A street's spacing and speed state the drag of the body that shed it

The street a steady calculation cannot draw built the Kármán street as a model of point vortices and bisected the ratio of width to spacing at which the staggered arrangement is neutrally stable: $h/a=0.28055$, against an exact $\ln(1+\sqrt2)/\pi$. It then recorded a list of what the model doesn’t contain, and the entry that matters here reads no drag. The connection between the street and the drag on a cylinder, it said, is real and is not in here.

It can be put in, and with very little effort, since Kármán himself did it. In 1912, in the second of his short notes on the street, written with Rubach, he took the argument to a further question: what does the street say about the force on the body? The result deserves far better than the footnote it usually gets. It treats a wake as a set of numbers that can be read from a photograph, and it returns a drag with no surface pressure in it and no account of what body made the wake.

A wake that keeps the drag and forgets the body took the same point at the far end, where only the total deficit survives. This essay takes it at the near end, where a wake has structure and each part of the structure counts.

The balance #

Hold a body at rest in a stream $U$, with a street forming behind it. The vortices have strength $\Gamma$, spacing $a$ along each row, and rows $h$ apart. An isolated street moves through the fluid at a speed fixed by its own induced velocity, in the direction opposite to the stream behind a body:

$$u_s=\frac{\Gamma}{2a}\tanh\frac{\pi h}{a}.$$

So the street travels downstream at $U-u_s$, and one new pair of vortices appears per period, so that the shedding frequency is $f=(U-u_s)/a$. The momentum in a long control volume round the body has two contributions at each pair’s arrival. The first is the impulse a pair of vortices carries, $\rho\Gamma h$ per pair. The second is a pressure term, from the way a non-steady flow with moving vortices has a varying pressure that does not cancel at the volume’s ends. Kármán’s result, for unit span, is

$$D = \rho\Gamma\frac{h}{a}\,(U-2u_s)+\frac{\rho\Gamma^2}{2\pi a}.$$

With $h/a$ at the stable 0.281, and in terms of the street speed $w=u_s/U$, it condenses to the published form:

$$C_D=\frac{2h}{d}\,(2.83\,w-1.12\,w^2),$$

with $d$ the body’s width. There’s a quantity needed that the street doesn’t itself give, $d$. It enters through the Strouhal number, which is measured: at $St=fd/U$,

$$\frac{a}{d}=\frac{1-w}{St}.$$

So at a Strouhal number of 0.21 and a street speed of 0.14, the spacing is 4.1 body widths, the width 1.15, the strength $1.62\,Ud$, and the drag coefficient 0.86. Kármán and Rubach compared such numbers with the measured drag on a cylinder and a plate and found them of the right size, and that’s a remarkable agreement, since nothing in the calculation knows that a pressure on a rounded surface has separated into a base suction. Everything is taken from far behind.

Why a control volume need not ask about the body #

There’s a real conceptual point behind the formula. D’Alembert’s paradox says a steady, inviscid flow has no drag, and the proof goes by drawing a control volume round the body and finding nothing passing through it with any net momentum. The street’s balance is the same control volume. What’s changed is that something does pass through: a train of pairs of vortices, each with impulse, in an otherwise ideal outer flow. So the proof of no drag and the formula for the drag are the same calculation, carried out with and without a wake. Nothing about viscosity appears in the formula, although it’s the source of the street, because viscosity’s job is finished by the time the vorticity has left the body. Where the flow lets go is what decides that a street forms and how strong it is, and from then on the balance doesn’t care.

This parallels the span of a lifting wing. There, too, the drag can be read from the far wake, the trailing vortices’ geometry alone, and there too the surface pressure gives the same answer by a very different integral. A wing’s induced drag has the kinetic energy of its wake as a clean equivalent; a street’s drag doesn’t, since the point vortices have infinite self-energy and its own energy account needs a core model. Its momentum account needs none.

A chimney as a worked case #

Consider a chimney a metre across in a wind of twenty metres a second, with a street speed of 0.14 of the wind. The shedding frequency is $0.21\times20/1=4.2$ hertz. The vortices have spacing 4.1 metres and rows 1.15 metres apart, and each has a circulation of $1.62\times20\times1$, about 32 square metres a second. The first term gives $\rho\Gamma(h/a)(U-2u_s)$, which is $1.2\times32\times0.281\times14.4$, or about 155 newtons per metre of height, and the second, $\rho\Gamma^2/2\pi a$, gives about 48. So the total is 203 newtons a metre, which is the 0.86 quoted: $\frac12\times1.2\times400\times0.86$. A chimney’s designer would use a drag coefficient closer to 1.2 or less, depending on the Reynolds number; here the point is what kind of quantity this is. A camera, a frequency and a pair of probes have delivered a force that nobody measured.

The sideways force is another matter. The same frequency drives a fluctuating lift at 4.2 hertz, whose size depends on the circulation shed at each side and on how well it’s correlated along the height. That is what shakes a chimney, and a street’s spacing and speed say nothing about that correlation: the calculation assumes a street that’s perfectly two-dimensional. So a real structure designer needs the frequency a wake chooses from this argument and a correlation length from a measurement.

Where the drag sits #

Most of the drag is the first term. A vortex pair is an impulse, and each new pair deposits one per period. That is what the essay on what a pair carries meant by treating the momentum deposited as a statement about impulse: the rate of creation of impulse is the force. Its factor $(U-2u_s)$ has a clean meaning. The vortices are carried downstream at $U-u_s$, and the impulse each pair has is measured in a frame that has to be chosen, and the balance over a fixed volume, with fluid passing through at $U$, includes that bookkeeping.

The second term is a pressure correction, and it goes as the square of the strength. It’s small at low street speeds and catches up by a street speed of about 0.3. Its presence is a warning against thinking of a wake survey by the steady momentum theorem. A pitot rake averaged over time in the wake’s near part misses that term. It’s a part of the force that doesn’t appear in the mean velocity deficit at a stated station, and is why wake surveys need to be taken well downstream, where the fluctuations have died, to give a good drag.

The next thing to ask is how much the answer cares about $h/a$, since nobody expects the real street’s ratio to be exactly stable.

It cares very little, because it’s a ratio that’s already been absorbed in a strength: with the frequency and street speed fixed, a wider street has a smaller $\Gamma$ for the same speed and each pair’s impulse $\rho\Gamma h$ stays about the same. So the argument doesn’t depend on the stability calculation at all. Real wakes run from about 0.25 at formation to about 0.35 or more farther downstream, and the result is unaffected.

The sensitivity lies elsewhere. Two measured inputs matter: the street’s speed and its frequency. A change of street speed from 0.14 to 0.2 raises the coefficient by 30 per cent. Measuring $u_s$ is hard, since it’s a few per cent of $U$ in some wakes and it varies by a good deal in the first few diameters. Measurement of the Strouhal number is easy. So Kármán’s formula has a useful practical form: read a shedding frequency from a hot wire, read the convection speed from two hot wires a known distance apart, and the drag follows.

The strength that does not survive #

A very striking inference is possible by working the argument the other way round. The vorticity the street carries comes from the boundary layers on the body, and these have a known rate of supply. A separated shear layer with a speed $U_s$ outside it and nothing much on the inside carries circulation downstream at a rate $\frac12U_s^2$ per unit time. It’s simply the mean velocity times the vorticity integrated across the layer. In one shedding period each layer feeds $\frac12U_s^2/f$, and that’s the strength available to one vortex.

For a cylinder with the separated speed about 1.3 times the stream’s, each layer feeds a little over $4\,Ud$ per period, and a street with a strength of 1.6 to 2.2 keeps less than half or so of it. The rest has gone. This isn’t a loss in the sense that it dissipated. It’s been cancelled. Roshko’s measurements of 1954 first quantified that. Gerrard in 1966 gave the mechanism: as one side’s growing vortex draws the opposite layer across the wake, opposite-signed vorticity is carried into it. That’s what ends each vortex’s growth, and it destroys a substantial share of the supply. Each layer is partly consumed by the other layer’s vortex, and viscosity at the interface finishes the job. Vorticity has signs, and a cancellation at the base of a wake is all that the budget needs.

That is an excellent example of the kind of information a control-volume argument reveals. Nobody needs to look inside the formation region to find out that half the vorticity vanishes there, and it follows by subtraction.

And it explains some otherwise puzzling design facts. A splitter plate, which stops one side’s layer from reaching the other, removes shedding and reduces drag considerably, sometimes by a quarter or more, because the cross-feed that sets the rhythm is cut. Blowing a little air out of the base, base bleed, delays the interaction and has a comparable effect. Neither touches the separation, and both change the street, which is exactly what the formula says will change the drag. It’s a satisfying feature that the street’s own terms make those connections, when a surface-pressure calculation would have been at a loss to say why a plate behind a body should affect a pressure in front of the plate.

The instability that does not matter #

The one result the street’s fame depends on is its stability ratio, so there’s a natural question about what actually happens to a neutrally stable street. A linear analysis at the neutral ratio says nothing about what grows at second order, and Domm showed in 1956 that the ratio isn’t stable to nonlinear terms, so there’s a finite-amplitude story.

The march is at most a suggestion, since it’s in a strip of eight periods and it can only show the modes that fit. But it gives the right size for the question. A kick of three per cent of the spacing, a good deal more than the street’s own smooth formation would impose, takes 237 units to break the street, and one such unit is about ten body widths of travel at this strength. That’s about 2400 widths. Smaller kicks outlast the run.

Real streets die in tens to hundreds of widths. Their cores diffuse, the two rows’ opposite vorticity cross-cancels, the wake thickens by entrainment, and $h/a$ changes along it. Hooker in 1936 and Bénard before him saw how real streets spread. So the famous stability calculation, so clean and so widely repeated, describes something too slow to be the actual fate of a street. It remains an exact and elegant result about point vortices, and nobody should conclude that the street’s persistence comes from it. The later reinterpretation has also gone further: as the frequency a wake chooses notes, a real wake’s frequency comes from a global instability near the formation region, and the street is what that leaves behind it. The numbers 0.28 and $h/a$ then look like an interesting coincidence, and the same value of the ratio has been observed in wakes where the underlying reasons differ. Saffman and Schatzman found in 1982 that a street of finite cores has a range of stable spacing near the same value. That helps a bit, and it doesn’t change this conclusion.

What the picture cannot show #

The formation region. Everything comes from a street that has already formed. A real wake takes two or three diameters to roll up, and in that region the balance includes more than the formula has. The control volume must be taken beyond it, and at that distance the street has already begun to spread.

Turbulence and three dimensions. A real cylinder’s street at Reynolds numbers above about 300 is turbulent, with cores of finite size, oblique shedding and strong span-wise variation. Roshko’s measurements show the agreement with this formula worsening at higher Reynolds numbers, where the Strouhal number stays much the same while the drag does not.

The body’s own drag crisis. Above a Reynolds number of a few hundred thousand a cylinder’s drag falls by three quarters while its wake narrows, and the street’s characteristics change in the same direction. A formula from the street could follow that in principle. In practice, measurements at those speeds are rarely of the street, and the wake there is much less organised.

Shape. The formula has none, which is its beauty and also its limit. Two different bodies with the same $d$, Strouhal number and street speed have the same drag coefficient by this reckoning. Roshko’s universal Strouhal number, based on a wake width rather than the body’s width, is the proper way to compare bodies, and it would change these numbers a little.

The convention #

Lengths are in the body’s width $d$; speeds in the free stream $U$; vortex strengths in $Ud$; times in the march in $a^2/\Gamma$. The street is staggered and infinite, of equal and opposite vortices with no cores. The Strouhal number’s value of 0.21 is the cylinder’s, near a Reynolds number of a thousand or so. The separated speed in the circulation budget comes from Roshko’s measured base pressure, $U_s=\sqrt{1-C_{pb}}\,U$ with a base pressure coefficient near $-0.7$. In the kick march the mode is one with a wavelength of twice the spacing, on eight periods, and departure is measured as the root mean square of each vortex’s height from its row.

How it was checked #

The periodic velocity sum reproduces both closed-form speeds to fourteen figures, which tests the velocity kernel used in the march. The formula’s two forms agree: Kármán’s two-term form as computed here and the condensed form with coefficients 2.83 and 1.12 give the same value. The model refuses a street speed outside what’s physical for a wake, a zero spacing and a strip of one period.

Who found it, and when #

Von Kármán’s stability paper is from 1911 and Kármán and Rubach’s drag paper from 1912. Hooker’s measurements of a real street’s spreading date from 1936, Roshko’s budget of the vorticity from 1954, Domm’s nonlinear stability analysis from 1956, Gerrard’s account of the formation region from 1966, and Saffman and Schatzman’s analysis of finite cores from 1982.

Still open: what the wake would say if the budget were closed #

The circulation budget here came from two separately measured inputs and a difference. The next calculation would march a real two-dimensional cylinder wake, with a viscous solver at a Reynolds number of a hundred or two, integrate the vorticity flux through a line across each side’s shear layer through a period, integrate the circulation of each shed vortex by its own contour, and take the difference, station by station downstream. It would ask whether the cancellation is concentrated in the formation region, how much goes on happening between vortices after that, and whether the street’s speed and spacing at a fixed distance give a coefficient that matches the pressure integral on the same cylinder to within the formula’s own terms.