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Non-parallel effects in the instability of Long's vortex

Published online by Cambridge University Press:  26 April 2006

M. R. Foster  and
David Jacqmin
M. R. Foster
Affiliation:
Ohio Aerospace Institute, Brook Park, OH 44142, USA Permanent address: Department of Aeronautical and Astronautical Engineering, The Ohio State University, Columbus, OH 43210-1276, USA.
David Jacqmin
Affiliation:
Ohio Aerospace Institute, Brook Park, OH 44142, USA Permanent address: NASA Lewis Research Center, Mail Stop 5-11, 21000 Brook Park Road, Cleveland, OH 44135, USA.
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Abstract

As shown in Foster & Smith (1989), at large flow force M, Long's self-similar vortex is in the form of a swirling ring-jet, whose axial velocity profile is of sech2 form. At azimuthal wavenumber n of comparable order to the axial wavenumber, linear helical modes of instability are essentially those of the Bickley jet varicose and sinuous modes. However, at small axial wavenumbers, the three-dimensionality of the vortex is important, and the instabilities depend heavily on the effects of the swirl. We explore here the effects of finite Reynolds number Re on these long-wave inertial modes. It is shown that, because the radial velocity scales with Re−1M, the non-parallelism of the flow is more important than the viscous terms in determining the finite-Re behaviour. The three-layer structure of the parallel-flow instability modes remains, but with a critical layer considerably modified by radial velocity. In investigating the critical range Re = O(M3), we find the following: for n > 1, the non-parallelism stabilizes the unstable inertial modes, leading to determination of neutral curves; for n < − 1, the non-parallel effects always destabilize the vortex to these helical modes. Determination of the unstable modes and neutral curves for the n > 1 case requires a computational scheme that accounts for the presence of viscosity. It turns out that the n < 1 (n > − 1) modes are prograde (retrograde) with respect to the rotation of the main vortex.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 244 , November 1992 , pp. 289 - 306
Copyright
© 1992 Cambridge University Press

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