The frequency a wake chooses

A wake sheds at a frequency, and making that frequency dimensionless is the obvious thing to do:

$$\mathrm{St} = \frac{f d}{U}.$$

The claim attached to it is the strongest collapse in bluff-body aerodynamics. Over four decades of Reynolds number a circular cylinder sheds at about 0.2, whatever the fluid, whatever the size, whatever the speed — which is why a vortex flowmeter is a piece of steel in a pipe with a pressure transducer behind it and no calibration curve worth speaking of.

The claim is also, read carefully, three separate claims. That the number does not depend on the Reynolds number. That it does not depend on the body. And that it is a property of the flow rather than of anything else in the apparatus. All three are worth measuring, and the third one fails hardest.

What the street settles exactly #

This collection already has the point-vortex street, and is explicit about what it does not contain: it is a model of a wake rather than a solution of one, and this site’s grid does not shed. What that model does settle, exactly, is the geometry.

A double row of vortices is neutrally stable at one width-to-spacing ratio and unstable at every other, and the condition is $\cosh(\pi h/a) = \sqrt 2$.

Bisection returns $h/a = 0.280549926$, and the closed form $\ln(1+\sqrt2)/\pi$ agrees to a part in 10¹⁶. Two more numbers follow and neither is approximate. Because $\cosh = \sqrt2$ forces $\sinh = 1$, the hyperbolic tangent at that ratio is exactly $1/\sqrt2$ — which drops straight into Kármán’s self-induced speed,

$$u = \frac{\Gamma}{2a}\tanh\frac{\pi h}{a} = \frac{\Gamma}{2\sqrt2\,a}.$$

So the street’s geometry and the street’s speed are one statement, and the shedding frequency follows from both: one vortex leaves each row per period, the pattern convects at $U - u$, and $f = (U-u)/a$. The Strouhal number formed with the body’s width is then $(d/a)(1 - u/U)$ — a ratio of two lengths multiplied by a ratio of two speeds, and the body enters only through the first of them.

That is the whole shape of what follows. If the number varies between bodies, the place to look is the length.

What the street’s numbers are worth as an estimate #

It is tempting to go one step further and compute a Strouhal number outright from the street model, and the attempt is worth making because of how it fails.

The model supplies $u/\Gamma a$ exactly and supplies nothing else. To get a frequency out of it two measurements have to be imported: the streamwise spacing $a$ in units of the body’s width, and the circulation of each vortex in units of $Ua$. With $a/d = 4.3$ and $\Gamma/Ua = 0.4$ — both taken from photographs of laminar streets rather than from any theory — the model returns $\mathrm{St} = 0.1997$, which is within a per cent and a half of the cylinder’s measured 0.212.

That agreement is a coincidence of the two imported numbers and not a prediction. Change $a/d$ by ten per cent and the answer changes by ten per cent, because $\mathrm{St} = (d/a)(1 - u/U)$ and the second factor is 0.859 whatever happens. The model contributes the 0.859 and the measurement contributes the 0.233, and the second is the larger and less certain half.

This is worth stating plainly because a number that comes out right for the wrong reason is the most durable kind of error in this subject, and the same trap sits behind every explanation of lift that gets the answer by arranging its inputs. What the street model settles is geometry. Frequency needs a length and a strength, and it has neither.

Three bodies, and a factor of one and a half #

A circular cylinder sheds at 0.212, a ninety-degree wedge at 0.200 and a flat plate normal to the flow at 0.145. That is a spread of 1.462 — half as big again — between three shapes in the same fluid at the same Reynolds number, and it is the reason a vortex flowmeter has to be calibrated for its own bluff-body geometry after all.

Roshko’s move was to change both quantities in the number. The wake behind a bluff body is bounded by free streamlines on which the speed is not $U$: Bernoulli along such a streamline from the free stream to the separation point gives

$$\frac{U'}{U} = \sqrt{1 - C_{pb}},$$

with $C_{pb}$ the measured base pressure — and that is computed here rather than quoted, from base pressures of −1.05, −1.10 and −1.66. And the width that matters is the wake’s, $d'$, rather than the body’s. The universal number is then $\mathrm{St}^* = \mathrm{St}\,(d'/d)\,/\,(U'/U)$.

It comes out at 0.16287, 0.16286 and 0.16270 — a spread of 1.001, one part in a thousand. The shedding was never body-dependent. The length in the number was.

The wake widths are Roshko’s and are not computed here. They come from his free-streamline calculation at the measured base pressure, and this collection’s own free-streamline machinery is not asked to reproduce them. What is demonstrated is that the collapse works; what is not demonstrated is that the widths were derived here.

A constant that is only constant high up #

The second claim — independence of the Reynolds number — is the one everybody makes without thinking, and it is measurably false over most of the range in which vortex streets get drawn.

Roshko’s own fits are $\mathrm{St} = 0.212 - 4.5/\mathrm{Re}$ below Re = 150 and $0.212 - 2.7/\mathrm{Re}$ from 300 to 2,000. At Re = 60 the first gives 0.137, which is thirty-five per cent below the asymptote. Solving for where the second is within one per cent of 0.212 gives Re = 1274.

So “St ≈ 0.2” is a statement about the top of that plot. Below about a thousand the number is a function of the Reynolds number, and the range 47 to 200 — where the street is laminar and periodic and where almost every photograph of one was taken — is precisely where the collapse is worst. That is the same lesson the fourteen-group survey drew: a group’s asymptotic value and its value in the range anybody works in are two different numbers.

What the collapse has no opinion about #

The third claim is the interesting one. A Strouhal number says the frequency is set by $U/d$. Let the body move, and it is not.

A wake is a self-excited oscillator: it has its own amplitude and its own frequency, and it does not need to be driven. Systems of that kind lock to a nearby driving frequency over a band, and inside the band they oscillate at the driver’s frequency rather than their own. That is not a property of fluids; it is a property of self-excited oscillators, which is exactly why the right model to reason with here is the smallest one that has the property and no fluid in it at all.

$$\ddot x - \varepsilon(1 - x^2)\dot x + x = A\cos\Omega t.$$

The plateau is lock-in. Inside it the oscillator has abandoned its own frequency, and every quantity a Strouhal number would predict is wrong: the shedding is at the frequency of whatever is moving the body, the phase is fixed rather than drifting, and — the part that matters structurally — the fluctuating force is coherent over the whole span instead of being decorrelated along it.

The band, measured #

Integrating the equation across a range of driving frequencies and counting zero crossings gives the width of that band directly.

Zero at zero amplitude, 0.01 at A = 0.05, 0.09 at 0.2 and 0.34 at 0.5. That is the Arnold tongue, and it is computed rather than quoted from perturbation theory — which matters here because the amplitudes of interest are not small.

The engineering consequence is the one that costs money. A cylinder free to vibrate is not a cylinder being driven from outside; it is a cylinder being driven by its own wake, and the coupled system locks over a range of flow speeds. Within that range the structure and the wake share a frequency, the oscillation amplitude grows until the damping catches it, and the fatigue life of the thing is decided by a number the Strouhal collapse does not contain. This is the same failure of a quasi-steady picture that Theodorsen’s function measures for a wing, arrived at from the other side.

The range where a street exists at all #

The Reynolds-number dependence has a lower end as well as a shape, and it is worth naming because the number’s failure below a thousand is often read as scatter.

Below about Re = 47 a cylinder’s wake is steady: there is a pair of standing eddies behind it, they grow with Reynolds number, and nothing sheds. The onset of shedding is a Hopf bifurcation of that steady wake rather than anything the street model can speak about — the model assumes a street already exists and asks which spacing survives. So the lower limit of the Strouhal number’s range is a linear-stability threshold in a different calculation entirely.

Between 47 and about 190 the street is laminar, two-dimensional and periodic, and that is the range in which the 4.5/Re correction is largest. Above 190 the wake goes three-dimensional, first in one mode and then another, and the frequency picks up a small discontinuity at each transition. Above about $2\times10^5$ the boundary layer on the cylinder becomes turbulent before it separates, the wake narrows, the base pressure recovers, and the Strouhal number jumps to around 0.45 — which is the drag crisis seen through the frequency instead of through the force.

So the range over which the number really is 0.212 ± 0.01 runs from about a thousand to about two hundred thousand: a little over two decades out of the six the collection draws cylinders across. That is a good collapse. It is not the four decades the folklore claims.

What the model is and is not #

The van der Pol equation is a caricature and this essay says so at every use of it. It contains no fluid: no Reynolds number, no separation, no vorticity, no wake width. Nothing in it can be used to predict a lock-in bandwidth for a real cylinder, and the numbers above are properties of that equation.

What it is good for is the shape of the answer. The existence of a band, the growth of the band with forcing amplitude, and the abandonment of the oscillator’s own frequency inside it are consequences of self-excitation rather than of the particular equation, and they hold for every model of that class. Reasoning with the smallest such model, and refusing to quote its numbers as physics, is the honest use of it — the same discipline this collection applies to the point-vortex street and to three equations that stopped describing convection.

Two limits of the street model are worth restating for the same reason. It has no viscosity, so it cannot say at what Reynolds number a street forms — that is Re ≈ 47, and it is a stability result about the steady wake rather than about the street. And its stability analysis is two-dimensional, whereas a real wake goes three-dimensional above Re ≈ 190 and the neat spacing ratio stops being observable long before the shedding does.

Two frequencies, not one #

Strouhal’s wire sang, and the sound is the fluctuating force the shedding puts on the body — which does not oscillate at one frequency but at two, for a reason that is pure bookkeeping.

Vortices leave alternately, one side then the other, so the transverse force reverses once per pair and oscillates at the shedding frequency itself. The streamwise force does not care which side a vortex came from: every shedding event pulls back on the body in the same direction, so the drag fluctuates at twice the shedding frequency, and at a much smaller amplitude.

A force spectrum therefore has a peak at $\mathrm{St}\,U/d$ and another at twice it, and which one an instrument sees depends on which component it is measuring — a useful diagnostic, and a common source of a factor of two in a reported Strouhal number.

Structurally it means there are two lock-in windows rather than one. The in-line response is excited at the doubled frequency, so it is reached at about half the flow speed of the cross-flow one, and it arrives first as the current builds. Its amplitude is the smaller of the two, which is exactly why it is the one that gets missed.

And the tone stays at the shedding frequency, because the transverse force is the efficient radiator.

What an engineer does with a number that moves #

The Strouhal number is used to answer one question — will this thing sing, or shake, at a speed it will actually see — and the two residuals change the answer in opposite directions, so it is worth saying which way each one cuts.

The length residual makes the raw number pessimistic in an unhelpful place. A bluff body that is not a cylinder does not have a Strouhal number near 0.21: a flat plate normal to the flow is at 0.14, and using 0.21 for it puts the predicted shedding frequency fifty per cent high. Roshko’s width fixes it when the wake width is known, and the wake width is exactly what a designer does not have for a novel shape. So the number is quoted per geometry, from tables, and the table is the residual written out.

The lock-in residual makes it optimistic everywhere. A structure whose natural frequency is placed safely away from $\mathrm{St}\,U/d$ is safe only if the wake declines to move. It does not: within the band the wake abandons its own frequency and drives the structure at the structure’s, which is precisely the case the calculation was performed to avoid. The design rule that follows is not a frequency separation at all — it is a limit on the response amplitude, or a helical strake that destroys the correlation along the span so no coherent street forms.

That is why chimneys and marine risers carry strakes and cables carry dampers. The number told everyone where the danger was; the residual is why the fix is not to move away from it.

Where the length came from #

Vincenz Strouhal measured the tone of a wire in a moving airstream in 1878 and found the frequency proportional to speed over diameter — the number is named for that experiment rather than for any theory. Bénard photographed the alternating rows in 1908, and Kármán analysed their stability in 1911 and 1912, which is where the $\ln(1+\sqrt2)/\pi$ comes from.

Roshko’s 1954 report is the one that matters for this essay. His argument was not that the measurements disagreed but that they had been made dimensionless with the wrong length: the frequency belongs to the wake, so the wake’s width and the wake’s velocity are what it should be scaled with. His universal number of 0.163 is what the three bodies here collapse to.

There is a second reading of the same work that is worth separating out. Roshko’s collapse uses the base pressure, and the base pressure is the hardest thing about a bluff body to predict — it is the one quantity that free-streamline theory gets badly wrong on its own, and the whole notched-hodograph apparatus exists to import a measured value of it. So the universal Strouhal number is universal in the sense that it removes one unknown by importing another, and the imported one is a measurement.

That is not a criticism. It is the reason the collapse is a statement about scales rather than a theory of shedding: given the base pressure and the wake width, the frequency follows; predicting either of those from the shape is a separate and much harder problem this collection does not solve.

That is a general moral about dimensionless groups and this collection has met it twice before. A group that fails to collapse is not evidence that the physics varies. It is evidence about the choice of scales, and the first place to look is the length — which is what the Reynolds number’s own length turns out to be about, and what dynamic similarity is unable to fix when two groups demand contradictory ones.

What this leaves #

Two residuals, then, and they are different in kind. The first is a length, and Roshko removed it: with the wake’s own width the number really is nearly universal. The second is the body’s own motion, and nothing removes it, because a self-excited oscillator locked to a driver is not doing the thing the number describes.

The next number along has the same shape and a sharper edge: the capillary number, which collapses drop deformation beautifully and cannot say whether the drop breaks.