In 1911 a doctoral student named Karl Hiemenz was handed a simple job at Göttingen: measure the pressure around a cylinder in a steady stream of water. Ludwig Prandtl wanted numbers. Hiemenz kept getting a wobble.
He reported it. Prandtl said the cylinder probably wasn’t round. Hiemenz polished it with full German precision, put it back, got the wobble. Prandtl suggested the channel wasn’t symmetric; Hiemenz set about perfecting that too.
Prandtl’s graduate assistant, a Hungarian named Theodore von Kármán, took to asking each morning: Herr Hiemenz, is the flow steady now? For weeks the answer was the same. It always oscillates.
Nothing was wrong with either. Steady water, a rigid cylinder, nothing doing the shaking — and downstream, the flow keeps time.
Put anything blunt in a stream and the fluid gives up on the back half. It separates, rolls into a vortex on one side, lets go, then repeats on the other. Each vortex drifts off downstream and dies. What it leaves is a staggered double row — clockwise whirls in one line, counter-clockwise in the other, evenly spaced, receding like lamps down a street. The French call it the Bénard–Kármán street; Bénard photographed it first.
The street does not travel. Every vortex in it is leaving; the arrangement sits still.
What stands here is not a shape. It is a rate. In 1878 Vincenc Strouhal, measuring a whirled wire’s hum, found the pitch obeyed something almost embarrassing: frequency rises with speed, falls with thickness, and fD/U barely moves. He got about 0.185; Roshko refined it in a 1954 wind tunnel. For most blunt bodies at most speeds it sits near 0.2. The wake has a clock, and its rate is set by the body’s width and the stream’s speed, nothing else.
I promised last time that this lived in a band of speeds narrower than you’d guess. Half right; the wrong half is the interesting one. Below a Reynolds number near 47 the wake is steady — a fixed pair of eddies stuck to the cylinder’s back. At 47 that stops being stable and shedding switches on. Williamson’s map of the regimes gives the laminar street roughly 49 to 194, after which the vortex tubes buckle along their length and it goes three-dimensional.
But the shedding does not stop. It runs on past a hundred thousand and a million, over chimneys, cables and islands. The clouds trailing Alexander Selkirk Island are the same object a million times the size of one you can raise in a bucket with a pen. The tidy street is narrow-band. The clock is not.
Now the part I did not expect.
Kármán’s contribution, worked out over a weekend, was a stability calculation. He showed the symmetric arrangement — vortices in facing pairs — cannot survive, and the staggered one can, but only at one exact aspect ratio: row separation over spacing along a row, 0.28056. Measured streets sit close to it — among the tidiest results in fluid mechanics, and why his name is on it.
It is also a knife edge. The model is an infinite array of point vortices in a fluid with no viscosity, and at 0.281 what it yields is not stability but the absence of instability — neutral, and only against disturbances of vanishing size. Every other spacing is unstable outright. Domm, in the 1950s, found it holds for vanishing viscosity and vanishing time.
So the street does not stand because it is stable. It stands because it is being made, continuously, at the cylinder, on a metronome, and carried off at a steady speed. Things released at a fixed rate into a moving stream lay themselves out evenly whether or not the arrangement could survive alone. The evenness is a print, not a structure. Shut the tap and the street is gone.
Which sharpens something this publication had let slide. Several vortices have appeared here and every one stayed put: No. 21’s bathtub funnel, No. 10’s Great Red Spot, No. 33’s hexagon keeping six corners since Voyager. Those persist by not leaving. A street is the first made entirely of departures — the whirls disposable by design, the only thing with an address the gap between them.
Tacoma Narrows usually gets blamed on the street. The 1940 collapse is taught almost everywhere as vortex shedding driving a bridge at its resonant frequency; Billah and Scanlan checked in 1991 and found the torsional motion nowhere near it. It was flutter: the bridge’s own movement generating the forces that moved it further. Forced resonance and self-excitation are different animals, and this is the second — which is why Hiemenz could not polish it away.
Kármán first met the pattern in a fourteenth-century painting in a Bologna church — St Christopher fording a river with the infant Christ on his shoulder, eddies alternating off his legs. “The problem for historians may have been why Christopher was carrying Jesus through the water. For me it was why the vortices.” You can count them a long way downstream, and every one is on its way out. The row is there because something upstream has not stopped paying.
Roshko, late in a career spent on it, allowed that most of what we know about vortex streets “remains almost entirely in the empirical, descriptive realm of knowledge.” A hundred and fifteen years after a student could not get his water to hold still.
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One loop I’m watching
Next: a sound. Certain sand dunes, when their faces avalanche, give off a low note — loud, sustained, held for minutes — at a pitch that has almost nothing to do with how big the dune is. The grains making the sound are falling away from it the entire time it lasts. Next time.