Fluid dynamics · three-dimensional coherent structures

Vortex Rings

A transparent nozzle delivers one finite stroke into still fluid. Orbit the three-dimensional tank to follow shear-layer roll-up, self-induced translation, and pinch-off when the stroke becomes too long; luminous particles reveal rotation through the toroidal core.

Time after launch $0.00\,\mathrm{s}$

Formation number

L/D

$4.0$

Ring speed

U_{\mathrm{ring}}

$0.000\,\mathrm{m\,s^{-1}}$

Core radius

a

$11.9\,\mathrm{mm}$

Leading circulation

\Gamma

$0.000\,\mathrm{m^2\,s^{-1}}$

Ring Reynolds number

\mathrm{Re}_{\Gamma}

$0$

Formation regimeRolling up

Vortex generator

Shape one finite stroke

Luminous ring coreHighlight the coherent toroidal core and its secondary ring.Entrained tracersShow analytic particles rotating through each toroidal core.Shear-layer wakeKeep the rolled-up filaments and excess-stroke jet visible.Velocity vectorsReveal poloidal rotation and axial self-induced motion.Axial analysis planeAdd a restrained center slice behind the three-dimensional structure.Nozzle diameter

D

0.14\,\mathrm{m}

Piston stroke ratio

L/D

4.0

Source circulation

\Gamma_0

0.17\,\mathrm{m^2\,s^{-1}}

Kinematic viscosity

\nu

1.8\times10^{-5}\,\mathrm{m^2\,s^{-1}}

Drag to orbit, scroll to zoom, and relaunch after changing the stroke.

This Lab combines the thin-core self-induced ring speed $U_{\mathrm{ring}}\approx\frac{\Gamma}{4\pi R}\left[\ln\left(\frac{8R}{a}\right)-\frac14\right]$ , viscous core spreading, and an empirical formation number near $L/D\approx4$ , then constructs a legible three-dimensional structure from analytic toroidal particles. It is not a full three-dimensional Navier–Stokes or turbulence solve; particles and wake geometry visualize the model rather than report particle-resolved CFD.

Physics tutorial

Vortex Rings: why a smoke ring propels itself

BackgroundA piston pushes a finite slug of fluid through a circular nozzle. The velocity jump at the rim creates an annular shear layer; that vorticity sheet rolls inward and closes into a toroidal vortex tube.

Why it mattersA vortex ring concentrates local rotation, circulation, entrainment, and coherent structure into one trackable object. Related mechanisms appear in jellyfish propulsion, ventricular jets, pulsed mixing, and industrial nozzles.

Start with the essentials

Focus question: Fluid inside the core mostly rotates around the torus, so why does the whole ring travel along the nozzle axis?
One-sentence intuition: Every segment of the vortex tube induces velocity. Adding contributions around the closed loop produces a shared axial velocity through the center, so the ring translates under its own circulation field.

Core mathematical model

Thin-core self-induced speed

U_{\mathrm{ring}}\approx\frac{\Gamma}{4\pi R}\left[\ln\left(\frac{8R}{a}\right)-\frac14\right]

Larger circulation $\Gamma$ speeds the ring up, while major radius $R$ , core radius $a$ , and the thin-core logarithmic correction set the remaining scale.

Viscous core spreading

a(t)\approx\sqrt{a_0^2+4\nu t}

Kinematic viscosity diffuses vorticity out of the thin core. The core thickens and self-induced speed gradually falls.

Formation number

F=\frac{L}{D},\qquad F_{\mathrm{pinch}}\approx4

The ratio of piston stroke to nozzle diameter controls whether the shear layer can roll into one leading ring. Beyond the empirical limit, excess stroke feeds a wake or secondary structure.

Circulation Reynolds number

\mathrm{Re}_{\Gamma}=\frac{\Gamma}{\nu}

This dimensionless ratio compares ring circulation with viscous diffusion. A larger value lets vorticity remain concentrated for longer.

Common difficulties

Particles are not tiny independent satellites

Typical misconceptionBecause luminous particles orbit the torus, each particle must be constrained to a prescribed circular track.

Better mental modelThe particles are a material illustration of the analytic ring structure, exposing rotation and entrainment through the toroidal core. This Lab does not solve a full three-dimensional particle velocity field.

A longer stroke does not strengthen one ring forever

Typical misconceptionContinuing to push the piston sends every increment of circulation into the same leading ring.

Better mental modelNear the formation limit, the leading ring pinches away from its feeding shear layer. Increasing $L/D$ beyond that point mainly lengthens the wake and can create a secondary ring.

Run the experiment

01
Form a compact ring first
Choose Compact ring, relaunch, and orbit the view around the core.
What to observe: A short stroke creates a weaker but clean leading ring with little persistent wake between the nozzle and the torus.
02
Approach the formation limit
Choose Formation limit, enable velocity vectors, and compare core rotation with axial translation.
What to observe: Local poloidal rotation coexists with centerline translation, while the leading ring captures nearly the maximum fraction of source circulation.
03
Create a pinch-off wake
Choose Pinch-off wake and wait for the formation stage to finish.
What to observe: The leading ring separates from its feeding shear layer. Excess stroke remains in a trailing jet and begins organizing into a smaller secondary structure.

Core mathematical model

Thin-core self-induced speed

Viscous core spreading

Formation number

Circulation Reynolds number

Common difficulties

Particles are not tiny independent satellites

A longer stroke does not strengthen one ring forever

Run the experiment

Form a compact ring first

Approach the formation limit

Create a pinch-off wake