Super-harmonically resonant swirling waves in longitudinally forced circular cylinders*

1. Resonant sloshing in circular cylinders was studied by Faltinsen et al. (2016), whose theory was used to describe steady-state resonant waves due to time-harmonic elliptic container orbits. In the limit of longitudinal motions, a symmetry-breaking of the planar wave solution occurs, with clockwise and anti-clockwise swirling equally likely. In addition to this harmonic dynamics, previous experiments have unveiled that diverse super-harmonic dynamics are observable far from primary resonances. Among these, the double-crest (DC) dynamics, first observed by Reclari et al. (2014) for rotary sloshing, is particularly relevant, as its manifestation is naturally favored by the structure of the external driving. In this work we develop a weakly nonlinear analysis to describe the system response to longitudinal forcing. The resulting system of amplitude equations predicts that a planar wave symmetry-breaking via stable swirling may also occur under super-harmonic excitation. This finding is confirmed by our experiments, which identify three possible regimes, i.e. (i) stable planar DC waves, (ii) irregular motion and (iii) stable swirling DC waves, whose corresponding stability boundaries in the forcing frequency-amplitude plane quantitatively match the present theoretical estimates.