Smoke ring physics
A vortex ring is a donut-shaped region of spinning fluid that propels itself through still air. This simulation shows the cross-section — two counter-rotating vortex cores that carry a pocket of fluid forward. Click anywhere to fire a new ring. When two rings travel along the same axis, they undergo leapfrogging: the trailing ring contracts and accelerates through the leading ring, which expands and slows, and they trade places endlessly.
ω = ∇ × v | dΓ/dt = ν∇²ω | Vring = Γ/(4πR) · [ln(8R/a) − ¼]
What is a vortex ring?
A vortex ring (or smoke ring) is a toroidal region of rotating fluid that travels through a surrounding medium. In cross-section, it appears as two counter-rotating vortex cores. The fluid circulates around each core, and the mutual induction between the two halves propels the ring forward. The classic example is a smoke ring puffed from a smoker’s lips, but vortex rings appear in dolphins’ bubble play, volcanic eruptions, and cardiac blood flow.
The leapfrog instability
When two vortex rings travel along the same axis, something remarkable happens. The velocity field of the leading ring causes the trailing ring to contract radially and speed up. Simultaneously, the trailing ring’s field causes the leading ring to expand and slow down. The trailing ring shoots through the center of the leading ring, and they swap roles. This leapfrogging was first described by Helmholtz in 1858 and can repeat indefinitely in an inviscid fluid, though viscosity eventually dissipates the rings in reality.
The mathematics
The self-induced velocity of a thin vortex ring of radius R, core radius a, and circulation Γ is given by Kelvin’s formula: V = Γ/(4πR) · [ln(8R/a) − 1/4]. Smaller rings move faster (inversely proportional to R), which is the key mechanism behind leapfrogging. The simulation uses a simplified 2D cross-section model where each ring is represented by a pair of point vortices with opposite circulation.
Viscous decay
In a real fluid, viscosity diffuses the concentrated vorticity outward, causing the core radius to grow as a ~ √(νt). As the core spreads, the ring slows and eventually dissipates. The viscosity slider controls this decay rate — at zero viscosity, rings persist indefinitely; at high viscosity, they fade quickly.