Fluid dynamics · three-dimensional filament studio

Vortex Rings

Two already-formed vortex rings shrink, expand, and exchange axial order in their mutually induced velocity field. Orbit the numerical filaments and passive tracers, then connect the motion to position and radius histories, invariant drift, and an independent single-ring speed baseline.

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, not a prescribed animation. Its ring-speed reference is not applicable; retain the impulse, energy, curvature, and core-gap audits. The model stops at core contact instead of reconnecting the knot.

Choose Single ring, then refine segments. Its reference integrates a smooth circle with the same core kernel; it is not a thin-core formula with a different core convention.

The solver pauses at $d_{\min}<2a$ or unresolved bending. No reconnection or viscous collision is inferred.

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.

Inviscid regularized filaments: every segment contributes through the Biot–Savart kernel, with fourth-order Runge–Kutta stepping. The fixed core scale is numerical regularization, not solved smoke density or viscous spreading. The model pauses at core contact or insufficient resolution; it does not simulate nozzle formation, viscosity, turbulence, or reconnection.

Physics tutorial

Why vortex rings leapfrog

BackgroundA smoke ring contains a closed vortex tube. This experiment begins with formed rings, not nozzle roll-up. Each centerline is a closed three-dimensional polygon; every segment induces velocity at all other nodes and passive tracers.

Why it mattersTwo co-directed rings are neither rigid objects nor independent circular tracks. The rear ring can shrink and accelerate while the front expands and slows, then exchange roles. The three-dimensional view, position plot, and radius plot must explain 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 exchanges axial order, not filament topology. Mutual induction changes each ring’s radius. Closed filaments are never artificially reconnected in this model.

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}}

Interpolate nodes with periodic cubic segments, integrate the Rosenhead–Moore kernel with three-point Gauss quadrature per segment, then move nodes with fourth-order Runge–Kutta. The fixed scale $a$ is regularization, 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

Integrate independently over a smooth circle, rather than the solver’s polygon. Refinement should bring single-ring speed toward this same-kernel result. This tests discretization consistency, not experimental fluid accuracy.

Closed-filament impulse

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

Report the change in the impulse vector normalized by the sum of the initial individual-ring impulse scales. Opposing rings may have zero total impulse, so dividing by initial total impulse would be invalid.

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}}

Monitor energy with three-point Gauss double quadrature on segments. Fixed node count, polygon geometry, and time stepping all introduce error; drift during strong deformation is not physical viscous dissipation.

Common difficulties

A core tube is not smoke density

Typical misconceptionBrighter or thicker tubes indicate numerically resolved denser smoke.

Better mental modelTube thickness marks regularization; colors identify rings. Passive particles follow the computed velocity field without feeding back on it or resolving compressible smoke density. Hiding particles does not change filament dynamics.

Opposing rings do not imply a reconnection simulation

Typical misconceptionThese rings must eventually merge, annihilate, or reconnect into smaller rings here.

Better mental modelAs cores approach, the thin-filament description becomes unreliable. The experiment pauses explicitly. Reconnection needs an additional viscous or quantum-fluid model, not an invented visual transition.

Run the experiment

01
Match three readings of one exchange
Choose Leapfrogging and play or advance. Ring A is teal; ring B is amber.
What to observe: When axial curves cross, the order counter increases and the smaller ring passes through the larger ring. Disable center-following to observe overall translation.
02
Audit segments, not smooth-looking graphics
Choose Single ring, compare coarse and fine segment counts, and reveal the numerical nodes.
What to observe: The gap between polygon speed and smooth-circle speed decreases. Reducing core scale without refining can increase error instead.
03
Find where the model should stop explaining
Compare Opposing rings with Wavy ring while keeping other parameters fixed.
What to observe: Opposing rings approach and expand; a wavy seed changes local curvature. Core contact or unresolved bending pauses the model with a reason, without invented reconnection.

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