Skip to main content
SpringerLink
Log in

Self-Similar Collapse of n Point Vortices

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Abstract

A system of a finite number of linear vortices in self-similar motion and under suitable conditions that are related to the initial positions and circulations of the vortices can collapse to the point with finite time. It is shown how the initial positions that lead to the collapse can be found numerically. An explicit solution for the self-similar collapse of the trajectories is derived. Examples of a collapsing system of 3, 7, and 30 vortices are given. A description is given of how to obtain the curves of collapsing positions of vortices if one particular system of vortices has already been found. Examples of such curves are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

References

  • Aref, H.: Motion of three vortices. Phys. Fluids 22(3), 393–400 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  • Aref, H.: Integrable, chaotic, and turbulent vortex motion in two-dimensional flows. Annu. Rev. Fluid Mech. 15, 345–389 (1983)

    Article  MathSciNet  Google Scholar 

  • Aref, H.: Point vortex dynamics: recent results and open problems. Fluid Dyn. Res. 3, 63–74 (1988)

    Article  Google Scholar 

  • Aref, H., Rott, N., Thomann, H.: Gröbli’s solution of the three-vortex problem. Annu. Rev. Fluid Mech. 24, 1–20 (1992)

    Article  MathSciNet  Google Scholar 

  • Aref, H.: Point vortex dynamics: a classical mathematics playground. Phys. Fluids 48, 065401 (2007)

    MathSciNet  Google Scholar 

  • Aref, H.: Vortices and polynomials. Fluid Dyn. Res. 39, 5–23 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Aref, H.: Self-simialr motion of three point vortices. Phys. Fluids 22, 057104 (2010)

    Article  MathSciNet  Google Scholar 

  • Borisov, A.V., Mamaev, I.S.: Mathematical Methods in the Dynamics of Vortex Structures. Institue of Computer Science, Izhevsk (2005). (in Russian)

    Google Scholar 

  • Barreiro, A., Bronski, J., Newton, P.K.: Spectral gradient flow and equilibrium configurations of point vortices. Proc. R. Soc. A (2012). doi:10.1098/rspa.2009.0419

    MATH  Google Scholar 

  • Catracis, H.J.: The logaritimic spiral: mathematical aspects and modeling in turbulence. J. Math. Res. 3(3), 3–11 (2011)

    Google Scholar 

  • Gallagher, I., Gally, T.: Uniqueness for the two-dimensional Navier–Stokes equation with a measure as initial vorticity. Math. Ann. 332(2), 287–327 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Goodman, J., Hou, T.Y., Lowengrub, J.: Convergence of the point vortex method for the 2-D Euler equations. Comm. Pure Appl. Math. 5, 415–430 (1990)

    Article  MathSciNet  Google Scholar 

  • Gröbli, W.: Specielle probleme über die bewegung geradliniger paralleler wirbelfäden, pp. 86. Zurich und Furrer, Zurich (1877)

  • Helmholtz, H.: On integrals of the hydrodynamical equations which express vortex-motion. Trans. PG. Tait. Phil. Mag. 33(4), 485–512 (1867)

    Google Scholar 

  • Hernandez-Gaduno A., Lacomba E.A.: Collision and regularization for the 3-vortex problem, arXiv:math-ph/0412024v1 (2004)

  • Hernandez-Gaduno A., Lacomba E.A.: Collision of four point vortices in the plane, arXiv:math-ph/0609016v1 (2006)

  • Kochin, N.E., Kibel, I.A., Roze, N.V.: Theoretical hydromechanics. Interscience Publishers, New York (1965)

    Google Scholar 

  • Kozlov, V.V.: Dynamical Systems X. General Theroy of Vortices, Encyklopedia of Mathematical, Science, vol. 67. Springer, New York (2003)

    Book  Google Scholar 

  • Kimura, Y.: Similarity solution of two-dimensional point vortices. J. Phys. Soc. Jpn. 56(6), 2024–2030 (1987)

    Article  Google Scholar 

  • Kirchhoff, G.R.: Vorlesungen über mathematische Physik, Bd. 1: Mechanik. Teubner, Leipzig (1876)

  • Kudela, H., Malecha, Z.: Eruption of a boundary layer induced by a 2D vortex patch. Fluid Dyn. Res. 41, 1–18 (2009)

    Article  MathSciNet  Google Scholar 

  • Leoncini, X., Kuznetsov, L., Zaslasky, G.M.: Motion of threee vortices near collapse. Phys. Fluids 12, 1911–1927 (2000)

    Article  MathSciNet  Google Scholar 

  • Moffat, H.K.: Spiral Structures in Turbulent Flow. New Approaches and Concepts in Turbulence. Brikhäuser, Monte Verità (1993)

    Google Scholar 

  • NASA/GSFC (2012) Credit: Jacques Descloitres, MODIS Rapid Response Team. http://visibleearth.nasa.gov/view.php?id=68992

  • Novikov, E.A.: Dynamics and statistics of a system of vortices. Zh. Eksp. Teor. Fiz. 68, 1868–1882 (1975)

    Google Scholar 

  • Novikov, E.A.: Stochastization and collaspe of vortex systems. Ann. N. Y. Acad. Sci. 357, 47–54 (1980)

    Article  Google Scholar 

  • Novikov, E.A., Sedov, YuB: Vortex collapse. Zh. Eksp. Teor. Fiz. 77, 588–597 (1979)

    Google Scholar 

  • Newton, P.K.: The N-vortex porblem. Analytical Techniques. Springer, New York (2001)

    Google Scholar 

  • O’Neil, K.A.: Stationary configurations of points vortices. Trans. Am. Math. Soc. 302, 383–425 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  • Palmore, J.I.: Relative equilibria of vortices in two dimensions. Proc. Natl. Acad. Sci. 79, 716–718 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  • Ruskeepää, H.: Mathematica Navigator, Mathematics, Statiscics, and Graphics. Academic Press, Elsevier (2009)

    Google Scholar 

  • Synge, J.L.: On the motion of three vortices. Can. J. Math. 1, 257–270 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  • Wayne, C.E.: Vortices and two-dimensional fluid motion. Not. AMS 58(1), 10–19 (2011)

    MathSciNet  MATH  Google Scholar 

  • Wolibner, W.: Un theorème sur l’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment long. Math. Z 37, 698–726 (1933)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author would like to thank Prof. Grzegorz Karch from the Institute of Mathematics of Wrocaw University for the initial review of the manuscript.

Author information

Authors and Affiliations

  1. Wrocaw University of Technology, Wrocław, Poland

    Henryk Kudela

Authors
  1. Henryk Kudela
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Henryk Kudela.

Additional information

Communicated by Paul Newton.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 42 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kudela, H. Self-Similar Collapse of n Point Vortices. J Nonlinear Sci 24, 913–933 (2014). https://doi.org/10.1007/s00332-014-9207-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00332-014-9207-8

Keywords

Mathematics Subject Classification

Search

Navigation