Navier–Stokes

What is actually being simulated?

This is a numerical Burgers-type strained vortex: an axisymmetric reduction of the incompressible Navier–Stokes equations. A prescribed axial strain draws fluid inward; swirl evolves under transport and viscosity. The bright tracers are integrated through that computed velocity field. The tubes are instantaneous streamlines, not moving fluid parcels.

The solver advances angular momentum L = r uθ on 256 radial intervals using centered differences and Crank–Nicolson time stepping. Viscosity is 0.04 in nondimensional units. Strain smoothly increases from 0.2 to 2.4; radial inflow and axial outflow satisfy incompressibility. The radial boundary maintains circulation.

Readouts are measured from the computed field: maximum azimuthal speed, radius enclosing 63.2% of circulation, and estimated axial vorticity at the axis, each relative to its initial value. At fixed timeline positions, tracers explore the frozen velocity field.

Grid refinement is checked against the exact transient Gaussian-vortex solution. This validates this reduced model, not the OpenAI construction. The imposed strain extends beyond the viewing window and does not have finite whole-space energy. This simulation remains finite.

OpenAI’s proposed blowup requires different profiles, oscillatory corrections, and smooth forcing. Their shrinking-core scaling and bounded whole-flow energy are claims of that construction, not outcomes measured here.

See the illustrated version ↗
Numerical validation results ↗
Read the paper · §2 ↗