Curl Noise Particle Field

Lighting the gallery...

Presets

Noise field

Obstacle

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Colour by

Cube frame and floor Auto-orbit camera

Live readouts

Graphs

Speed distribution of the field (sampled at 600 random points)

Kinetic energy (orange), enstrophy (blue), helicity (green) of the field vs time

Flow on the obstacle surface along a great circle: tangential speed (orange) and normal speed (blue, must be 0)

Frame time (ms), green line = 16.7 ms

Numerical checks

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How to use

Every glowing streak is a particle carried by a velocity field that is, by construction, divergence-free: it is the curl of a vector potential made of travelling waves, so particles neither clump nor spread out (an incompressible fluid). Raise the noise frequency for smaller eddies, change the evolution rate to let the field slowly morph, and pick an obstacle: the potential is ramped to zero near its surface (Bridson's method), so the flow slides around the sculpture without ever entering it. Drag to orbit, wheel to zoom.

The science

Curl noise (Bridson, Houriham and Koenig, 2007) builds a velocity field from a vector potential A: u = curl A. Because div(curl A) = 0 identically, the flow is incompressible. Here A is a sum of N = 8 plane waves, each with wave vector k_i, a unit polarisation e_i perpendicular to k_i and a phase speed w_i:

A(x,t) = sum_i a_i e_i sin(f k_i . x + phi_i + w_i t), u0 = curl A = sum_i a_i f (k_i x e_i) cos(f k_i . x + phi_i + w_i t)

The derivatives are analytic. To respect a solid obstacle with signed distance d(x), the potential is multiplied by a smooth ramp r(d) that is 0 on the surface and 1 beyond a distance delta: u = curl(r A) = r u0 + r'(d) grad d x A. On the surface r = 0, so u = r'(d) grad d x A is perpendicular to the surface normal: no flow through the wall, and the divergence is still exactly zero. With the smoothstep ramp r'(0) = 0 as well, so the velocity vanishes on the wall (a no-slip wall) and grows away from it. 2D curl noise is the special case A = (0,0,psi), u = (dpsi/dy, -dpsi/dx) used by the 2D version.

A flow with div u = 0 preserves volume: the Jacobian determinant of the flow map is exactly 1 (Liouville), which the check measures. Particles are advanced by a second-order midpoint scheme in a fragment shader (position and age in a float texture, up to 262,144 particles), and each is drawn as a streak from x to x - trail u. Respawn is random inside the cube when a particle leaves, ages out or touches the obstacle. Simplifications: the field is a kinematic model, not a solution of the Navier-Stokes equations; the sculpture shapes are analytic signed-distance fields, matching the Blender meshes; the cube is a visual frame only. The gallery, frame, torus and sphere were modelled and baked in Blender 5.2.

What the checks mean

Divergence uses central differences with the obstacle ramp active. Curl compares the analytic velocity with a finite-difference curl of the ramped potential. No penetration samples points on the obstacle surface and tests u dot n = 0. Volume preservation computes the flow-map Jacobian by finite differences of RK4 trajectories. GPU vs CPU runs 64 particles for 100 steps through the real fragment shader and compares with a double-precision CPU integration of the same field. The sweep checks whether trajectories ever enter the obstacle for several boundary-layer widths.

Keyboard and accessibility

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