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A sufficient condition for the instability of columnar vortices

Published online by Cambridge University Press:  20 April 2006

S. Leibovich
Affiliation:
Sibley School of Mechanical and Aerospace Engineering. Cornell University, Ithaca, NY 14853
K. Stewartson
Affiliation:
Department of Mathematics, University College London

Abstract

The inviscid instability of columnar vortex flows in unbounded domains to three-dimensional perturbations is considered. The undisturbed flows may have axial and swirl velocity components with a general dependence on distance from the swirl axis. The equation governing the disturbance is found to simplify when the azimuthal wavenumber n is large. This permits us to develop the solution in an asymptotic expansion and reveals a class of unstable modes. The asymptotic results are confirmed by comparisons with numerical solutions of the full problem for a specific flow modelling the trailing vortex. It is found that the asymptotic theory predicts the most-unstable wave with reasonable accuracy for values of n as low as 3, and improves rapidly in accuracy as n increases. This study enables us to formulate a sufficient condition for the instability of columnar vortices as follows. Let the vortex have axial velocity W(r), azimuthal velocity V(r), where r is distance from the axis, let Ω be the angular velocity V/r, and let Γ be the circulation rV. Then the flow is unstable if $ V\frac{d\Omega}{dr}\left[ \frac{d\Omega}{dr}\frac{d\Gamma}{dr} + \left(\frac{dW}{dr}\right)^2\right] < 0.$

Type
Research Article
Copyright
© 1983 Cambridge University Press

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References

Barston, F. M. 1980 Circle theorem for inviscid steady flows Int. J. Eng. Sci. 18, 477489.Google Scholar
Brown, S. N. & Stewartson, K. 1978 The evolution of a small inviscid disturbance to a marginally stable stratified flow: stage two. Proc. R. Soc. Land A 363, 175194.Google Scholar
Brown, S. N. & Stewartson, K. 1900 On the secular stability of a regular Rossby neutral mode Geophys. Astrophys. Fluid Dyn. 14, 118.Google Scholar
Cotton, F. W. & Salwen, H. 1981 Linear stability of rotating Hagen–-Poiseuille flow J. Fluid Mech. 108, 101125.Google Scholar
Duck, P. W. & Foster, M. R. 1980 The inviscid stability of a trailing line vortex Z. angew. Math. Phys. 31, 523530.Google Scholar
Escudier, M. P., Bornstein, J. & Zehnder, N. 1980 Observations and LDA measurements of confined vortex flow J. Fluid Mech. 98, 4963.Google Scholar
Faler, J. H. & Leibovich, S. 1977 An experimental map of the internal structure of a vortex breakdown J. Fluid Mech. 86, 313335.Google Scholar
Foster, M. R. & Duck, P. W. 1982 The inviscid stability of Long's vortex. Submitted to Phys. Fluids.Google Scholar
Garg, A. K. & Leibovich, S. 1979 Spectral characteristics of vortex breakdown flowfields Phys. Fluids 22, 20532064.Google Scholar
Hall, M. G. 1972 Vortex breakdown Ann. Rev. Fluid Mech. 4, 195218.Google Scholar
Howard, L. N. 1961 Note on a paper by John W. Miles J. Fluid Mech. 10, 509512.Google Scholar
Howard, L. N. & Gupta, A. S. 1962 On the hydrodynamic and hydromagnetic stability of swirling flows J. Fluid Mech. 14, 463476.Google Scholar
Joseph, D. D. 1976 Stability of Fluid Motions I. Springer.
Leibovich, S. 1978 The structure of vortex breakdown Ann. Rev. Fluid Mech. 10, 221246.Google Scholar
Lessen, M. & Paillet, F. 1974 The stability of a trailing line vortex. Part 2. Viscous theory J. Fluid Mech. 65, 769779.Google Scholar
Lessen, M., Singh, P. J. & Paillet, F. 1974 The stability of a trailing line vortex J. Fluid Mech. 63, 753763.Google Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Long, R. R. 1958 Vortex motion in a viscous fluid J. Met. 15, 108112.Google Scholar
Long, R. R. 1961 A vortex in an infinite viscous fluid J. Fluid Mech. 11, 611624.Google Scholar
Ludwieg, H. 1961 Ergänzung zu der Arbeit: ‘Stabilität der Strömung in einem zylindrischen Ringraum’. Z. Flugwiss. 9, 359361.Google Scholar
Maslowe, S. A. 1974 Instability of rigidly rotating flows to non-axisymmetric disturbances J. Fluid Mech. 64, 303317.Google Scholar
Maslowe, J. A. & Stewartson, K. 1982 On the linear inviscid stability of rotating Poiseuille flow. Phys. Fluids (to appear).Google Scholar
Pedley, T. J. 1968 Instability of rapidly rotating shear flows to non-axisymmetric disturbances J. Fluid Mech. 31, 603614.Google Scholar
Singh, P. I. & Uberoi, M. S. 1976 Experiments on vortex stability Phys. Fluids 19, 18581863.Google Scholar
Stewartson, K. 1982 The stability of swirling flows at large Reynolds number when subjected to disturbances with large azimuthal wavenumber. Phys. Fluids 19531957.Google Scholar
Warren, F. W. 1978 Hermitian forms and eigenvalue bounds Stud. Appl. Math. 59, 249281.Google Scholar
Watts, H. A., Scott, M. R. & Lord, M. E. 1978 Computational solution of complex valued boundary problems. Sandia Labs, Albuquerque, NM, Rep. SAND 78–1501.Google Scholar