Chaotic scattering of the point vortex by falling circular cylinder

S. V. Sokolov   ; I. S. Koltsov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 25 (2015) no. 2, pp. 184-196

Résumé

We consider a system which consists of a circular cylinder subject to gravity interacting with a point vortex in a perfect fluid. In contrast to previous works, in this paper the circulation about the cylinder is assumed to be zero. The governing equations are Hamiltonian and admit evident integrals of motion: the horizontal and vertical components of the momentum; the latter is obviously non-autonomous. Using autonomous integral we reduce the order of the system by one degree of freedom in a case of zero circulation which early was not considered. Unlike nonzero circulation in the absence of point vortices when the cylinder moves inside a certain horizontal stripe it is shown that in the presence of vortices and with circulation equal to zero a vertical coordinate of the cylinder is unbounded decreasing. We then focus on the numerical study of dynamics of our system. In a case of zero circulation trajectories are noncompact. The different kinds of the scattering function of the vortex by cylinder were obtained. The form of these functions argues to chaotic behavior of the scattering which means that an additional analytical integral is absent.

Détail

Keywords: point vortices, rigid body, chaotic scattering, Hamiltonian systems, reduction.

Bibliographie

[1] Maxwell J. K., “On a particular case of descent of a heavy body in a resisting medium”, Camb. and Dubl. Math. Journ., 9 (1854), 145–148

[2] Zhukovski N. E., “On light elongated bodies that fall in the air while rotating around the longitudinal axis, I”, Complete Works in 7 vol., v. 5, Glav. Red. Aviats. Lit., M.–L., 1937, 72–80 (in Russian) | Zbl

[3] Zhukovski N. E., “On light elongated bodies that fall in the air while rotating around the longitudinal axis, II”, Complete Works in 7 vol., v. 5, Glav. Red. Aviats. Lit., M.–L., 1937, 100–115 (in Russian)

[4] Kozlov V. V., “On the problem of a heavy rigid body falling in a resistant medium”, Vestn. Mosk. Univ., Ser. Mat. Mekh., 1990, no. 1, 79–86 (in Russian)

[5] Borisov A. V., Mamaev I. S., “On the motion of a heavy rigid body in an ideal fluid with circulation”, Chaos, 16:1 (2006), 013118 | MR | Zbl

[6] Borisov A. V., Kozlov V. V., Mamaev I. S., “Asymptotic stability and associated problems of failing rigid body”, Regular and Chaotic Dynamics, 12:5 (2007), 531–565 | MR | Zbl

[7] Chaplygin S. A., “On the effect of a plane-parallel air flow on a cylindrical wing moving in it”, Complete Works, v. 3, Izd. Akad. Nauk SSSR, L., 1933, 3–64 (in Russian)

[8] Kozlov V. V., “On a heavy cylindrical body falling in a fluid”, Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, 1993, no. 4, 113–117 (in Russian) | MR

[9] Ramodanov S. M., “The effect of circulation on the fall of a heavy rigid body”, Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, 1996, no. 5, 19–24 (in Russian)

[10] Kirchhoff G. R., Vorlesungen über mathematische physik, v. I, Teubner, Leipzig, 1876

[11] Borisov A. V., Mamaev I. S., “An integrability of the problem on motion of cylinder and vortex in the ideal fluid”, Regular and Chaotic Dynamics, 8:2 (2003), 163–166 | MR | Zbl

[12] Borisov A. V., Mamaev I. S., Ramodanov S. M., “Dynamics of a circular cylinder interacting with point vortices”, Discrete and Contin. Dyn. Syst. B, 5:1 (2005), 35–50 | MR | Zbl

[13] Borisov A. V., Mamaev I. S., Ramodanov S. M., “Dynamic interaction of point vortices and a two-dimensional cylinder”, J. Math. Phys., 48:6 (2007), 065403 | MR | Zbl

[14] Shashikanth B. N., “Poisson brackets for the dynamically interacting system of a 2D rigid cylinder and $N$ point vortices: the case of arbitrary smooth cylinder shapes”, Regular and Chaotic Dynamics, 10:1 (2005), 1–14 | MR | Zbl

[15] Michelin S., Smith S. G. L., “Falling cards and flapping flags: understanding fluid-solid interaction using an unsteady point vortex model”, Theor. Comput. Fluid Dyn., 24 (2010), 195–200 | Zbl

[16] Jones M. A., Shelly M. J., “Falling cards”, J. Fluid Mech., 540 (2005), 393–425 | MR | Zbl

[17] Kadtke J. B., Novikov E. A., “Chaotic capture of vortices by a moving body. I: The single point vortex case”, Chaos, 3:4 (1993), 543–553 | MR | Zbl

[18] Luithardt H. H., Kadtke J. B., Pedrizzetti G., “Chaotic capture of vortices by a moving body. II: Bound pair model”, Chaos, 4:4 (1994), 681–691 | MR | Zbl

[19] Sokolov S. V., Ramodanov S. M., “Falling motion of a circular cylinder interacting dynamically with a point vortex”, Nelineinaya dinamika, 8:3 (2012), 617–628 (in Russian)

[20] Sokolov S. V., Ramodanov S. M., “Falling motion of a circular cylinder interacting dynamically with a point vortex”, Regular and Chaotic Dynamics, 18:1–2 (2013), 184–193 | MR | Zbl

[21] Sokolov S. V., “Falling motion of a circular cylinder interacting dynamically with $N$ point vortices”, Nelineinaya dinamika, 10:1 (2014), 1–14

[22] Sokolov S. V., “Falling motion of a circular cylinder interacting dynamically with $N$ point vortices”, Nonlinear Dynamics and Mobile Robotics, 2:1 (2014), 99–113

[23] Sokolov S. V., “Falling motion of a circular cylinder interacting dynamically with a vortex pair in a perfect fluid”, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2014, no. 2, 86–99 (in Russian) | MR

[24] Föppl L., “Wirbelbewegung hinter einem Kreiszylinder”, Sitzungsberichte der Baäyrischen Akademie der Wissenschaften, 1913

[25] Manakov S. V., Shchur L. N., “Stochastic aspect of two-particle scattering”, JETP Lett., 37:1 (1983), 54–57

[26] Tophøj L., Aref H., “Chaotic scattering of two identical point vortex pairs revisited”, Phys. of Fluids, 20 (2008), 093605 | Zbl