Hostname: page-component-f554764f5-fnl2l Total loading time: 0 Render date: 2025-04-17T16:58:25.904Z Has data issue: false hasContentIssue false
  • English
  • Français

On the equilibrium and stability of a row of point vortices

Published online by Cambridge University Press:  26 April 2006

Hassan Aref
Hassan Aref
Affiliation:
Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The equilibrium and stability of a single row of equidistantly spaced identical point vortices is a classical problem in vortex dynamics, which has been addressed by several investigators in different ways for at least a century. Aspects of the history and the essence of these treatments are traced, stating some in more accessible form, and pointing out interesting and apparently new connections between them. For example, it is shown that the stability problem for vortices in an infinite row and the stability problem for vortices arranged in a regular polygon are solved by the same eigenvalue problem for a certain symmetric matrix. This result also provides a more systematic enumeration of the basic instability modes. The less familiar theory of equilibria of a finite number of vortices situated on a line is also recalled.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 290 , 10 May 1995 , pp. 167 - 181
Copyright
© 1995 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.)

Article purchase

Temporarily unavailable

References

Ahmed, S., Bruschi, M., Calogero, F., Olshanetsky, M. A. & Perelomov, A. M. 1979 Properties of the zeros of the classical polynomials and of the Bessel functions. Nuovo Cim. 49, 173–199.Google Scholar
Aref, H. 1985 Chaos in the dynamics of a few vortices - fundamentals and applications. In Theoretical and Applied Mechanics (ed. F. I. Niordson & N. Olhoff), pp. 43–68. North-Holland.
Aref, H. 1989 Three-vortex motion with zero total circulation: Addendum. Z. Angew. Math. Phys. 40, 495–500.Google Scholar
Birkhoff, G. & Fisher, J. 1959 Do vortex sheets roll up. Rend. Circ. Mat. Palermo 8, 77–90.Google Scholar
Blomberg, D. D. 1984 Point vortex models of a forced shear layer. MSc thesis, Brown University.
Brown, G. L. & Roshko, A. 1974 On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775–816.Google Scholar
Bruschi, M. 1979 On the algebra of certain matrices related to the zeros of Hermite polynomials. Lett. Nuovo Cim. 24, 509–511.Google Scholar
Calogero, F. 1977 On the zeros of Hermite polynomials. Lett. Nuovo Cim. 20, 489–490.Google Scholar
Calogero, F. & Perelomov, A. M. 1978 Properties of certain matrices related to the equilibrium configuration of the one-dimensional many-body problems with the pair potentials V1(x) = -logsin x and V2(x) = I/sin2x. Commun. Math. Phys. 59, 109–116.Google Scholar
Calogero, F. & Perelomov, A. M. 1979 Some Diophantine relations involving circular functions of rational angles. Lin. Alg. Appl. 25, 91–94.Google Scholar
Dritschel, D. G. 1985 The stability and energetics of corotating uniform vortices. J. Fluid Mech. 157, 95–134.Google Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 1980 Tables of Integrals, Series, and Products. Corrected and enlarged edition. Academic.
Havelock, T. H. 1931 The stability of motion of rectilinear vortices in ring formation. Phil. Mag. ((7) 11, 617–633.Google Scholar
Ho, C.-M. & Huang, L.-S. 1982 Subharmonics and vortex merging in mixing layers. J. Fluid Mech. 119, 443–473.Google Scholar
Ho, C.-M. & Huerre, P. 1984 Perturbed free shear layers. Ann. Rev. Fluid Mech. 16, 365–424.Google Scholar
Kármán, T. von 1911 Über den Mechanismus des Widerstandes, den ein bewegter Körper in einer Flüssigkeit erfährt. 1. Teil. Nachr. K. Ges. Wiss. Göttingen, Math.-phys. Klasse, 509–517.Google Scholar
Kármán, T. von 1912 Über den Mechanismus des Widerstandes, den ein bewegter Körper in einer Flüssigkeit erfährt. 2. Teil. Nachr. K. Ges. Wiss. Göttingen, Math.-phys. Klasse, 547–556.
Kármán, T. von 1956 Collected Works of Theodore von Kármán, Vol. I. Butterworths.
Káarmáan, T. Von & Rubach, H. 1912 Über den Mechanismus des Flüssigkeits- und Luft-widerstandes. Phys. Z. 13, 49–59.Google Scholar
Khazin, L. G. 1976 Regular polygons of point vortices and resonance instability of steady states. Sov. Phys. Dokl. 21, 567–569.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Dover.
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics, 2nd edn. Pergamon.
Lin, C. C. 1943 On the Motion of Vortices in Two Dimensions. Toronto University Press.
Mayer, A. M. 1878a Floating magnets. Nature 17, 487–488.Google Scholar
Mayer, A. M. 1878b Floating magnets.. Nature 18, 258–260.Google Scholar
Mertz, G. J. 1978 Stability of body-centered polygonal configurations of ideal vortices. Phys. Fluids 21, 1092–1095.Google Scholar
Morton, W. B. 1935 Vortex polygons. Proc. R. Irish Acad. 42, 21–29.Google Scholar
Rosenhead, L. 1931 The formation of vortices from a surface of discontinuity. Proc. R. Soc. Lond., A 134, 170–192.Google Scholar
Roshko, A. 1976 Structure of turbulent shear flows. AIAA J. 14, 1349–1357.Google Scholar
Rott, N. 1989 Three-vortex motion with zero total circulation. Z. Angew. Math. Phys. 40, 473–494.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.
Saffman, P. G. & Baker, G. R. 1979 Vortex interactions. Ann. Rev. Fluid Mech. 11, 95–122.Google Scholar
Stuart, J. T. 1986 Leon Rosenhead 1906-1984. Biograph. Mem. Fell. R. Soc. 32, 407–420.Google Scholar
Szegö, G. 1939 Orthogonal Polynomials. Amer. Math. Soc. Coll. Publ. vol. 23, Providence, RI.
Tengara, I. 1981 Some mathematical problems in waves and vortices. Thesis. Imperial College of Science and Technology.
Thomson, J. J. 1883 On the Motion of Vortex Rings (Adams Prize Essay). Macmillan.
Thomson, W. 1878 Floating magnets. Nature 18, 13–14.Google Scholar
Williamson, C. H. K. & Roshko, A. 1988 Vortex formation in the wake of an oscillating cylinder. J. Fluids Struct. 2, 355–381.Google Scholar
Winant, C. D. & Browand, F. K. 1974 Vortex pairing: the mechanism of turbulent mixing-layer growth at moderate Reynolds number. J. Fluid Mech. 63, 237–255.Google Scholar