Linear stability analysis of a counter rotating vortex pair of unequal strength

Authors

  • Joine So
  • Kris Ryan
  • Gregory J Sheard

DOI:

https://doi.org/10.21914/anziamj.v50i0.1455

Abstract

An elliptic type instability of a counter rotating vortex pair of unequal strength was numerically investigated with a linear stability analysis method. The peak growth rates of the unstable modes were predicted. The instability characteristics were found to differ from an equal strength vortex pair, of either co-rotating or counter rotating vortices. This investigation serves as a fundamental model to the flow of two unequal strength vortices, which can be generated from the ends of aerodynamic surfaces of an aircraft, such as wing tips and ailerons. These results provide predictions of the vortex arrangements likely to develop three dimensional instabilities, which is known to promote the dissipation of the underlying vortex structure. References
  • Batchelor, G. K., An Introduction to Fluid Dynamics, Cambridge University Press, 1967.
  • Crow, S. C., Stability theory for a pair of trailing vortices, AIAA, 8(12), 2172--2179, 1970.
  • Kelvin, L., Vibratons of a columnar vortex, Phil. Mag., 10, 155--168, 1880.
  • Kerswell, R. R., Elliptical instability, Annu. Rev. Fluid Mech., 34, 83--113, 2002. doi:10.1146/annurev.fluid.34081701.171829
  • Lacaze, L., Ryan. K. and Le Dizes, S., Elliptic instability in a strained Batchelor vortex, J. Fluid Mech., 577, 345--361, 2007. doi:10.1017/S0022112007004879
  • Laporte, F. and Corjon, A. Direct numerical simulations of the elliptic instability of a vortex pair, {Phys. Fluids}, 12(5), 1016--1031, 2000. doi:10.1063/1.870357
  • Leweke, T. and Williamson, C. H. K., Cooperative elliptic instability of a vortex pair, J. Fluid Mech., 360, 85--119, 1998.
  • Le Diz{e}s, S. and Verga, A., Viscous interactions of two co-rotating vortices before merging, J. Fluid Mech., 467, 389--410, 2002. doi:10.1017/S0022112002001532
  • Meunier, P. and Leweke, T., Elliptic instability of a co-rotating vortex pair, J. Fluid Mech., 533, 125--159, 2005. doi:10.1017/S0022112005004325
  • Moore, D. W. and Saffman, P. G., The instability of a straight vortex filament in a strain field, Proc. R. Soc. Lond. A., 346, 413--425, 1975.
  • Sheard, G. J., Thompson, M. C. and Hourigan, K., From spheres to circular cylinders: The stability and flow structures of bluff ring wakes, J. Fluid Mech., 492, 147--180, 2003. doi:10.1017/S002211200300512
  • Sheard, G. J., Leweke, T., Thompson, M. C. and Hourigan, K., Flow around an impulsively arrested circular cylinder, Phys. Fluids, 19(8), 2007, 083601. doi:10.1063/1.2754346
  • Sheard, G. J. and Ryan, K., Pressure-driven flow past spheres moving in a circular tube, J. Fluid Mech., 592, 233--262, 2007. doi:10.1017/S0022112007008543
  • So, J., Ryan, K. and Sheard, G. J., Interaction of an unequal-strength vortex pair, In Proceedings of the 16th Australasian Fluid Mechanics Conference, Crown Plaza, Gold Coast, Queensland, Australia, 3--7 Dec 2007, 1457-1462.
  • Spalart, P. R., 1998, Airplane trailing vortices, Annu. Rev. Fluid Mech., 30, 107--138.
  • Widnall, S. E. and Sullivan, J. P., On the stability of vortex rings, Proc. Roy. Soc. A, 332, 335--353, 1973.
  • Widnall, S. E., Bliss, D. B. and Tasi, C.-Y., The instability of short waves on a vortex ring, J. Fluid Mech., 66, part 1, 35--47, 1974.

Published

2008-10-29

Issue

Section

Proceedings Computational Techniques and Applications Conference