Fluid Flow Field - 3D Model

Starting the wind tunnel...

Presets

Vortex ring and wind

Free ring (self-propelled, advected by the flow)

Probe, view and look

Mid-plane speed slice (the 2D picture) Vortex filament Glass walls Auto-orbit camera

Live readouts

Graphs

On-axis velocity of the ring: numeric Biot-Savart (dots) vs closed form (line)

Tracer speed histogram

Ring radius (blue) and impulse / (pi a^2 Gamma) (orange) vs time

Frame time (ms), green line = 16.7 ms

Numerical checks

running...

How to use

The 2D page shows a counter-rotating vortex pair in a stream; here the same pair is the cross-section of a vortex ring. Tracers are carried by the flow of the ring plus the wind (the 2D picture is the orange slice). Pin the ring and watch the stream wrap around it, or untick the pin to let it self-propel; a tilted ring in a cross-wind curves away. Move the probe to read local speed, dynamic pressure and vorticity. Drag to orbit, wheel to zoom.

The science

A vortex filament of circulation Gamma induces the velocity given by the Biot-Savart law. For a closed ring discretised into segments dl_i with regularisation radius sigma (Rosenhead-Moore kernel):

u(x) = U + (Gamma / 4 pi) sum_i dl_i x r_i / (|r_i|^2 + sigma^2)^(3/2), r_i = x - x_i

This is the curl of a vector potential, so div u = 0 exactly. On the ring axis the continuous ring gives u(z) = Gamma a^2 / (2 (a^2 + z^2 + sigma^2)^(3/2)), and the circulation along the whole axis is Gamma a^2/(a^2 + sigma^2). A thin ring translates at about Gamma/(4 pi a) (ln(8a/sigma) - c); Kelvin's value c = 1/4 applies to a uniform-vorticity core, other smoothing kernels give a different constant, which the core-size sweep measures. The linear impulse of the ring, I = (Gamma/2) integral x cross dl = Gamma pi a^2 along the axis, is conserved by the ideal dynamics, which the free-ring mode lets you watch.

Tracers are integrated with a second-order midpoint (RK2) scheme; the check verifies its order of convergence. The dynamic pressure at the probe is 1/2 |u|^2 per unit density (Bernoulli for steady flow; the unsteady free ring is only quasi-steady). The ring is a discrete filament: the Crow-like azimuthal waviness you may see in free mode is partly numerical noise, and viscosity, core stretching and turbulence are absent. Fan speed is purely decorative (proportional to U). The wind tunnel, fan, frame and floor are modelled and baked in Blender 5.2.

What the checks mean

Divergence uses central differences of the live field at random points. On-axis profile compares the discrete sum with the closed form. Axis circulation integrates u along the axis from -60 to 60 (plus the analytic tails) and compares with Gamma a^2/(a^2+sigma^2). Impulse runs a clean free ring for 1.5 time units and tests conservation of the impulse. RK2 order halves the step on a tracer orbit and expects an error ratio near 4. The core-size sweep measures the free-ring speed for three core radii and the constant c it implies in the thin-core law (c drifts with sigma away from the thin-core limit).

Keyboard and accessibility

Space pauses, R resets, C re-runs the checks (while the page has focus).
All controls are native inputs with labels.