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The stability of a trailing-line vortex in compressible flow

Published online by Cambridge University Press:  26 April 2006

Jillian A. K. Stott  and
Peter W. Duck
Jillian A. K. Stott
Affiliation:
Department of Mathematics, University of Manchester, Manchester, M13 9PL, UK Present address: School of Mathematics, University of New South Wales, Australia.
Peter W. Duck
Affiliation:
Department of Mathematics, University of Manchester, Manchester, M13 9PL, UK
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Abstract

We consider the inviscid stability of the Batchelor (1964) vortex in a compressible flow. The problem is tackled numerically and also asymptotically, in the limit of large (azimuthal and streamwise) wavenumbers, together with large Mach numbers. The nature of the solution passes through different regimes as the Mach number increases, relative to the wavenumbers. At very high wavenumbers and Mach numbers, the mode which is present in the incompressible case ceases to be unstable, whilst a new ‘centre mode’ forms, whose stability characteristics are determined primarily by conditions close to the vortex axis. We find that generally the flow becomes less unstable as the Mach number increases, and that the regime of instability appears generally confined to disturbances in a direction counter to the direction of the rotation of the swirl of the vortex.

Throughout the paper comparison is made between our numerical results and results obtained from the various asymptotic theories.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 269 , 25 June 1994 , pp. 323 - 351
Copyright
© 1994 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1970 Handbook of Mathematical Functions. Dover.
Batchelor, G. K. 1964 Axial flow in trailing line vortices. J. Fluid Mech. 20, 645.Google Scholar
Berry, M. V. 1989 Uniform asymptotic smoothing of Stokes's discontinuities. Proc. R. Soc. Lond. A 422, 7.Google Scholar
Coleman, C. S. 1989 The stability of swirling jets. Astro. Soc. Austral. Proc. 8, 38.Google Scholar
Duck, P. W. 1986 The inviscid stability of a swirling flow: large wavenumber disturbance. 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
Duck, P. W. & Khorrami, M. R. 1991 A note on the effects of viscosity on the stability of a trailing line vortex. ICASE Rep. 91–6. (Also J. Fluid Mech. 245, 175, 1992.)
Foster, M. R. & Duck, P. W. 1982 Inviscid stability of Long's vortex. Phys. Fluids 25, 1715.Google Scholar
Foster, M. R. & Smith, F. T. 1989 Stability of Long's vortex at large flow force. J. Fluid Mech. 206, 405.Google Scholar
Gouml;rtler, H. 1954 Theoretical investigations of laminar boundary-layer problems II –- theory of swirl in an axially symmetric jet, far from the orifice. Air Force Contract AF 61(514) 627C.Google Scholar
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. I. 1989 Application of spectral collocation techniques to the stability of swirling flows. J. Comput. Phys. 81, 206.Google Scholar
Leibovich, 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. 65, 769.Google Scholar
Lessen, M., Singh, P. & Paillet, F. 1974 The stability of a trailing line vortex. Part 1. Inviscid theory. J. Fluid Mech. 63, 753.Google Scholar
Loitsyanskii, L. G. 1953 Propagation of a rotating jet in an infinite space surrounded by the same liquid. Prik. Mat. Mehk. 17, 3.Google Scholar
Long, R. R. 1961 A vortex in an infinite fluid. J. Fluid Mech. 11, 611.Google Scholar
Mayer, E. W. & Powell, K. G. 1992 Viscous and inviscid instabilities of a trailing vortex. J. Fluid Mech. 245, 91.Google Scholar
Michalke, A. 1971 Instabilität eines kompressiblen runden Freistrahls unter Berucksichtigung des Einflusses den Strahlgrenzschtdicke. J. Flugwiss 19, 319.Google Scholar
Michalke, A. 1984 Survey on jet instability theory. Prog. Aerospace Sci. 21, 519.Google Scholar
Smith, F. T. & Brown, S. N. 1990 The inviscid instability of a Blasius boundary layer at large values of the Mach number. J. Fluid Mech. 219, 499.Google Scholar
Stewartson, K. & Brown, S. N. 1984 Inviscid centre-modes and wall-modes in the stability theory of swirling Poiseuille flow. IMA J. Appl. Maths 32, 311.Google Scholar
Stewartson, K. & Brown, S. N. 1985 Near neutral centre modes as inviscid perturbations to a trailing line vortex. J. Fluid Mech. 156, 387.Google Scholar
Stewartson, K. & Capell, S. 1985 On the stability of ring modes in a trailing line vortex: the upper neutral points. 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