Fluid dynamics · three-dimensional filament studio

Vortex Rings

Why does the rear vortex ring shrink and slip through the front one? The two rings trade size and speed. The front ring's flow pulls the rear ring inward, so it shrinks and speeds up; the rear ring's flow pushes the front ring outward, so it widens and slows. The narrower, faster rear ring then slips through the front one. Throughout, the vortex cores never cut or reconnect.

Two vortex rings that have already formed shrink, expand and swap places in the velocity field they induce in each other. Orbit the computed filaments and passive tracers, then compare the motion with the position and radius plots, the drift of conserved quantities and an independent speed baseline for a single ring.

Simulation time $0.00\,\mathrm{s}$

Ring A speed

U_A

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

Order exchanges $0$

Normalized impulse drift $0.000\%$

Energy-quadrature drift $0.00\%$

Nonlocal gap

d_{\min}/a

$0.00$

Maximum curvature

\kappa_{\max}R_0

$1.00$

Advance + tracer cost $0.0\,\mathrm{ms}$

Model boundaryInviscid · fixed regularization

A controlled interaction

Let the rings move each other

Material pathsPassive particles advected in the computed velocity field, not prescribed circular orbits.Axial slice vectorsSample the same three-dimensional field through the ring centers.Numerical nodesReveal the polygon underlying the displayed core tube.Follow the ring centerKeep the interaction in view; disable to observe absolute translation.Initial ring radius

R_0

0.16\,\mathrm{m}

Circulation magnitude

|\Gamma|

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

Initial axial separation

d_0/R_0

0.70

Regularization scale

a/R_0

0.09

Segments per ring

N

64

Four-wave seed amplitude

\varepsilon/R_0

0.06

Playback rate

0.45

The trefoil is a closed, knotted filament evolved by the same Biot–Savart solver as the rings. The ring-speed reference does not apply to it, so watch the impulse, energy, curvature and core-gap checks instead. The model stops when two cores touch; it does not reconnect the knot.

Choose Single ring, then increase the segments per ring. Its reference integrates a smooth circle with the same core kernel, instead of using a thin-core formula with a different core convention.

The solver pauses when $d_{\min}<2a$ or when the bending is too sharp for the segments to resolve. It does not infer reconnection or a viscous collision.

Conceptual inspiration: Bugman123 Fluid Motion. Softened-filament model: Rosenhead–Moore; see Ruban, Physics of Fluids 30, 084104 (2018), nonlocal term. This is not that paper’s quantum-core model.

The rings are inviscid, regularized filaments. Every segment contributes to the velocity through the Biot–Savart kernel, and fourth-order Runge–Kutta advances them in time. The fixed core scale is a numerical regularization; the model does not solve for smoke density or for viscous spreading of the core. It pauses when the cores touch or the resolution is insufficient. It does not simulate ring formation at a nozzle, viscosity, turbulence or reconnection.

Physics tutorial

Why vortex rings leapfrog

BackgroundAt the heart of a smoke ring is a closed vortex tube. This experiment starts with rings that have already formed; it does not simulate the roll-up at a nozzle. Each ring centerline is a closed three-dimensional polygon, and every segment induces a velocity at all the other nodes and at the passive tracers.

Why it mattersTwo rings moving the same way are not rigid objects, and they do not follow separate circular tracks. The rear ring can shrink and speed up while the front ring expands and slows down, and then they swap roles. The three-dimensional view, the position plot and the radius plot should all describe the same exchange.

Start with the essentials

Focus question: When the rear ring passes through the front ring, do their cores cut each other, or do their radii and speeds exchange?
One-sentence intuition: Leapfrogging swaps which ring is in front; it does not change how the filaments are connected. Each ring changes radius in the velocity field induced by the other, and this model never reconnects the closed filaments artificially.

Core mathematical model

Regularized Biot–Savart velocity

\mathbf u(\mathbf x)=\sum_j\frac{\Gamma_j}{4\pi}\oint_{C_j}\frac{d\mathbf X\times(\mathbf x-\mathbf X)}{\left(|\mathbf x-\mathbf X|^2+a^2\right)^{3/2}}

The nodes are joined by periodic cubic curves. On each segment, three-point Gauss quadrature integrates the Rosenhead–Moore kernel, and fourth-order Runge–Kutta then moves the nodes. The fixed scale $a$ is a regularization length, not a resolved viscous core radius.

Independent smooth-circle speed baseline

U_{\rm ref}=\frac{\Gamma}{4\pi}\int_0^{2\pi}\frac{R^2(1-\cos\theta)}{\left[2R^2(1-\cos\theta)+a^2\right]^{3/2}}\,d\theta

This baseline is integrated over a smooth circle, not over the solver polygon. With more segments, the single-ring speed should approach this result, which uses the same kernel. It tests whether the discretization is consistent; it does not show agreement with real fluid experiments.

Closed-filament impulse

\frac{\mathbf I}{\rho}=\frac12\sum_j\Gamma_j\oint_{C_j}\mathbf X\times d\mathbf X

The readout shows the change in the impulse vector, divided by the sum of the initial impulse scales of the individual rings. Two opposing rings can have zero total impulse, so dividing by the initial total impulse would not work.

Same-kernel energy integral

\frac{H}{\rho}=\frac{1}{8\pi}\sum_{i,j}\Gamma_i\Gamma_j\oint\!\oint\frac{d\mathbf X_i\cdot d\mathbf X_j}{\sqrt{|\mathbf X_i-\mathbf X_j|^2+a^2}}

Energy is monitored with three-point Gauss double quadrature over the segments. The fixed number of nodes, the polygon shape and the time stepping all add error; drift during strong deformation is not real viscous dissipation.

Common difficulties

A core tube is not smoke density

Typical misconceptionA brighter or thicker tube means the simulation found denser smoke there.

Better mental modelTube thickness only marks the regularization scale, and the colors tell the rings apart. Passive particles move with the computed velocity field, but they do not act back on the flow and do not represent compressible smoke density. Hiding them does not change how the filaments move.

Approaching rings are not a reconnection simulation

Typical misconceptionIn this model, two opposing rings must eventually merge, annihilate or reconnect into smaller rings.

Better mental modelOnce the cores get close, the thin-filament description is no longer reliable, and the experiment pauses and says so. Reconnection needs a separate viscous or quantum-fluid model; an animation cannot stand in for it.

Run the experiment

01
See one exchange three ways
Choose Leapfrogging and play or advance. Ring A is teal; ring B is amber.
What to observe: When the two axial-position curves cross, the exchange count goes up, and the smaller ring is passing through the larger one. Turn off Follow the ring center to see the overall translation.
02
Check the segment count, not just the smooth picture
Choose Single ring, compare coarse and fine segment counts, and reveal the numerical nodes.
What to observe: The polygon speed moves closer to the smooth-circle speed. Shrinking the core scale without adding segments can make the error larger instead.
03
Find where the model stops being valid
Compare Opposing rings with Wavy ring while keeping other parameters fixed.
What to observe: Opposing rings approach each other and expand; a wavy start changes the local curvature. When the cores touch or the bending is too sharp for the segments, the model pauses and gives the reason. It does not draw an imagined reconnection.

Conceptual inspiration: FluidMotion. Original simulations and graphics; the model limits are described above.