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Viscous and inviscid instabilities of a trailing vortex

Published online by Cambridge University Press:  26 April 2006

Ernst W. Mayer  and
Kenneth G. Powell
Ernst W. Mayer
Affiliation:
Department of Aerospace Engineering, The University of Michigan, Ann Arbor. MI 48109, USA
Kenneth G. Powell
Affiliation:
Department of Aerospace Engineering, The University of Michigan, Ann Arbor. MI 48109, USA
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Abstract

A spectral collocation and matrix eigenvalue method is used to study the linear stability of the trailing line vortex model of Batchelor. For both the inviscid and viscous stability problem, the entire unstable region in the swirl/axial wavenumber parameter space is mapped out for various azimuthal wavenumbers m. In the inviscid case, the non-axisymmetric perturbation with azimuthal wavenumber m = 1 has an unstable region of larger extent than any other, with an unusual two-lobed structure; also, the location and numerical value of the maximum disturbance growth rate previously reported for this case are shown to be incorrect. Exploiting the increasingly localized structure of perturbation eigenfunctions allows accurate results to be obtained up to values of m more than 3 orders of magnitude larger than previously, and the results for the most unstable mode are in excellent agreement with the asymptotic theory of Leibovich & Stewartson. A viscous analysis of these fundamentally inviscid modes reveals that the critical Reynolds number at which instability first occurs increases as O(m2) for m [Gt ] 1, and finds the critical values of swirl and wavenumber, which approach limiting values as m → ∞.

In the viscous case, the instabilities for m = 0 and 1 recently reported by Khorrami are found via a simplified numerical approach and the entire unstable region for each of these modes is mapped out over a wide range of Reynolds numbers. The critical Reynolds numbers for these modes are found to be 322.42 and 17.527, respectively, the latter having been unreported previously. The instabilities persist in the limit of large Reynolds number, with corresponding disturbance growth rates decreasing roughly as 1/Re. In addition to the primary mode, a new family of long-wave viscous instabilities is found for the m = 1 case.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 245 , December 1992 , pp. 91 - 114
Copyright
© 1992 Cambridge University Press

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References

Barston F. M. 1980 A circle theorem for inviscid steady flows. Intl J. Engng Sci. 18, 477–489.Google Scholar
Batchelor G. K. 1964 Axial flow in trailing line vortices. J. Fluid Mech. 20, 645–658.Google Scholar
Batchelor, G. K. & Gill A. E. 1962 Analysis of the stability of axisymmetric jets. J. Fluid Mech. 14, 529–551.Google Scholar
Boyd J. P. 1989 Chebyshev and Fourier Spectral Methods. Springer.
Duck, P. W. & Foster M. R. 1980 The inviscid stability of a trailing line vortex Z. Angew. Math. Phys. 31, 524–532.Google Scholar
Duck P. W. 1986 The inviscid stability of swirling flows: large wavenumber disturbances. Z. Angew. Math. Phys. 37, 340–360.Google Scholar
Gary, J. & Helgason R. 1970 A matrix method for ordinary differential eigenvalue problems. J. Comput. Phys. 5, 169–187.Google Scholar
Golub, G. & VanLoan C. 1989 Matrix Computations, 2nd edn. The Johns Hopkins University Press.
Gottlieb, D. & Orszag S. A. 1977 Numerical Analysis of Spectral Methods: Theory and Applications. Philadelphia: Soc. Indus. & Appl. Maths.
Howard, L. N. & Gupta A. A. 1962 On the hydrodynamic and hydromagnetic stability of swirling flows. J. Fluid Mech. 14, 463–476.Google Scholar
Khorrami M. R. 1991 On the viscous modes of instability of a trailing line vortex. J. Fluid Mech. 255, 197–212.Google Scholar
Khorrami M. R., Malik, M. R. & Ash R. L. 1989 Application of spectral collocation techniques to the stability of swirling flows. J. Comput. Phys. 81, 206–229.Google Scholar
Lin C. C. 1966 The Theory of Hydrodynamic Stability. Cambridge University Press.
Lessen, M. & Paillet F. 1974 The stability of a trailing line vortex. Part 2. Viscous theory. J. Fluid Mech. 65, 769–779.Google Scholar
Lessen M., Singh, P. J. & Paillet F. 1974 The stability of a trailing line vortex. Part 1. Inviscid theory. J. Fluid Mech. 63, 753–763.Google Scholar
Leibovich, S. & Stewartso K. 1983 A sufficient condition for the instability of columnar vortices. J. Fluid Mech. 126, 335–356.Google Scholar
Stewartson K. 1982 The stability of swirling flows at large Reynolds number when subjected to disturbances with large azimuthal wavenumber. Phys. Fluids 25, 1953–1957.Google Scholar
Stewartson, K. & Capell K. 1985 On the stability of ring modes in a trailing line vortex: the upper neutral points. J. Fluid Mech. 156, 369–386.Google Scholar
Stewartson, K. & Leibovich S. 1987 On the stability of a columnar vortex to disturbances with large azimuthal wavenumber: the lower neutral points. J. Fluid Mech. 178, 549–566.Google Scholar
Wilkinson J. H. 1965 The Algebraic Eigenvalue Problem. Oxford University Press.