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Collision of two vortex rings

Published online by Cambridge University Press:  26 April 2006

S. Kida ,
M. Takaoka  and
F. Hussain
S. Kida
Affiliation:
Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA
M. Takaoka
Affiliation:
Department of Physics, Faculty of Science, Kyoto University, Kyoto 606, Japan Present address: Department of Mechanical Engineering, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560, Japan.
F. Hussain
Affiliation:
Department of Mechanical Engineering, University of Houston, Houston, TX 77004, USA
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Abstract

The interaction of two identical circular viscous vortex rings starting in a side-by-side configuration is investigated by solving the Navier–Stokes equation using a spectral method with 643 grid points. This study covers initial Reynolds numbers (ratio of circulation to viscosity) up to 1153. The vortices undergo two successive reconnections, fusion and fission, as has been visualized experimentally, but the simulation shows topological details not observed in experiments. The shapes of the evolving vortex rings are different for different initial conditions, but the mechanism of the reconnection is explained by bridging (Melander & Hussain 1988) except that the bridges are created on the front of the dipole close to the position of the maximum strain rate. Spatial structures of various field quantities are compared. It is found that domains of high energy dissipation and high enstrophy production overlap, and that they are highly localized in space compared with the regions of concentrated vorticity. The kinetic energy decays according to the same power laws as found in fully developed turbulence, consistent with concentrated regions of energy dissipation. The main vortex cores survive for a relatively long time. On the other hand, the helicity density which is higher in roots of bridges and threads (or legs) changes rapidly in time. The high-helicity-density and high-energy-dissipation regions overlap significantly although their peaks do not always do so. Thus a long-lived structure may carry high-vorticity rather than necessarily high-helicity density. It is shown that the time evolution of concentration of a passive scalar is quite different from that of the vorticity field, confirming our longstanding warning against relying too heavily on flow visualization in laboratory experiments for studying vortex dynamics and coherent structures.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 230 , September 1991 , pp. 583 - 646
Copyright
© 1991 Cambridge University Press

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References

Anderson, C. & Greengard, C 1988 The vortex ring merger problem at infinite Reynolds number (Preprint).
Ashurst, W. T. & Meiron, D. 1987 Numerical study of vortex reconnection. Phys. Rev. Lett. 58, 1632–1635.Google Scholar
Beale, J. T., Kato, T. & Majda, A. 1984 Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66.Google Scholar
Brissaud, A., Fhish, U., Leorat, J., Lesieur, M. & Mazure, A. 1973 Helicity cascades in fully developed isotropic turbulence. Phys. Fluids 16, 1366–1367.Google Scholar
Chamberlain, J. P. & Liu, C. H. 1985 Navier-Stokes calculation for unsteady three-dimensional vortical flows in unbounded domains,. AIAA J. 23, 868–874.Google Scholar
Chorin, A. J. 1982 The evolution of a turbulent vortex. Commun. Math. Phys. 83, 517–535.Google Scholar
Comte-Bellot, G. & Corrsin, S. 1966 The use of a contraction to improve the isotropy of grid-generated turbulence. J. Fluid Mech. 25, 657–682.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Fohl, F. & Turner, J. S. 1975 Colliding vortex rings. Phys. Fluids 18, 433–36.Google Scholar
Ginevskh, A. S., Kolesnikov, A. V. & Ukhanova, L. N. 1979 Decay of turbulence behind adouble grid of cylinders with opposite motion of grids. lsv. Acod. Nauk. SSSR, Mekh. Zhidk. Gaza 14, 17–25.Google Scholar
Hussain, F. 1983 Coherent structure - reality and myth. Phys. Fluids 26, 2816–2850.Google Scholar
Hussain, F. 1986 Coherent structures and turbulence. J. Fluid Mech. 173, 303–356.Google Scholar
Kerr, R. M. & Hussain, F. 1989 Simulation of vortex reconnection. Physica D 37, 474–484.Google Scholar
Kida, S. 1981 Similar solutions in modified cumulant expansion. Phys. Fluids 24, 604–614.Google Scholar
Kida, S. 1983 Turbulence and singularity of the velocity field. Publ. RIMS, Kyoto University 19, 1033–1048.Google Scholar
Kida, S. 1985 Numerical simulation of two-dimensional turbulence with high-symmetry. J. Phys. Soc. Japan 54, 2840–2854.Google Scholar
Kida, S. & Murakami, Y. 1988 Numerical study of energy decay of turbulence. J. Phys. Soc.Japan 57, 3657–3660.Google Scholar
Kida, S. & Murakami, Y. 1989 Statistics of velocity gradients in turbulence at moderate Reynolds numbers. Fluid Dyn. Res. 4, 347–370.Google Scholar
Kida, S. & Takaoka, M. 1987 Bridging in vortex reconnection. Phys. Fluids 30, 2911–1924.Google Scholar
Kida, S. & Takaoka, M. 1988 Reconnection of vortex tubes. Fluid Dyn. Res. 3, 257–261.Google Scholar
Kida, S. & Takaoka, M. 1991 Breakdown of frozen motion of vorticity field and vorticity reconnection. J. Phys. Soc. Japan 60 (in press).Google Scholar
Kida, S., Takaoka, M. & Hussain, F. 1989 Reconnection of two vortex rings. Phys. Fluids A 1, 630–632.Google Scholar
Kida, S., Takaoka, M. & Hussain, F. 1991 Formation of head-tail structure in a two-dimensional uniform straining flow. Phys. Fluids (submitted).Google Scholar
Kolmogorov, A. N. 1941 On degeneration of isotropic turbulence in incompressible viscous liquid. Dokl. Akad. Nauk. URSS 31, 538–540.Google Scholar
Leonard, A. 1974 Numerical simulation of interacting three-dimensional vortex filaments. In Proc. 4th Intl Conf. Numerical Method in Fluid Dynamics, pp. 245–250. Springer.
Leonard, A. 1980 Vortex methods for flow simulation. J. Comput. Phys. 37, 289–335.Google Scholar
Lesieur, M. & Schertzer, D. 1978 Autosimilar extinction of turbulence with large Reynolds number. J. MeAc. 17, 609–646.Google Scholar
Levich, E. 1987 Certain problems in the theory of developed hydrodynamical turbulence. Phys. Rep. 151, 129–238.Google Scholar
Lino, S. C. & Wan, C. A. 1972 Decay of isotropic turbulence generated by a mechanically agitated grid. Phys. Fluids 15, 1363–1369.Google Scholar
Melander, M. V. & Hussain, F. 1988 Cut-and-connect of two antiparallel vortex tubes. Center for Turbulence Research Proc. Summer Program 1988 (referred to herein as MH).
Melander, M. V. & Hussain, F. 1990 Topological aspects of vortex reconnection. In Topological Fluid Mechanics (ed. H. K. Moffatt & A. Tsinober), pp. 485–499. Cambridge University Press.
Melander, M. V. & Zabusky, N. J. 1988 Interaction and ‘apparent’ reconnection of 3D vortex tubes via direct numerical simulations. Fluid Dyn. Res. 3, 247–250.Google Scholar
Moffatt, K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 159, 117–129.Google Scholar
Moffatt, K. 1985 Magnetostatic equilibria and analogous Euler flows of arbitrary complex topology. Part 1. Fundamentals. J. Fluid Mech. 159, 359–378.Google Scholar
Orszag, S. A. 1971 Numerical simulation of incompressible flows within simple boundaries. I. Galerkin (spectral) representations. Stud. Appl. Maths 50, 293–327.Google Scholar
Oshima, Y. & Asaka, S. 1977 Interaction of two vortex rings along parallel axes in air. J. Phys.Soc. Japan 42, 708–713.Google Scholar
Oshima, Y. & Izutsu, N. 1988 Cross-linking of two vortex rings. Phys. Fluids 31, 2401–2403.Google Scholar
Pumir, A. & Kerr, R. M. 1987 Numerical simulation of interacting vortex tubes. Phys. Rev. Lett.58, 1636–1639.Google Scholar
Pumir, A. & Siggia, E. D. 1987 Vortex dynamics and the existence of solutions to the Navier-Stokes equations. Phys. Fluids 30, 1606–1626.Google Scholar
Saffman, P. G. 1967 The large-scale structure of homogeneous turbulence. J. Fluid Mech. 27, 581–593.Google Scholar
Schatzle, P. 1987 An experimental study of fusion of vortex rings. Ph.D. thesis, California Institute of Technology.
Schwahz, K. W. 1985 Three-dimensional vortex dynamics in superfluid 4He: line-line and line-boundary interactions. Phys. Rev. B 31, 5782–5804.Google Scholar
Shariff, K., Leonard, A., Zabusky, N. J. & Ferziger, J. H. 1988 Acoustics and dynamics of coaxial interacting vortex rings. Fluid Dyn. Res.3, 337–343.Google Scholar
Shirayama, S. & Kuwahara, H. 1984 Vortex method in three-dimensional flow. In Proc. 9th Intl. Conf. Numerical Methods in Fluid Dynamics, (ed. Soubbaramayer and J. P. Boujot), pp. 503–507. Springer.
Siggia, E. D. 1985 Collapse and amplification of a vortex filament. Phys. Fluids 28, 794–805.Google Scholar
Siggia, E. D. & Pumir, A. 1985 Incipient singularities in the Navier-Stokes equation. Phys. Rev. Lett. 55, 1749–1752.Google Scholar
Stanaway, S., Shariff, K. & Hussain, F. 1988 Head-on collision of viscous vortex rings. Center for Turbulence Research: Proc. Summer Program 1988.
Takaki, R. & Hussain, F. 1985 Recombination of vortex filaments and its role in aerodynamic noise. In. Proc. 5th Symp. on Turbulent Shear Flows, Cornell University, pp. 3.19–3.25.
Tatsumi, T., Kida, S. & Mizushima, J. 1978 The multiple-scale cumulant expansion for isotropic turbulence. J. Fluid Mech. 85, 97–142.Google Scholar
Uberoi, M. S. 1963 Energy transfer in isotropic turbulence. Phys. Fluids 6, 1048–1056.Google Scholar