Fig. 3: Oscillating behavior of bubble ring.

A Under the action of the initial pressure, the bubble ring expands and contracts, and its radius oscillate repeatedly. B Schematic representation of a ring bubble interacting with a vortex fluid, the high-speed camera images show the ring formed around the ring core by the gas following the vortex, providing a visible representation of the vortex structure. C Mean circulation (Γ_m) versus output energy, comparing different generation methods, including gravity-driven, piston-driven, and compressed gas-driven. D The decay of the vortex-ring-circulation under rebound and jetting, with distinct stages of deflation, oscillation. In the initial stage (0–2 ms), the bubble undergoes rapid expansion and acceleration influenced by the primary bubble’s flow field, shock waves, and surrounding fluid. This is followed by deflation and deceleration. After 2 ms (yellow region), the bubble stabilizes and migrates at an approximately constant velocity for the rebound, including experimental data (solid line) and theoretical attenuation model (dashed line). E Oscillatory behavior of the ring bubble radius (R) over time. F Oscillatory behavior of the core radius (R_c) over time. G Under different constraint tube, the energy increases, the circulation increases, and the state of the vortex ring becomes more stable. H It becomes difficult for the electrode to discharge under the slender tube, and part of the energy is converted into arc discharge, which cannot be injected into the vortex ring to increase the circulation. I Experimental and theoretical data on the frequency of vortex ring oscillation. As the energy increases, the vortex ring oscillation frequency decreases and the oscillation amplitude increases, which is conducive to running longer distances. Data represented as mean ± SD in Supplementary Table S3, S4, S5. *p < 0.05. Scale bars, 10 mm. Source data are provided as a Source Data file.