Regular and Chaotic Dynamics

, Volume 12, Issue 1, pp 68–80 | Cite as

Interaction between Kirchhoff vortices and point vortices in an ideal fluid

  • A. V. Borisov
  • I. S. Mamaev
Research Articles

Abstract

We consider the interaction of two vortex patches (elliptic Kirchhoff vortices) which move in an unbounded volume of an ideal incompressible fluid. A moment second-order model is used to describe the interaction. The case of integrability of a Kirchhoff vortex and a point vortex by the variable separation method is qualitatively analyzed. A new case of integrability of two Kirchhoff vortices is found. A reduced form of equations for two Kirchhoff vortices is proposed and used to analyze their regular and chaotic behavior.

MSC2000 numbers

37N05 76M23 

Key words

vortex patch point vortex integrability 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Kirchhoff, G., Vorlesungen über mathematische Physik. Mechanik, Leipzig: Teubner, 1876.Google Scholar
  2. 2.
    Kida, S., Motion of an Elliptical Vortex in a Uniform Shear Flow, J. Phys. Soc. Japan, 1981, vol. 50, pp. 3517–3520.CrossRefGoogle Scholar
  3. 3.
    Neu, J., The Dynamics of Columnar Vortex in an Imposed Strain, Phys. Fluids, 1984, vol. 27, pp. 2397–2402.MATHCrossRefGoogle Scholar
  4. 4.
    Chaplygin, S.A., On a Pulsating Cylindrical Vortex, Trudy Otd. Fiz. Nauk Mosk. Obshch. Lyub. Estest., 1899, vol. 10, no. 1, pp. 13–22. Reprinted in: Sobranie sochinenii (Collected Papers), Moscow-Leningrad: Gostekhizdat, 1948, vol. 2, pp. 138–154. English translation in: Regul. Chaotic Dyn., 2007, vol. 12, pp. ?-?.Google Scholar
  5. 5.
    Meleshko, V.V. and van Heijst, G.J.F., On Chaplygin’s Investigations of Two-Dimensional Vortex Structures in an Inviscid Fluid, J. Fluid Mech., 1994, vol. 272, pp. 157–182.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Newton, P.K., The N-Vortex Problem. Analytical Techniques, New York: Springer, 2001.MATHGoogle Scholar
  7. 7.
    Riccardi, G. and Piva, R., Motion of an Elliptical Vortex under Rotating Strain: Condition for Asymmetric Merging, Fluid Dyn. Res., 1998, vol. 23, pp. 63–88.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dhanak, M.R. and Marshall, M.P., Motion of an Elliptical Vortex under Applied Periodic Strain, Phys. Fluids A., 1993, vol. 5, pp. 1224–1230.MATHCrossRefGoogle Scholar
  9. 9.
    Ide, K. and Wiggins, S., The Dynamics of Elliptically Shaped Regions of Uniform Vorticity in Time-Periodic, Linear External Velocity Fields, Fluid Dyn. Res., 1995, vol. 15, pp. 205–235.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Bertozzi, A.L., Heteroclinic orbits and chaotic dynamics in planar fluid flows, SIAM J. Math. Anal., 1988, vol. 19, pp. 1271–1294.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Polvani, L. and Wisdom, J., Chaotic Lagrangian Trajectories around an Elliptical Patch Embedded in a Constant and Uniform Background Shear Flow, Phys. Fluids A, 1990, vol.2, pp. 123–126.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Dritschel, D.G. The Stability of Elliptical Vortices in an External Straining Flow, J. Fluid. Mech., 1990, vol. 210, pp. 223–261.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Polvani, L.M. and Flierl, G.R., Generalized Kirchhoff Vortices, Phys. Fluids, 1986, vol. 29, pp. 2376–2379.MATHCrossRefGoogle Scholar
  14. 14.
    Melander, M.V., Zabusky, N.J. and Styczek, A.S., A Moment Model For Vortex Interactions of Two-Dimensional Euler Equation. Part I: Computational Validation of Hamiltonian Elliptical Representation, J. Fluid. Mech., 1986, vol. 167, p. 95–115.MATHCrossRefGoogle Scholar
  15. 15.
    Melander, M.V., Zabusky, N.J. and McWilliams, J.C., Symmetric Vortex Merger in Two Dimensions: Causes and Conditions, J. Fluid. Mech., 1988, vol. 195, pp. 303–340.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Melander, M.V., Zabusky, N.J. and McWilliams, J.C., Asymmetric Vortex Merger in Two Dimensions: Which Vortex Is ‘Victorious’? Phys. Fluids, 1987, vol. 30, pp. 2610–2612.CrossRefGoogle Scholar
  17. 17.
    French Villat, H., Leçons sur la théorie des tourbillons, Paris: Gauthier-Villars, 1930.Google Scholar
  18. 18.
    Lamb, H., Hydrodynamics, Cambridge: Cambridge Univ. Press, 2nd ed., 1895.MATHGoogle Scholar
  19. 19.
    Legras, B. and Dritschel, D., The Elliptical Model of Two-Dimensional Vortex Dynamics. I. The basic State, Phys. Fluids A, 1991, vol. 3, pp. 845–854.CrossRefMathSciNetGoogle Scholar
  20. 20.
    Dritschel, D. and Legras, B. The Elliptical Model of Two-Dimensional Vortex Dynamics. II. Disturbance Equations, Phys. Fluids A, 1991, vol.3, pp. 855–869.CrossRefMathSciNetGoogle Scholar
  21. 21.
    Lebedev, V.G. A Qualitative Analysis of a Joint Dynamics of Kirchhoff and a Point Vortices, Regul. Chaotic Dyn., 1999, no. 3, Vol. 4, pp. 70–81.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Riccardi, G. and Piva, R. The Interaction of an Elliptical Path with a Point Vortex, Fluid Dyn. Res., 2000, vol. 27, pp. 269–289.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Borisov, A.V. and Mamaev, I.S., Matematicheskie metody dinamiki vikhrevykh struktur (Mathematical Methods in the Dynamics of Vortex Structures), Izhevsk: Inst. Kompyut. Issled., 2005.MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2007

Authors and Affiliations

  • A. V. Borisov
    • 1
  • I. S. Mamaev
    • 1
  1. 1.Institute of Computer ScienceUdmurt State UniversityIzhevskRussia

Personalised recommendations