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
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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