Shear

the claim

Helmholtz in 1868 and Kelvin in 1871 worked out what happens where two fluids slide past each other: nothing, if the boundary is perfectly flat, and everything otherwise. Flat is a solution, but an unstable one, which makes it a solution nothing can actually be in. Every ripple grows. The layer chooses which ripple to grow and the choice is sharp: ripples finer than the layer's own thickness are ignored entirely, and of the rest the fastest has a wavelength of about seven times the layer's vorticity thickness.

This page solves the two-dimensional incompressible Euler equations in vorticity form to get there. Nothing about spiral spacing is in the code. Before any of it was drawn, five gates were written down in the project notes and then run headless: the Poisson inversion round trips to four parts in a thousand trillion; an undisturbed layer sits still instead of inventing its own instability; ripples above the cutoff decay while ripples below it grow; the fastest mode grows at 0.0932 against a predicted 0.0949, which is under two percent out; and a layer given nothing but broadband noise picked a spacing of 16.36 thicknesses against a predicted 14.13.

That last one is the interesting failure. Against the thickness the layer started with it missed by 21 percent, and it missed long, in exactly the direction the notes had said an error would have to go if it came from the scheme's own smearing. Chasing it turned up something better than a fix: a boundary spreads while it waits, and the instability selects for the thickness it has at the moment it commits, not the one it was born with. Measured against its real thickness the miss falls to 16 percent, one Fourier mode away from the predicted one, and still long.

what is solved and what is chosen

Solved: incompressibility, exactly, because this formulation has no way to write a compressible flow down. The transport of vorticity. The recovery of the entire velocity field from its spin, by a Fourier transform along the stream and a tridiagonal solve across it, which is an inversion and not an iteration stopped when it looked good enough. The instability. The spacing. The roll-up. The way neighbouring spirals lean into each other and merge. Your finger, which enters as a vortex dipole, because that is what pushing incompressible fluid is.

Chosen, and worth saying plainly. A closed layer mixes itself out and stops, so this one is not closed: it is a river. A narrow strip at the far right, off the edge of what you are shown, scrubs the water back to its pristine state before it wraps around to the left, which is the standard way to run a flow that develops along its length inside a box that repeats. The layer is tapped where it enters, very slightly, the way an experiment taps the plate that separates the two streams. And the picture is the magnitude of the dye gradient, so what is lit is the boundary and only the boundary: unmixed water has no gradient and stays dark, and mixing puts the light out.

Everything between the inlet and that strip is the equation's business, and it does not stop. The layer has now been measured running the same way for hundreds of times longer than it takes one spiral to form, and it neither settles down nor fills the frame with fog. It arrives straight and it leaves in pieces, continuously, which is the only steady state a boundary like this one has.

Method: 2D incompressible Euler in vorticity-streamfunction form on a 256 by 96 grid, periodic along the stream and walled across it; second-order semi-Lagrangian transport; spectral-tridiagonal Poisson inversion; explicit viscosity at Reynolds 556, with the dye diffusing at two thirds of that. The pre-registration, the solver and the five gates are in the repository: shear.ts, shear-probe.ts, and shear-preregistration-day211.md.