Azimuthal instability of vortex rings generated by an oscillating disk

We report the instabilities of vortex rings generated by an oscillating disk. Assuming sinusoidal variation in the azimuthal direction with mode number, $m$, a Floquet linear stability analysis is performed. We study the dynamics for a range of the two control parameters, the Keulegan-Carpenter number $KC=2\pi A/c$ and the Stokes number $\beta=fc^2/\nu$, where $A$ is the amplitude of oscillation, $f$ is the frequency of oscillation, $c$ is the diameter of the disk, and $\nu$ is the kinematic viscosity of the fluid. We observe two distinctive flow regions in the ($KC,\beta$) parameter space. First, in the low $\beta$ region, the flow breaks its symmetry with a single wavenumber mode getting a positive growth rate. Second, in the high $\beta$ region, high-order unstable modes emerge, with the highest mode number $m=9$ recorded. Furthermore, we carry out Direct Numerical Simulations (DNS) on the fully three-dimensional Navier-stokes equations. The results reproduce the main features of the high-order unstable modes predicted by the Floquet analysis, exhibiting the highest mode number $m=6$. We conjecture that the inconsistence in the highest mode number between the Floquet linear stability analysis and the DNS implies the non-linear characteristic of the current problem.