How it Works
A vortex ring is represented in cross-section as two counter-rotating point vortices (one at each side of the ring diameter). The Biot-Savart law gives the velocity induced at any point by a vortex filament: each vortex core induces a rotational velocity field that decays as 1/r from the centre. The two cores together produce a net upward (axial) self-induced velocity that propels the ring.
We track N tracer particles released into the flow field to visualise the induced velocity. At each timestep the velocity at every tracer is computed as the vector sum of contributions from both vortex cores. A Rankine vortex model smooths the singularity inside the core radius a.
v_θ(r) = Γ/(2πr) for r > a
v_θ(r) = Γ·r/(2πa²) for r ≤ a (Rankine core)
Frequently Asked Questions
What is a vortex ring?
A vortex ring is a torus-shaped region of rotating fluid that self-propels through a medium. Smoke rings and underwater bubble rings are everyday examples.
Why does a vortex ring move forward?
The circular vortex filament induces a velocity field on itself via the Biot-Savart law. The net induced velocity at the ring centre points in the axial direction, propelling the ring forward.
What is circulation in fluid dynamics?
Circulation Γ is the line integral of velocity around a closed loop. For a vortex ring it quantifies the strength of rotation and directly sets the self-propagation speed.
How does ring radius affect speed?
For a thin-core vortex ring the propagation speed scales as Γ/(4πR) × [ln(8R/a) − 1/4], so larger rings travel more slowly while smaller rings with the same circulation move faster.
What is the Biot-Savart law in fluid mechanics?
Analogous to electromagnetism, the Biot-Savart law relates the vorticity distribution to the induced velocity field: dv = (Γ/4π) × (dl × r) / |r|³, where dl is a vortex filament element.
What is core thickness and why does it matter?
The core radius a is the cross-sectional radius of the vortex tube. A thinner core gives a faster ring; a very thin core can become unstable to Kelvin waves.
Do vortex rings decay over time?
Yes. Viscosity diffuses the core, slowly increasing its effective radius and reducing circulation. The ring slows and eventually dissipates.
What are Kelvin waves on a vortex ring?
Kelvin waves are azimuthal perturbations that travel around the core of the vortex ring. They can grow and cause the ring to break up into smaller vortex structures.
Can two vortex rings interact?
Yes. Two co-axial same-sign rings undergo leap-frogging: the rear ring threads through the front ring, which in turn threads back through, repeating in a periodic exchange.
What practical applications involve vortex rings?
Vortex rings appear in jet engine exhausts, cardiac blood flow through the mitral valve, dolphin bubble play, and targeted drug delivery using acoustic vortex rings.