The onset of instability in exact vortex rings with swirl

We study the time-dependent behavior of disturbances to inviscid vortex rings with swirl, using two different approaches. One is a linearized stability analysis for short wavelengths, and the other is direct flow simulation by a computational vortex method. We begin with vortex rings which are exact solutions of the Euler equations of inviscid, incompressible fluid flow, axisymmetric, and traveling along the axis; swirl refers to the component of velocity around the axis. Exact vortex rings with swirl can be computed reliably using a variational method. Quantitative predictions can then be made for the maximum growth rates of localized instabilities of small amplitude, using asymptotic analysis as in geometric optics. The predicted growth rates are compared with numerical solutions of the full, time-dependent Euler equations, starting with a small disturbance in an exact ring. These solutions are computed by a Lagrangian method, in which the three-dimensional flow is represented by a collection of vortex elements, moving according to their induced velocity. The computed growth rates are typically found to be about half of the predicted maximum, and the dependence on location and ring parameters qualitatively matches the predictions. The comparison of these two very different methods for estimating the growth of instabilities serves to check the realm of validity of each approach. © 1996 Academic Press, Inc.

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Author J. Thomas Beale Mathematics