The interaction of perturbed vortex rings and its sound generation. Part II

The present work extends the study of interactions of two coaxial vortex rings of thin cores with azimuthal instability to wavenumber range 6 ≤ m ≤ 11. Unstable modes are found at 6 ≤ m ≤ 10. Stable mode is found only at m = 11. The present range of unstable wavenumbers m = 1, 2, 5 to 10 of the two interacting vortices is wider than m* = 8 of a single vortex ring of the same property of core structure. Besides the category of stable mode, two other categories of the unstable mode, namely the odd-m and even-m modes, are classified. The odd-m unstable mode of m = 5, 7 and 9 concerns the evolution of star-like structure of one vortex ring with tail sheaths trailing behind. This unstable mode has slower evolution and becomes unstable after the third slip through. The even-m unstable mode of m = 6, 8 and 10 more rapid evolution and becomes unstable after the second slip through. Either one (m = 8 and 10) or two (m = 6) interacting vortex rings evolve into star-like structure with roll-shape sheaths. At m = 8, the unstable mode m* = 8 of a single vortex ring, besides the evolution of the roll-shape structure, sheaths wrapping the rolls are also found. Although quadrupole sound radiation is still observed, the far field sound pressure fluctuations depend on the interaction modes, with the roll-shape one (even-m mode) generally the highest, followed by the tail shape one (odd-m mode). The criterion for the onset of unstable mode established in the previous study is also confirmed for the higher wavenumbers.