M. Coxeter, Introduction to Geometry, 2nd. ed. (Wiley, New York, 1969).
  13. A. V. Borisov and A. E. Palmov, Dynamics and statics of vortices on a plane and a sphere I, Regul. Chaotic Dyn. 3, 28 (1998).
  14. H. Aref, Integrable, chaotic and turbulent vortex motion in two-dimensional flows, Annu. Rev. Fluid Mech. 15, 345 (1983).
  15. H. Goldstein, C. P. Poole, and J. L. Safko, Classical Mechanics, 3rd. ed. (Addison-Wesley, New York, 2002).
  16. L. D. Landau and E. M. Lifschitz, Mechanics (Pergamon, New York; reprinted by Butterworth-Heinemann, Oxford, 1960).
  17. E. Laura, Sulle equazioni differenziali canoniche del moto di un sistema di vortici elementari, rettilinei e paralleli in un fluido incompressibile indefinito, Atti dell'Accademia delle Scienze di Torino, Classe di Scienze Fisiche, Matematiche e Naturali 40, 296 (1905).
  18. N. Rott, Three-vortex motion with zero total circulation, J. Appl. Math. Phys. (ZAMP) 40, 473 (1989) [with an Addendum by H. Aref (1989)].
  19. H. Aref, Three-vortex motion with zero total circulation, J. Appl. Math. Phys. (ZAMP) (Addendum) 40, 495 (1989).
  20. H. Aref, P. K. Newton, M. A. Stremler, T. Tokieda, and D. L. Vainchtein, Vortex crystals, Adv. Appl. Mech. 39, 1 (2003).
  21. H. Aref, Self-similar motion of three point vortices, Phys. Fluids 22, 057104 (2010).
  22. Y. Kimura, Similarity solution of two-dimensional point vortices, J. Phys. Soc. Jpn. 56, 2024 (1987).
  23. F. Jing, E. Kanso, and P. K. Newton, Viscous evolution of point vortex equilibria: The collinear state, Phys. Fluids 22, 123102 (2010).
  24. S.-C. Kim and S.-I. Sohn, Interactions of three viscous point vortices, J. Phys. A: Math. Theor. 45, 455501 (2012).
  25. D. D. Holm, J. E. Marsden, and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math. 137, 1 (1998).
  26. T. Gotoda and T. Sakajo, Distributional enstrophy dissipation via the collapse of three point vortices, J. Nonlinear Sci. 26, 1525 (2016).
  27. V. Krishnamurthy and H. Aref, Evolving geometry of a vortex triangle, Bull. Am. Phys. Soc. 56(18) (2011).

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Phys. Rev. Fluids 3, 024702– Published 23 February, 2018
  • Received 25 May 2017
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