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Stability of Long's vortex at large flow force

Published online by Cambridge University Press:  26 April 2006

M. R. Foster
Affiliation:
Department of Aeronautical and Astronautical Engineering, The Ohio State University, 203 Neil Avenue Mall, Columbus, OH 43210-1276, USA
F. T. Smith
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK

Abstract

Long's self-similar vortex is known to have two solutions for each supercritical value of the flow force. Each of those solutions is shown here to have a double structure if the flow force is large. We then investigate the inertial instabilities of one of those large-flow-force limit solutions, and find them to be related to the instabilities of the Bickley jet in one régime. However, the swirl in the vortex becomes important for long waves, very strongly modifying the sinuous and varicose Bickley modes. We find in particular that the asymptotic results obtained agree well with our numerical solutions for the sinuous mode, but not for the varicose mode, the difficulty in the latter case being apparently due to mode jumping. The asymptotics show a varicose long-wave neutral mode for positive azimuthal wavenumber, and two such modes for negative wavenumbers. The upper neutral sinuous mode occurs at much, larger wavenumber than in the Bickley case, and its structure is also presented. The study overall is aimed at providing a basis for the investigation of strongly nonlinear effects.

Type
Research Article
Copyright
© 1989 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Batchelor, G. 1964 Axial flow in trailing line vortices. J. Fluid Mech. 20, 645.Google Scholar
Burggraf, O. R. & Foster, M. R. 1977a The stability of tornado-like vortices. Final Rep., Grant No. 04-6-022-44004. US Department of Commerce.
Burggraf, O. R. & Foster, M. R. 1977b Continuation or breakdown in tornado-like vortices. J. Fluid Mech. 80, 685.Google Scholar
Church, C. R. & Ehresman, C. M. 1971 A brief report on the Purdue waterspout research program. Purdue Tornado Project Rep.Google Scholar
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic stability of parallel flow of inviscid fluids. Adv. Appl. Math. 7, 1.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
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, 524.Google Scholar
Foster, M. R. & Duck, P. W. 1982 Inviscid stability of Long's vortex. Phys. Fluids 25, 1715.Google Scholar
Golden, J. H. 1973 Life cycle of the Florida Keys waterspout as a result of five interacting scales of motion. Ph.D. thesis, Florida State University, Tallahassee.
Hoecker, W. H. 1960 Windspeed and air flow patterns in the Dallas tornado of April 7, 1957. Mon. Weather Rev. 88, 167.Google Scholar
Howard, L. N. & Gupta, A. S. 1962 On the hydrodynamic and hydromagnetic stability of swirling flows. J. Fluid Mech. 14, 473.Google Scholar
Leibovich, S. & Stewartson, K. 1983 A sufficient condition for instability of columnar vortices. J. Fluid Mech. 126, 335.Google Scholar
Lessen, M., Desphande, N. V. & Hadji-Ohanes, B. 1973 Stability of a potential vortex with a non-rotating and rigid-body rotating top-hat jet core. J. Fluid Mech. 60, 459.Google Scholar
Lessen, M. & Singh, P. J. 1973 The stability of axisymmetric free shear layers. J. Fluid Mech. 60, 433.Google Scholar
Lessen, M., Singh, P. J. & Paillet, F. 1974 The stability of a trailing line vortex. J. Fluid Mech. 63, 753.Google Scholar
Long, R. R. 1961 A vortex in an infinite fluid. J. Fluid Mech. 11, 611.Google Scholar
Maslowe, S. & Stewartson, K. 1982 On the linear inviscid stability of rotating Poiseuille flow. Phys. Fluids 25, 1517.Google Scholar
Mulholland, H. P. & Goldstein, S. 1929 The characteristic numbers of the Mathieu equation with purely imaginary parameters. Phil. Mag. 8, 834.Google Scholar
Pedley, T. 1968 On the instability of rapidly rotating shear flows to non-axisymmetric disturbances. J. Fluid Mech. 31, 603.Google Scholar
Pedley, T. 1969 On the instability of viscous flow in a rapidly rotating pipe. J. Fluid Mech. 35, 97.Google Scholar
Smith, F. T. & Bodonyi, R. J. 1982 Non-linear critical layers and their development in streaming-flow stability. J. Fluid Mech. 118, 165.Google Scholar
Snow, J. T. 1978 On inertial instability as related to the multiple vortex phenomenon. J. Atmos. Sci. 35, 1660.Google Scholar
Squire, H. 1951 The round laminar jet. Q. J. Mech. Appl. Maths 4, 321.Google Scholar
Staley, D. O. & Gall, B. L. 1979 Barotropic instability in a tornado vortex. J. Atmos. Sci. 36, 973.Google Scholar
Stewartson, K. 1982 The stability of swirling flow at large Reynolds number when subjected to disturbances with large azimuthal wavenumber. Phys. Fluids 25, 1953.Google Scholar
Stewartson, K. & Capell, K. 1985 Marginally stable ring modes in a trailing line vortex: 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