developed recently by . They use approximations of WKB type, as in geometric optics,
We study the time-dependent behavior of disturbances to inviscid vortex rings with swirl, using two different approaches. One is a valid for small amplitude, short wavelength disturbances. linearized stability analysis for short wavelengths, and the other is The predicted growth rates can be compared with numeridirect flow simulation by a computational vortex method. We begin cal solutions of the full, time-dependent Euler equations.
with vortex rings which are exact solutions of the Euler equations
We compute solutions of the perturbed problems using of inviscid, incompressible fluid flow, axisymmetric, and traveling a three-dimensional vortex method, in which the flow is along the axis; swirl refers to the component of velocity around the axis. Exact vortex rings with swirl can be computed reliably using represented by a collection of Lagrangian vortex elements, a variational method. Quantitative predictions can then be made moving according to their induced velocity. The asymptotic for the maximum growth rates of localized instabilities of small analysis and the numerical solution are two very different amplitude, using asymptotic analysis as in geometric optics. The means for describing the dynamics of the perturbed ring;
predicted growth rates are compared with numerical solutions of the comparison of results serves to check the realm of the full, time-dependent Euler equations, starting with a small disturbance in an exact ring. These solutions are computed by a Lagran-validity of each approach. In addition, the time-dependent gian method, in which the three-dimensional flow is represented by simulation of numerically exact solutions of the Euler a collection of vortex elements, moving according to their induced equations provides a test for the quantitative performance velocity. The computed growth rates are typically found to be about of the vortex method. half of the predicted maximum, and the dependence on location Vortex rings are the primary class of steady, inviscid, and ring parameters qualitatively matches the predictions. The comparison of these two very different methods for estimating the three-dimensional fluid flow with vorticity of limited exgrowth of instabilities serves to check the realm of validity of each tent. The study of instabilities in vortex tubes and rings can approach.