Vortex shedding
A Socratic walk-through of vortex shedding — reasoned out one step at a time, not lectured.
The question we started with
THE QUESTION #Why does a steady, tuneless wind make a wire or a mast sing at one definite pitch?
Wind over a fence wire produces a note. Not a rush, not a hiss — a pitch, steady enough to hum along with. Yet the wind carries no rhythm of its own, and the wire is not being plucked. Where does a frequency come from when neither the driver nor the driven object supplies one?
The tempting answer is that the wire is resonating, the way a guitar string does. That answer has a test attached, and it fails it: change the wind speed and the note changes, whereas a string's pitch does not care how hard you pluck it. So the periodicity is manufactured somewhere in the airflow, and the wire only broadcasts it.
Reasoning it through
REASONING #Before asking how, ask what the frequency could possibly depend on — a question dimensional analysis answers without any fluid mechanics at all.
The ingredients are the wind speed U, in metres per second; the diameter D, in metres; and the air's density and viscosity. We want a frequency, in units of one per second. From U and D alone there is exactly one combination with those units: U/D. Any correct answer therefore has the form
f = St × U / D
where St is a pure number. That is not a model but an inevitability — the only freedom nature retains is what St is, and how it depends on the one remaining dimensionless group, the Reynolds number Re = UD/ν. Measurement supplies the answer: for a circular cylinder St sits near 0.2 and stays remarkably flat from Reynolds numbers of a few hundred to a couple of hundred thousand. That 0.2 is recalled and empirical, but the form of the law is derived.
Now check it against what you hear. A 2 mm fence wire in a 10 m/s breeze: f = 0.2 × 10 / 0.002 = 1000 Hz, a clear high whistle. A 100 mm mast in the same wind: f = 0.2 × 10 / 0.1 = 20 Hz — a felt throb rather than a note. The formula gets both the pitch and the character right, from geometry alone.
So what makes the flow periodic? Follow the air round the cylinder. It accelerates over the shoulders, where pressure drops, then must decelerate towards the rear, where pressure rises again. The fluid in the thin boundary layer has already lost energy to friction and cannot climb that rising pressure, so it stalls and the flow separates, leaving two shear layers trailing off the shoulders into a low-pressure wake.
Those two shear layers do not settle down side by side. The standard account — and the clean version of it is a stability calculation rather than a hand argument — is that the symmetric arrangement is unstable above a Reynolds number of about 47. Once one side's roll-up grows slightly larger, its circulation reaches across the wake and draws the opposite shear layer over the centreline, cutting off the supply feeding the first vortex. That vortex, no longer fed, detaches and floats downstream, and the process repeats on the other side. Alternation is forced: the growth of one vortex is what strangles it.
Each shed vortex is a low-pressure core hugging one side, so it pulls the cylinder sideways — and since they alternate, the side force alternates too. That is your oscillator. The wire is not vibrating at its own frequency; it is pushed back and forth by the wake it is making.
Which gives the account a sharp test. The claim is that the frequency belongs to the flow, so it must be proportional to wind speed and inversely proportional to diameter. Double the wind and the note should rise an octave; double the wire and it should fall one. Both are readily observed on a windy day with a set of wires, and any tone that stayed put as the wind freshened would refute the whole story.
Except that sometimes it does stay put, which is the most interesting part of the subject.
The analogy
THE ANALOGY #Think of a flag rope slapping against a pole. Nobody is beating time, yet a rhythm establishes itself, because each slap sets up the condition that produces the next. The tempo belongs to neither rope nor wind but to the interaction — feed in a stronger wind and it rises.
the rope's rhythm requires it to strike something, whereas the cylinder's wake keeps its rhythm perfectly well if the cylinder is rigidly clamped and never moves at all — which is precisely why the frequency can be predicted before knowing anything about the structure's own dynamics.
Clarifying the model
THE MODEL #Three refinements make this honest.
The first is the exception that proves the mechanism. If the shedding frequency approaches the structure's own natural frequency, the two lock together: the structure's motion begins to organise the shedding, and the wake synchronises to the structure rather than the other way round. During this lock-in the frequency stops obeying St·U/D and stays pinned over a band of wind speeds while the amplitude grows dangerously. So the proportionality law is true of a wake left to itself and false exactly where an engineer cares most. This is not the famous Tacoma Narrows failure, generally attributed to torsional flutter, a different aeroelastic instability; conflating the two is a common slip.
The second concerns that flat Strouhal number. Its constancy makes the singing wire reliable, but it is a plateau, not a law: below Re of about 47 there is no shedding at all, only a pair of standing eddies, and above roughly 3 × 10⁵ the boundary layer turns turbulent before it separates, the separation point jumps rearward, and the shedding becomes broadband. The tone belongs to the middle of the range.
The third is a caution about "one definite pitch". A long wire does not see the same wind speed along its whole length, so different spans shed at slightly different rates and the note is a narrow band rather than a line. What sharpens it is usually the structure itself, responding preferentially near its own resonance — so the wire colours the sound even though it did not choose the pitch. This is also why the standard engineering fix is not stiffness but disruption: the helical strakes wound round chimneys work by keeping the shedding from staying correlated along the length, so no single frequency ever builds up.
A picture of it
THE PICTURE #How to readThis is a timeline repurposed — the axis is Reynolds number, not time, so read left to right as the wind speeding up or the cylinder growing, not as anything happening in sequence. Each stage names the flow's condition and then what the wake does in it. The band you can hear is the fourth one, where the Strouhal number holds near 0.2 and the shedding is periodic enough to make a note; the singing stops at both ends of the chart, for opposite reasons.
What became clearer
WHAT CLEARED #The pitch was never in the wind and never in the wire. It is manufactured by the wake: a bluff body cannot keep the flow attached, the two shear layers it sheds cannot coexist symmetrically, and each vortex's growth is what terminates it — so the flow keeps its own time. Dimensional analysis fixes the form of the answer before any physics is done, since U/D is the only frequency the situation can build, and measurement supplies the single number 0.2 that turns it into a prediction. What the structure contributes is amplification — except in lock-in, where it seizes control of the frequency altogether.
Where to go next
ONWARD #- How the same wake periodicity is turned to use in a vortex flowmeter, where counting the shedding rate measures the flow speed.
Key terms
TERMS #| Term | What it means |
|---|---|
| Strouhal number | the group St = fD/U, near 0.2 for a circular cylinder over a wide range of Reynolds number. |
| Lock-in | the regime in which a structure's own vibration captures the shedding frequency, holding it fixed over a band of wind speeds. |
Every term the collection defines is gathered in the glossary.