Jellyfish Bell Contraction 2D: Vortex-Ring Formation Number

This 2D companion to the 3D bell-contraction simulator swaps the momentum-flux thrust balance for a vortex-impulse formulation grounded in real vortex-ring-formation research: it integrates the vorticity flux shed through the orifice each stroke, tracks the dimensionless formation number that governs when a coherent ring pinches off versus spills into a trailing jet (the universal ≈4 threshold documented by Gharib, Rosenfeld and Dabiri), and derives the reaction thrust on the animal directly from the ring's growing hydrodynamic impulse and Saffman self-induced velocity. The side-view canvas renders the vortex ring's cross-section as a drifting pair of cores so you can watch circulation, formation number and swim speed respond live as you tune frequency, duty cycle, bell size and drag.