Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-18T18:38:59.004Z Has data issue: false hasContentIssue false
  • English
  • Français

Topological vortex dynamics in axisymmetric viscous flows

Published online by Cambridge University Press:  26 April 2006

Mogens V. Melander  and
Fazle Hussain
Mogens V. Melander
Affiliation:
Department of Mathematics, Southern Methodist University, Dallas, TX 75275, USA
Fazle Hussain
Affiliation:
Department of Mechanical Engineering, University of Houston, Houston, TX 77204, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The topology of vortex lines and surfaces is examined in incompressible viscous axisymmetric flows with swirl. We argue that the evolving topology of the vorticity field must be examined in terms of axisymmetric vortex surfaces rather than lines, because only the surfaces enjoy structural stability. The meridional cross-sections of these surfaces are the orbits of a dynamical system with the azimuthal circulation being a Hamiltonian H and with time as a bifurcation parameter μ. The dependence of H on μ is governed by the Navier–Stokes equations; their numerical solutions provide H. The level curves of H establish a time history for the motion of vortex surfaces, so that the circulation they contain remains constant. Equivalently, there exists a virtual velocity field in which the motion of the vortex surfaces is frozen almost everywhere; the exceptions occur at critical points in the phase portrait where the virtual velocity is singular. The separatrices emerging from saddle points partition the phase portrait into islands; each island corresponds to a structurally stable vortex structure. By using the flux of the meridional vorticity field, we obtain a precise definition of reconnection: the transfer of flux between islands. Local analysis near critical points shows that the virtual velocity (because of its singular behaviour) performs ‘cut-and-connect’ of vortex surfaces with the correct rate of circulation transfer - thereby validating the long-standing viscous ‘cut-and-connect’ scenario which implicitly assumes that vortex surfaces (and vortex lines) can be followed over a short period of time in a viscous fluid. Bifurcations in the phase portrait represent (contrary to reconnection) changes in the topology of the vorticity field, where islands spontaneously appear or disappear. Often such topology changes are catastrophic, because islands emerge or perish with finite circulation. These and other phenomena are illustrated by direct numerical simulations of vortex rings at a Reynolds number of 800.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 260 , 10 February 1994 , pp. 57 - 80
Copyright
© 1994 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ashurst, W. T. & Meiron, D. I. 1987 Numerical study of vortex reconnection. Phys. Rev. Lett. 58, 1632.Google Scholar
Boratav, O. N., Pelz, R. B. & Zabusky, N. J. 1992 Reconnection in orthogonally interacting vortex tubes: Direct numerical simulations and quantifications. Phys. Fluids A 4, 581–606.Google Scholar
Caflisch, R., Li, X. & Shelley, M. 1992 The collapse of an axisymmetric, swirling vortex sheet. Nonlinearity (submitted).Google Scholar
Fohl, F. & Turner, J. S. 1975 Colliding vortex rings. Phys. Fluids 18, 433.Google Scholar
Furth, H. P. 1985 Nonideal magnetohydrodynamic instabilities and toroidal magnetic confinement. Phys. Fluids 28, 1595–1611.Google Scholar
Grauer, R. & Sideris, T. C. 1991 Numerical computation of 3D incompressible ideal fluids with swirl. Phys. Rev. Lett. 67, 3511.Google Scholar
Greene, J. M. 1988 Geometrical properties of three-dimensional reconnecting magnetic fields with nulls. J. Geophys. Res. 93, 8583.Google Scholar
Greene, J. M. 1990 Vortex nulls and magnetic nulls. In Topological Fluid Mechanics (ed. H. K. Moffatt & A. Tsinober) pp. 478–484. Cambridge University Press.
Hussain, F. 1986 Coherent structures and turbulence. J. Fluid Mech. 173, 303.Google Scholar
Kkrr, R. M. 1992 Evidence for a singularity of the three-dimensional incompressible Euler equations. In Topological Aspects of the Dynamics of Fluids and Plasma (ed. H. K. Moffatt, G. M. Zaslavsky, P. Comte et al.), pp. 309–401. Kluwer.
Kerr, R. & Hussain, F. 1989 Simulation of vortex reconnection. Physica D 37, 474.Google Scholar
Kida, S. & Takaoka, M. 1991 Breakdown of frozen motion of vorticity field and vorticity reconnection. J. Phys. Soc. Japan 60, 2184.Google Scholar
Melander, M. V. & Hussain, F. 1991 Reconnection of two antiparallel vortex tubes; A new cascade mechanism. In Turbulent Shear Flows 7 (ed. F. Durst et al.). Springer.
Moffatt, H. K. & Tsinober, A. (ed.) 1990 Topological Fluid Mechanics. Cambridge University Press.
Moffatt, H. K., Zaslavsky, G. M., Comte, P. & Tabor, M. (ed.) 1992 Topological Aspects of the Dynamics of Fluids and Plasmas. Kluwer.
Newcomb, W. A. 1958 Motion of magnetic lines of ferce. Ann. Phys. N.Y. 3, 347.Google Scholar
Oshima, Y. & Izutzu, N. 1988 Cross-linking of two vortex rings. Phys. Fluids 31, 2401.Google Scholar
Pumir, A. & Siggiia, E. D. 1992 Development of singular solutions to the axisymmetric Euler equations. Phys. Fluids A 4, 1472.Google Scholar
Schatzle, P. 1987 An experimental study of fusion of vortex rings PhD Thesis, Graduate Aeronautical Laboratories, Calif. Inst. Tech.
Schwarz, K. 1985 Three-dimensional vortex dynamics in superfluid 4He line-line and lineboundary interaction. Phys. Rev. B 31, 5782.Google Scholar
Shelley, M. J., Meiron, D. I. & Orszag, S. A. 1993 Dynamical aspects of vortex reconnection of perturbed anti-parallel vortex tubes. J. Fluid Mech. 246, 613.Google Scholar
Truesdell, C. 1954 The Kinematics of Vorticity. Indiana University Publ., Science series No. 19.
Virk, D., Melander, M. V. & Hussain, F. 1994 Dynamics of a polarized vortex ring. J. Fluid Mech. 260, 23–55.Google Scholar
Winkelmans, G. & Leonard, A. 1989 Improved vortex methods for three-dimensional flows. In Mathematical Aspects of Vortex Dynamics (ed. R. E. Calflish), pp. 25–35. SIAM.
Zabusky, N. J. & Melander, M. v. 1989 Three-dimensional vortex tube reconnection: morphology for orthogonally-offset tubes. Physica D 37, 555.CrossRefGoogle Scholar