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Numerical evidence of breaking of vortex lines in an ideal fluid

  • Evgeniy A. Kuznetsov
  • Olga M. Podvigina
  • Vladislav A. Zheligovsky
Part of the Fluid Mechanics and Its Applications book series (FMIA, volume 71)

Abstract

Emergence of singularity of vorticity at a single point, not related to any symmetry of the initial distribution, has been demonstrated numerically for the first time. Behaviour of the maximum of vorticity near the point of collapse closely follows the dependence (t 0t)−1, where t 0 is the time of collapse. This agrees with the interpretation of collapse in an ideal incompressible fluid as of the process of vortex lines breaking.

Keywords

Euler Equation Vortex Line Vortex Tube Numerical Evidence Vortex Filament 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Arnold, V.I. 1981 Theory of Catastrophe. Znanie, Moscow (in Russian) [English transl.: Theory of Catastrophe 1986, 2nd rev. ed. Springer].Google Scholar
  2. Arnold, V.I. 1989 Mathematical Methods of Classical Mechanics. 2nd ed., Springer-Verlag, New York.Google Scholar
  3. Beale, J.T., Kato, T. & Majda, A.J. 1984 Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys.94, 61–66.CrossRefMathSciNetzbMATHADSGoogle Scholar
  4. Boratav, O.N. & Pelz, R.B. 1994 Direct numerical simulation of transition to turbulence from high-symmetry initial condition. Phys. Fluids6, 2757–2784.CrossRefADSzbMATHGoogle Scholar
  5. Constantin, P., Feferman, CH. & Majda, A.J. 1996 Geometric constrains on potentially singular solutions for the 3D Euler equations. Commun. Partial Diff. Eqs.21, 559–571.zbMATHGoogle Scholar
  6. Crow, S.C. 1970 Stability Theory for a pair of trailing vortices. Amer. Inst. Aeronaut. Astronaut. J.8, 2172–2179.Google Scholar
  7. Frisch, U. 1995 Turbulence. The legacy of A.N.Kolmogorov. Cambridge Univ. Press.Google Scholar
  8. Grauer, R., Marliani, C., & Germaschewski, K. 1998 Adaptive mesh refinement for singular solutions of the incompressible Euler equations. Phys. Rev. Lett.80, 4177–4180.CrossRefADSGoogle Scholar
  9. Kerr, R.M. 1993 Evidence for a singularity of the 3-dimensional, incompressible Euler equations Phys. Fluids A 5, 1725–1746.MathSciNetzbMATHADSCrossRefGoogle Scholar
  10. Kolmogorov, A.N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds number, Doklady AN SSSR30, 9–13 (in Russian) [reprinted in 1991 Proc. R. Soc. Lond. A 434, 9–13].Google Scholar
  11. Kuznetsov, E.A. & Ruban, V.P. 1998 Hamiltonian dynamics of vortex lines for systems of the hydrodynamic type, JETP Letters67, 1076–1081.CrossRefADSGoogle Scholar
  12. Kuznetsov, E.A. & Ruban, V.P. 2000 Collapse of vortex lines in hydrodynamics. JETP91, 776–785.ADSGoogle Scholar
  13. L’vov, V.S. 1991 Scale invariant theory of fully developed hydrodynamic turbulence — Hamiltonian approach. Phys. Rep.207, 1–47.Google Scholar
  14. Monin, A.S. & Yaglom, A.M. 1992 Statistical hydro-mechanics. 2nd ed., vol.2, Gidrometeoizdat, St.Petersburg (in Russian) [English transl.: 1975 Statistical Fluid Mechanics. Vol. 2, ed. J.Lumley, MIT Press, Cambridge, MA].Google Scholar
  15. Pelz, R.B. 1997 Locally self-similar, finite-time collapse in a high-symmetry vortex filament model. Phys. Rev. E, 55, 1617–1626.ADSCrossRefGoogle Scholar
  16. Zakharov, V.E. & Kuznetsov, E.A. 1986 Quasiclassical theory of three-dimensional wave collapse. Sov. Phys. JETP64, 773–780.Google Scholar
  17. Zheligovsky, V.A., Kuznetsov, E.A. & Podvigina, O.M. 2001 Numerical modeling of collapse in ideal incompressible hydrodynamics. Pis’ma v ZhETF (JET Letters) 74, 402–406.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Evgeniy A. Kuznetsov
    • 1
    • 2
  • Olga M. Podvigina
    • 3
    • 2
    • 4
  • Vladislav A. Zheligovsky
    • 3
    • 2
    • 4
  1. 1.L.D.Landau Institute for Theoretical PhysicsMoscowRussian Federation
  2. 2.Observatoire de la Côte d’Azur, CNRS UMR 6529Nice Cedex 4France
  3. 3.International Institute of Earthquake Prediction Theory and Mathematical GeophysicsMoscowRussian Federation
  4. 4.Laboratory of general aerodynamics, Institute of MechanicsLomonosov Moscow State UniversityMoscowRussian Federation

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