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.