Why do the two vortices move forward instead of just rotating?

Each vortex sits in the velocity field induced by the other one. Because their circulations have opposite sign, the field they impose on each other points the same way — forward — so the pair drifts together at V_self ≈ Γ/(4πa).

What is the role of the core radius r_c?

A pure point vortex has infinite velocity at the centre. The Lamb–Oseen profile smoothes that singularity over a viscous core of size r_c, giving a finite peak speed and a smooth tracer field — closer to a real, slightly viscous ring.

Why a 2D cross-section instead of a real torus?

The defining dynamics — circulation, mutual induction, and self-propagation — are visible already in the slice that cuts the ring through its symmetry plane. Rendering the full 3D torus would hide the two-vortex structure that drives the motion.

Vortex Ring (Smoke Ring)

A smoke ring is shown in cross-section as a pair of counter-rotating Lamb–Oseen vortices of strength ±Γ separated by 2a. Each core induces a velocity field uθ(r) = (Γ/2πr)·(1 − exp(−r²/r_c²)) that is regularised at the centre by the viscous core radius r_c. By symmetry, the two vortices push each other forward at the self-induced speed V_self ≈ Γ/(4πa) — exactly the mechanism that lets a real toroidal vortex propagate through still air. Tracer particles seeded around the cores are advected by the combined velocity field and reveal the rolled-up smoke pattern.

Who it's for: Intro fluid dynamics and aerodynamics; complements the airfoil and Kármán-vortex labs.

Key terms

vortex ring
Lamb–Oseen vortex
self-induced velocity
circulation Γ
toroidal vortex
inviscid limit

How it works

A vortex ring is two counter-rotating vortex tubes joined into a torus. In a 2D cross-section it looks like a vortex pair that propels itself through the surrounding fluid at V_self ≈ Γ / (4π a).

Key equations

V_self ≈ Γ / (4π a) (thin ring, inviscid)

v_θ(r) = (Γ / 2π r)·(1 − exp(−r²/r_c²)) (Lamb–Oseen core)

Frequently asked questions

Why do the two vortices move forward instead of just rotating?: Each vortex sits in the velocity field induced by the other one. Because their circulations have opposite sign, the field they impose on each other points the same way — forward — so the pair drifts together at V_self ≈ Γ/(4πa).
What is the role of the core radius r_c?: A pure point vortex has infinite velocity at the centre. The Lamb–Oseen profile smoothes that singularity over a viscous core of size r_c, giving a finite peak speed and a smooth tracer field — closer to a real, slightly viscous ring.
Why a 2D cross-section instead of a real torus?: The defining dynamics — circulation, mutual induction, and self-propagation — are visible already in the slice that cuts the ring through its symmetry plane. Rendering the full 3D torus would hide the two-vortex structure that drives the motion.

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