Strouhal Number (Vortex Shedding Frequency)

Vincenz Strouhal was investigating why a wire sings in the wind. In 1878 he found that the tone rose with wind speed and fell with wire diameter in such a way that $fd/v$ stayed nearly constant — around 0.2 — over everything he could test. That constancy is the whole reason the group is useful. It means one number, measured once for a shape, predicts a shedding frequency at any size and any speed.

The mechanism is the Kármán vortex street. Flow separating from either side of a bluff body does not do so symmetrically; a vortex grows on one side, sheds, and its departure triggers one on the other. The wake becomes a staggered train of vortices, and the alternation drives a periodic side force on the body at the shedding frequency $f$.

The frequency doubling that catches people. The cross-flow force alternates at $f$ — one vortex from each side per cycle. The along-flow force alternates at $2f$, because the drag rises whenever either vortex sheds. A resonance check that compares only $f$ to the natural frequencies has missed half the problem, and the in-line response of a heat exchanger tube or a marine riser is a real failure mode with its own literature.

0.2 is a band, and it belongs to a shape. A smooth circular cylinder sits between roughly 0.18 and 0.22 for Reynolds numbers from about 300 to $2\times10^5$. Above that, in the critical range where the boundary layer transitions before separating, the wake loses its regularity and there is no single shedding frequency at all until it re-forms near $3.5\times10^6$. A square section runs near 0.12, a flat plate normal to the flow near 0.15. Roughness and free-stream turbulence move all of them.

And the length is a convention with teeth here. For a cylinder it is the diameter and everyone agrees. For a rectangular building or a bridge deck there are two obvious candidates — the across-wind width and the along-wind depth — and both are used in print. The same structure has two different Strouhal numbers depending on which the author picked, so a value quoted without its reference dimension cannot be transferred to your problem.

The reason any of this matters is lock-in, and lock-in is where the equation stops being true. Ordinarily the flow sets $f$ and the structure responds. But if the structure starts moving appreciably at its own natural frequency, its motion organises the shedding, and the shedding synchronises to the structure over a range of wind speeds — typically some tens of percent wide — instead of continuing to scale with $v$. Inside that window the response can grow far beyond what a forced-vibration calculation predicts. The Tacoma Narrows collapse was not simple vortex lock-in but a related aeroelastic instability, and the lesson generalises: once the body talks back to the flow, a fixed Strouhal number is no longer the right model.

The practical remedy usually attacks the correlation rather than the frequency. Helical strakes on a chimney, shrouds on a riser, staggered tube pitches in an exchanger — all of them break up the spanwise coherence of the shedding so that different stations shed out of step and the net force never builds. Stiffening the structure only moves the critical wind speed; spoiling the correlation removes the excitation.