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A note on the effects of viscosity on the stability of a trailing-line vortex

Published online by Cambridge University Press:  26 April 2006

Peter W. Duck  and
Mehdi R. Khorrami
Peter W. Duck
Affiliation:
Department of Mathematics, University of Manchester, Manchester M13 9PL, UK
Mehdi R. Khorrami
Affiliation:
High Technology Corporation, 28 Research Drive, Hampton, VA 23666, USA
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Abstract

The linear stability of the Batchelor (1964) vortex is investigated. Particular emphasis is placed on modes found recently in a numerical study by Khorrami (1991). These modes have a number of features very distinct from those found previously for this vortex, including (i) exhibiting small growth rates at large Reynolds numbers and (ii) susceptibility to destabilization by viscosity. In this paper these modes are described using asymptotic techniques, producing results which compare very favourably with fully numerical results at large Reynolds numbers.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 245 , December 1992 , pp. 175 - 189
Copyright
© 1992 Cambridge University Press

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References

Batchelor G. K. 1964 Axial flow in the trailing line vortices. J. Fluid Mech. 20, 645.Google Scholar
Bridges, T. J. & Morris P. J. 1984 Differential eigenvalue problems in which the parameter appears nonlinearly. J. Comput. Phys. 55, 437.Google Scholar
Duck P. W. 1986 The inviscid stability of swirling flows: Large wavenumber disturbances. Z. Angew. Math. Phys. 37, 340.Google Scholar
Duck, P. W. & Foster M. R. 1980 The inviscid stability of a trailing line vortex. Z. Angew. Math. Phys. 31, 523.Google Scholar
Gottlieb D., Hussaini, M. Y. & Orszag S. A. 1984 Spectral Methods for Partial Differential Equations. SIAM.
Gottlieb, D. & Orszag S. A. 1977 Numerical Analysis of Spectral Methods: Theory and Applications. SIAM.
Ince E. L. 1956 Ordinary Differential Equations. Dover.
Khorrami M. R. 1991 On the viscous modes of instability of a trailing line vortex. J. Fluid Mech. 225, 197.Google Scholar
Khorrami M. R., Malik, M. R. & Ash R. L. 1989 Application of spectral collocation techniques to the stability of swirling flow. J. Comput. Phys. 81, 206.Google Scholar
Leibovich S., Brown, S. N. & Patel Y. 1986 Bending waves on inviscid columnar vortices. J. Fluid Mech. 173, 595.Google Scholar
Lbibovich, S. & Stewartson K. 1983 A sufficient condition for the instability of columnar vortices. J. Fluid Mech. 126, 335.Google Scholar
Lessen, M. & Paillet F. 1974 The stability of a trailing line vortex. Part 2. Viscous theory. J. Fluid Mech. 63, 769.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.Google Scholar
Lin C. C. 1945a On the stability of two-dimensional parallel flows. Part I. General theory. Q. Appl. Maths 3, 117.Google Scholar
Lin C. C. 1945b On the stability of two-dimensional parallel flows. Part II. Stability in an inviscid fluid. Q. Appl. Maths 3, 218.Google Scholar
Lin C. C. 1945c On the stability of two-dimensional parallel flows. Part III. Stability in a viscous fluid. Q. Appl. Maths 3, 277.Google Scholar
Lin C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Mack L. M. 1987 Review of linear compressible stability theory. ICASE Workshop on the Stability of Time Dependent and Spatially Varying Flows. Springer.
Malik, M. R. & Poll D. I. A. 1985 Effect of curvature on three-dimensional boundary layer stability. AIAA J. 27, 1362.Google Scholar
Metcalfe, R. W. & Orszag S. A. 1973 Numerical calculation of the linear stability of pipe flows. Flow Research Rep. 25, Contract N00014–72-C-0365.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.Google Scholar
Stewartson, K. & Brown S. N. 1985 Near-neutral center-modes as inviscid perturbations to a trailing line vortex. J. Fluid Mech. 156, 387.Google Scholar
Stewartson, K. & Capell K. 1985 On the stability of ring modes in a trailing line vortex: the upper neutral point. J. Fluid Mech. 156, 369.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.Google Scholar
Wilkinson J. H. 1965 The Algebraic Eigenvalue Problem. Oxford University Press.