Centrifugal instability in non-axisymmetric vortices. (13th March 2015) - Nagarathinam, David; Sameen, A.; Mathur, Manikandan
Abstract : We study the centrifugal instability of non-axisymmetric vortices in the presence of an axial flow ( $w$ ) and a background rotation ( ${\itΩ}_{z}$ ) using the local stability approach. Analytically solving the local stability equations for an axisymmetric vortex with $w$ and ${\itΩ}_{z}$, growth rates for wave vectors that are periodic upon evolution around a closed streamline are calculated. The resulting sufficient criterion for centrifugal instability in an axisymmetric vortex is then heuristically extended to non-axisymmetric vortices and written in terms of integral quantities and their derivatives with respect to the streamfunction on a streamline. The new criterion for non-axisymmetric vortices, which converges to the exact criterion of Bayly ( Phys. Fluids, vol. 31, 1988, pp. 56–64) in the absence of background rotation and axial flow, is validated by comparisons with numerically calculated growth rates for two different anticyclonic vortices: the Stuart vortex (specified by the concentration parameter ${\itρ}, ~0<{\itρ}\leqslant 1$ ) and the Taylor–Green vortex (specified by the aspect ratio $E, ~0<E\leqslant 1$ ). With no axial velocity and finite background rotation, the criterion predicts a lower and an upper threshold of $|{\itΩ}_{z}|$ between which centrifugal instability is present. We further demonstrate that the criterion represents an improvement over the criterion of Sipp & Jacquin ( Phys. Fluids, vol. 12, 2000, pp. 1740–1748). Finally, in the presence of both axial velocity and background rotation, the criterion is shown to be accurate for large enough ${\itρ}$ and $E$ .