Leapfrogging vortex rings: Hamiltonian structure, geometric phases and discrete reduction

Fluid Dynamics Research

Volumn 33, Issue 4, 2003, Pages 333-356

  Shashikanth, Banavara N ^a   Marsden, Jerrold E ^b  

^a Horeshoe Cir Jett Hall   (United States)

^b CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Axisymmetric vortex rings; Geometric mechanics; Geometric phases; Hamiltonian systems; Leapfrogging motions; Symmetry reduction

Indexed keywords

APPROXIMATION THEORY; FLUID DYNAMICS; HAMILTONIANS; MATHEMATICAL MODELS;

DISCRETE REDUCTION;

NEWTONIAN LIQUIDS;

GEOMETRICAL EFFECT; INCOMPRESSIBLE FLUID; MATHEMATICAL METHOD; NEWTONIAN FLUID; VORTEX RING;

EID: 0242369784     PISSN: 01695983     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.fluiddyn.2003.05.001     Document Type: Article

References (40)

1. Adams, M.R., Ratiu, T.S., 1988. The three point vortex problem: commutative and noncommutative integrability. In: Meyer, K. Saari, D. (Eds.), Hamiltonian Dynamical Systems, Contemp. Math. 81, 245-257.
2. Arnold V.I. Khesin B. Topological methods in hydrodynamics Ann. Rev. Fluid Mech. 24 1992 145-166.
3. Arnold, V.I., Khesin, B., 1998. Topological Methods in Hydrodynamics, Applied Mathematical Sciences 125, Springer, Berlin.
4. Blaom A.D. Reconstruction phases via Poisson reduction Differential Geom. Appl. 12 3 2000 231-252.
5. Constantin P. Fefferman C. Direction of vorticity and the problem of global regularity of the Navier-Stokes equations Indiana Univ. Math. J. 42 3 1993 775-789.
6. Dyson F. The potential of an anchor ring-Pt. II Philos. Trans. Roy. Soc. London Ser. A 184 1893 1041-1106.
7. Foias C. Holm D.D. Titi E.S. The three dimensional viscous Camassa-Holm equations and their relation to the Navier-Stokes equations and turbulence theory Dyn. Differential Equations 14 2002 1-36.
8. Gurzhii A.A. Konstantinov M.-Yu. Meleshko V.V. Interaction of coaxial vortex rings in an ideal fluid Fluid Dynam. 23 1988 224-229.
9. Hicks W.M. On the mutual threading of vortex rings Proc. Roy. Soc. London Ser. A 102 1922 111-131.
10. Holm D. Rasetti-Regge Dirac bracket formulation of Lagrangian fluid dynamics on vortex filaments Math. Comput. Simulation 62 2003 53-63.
11. Holm D.D. Marsden J.E. Ratiu T.S. The Euler-Poincaré equations and semidirect products with applications to continuum theories Adv. Math. 137 1998 1-8.
12. Klein R. Majda A.J. An asymptotic theory for the nonlinear instability of antiparallel pairs of vortex filaments Phys. Fluids A 5 1993 369-379.
13. Klein R. Ting L. Vortex filament with axial core structure variation Appl. Math. Lett. 5 1992 99-103.
14. Klein R. Majda A.J. Damodaran K. Simplified equations for the interaction of nearly parallel vortex filaments J. Fluid Mech. 288 1995 201-248.
15. Knio O.M. Klein R. Improved thin-tube models for slender vortex simulations J. Comput. Phys. 163 2000 68-82.
16. Konstantinov M.Yu. Chaotic phenomena in the interaction of vortex rings Phys. Fluids 6 1994 1752-1767.
17. Lamb H. Hydrodynamics 6th Edition 1932 Dover New York.
18. Langer J. Perline R. Poisson geometry of the filament equation J. Nonlinear Sci. 1 1991 71-94.
19. Leonard N.E. Marsden J.E. Stability and drift of underwater vehicle dynamics: mechanical systems with rigid motion symmetry Physica D 105 1997 130-162.
20. Lim, T.T., 1997. http://www.eng.nus.edu.sg/mpelimtt/TT_LIM.htm
21. Lim T.T. Nickels T.B. Instability and reconnection in the head-on collision of two vortex rings Nature 357 1992 225-227.
22. Love A.E.H. On the motion of paired vortices with a common axis Proc. London Math. Soc. 25 1894 185-194.
23. Marsden, J.E., 1992. Lectures on Mechanics, London Mathematical Society Lecture Note Series, Vol. 174. Cambridge University Press, Cambridge.
24. Marsden, J.E., Montgomery, R., Ratiu, T.S., 1990. Reduction, Symmetry and Phases in Mechanics, Memoirs, Vol. 436. American Mathematical Society, Providence, RI.
25. Marsden, J.E., Ratiu, T.S., 1999. Introduction to Mechanics and Symmetry, Texts in Applied Mathematics, 2nd Edition. Vol. 17, Springer, Berlin, 1999.
26. Marsden J.E. Ratiu T.S. Scheurle J. Reduction theory and the Lagrange-Routh equations J. Math. Phys. 41 2000 3379-3429.
27. Marsden J.E. Shkoller S. Global well-posedness of the LANS-α equations Proc. Roy. Soc. London 359 2001 1449-1468.
28. Marsden J.E. Shkoller S. The anisotropic averaged Euler and Navier-Stokes equations Arch. Rat. Mech. An. 166 2003 27-46.
29. Marsden J.E. Weinstein A. Reduction of symplectic manifolds with symmetry Rep. Math. Phys. 5 1974 121-130.
30. Marsden J.E. Weinstein A. Coadjoint orbits, vortices and Clebsch variables for incompressible fluids Physica D 7 1983 305-323.
31. Oliver, M., Shkoller, S., 2000. The vortex blob method as a second-grade non-Newtonian fluid; E-print http://xyz.lanl.gov/abs/math.AP/9910088/.
32. Pekarsky S. Marsden J.E. Abstract mechanical connection and abelian reconstruction for almost Kähler manifolds J. Appl. Math. 1 2001 1-28.
33. Pelz R. Locally self-similar finite time collapse in a high-symmetry vortex filament model Phys. Rev. E 55 1997 1617-1626.
34. Rayner J.M.V. A vortex theory of animal flight. Part 2 J. Fluid Mech. 91 1979 731-763.
35. Rowley C.W. Marsden J.E. Variational integrators for point vortices Proc. CDC 40 2002 1521-1527.
36. Saffman, P.G., 1992. Vortex Dynamics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, Cambridge.
37. Shashikanth B.N. Marsden J.E. Burdick J.W. Kelly S.D. The Hamiltonian structure of a 2-D rigid circular cylinder interacting dynamically with N point vortices Phys. Fluids 14 3 2002 1214-1227.
38. Shashikanth B.N. Newton P. Vortex motion and the geometric phase. Part I. Basic configurations and asymptotics J. Nonlinear Sci. 8 1998 183-214.
39. Ting L. Klein R. Viscous Vortical Flows Lecture Notes in Physics Vol. 374 1991 Springer Berlin.
40. Triantayfyllou M.S. Triantayfyllou G.S. An efficient swimming machine Sci. Am. 272 1995 64-70.