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A family of helically symmetric vortex equilibria

  • DAN LUCAS (a1) and DAVID G. DRITSCHEL (a1)

We present a family of steadily rotating equilibrium states consisting of helically symmetric vortices in an incompressible inviscid irrotational unbounded fluid. These vortices are described by contours bounding regions of uniform axial vorticity. Helical symmetry implies material conservation of axial vorticity (in the absolute frame of reference) when the flow field parallel to vortex lines is proportional to (1+ϵ2r2)−1/2, where ϵ is the pitch and r is the distance from the axis. This material conservation property enables equilibria to be calculated simply by a restriction on the helical stream function. The states are parameterized by their mean radius and centroid position. In the case of a single vortex, parameter space cannot be fully filled by our numerical approach. We conjecture multiply connected contours will characterize equilibria where the algorithm fails. We also consider multiple vortices, evenly azimuthally spaced about the origin. Stability properties are investigated numerically using a helical CASL algorithm.

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Alekseenko S. V., Kuibin P. A., Okulov V. L. & Shtork S. I. 1999 Helical vortices in swirl flow. J. Fluid Mech. 382, 195243.
Dritschel D. G. 1985 The stability and energetics of corotating uniform vortices. J. Fluid Mech. 157, 95134.
Dritschel D. G. 1988 Contour surgery: a topological reconnection scheme for extended integrations using contour dynamics. J. Comput. Phys. 77 (1), 240266.
Dritschel D. G. 1991 Generalized helical Beltrami flows. J. Fluid Mech. 222, 525541.
Dritschel D. G. 1995 A general theory for two-dimensional vortex interactions. J. Fluid Mech. 293, 269303.
Dritschel D. G. & Ambaum M. H. P. 1997 A contour-advective semi-Lagrangian algorithm for the simulation of fine-scale conservative fields. Q. J. R. Meteorol. Soc. 123, 10971130.
Fukumoto Y. & Okulov V. L. 2005 The velocity field induced by a helical vortex tube. Phys. Fluids 17 (107101).
Kelvin Lord 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155168.
Kirchhoff G. 1876 Vorlesungen über mathematische Physik. Mechanik.
Kuibin P. A. & Okulov V. L. 1998 Self-induced motion of helical vortices. In IUTAM Symposium on Dynamics of Slender Vortices, pp. 55–62.
Landman M. J. 1990 On the generation of helical waves in circular pipe flow. Phys. Fluids A 2, 738747.
Loiseleux T., Chomaz J. M. & Huerre P. 1998 The effect of swirl on jets and wakes: linear instability of the rankine vortex with axial flow. Phys. Fluids 10, 11201134.
Macaskill C., Padden W. E. P. & Dritschel D. G. 2003 The CASL algorithm for quasi-geostrophic flow in a cylinder. J. Comput. Phys. 488, 232251.
Moore D. W. & Saffman P. G. 1972 The motion of a vortex filament with axial flow. Phil. Trans. R. Soc. Lond. A 272, 403429.
Norbury J. 1973 A family of steady vortex rings. J. Fluid Mech. 57 (3), 417431.
Okulov V. L. 2004 On the stability of multiple helical vortices. J. Fluid Mech. 521, 319342.
Okulov V. L. & Sørensen J. N. 2007 Stability of helical tip vortices in a rotor far wake. J. Fluid Mech. 576, 125.
Ricca R. L. 1994 Effect of torsion on vortex filament motion. J. Fluid Mech. 273, 241259.
Saffman P. G. 1992 Vortex Dynamics. Cambridge University Press.
Walther J. H., Guènot M., Machefaux E., Rasmussen J. T., Chatelain P., Okulov V. L., Sørensen J. N., Bergdorf M. & Koumoutsakos P. 2007 A numerical study of the stability of helical vortices using vortex methods. In The Science of Making Torque from Wind, Journal of Physics: Conference Series, vol. 75.
Widnall S. E. 1972 The stability of a helical vortex filament. J. Fluid Mech. 54 (4), 641663.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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