Experimental confirmation of Kelvin's equilibria

Area of Science:

• Fluid Dynamics
• Vortex Dynamics
• Hydrodynamics

Background:

• Thomson's theorem, established 124 years ago, predicts the stability of vortex rings.
• Understanding vortex stability is crucial in various fluid mechanics applications.

Purpose of the Study:

• To experimentally verify Thomson's theorem regarding the stability of N-vortex systems.
• To investigate the stability of regular N-gons formed by water vortices.

Main Methods:

• Generating water vortices within a cylinder using a revolving disk.
• Observing and analyzing the stability of N-gon vortex patterns for varying N.

Main Results:

• Regular N-gons are stable for N ≤ 6 and unstable for N ≥ 8.
• The N ≤ 6 equilibria demonstrate high resilience and self-reforming capabilities.
• The heptagonal system (N=7) exhibits critical stability or stability within a narrow parameter range.
• Interfacial axial symmetry breaks through spectral development, not spontaneously.
• A simple functional relationship exists between polygon rotation and disk speed, influenced by Froude and wave numbers.

Conclusions:

• Experimental results largely corroborate Thomson's theorem on vortex ring stability.
• The study provides new insights into the dynamics of vortex systems, particularly the heptagonal case.
• The findings offer a simplified model for predicting vortex pattern behavior based on physical parameters.