Super-harmonically resonant swirling waves in longitudinally forced circular cylinders

Resonant sloshing in circular cylinders was studied by Faltinsen et al. (J. Fluid Mech., vol. 804, 2016, pp. 608–645), whose theory was used to describe steady-state resonant waves due to a time-harmonic container’s elliptic orbits. In the limit of longitudinal container motions, a symmetry breaking of the planar wave solution occurs, with clockwise and anti-clockwise swirling equally likely. In addition to this primary harmonic dynamics, previous experiments have unveiled that diverse super-harmonic dynamics are observable far from primary resonances. Among these, the so-called double-crest (DC) dynamics, first observed by Reclari et al. (Phys. Fluids, vol. 26, no. 5, 2014, 052104) for circular sloshing, is particularly relevant, as its manifestation is the most favoured by the spatial structure of the external driving. Following Bongarzone et al. (J. Fluid Mech., vol. 943, 2022, A28), in this work we develop a weakly nonlinear analysis to describe the system response to super-harmonic 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 experimental observations, which identify three possible super-harmonic 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.

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Caption: Images of the fluid-free surface while the container is subjected to a longitudinal harmonic forcing of amplitude $a_x=\overline{a}_x/R\approx0.23$ at various driving angular frequencies $\Omega$ close to $\omega_2/2$. The fluid-free surface is observed in the direction aligned with the container motion. For each driving frequency (a), (b) and (c), the time interval between two snapshots is about $T/4$, with $T=2\pi/\Omega$ the corresponding oscillation period. On each snapshot, the vertical middle axis is represented by a red dotted line. For a forcing frequency $\Omega\approx0.48\omega_2$ (a) and $\Omega\approx0.52\omega_2$ (c) the free-surface image at each time t is mirror symmetric with respect to the middle vertical axis, the signature of a planar wave regime, while the symmetry is broken for $\Omega\approx0.50\omega_2$ (c) revealing a swirling wave. Results are shown for (a) $\Omega/\omega_2\approx 0.48$, (b) $\Omega/\omega_2\approx0.50$ and (c) $\Omega/\omega_2\approx0.52$.