The dynamics of vortex rings: leapfrogging in an ideal and viscous fluid

We consider the problem of motion of axisymmetric vortex rings in an ideal incompressible and viscous fluid. Using the numerical simulation of the Navier–Stokes equations, we confirm the existence of leapfrogging of three equal vortex rings and suggest the possibility of detecting it experimentally. We also confirm the existence of leapfrogging of two vortex rings with opposite-signed vorticities in a viscous fluid.

[1]  E. S. Levchenko,et al.  Motion of two vortex rings , 1985 .

[2]  William Mitchinson Hicks,et al.  On the mutual threading of vortex rings , 1922 .

[3]  V. Meleshko.  Coaxial axisymmetric vortex rings: 150 years after Helmholtz , 2010 .

[4]  F. Dyson.  The Potential of an Anchor Ring , 2010 .

[5]  A. Borisov,et al.  Topology and stability of integrable systems , 2010 .

[6]  Cristopher Moore,et al.  Braids in classical gravity , 1993 .

[7]  Interaction of coaxial vortex rings in an ideal fluid , 1988 .

[8]  G. Pedrizzetti,et al.  Vortex Dynamics , 2011 .

[9]  F. Dyson,et al.  The Potential of an Anchor Ring. Part II , 1893 .

[10]  Sir William Thomson F.R.S..  II. On vortex atoms , 1867 .

[11]  R. Wood,et al.  Vortex Rings , 1901, Nature.

[12]  David P. Stevens,et al.  A note on leapfrogging vortex rings , 1993 .

[13]  P. Saffman,et al.  The Velocity of Viscous Vortex Rings , 1970 .

[14]  Yuko Oshima,et al.  Interaction of Two Vortex Rings Moving along a Common Axis of Symmetry , 1975 .

[15]  Moore,et al.  Braids in classical dynamics. , 1993, Physical review letters.

[16]  S. Kelly,et al.  Dynamics and self-propulsion of a spherical body shedding coaxial vortex rings in an ideal fluid , 2013 .

[17]  Walter Gröbli,et al.  Specielle Probleme über die Bewegung geradliniger paralleler Wirbelfäden , 1877 .

[18]  Richard Montgomery,et al.  A remarkable periodic solution of the three-body problem in the case of equal masses , 2000, math/0011268.

[19]  Alexey V. Borisov,et al.  The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem , 2013 .

[20]  Jifeng Peng,et al.  Leapfrogging of two thick-cored vortex rings , 2013 .

[21]  William Mitchinson Hicks,et al.  XV. Researches on the theory of vortex rings.—Part II , 1885, Philosophical Transactions of the Royal Society of London.

[22]  T. Maxworthy,et al.  Some experimental studies of vortex rings , 1977, Journal of Fluid Mechanics.

[23]  Reduction and chaotic behavior of point vortices on a plane and a sphere , 2005, nlin/0507057.

[24]  P. Weidman,et al.  Vortex ring pairs: numerical simulation and experiment , 1993, Journal of Fluid Mechanics.

[25]  C. Simó,et al.  Dynamical properties of the figure eight solution of the three-body problem , 2002 .

[26]  W. M. Hicks.  Researches in vortex motion. Part III. On spiral or gyrostatic vortex aggregates , 2022, Proceedings of the Royal Society of London.

[27]  W. M. Hicks.  II. Researches in vortex motion.—Part III. On spiral or gyrostatic vortex aggregates , 2022 .