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Dynamics of a vortex filament in a shear flow

Published online by Cambridge University Press:  20 April 2006

Hassan Aref  and
Edward P. Flinchem
Hassan Aref
Affiliation:
Division of Engineering, Brown University, Providence, R.I. 02912
Edward P. Flinchem
Affiliation:
Division of Engineering, Brown University, Providence, R.I. 02912
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Abstract

Motions of a single vortex filament in a background flow are studied by numerical simulation of a set of model equations. The model, which in essence is due to Hama, treats the self-interaction of the filament through the so-called ‘localized-induction approximation’ (LIA). Interaction with the prescribed background field is treated by simply advecting the filament appropriately. We are particularly interested in elucidating the evolution of sinuous vortices such as the ‘wiggle’ seen by Breidenthal in the transition to three-dimensionality in the mixing layer. The model studied embodies two of the simplest ingredients that must enter into any dynamical explanation: induction and advection. For finite-amplitude phenomena we make contact with the theory of solitons on strong vortices developed by Betchov and Hasimoto. In a shear, solitons cannot exist, but solitary waves can, and their interactions with the shear are found to be key ingredients for an understanding of the behaviour of the vortex filament. When sheared, a soliton seems to act as a ‘nucleation site’ for the generation of a family of waves. Computed sequences are shown that display a remarkable morphological similarity to flow-visualization studies. The present application of fully nonlinear dynamics to a model presents an attractive alternative to the extrapolations from linearized stability theory applied to the full equations that have so far constituted the theoretical basis for understanding the experimental results.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 148 , November 1984 , pp. 477 - 497
Copyright
© 1984 Cambridge University Press

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References

Aref, H. 1983 Integrable, chaotic, and turbulent vortex motion in two-dimensional flows. Ann. Rev. Fluid Mech. 15, 345–89.Google Scholar
Arms, R. J. & Hama, F. R. 1965 Localized-induction concept on a curved vortex and motion of an elliptic vortex ring. Phys. Fluids 8, 553–559.Google Scholar
Batchelor, G. K. 1967 Introduction to Fluid Dynamics, pp. 509–511. Cambridge University Press.
Betchov, R. 1965 On the curvature and torsion of an isolated vortex filament. J. Fluid Mech. 22, 471–479.Google Scholar
Breidenthal, R. 1978 A chemically reacting, turbulent shear layer. Ph.D. thesis, California Institute of Technology.
Breidenthal, R. 1979 Chemically reacting, turbulent shear layer. AIAA J. 17, 310–311.Google Scholar
Dhanak, M. R. & Bernardinis, B. de 1981 The evolution of an elliptic vortex ring. J. Fluid Mech. 109, 189–216.Google Scholar
Donnelly, R. J. 1967 Experimental Superfluidity. University of Chicago Press.
Eisenhart, L. P. 1960 A Treatise on the Differential Geometry of Curves and Surfaces, §§13–15. Dover.
Golden, J. H. & Purcell, D. 1978 Life cycle of the Union City, Oklahoma tornado and comparison with waterspouts. Mon. Weather Rev. 106, 3–11.Google Scholar
Hama, F. R. 1963 Progressive deformation of a perturbed line vortex filament. Phys. Fluids 6, 526–534.Google Scholar
Hardin, J. C. 1982 The velocity field induced by a helical vortex filament. Phys. Fluids 25, 1949–1952.Google Scholar
Hasimoto, H. 1972 A soliton on a vortex filament. J. Fluid Mech. 51, 477–485.Google Scholar
Ho, C.-M. & Huerre, P. 1984 Perturbed free shear layers. Ann. Rev. Fluid Mech. 16, 365–423.Google Scholar
Hopfinger, E. J. & Browand, F. K. 1982 Vortex solitary waves in a rotating, turbulent flow. Nature 295, 393–395.Google Scholar
Hopfinger, E. J., Browand, F. K. & Gagne, Y. 1982 Turbulence and waves in a rotating tank. J. Fluid Mech. 125, 505–534.Google Scholar
Kida, S. 1981 A vortex filament moving without change of form. J. Fluid Mech. 112, 397–409.Google Scholar
Kida, S. 1982 Stability of a steady vortex filament. J. Phys. Soc. Japan 51, 1655–1662.Google Scholar
Lamb, G. L. 1977 Solitons on moving space curves. J. Math. Phys. 18, 1654–1661.Google Scholar
Laufer, J. 1983 Deterministic and stochastic aspects of turbulence. Trans. ASME E: J. Appl. Mech. 50, 1077–1083.Google Scholar
Leibovich, S. & Ma, H. Y. 1983 Soliton propagation on vortex cores and the Hasimoto soliton. Phys. Fluids 26, 3173–3179.Google Scholar
Levi, D., Sym, A. & Wojciechowski, S. 1983 N-solitons on vortex filament. Phys. Lett. A 94, 408–411.Google Scholar
Maxworthy, T., Hopfinger, E. J. & Redekopp, L. G. 1985 Wave motions on vortex cores. J. Fluid Mech. (to be published).Google Scholar
McLaughlin, D. W. & Scott, A. C. 1978 Perturbation analysis of fluxon dynamics. Phys. Rev. A 18, 1652–1680.Google Scholar
Peterson, R. E., Minor, J. E., Golden, J. H. & Scott, A. C. 1979 A rare close up of an Australian tornado. Weatherwise 32, 188–193.Google Scholar
Pierrehumbert, R. T. & Widnall, S. E. 1982 The two- and three-dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech. 114, 59–82.Google Scholar
Robinson, A. C. & Saffman, P. G. 1982 Three-dimensional stability of vortex arrays. J. Fluid Mech. 125, 411–427.Google Scholar
Robinson, A. C. & Saffman, P. G. 1984 Three-dimensional stability of an elliptical vortex in a straining field. J. Fluid Mech. 142, 451–466.Google Scholar
Shampine, L. F. & Gordon, M. K. 1975 Computer Solution of Ordinary Differential Equations. Freeman.
Spiegel, E. A. 1980 Fluid dynamical form of the linear and nonlinear Schrödinger equations. Physica D 1, 236–240.Google Scholar
Wilson, L., Pearson, A. & Ostby, F. P. 1979 Tornado! Weatherwise 32, 27–33.Google Scholar
Zabusky, N. J. 1981 Recent developments in contour dynamics for the Euler equations. Ann. NY Acad. Sci. 373, 160–170.Google Scholar
Zakharov, V. E. & Shabat, A. B. 1972 Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62–69.Google Scholar