Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-20T07:37:17.837Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuous parametric families of stationary and translating periodic point vortex configurations

Published online by Cambridge University Press:  30 October 2007

KEVIN A. O'NEIL
KEVIN A. O'NEIL*
Affiliation:
Department of Mathematics, The University of Tulsa, OK 74104, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The number of periodic arrangements of point vortices – point vortex streets – in two-dimensional fluid flow that are stationary is known to be finite for a generic choice of vortex circulations. When all circulations are the same in absolute value, however, stationary vortex street configurations have been associated with the zeros of certain trigonometric polynomials containing free complex parameters. The presence of these parameters may prove useful in constructing point vortex models of shear layers and wakes. In this paper it is shown that such a continuum of stationary configurations exists in a much wider class of point vortex street systems. The circulations may take on many values, not just two, providing increased flexibility in the modelling context. A simple method for computing these configurations is derived. The effects of symmetries on the solution polynomials are described, and illustrated with examples. In addition, novel translating vortex street configurations are found having arbitrary translation velocity and containing free parameters for vortex circulations ±1 and also for vortex circulations +1, −2.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 591 , 25 November 2007 , pp. 393 - 411
Copyright
Copyright © Cambridge University Press 2007

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

REFERENCES

Adler, M. & Moser, J. 1978 On a class of polynomials connected with the Korteweg-deVries equation. Commun. Math. Phys. 61, 1–30.CrossRefGoogle Scholar
Aref, H., Newton, P. K., Stremler, M., Tokieda, T. & Vainchtein, D. 2003 Vortex crystals. Adv. Appl. Mech. 39, 1–79.CrossRefGoogle Scholar
Aref, H., Stremler, M. A. & Ponta, F. L. 2006 Exotic vortex wakes – point vortex solutions. J. Fluids Struct. 22, 929–940.CrossRefGoogle Scholar
Aref, H. & Van Buren, M., 2005 Vortex triple rings. Phys. Fluids 17, 057104.CrossRefGoogle Scholar
Bartman, A. B. 1983 A new interpretation of the Adler-Moser KdV polynomials: Interaction of vortices. In Nonlinear and Turbulent Processes in Physics (ed. Sagdeev, R. Z.), vol. 3, pp. 1175–1181. Harwood Academic Publishers.Google Scholar
Burchnall, J. L. & Chaundy, T. W. 1930 A set of differential equations which can be solved by polynomials. Proc. Lond. Math. Soc. 2, 30, 401–414.CrossRefGoogle Scholar
Kadtke, J. & Campbell, L. 1987 Method for finding stationary states of point vortices. Phys. Rev. A 36, 4360–4370.CrossRefGoogle ScholarPubMed
Loutsenko, I. 2003 Integrable dynamics of charges related to bilinear hypergeometric equation. Commun. Math. Phys. 242, 251–275.CrossRefGoogle Scholar
Loutsenko, I. 2004 Equilibrium of charges and differential equations solved by polynomials. J. Phys. A: Math. Gen. 37, 1309–1322.CrossRefGoogle Scholar
Montaldi, J., Soulière, A. & Tokieda, T. 2003 Vortex dynamics on a cylinder. SIAM J. Appl. Dyn. Syst. 2, 417–430.CrossRefGoogle Scholar
O'Neil, K. 1987 Symmetric configurations of vortices. Phys. Lett. A 124 503–507.CrossRefGoogle Scholar
O'Neil, K. 2006 Minimal polynomial systems for point vortex equilibria. Physica D 219, 69–79.Google Scholar
Stremler, M. 2003 Relative equilibria of singly periodic point vortex arrays. Phys. Fluids 15, 3767–3775.CrossRefGoogle Scholar
Tkachenko, V. K. 1964 Thesis, Institute of Physical Problems, Moscow.Google Scholar
Wang, Z. J. 2005 Dissecting insect flight. Annu. Rev. Fluid Mech. 37, 183–210.CrossRefGoogle Scholar