The wind that brings its own frequency

A steel chimney 1.2 m across stands on a factory roof. It has been checked for the fifty-year wind, its foundation bolts are sized, and it is entirely adequate.

In a light breeze it swings sideways — across the wind, not along it — by about 180 mm, and does so for hours at a time, several times a month.

Nothing about that behaviour is visible in a wind pressure. The load that causes it is small, it is at right angles to the wind, and the speed at which it matters is one that a design calculation never looks at.

Why a shedding frequency exists at all #

Flow past a bluff body separates, and behind it the two shear layers roll up into vortices — alternately, one from each side, in a regular street. Each vortex leaves a low pressure behind it as it forms, so the sideways force on the body reverses with every shed vortex, and the body is pushed alternately left and right at the shedding frequency.

That frequency is remarkably orderly across a wide range of conditions:

$$f_s = \frac{St \cdot U}{D}$$

with $St$ the Strouhal number, about 0.2 for a circular cylinder over several decades of Reynolds number, and about 0.11 to 0.15 for the rectangular shapes buildings are. The number is a property of the shape, not of the fluid or the speed, which is what makes it useful.

Two consequences follow immediately and neither has an analogue anywhere else on this site.

The excitation frequency is proportional to the wind speed. Every other load’s timing is set by something outside the structure — a machine’s speed, a person’s pace, a fault’s rupture. This one is set by a variable that sweeps continuously through every value from zero upwards as the weather changes.

Therefore the coincidence is guaranteed. Setting $f_s = f_0$ gives the critical speed $U_{\text{crit}} = f_0 D / St$, and for this chimney that is 6.0 m/s — a fresh breeze. It is not a question of whether the structure meets its critical speed but of how often, and the answer is measured in hundreds of times a year.

Lock-in, and why it is not resonance #

The flat section in that figure is the part that makes vortex shedding a separate subject.

As the wind approaches the critical speed the cylinder begins to move, and a moving cylinder organises its own wake: the vortices shed in step with the motion rather than in step with the flow. Over a band of roughly ±25% around the critical speed the shedding abandons the Strouhal relationship entirely and follows the structure. Wind tunnel measurements show this as a plateau, and full-scale measurements show it as a structure that oscillates steadily while the wind speed wanders.

That is not a resonance in the sense the earlier essays used. In a resonance the excitation is fixed and the structure happens to respond strongly; here the structure changes the excitation. The load is a function of the response, which means:

• the response cannot be found by multiplying a force by a magnification factor, because there is no force independent of the response to multiply;
• the amplitude is limited by nonlinearity in the fluid rather than by damping alone, so it does not grow as $1/2\zeta$ indefinitely;
• the band is wider than any resonance peak, because the structure drags the excitation along with it rather than waiting for it to arrive.

It is the same disqualification that removes the magnification factor from the crowd on a footbridge, and lock-in is the mildest member of a family whose other members are considerably worse — the essay on self-excitation takes that family as a whole.

The Scruton number, which is the whole design answer #

The amplitude that results is inversely proportional to one dimensionless group:

$$Sc = \frac{4\pi m \zeta}{\rho D^2}$$

with $m$ the mass per unit length and $\rho$ the air density. It compares how much the structure weighs and dissipates against how much air it is disturbing, and the practical rule is that above about 20 the response is small and below about 10 it is not.

Mass and damping enter only as their product, which is the design statement and is not obvious from the equation of motion. It means a heavy structure needs no damping and a light one cannot be rescued by stiffening:

Stiffening changes $f_0$, which changes the critical speed and moves the problem to a different wind — it does not remove it. That is a genuinely different design logic from everything else in this collection, where stiffness is the usual lever and depth is the cheapest thing to buy.

There is one exception, and it is worth naming because it is the design decision most often taken. Raising the frequency far enough moves the critical speed above the range the site actually experiences — for a short, stocky mast the critical speed can be pushed to 25 m/s, which occurs rarely and briefly, and the fatigue count collapses. So stiffness does help, not by reducing the response but by making the episodes rare. That is a different argument from the one stiffness usually wins, and it depends entirely on the wind statistics at the site rather than on the structure.

The free body, and where the force comes from #

It is worth drawing the free body explicitly, because the load has an unusual origin and the unusual part is easy to lose.

Cut a metre of the chimney and draw it. On it act: the drag, along the wind, roughly steady; the lift, across the wind, alternating at the shedding frequency; the inertia force of the metre of steel; and the shear and moment from the chimney above and below. The sums cancel, as ever.

The alternating lift is the whole subject, and its magnitude is modest — a lateral force coefficient of about 0.7 for a circular cylinder in the subcritical range, which on this chimney at 6 m/s is roughly 20 N per metre. Twenty newtons. What turns twenty newtons per metre into 180 mm of motion is not the force but the multiplier:

So the mechanism is a small force meeting an enormous multiplier, and the multiplier exists because a steel chimney is the most lightly damped structure engineers build. Every other structure on this site has partitions, cladding, bolted joints and finishes rubbing against a frame; a chimney has a weld and a coat of paint.

The multiplier also takes its time. A 3 kN cross-wind force at 0.4% damping reaches 125 times its static deflection, and it needs about forty cycles to get there — which is the same reciprocal read as a count rather than as a height, since a magnification of 125 is $1/2\zeta$ and the cycles to build up are the same quantity. That is roughly three quarters of a minute at 0.9 Hz, and it is why lock-in needs a sustained breeze rather than a gust: the wind must hold within the band long enough for the multiplier to be collected.

What is actually done about it #

Three remedies, in increasing order of admission that the problem is real.

Accept the motion and check the fatigue. A chimney swinging 180 mm at 0.9 Hz for hours at a time accumulates cycles quickly: three thousand an hour, ten million in a few years. The limit state is not strength but fatigue at the welded details, and the stress range at the base is what has to be checked. This is the commonest outcome and it is why vortex shedding is a fatigue problem more often than a strength one.

Add damping. A tuned mass damper, a chain damper hanging inside the flue, or a viscous device at the top. Because the response goes as $1/Sc$ and $Sc$ contains $\zeta$ linearly, a small amount of added damping is worth a great deal — which is why an added mass is the standard fix for slender steel structures.

The critical speed has not moved, and it cannot: 6 m/s is St, U and D, and none of the three knows anything about damping. The wind will still find this chimney as often as it did before, and it will still find it at a breeze. What the damper changes is only what happens when it does — which is the honest way to describe every remedy of this kind. A device that halves an amplitude has not made the event rarer, and the fatigue calculation still counts the same cycles at a smaller stress range.

That linearity is also what makes the remedy cheap. Nothing else in this field returns a fifth of the response for a fifth of a per cent of critical damping; a resonance under an ordinary load returns the square root of the effort, and here the whole return is first order because the excitation itself is proportional to nothing the damper touches.

Spoil the shedding. Helical strakes wound around the top third of a chimney break the vortices’ spanwise correlation so that they no longer shed in phase along the length. It is the only remedy that attacks the excitation rather than the response, it costs about 25% more drag, and it is the reason so many industrial chimneys have a spiral fin near the top. It is also the only one of the three that would still work if the structure’s damping turned out to be a tenth of what was assumed — which, given how damping is arrived at, is not a hypothetical.

How often, and for how long #

The design question a chimney raises is not “how big” but “how often”, and the wind statistics answer it in a way no other load on this site does.

Wind speed at a site has a distribution, and the middle of it is where structures spend their lives. A speed of 6 m/s is exceeded for a large fraction of the year at almost any exposed location and occurs — in the sense of sitting inside a ±25% band around it — for hundreds of hours annually. Every one of those hours, this chimney is oscillating at 0.9 Hz, which is 3,240 cycles an hour.

Ten million cycles in a few years is entirely ordinary, and that number is the reason vortex shedding shows up in inspection reports as cracked welds at the base rather than as a chimney lying on the ground. The failure mode is fatigue and the design case is a breeze, which between them make this the least intuitive load case in structural engineering.

Ferrybridge, and the cylinders that were not alone #

The most instructive wind failure of a group of bluff bodies happened at Ferrybridge power station in Yorkshire on 1 November 1965, when three of eight cooling towers collapsed in a gale.

The towers had been designed for a wind loading based on a single isolated tower. In a group, the downwind ones stand in the wake of the upwind ones, where the flow is more turbulent and the pressure distribution is different — and the loading they actually received was outside what the design had contemplated. The investigation that followed changed British wind loading practice substantially, and one of its lasting effects was the recognition that a structure’s wind loading is a property of its surroundings and not only of itself.

That is the same lesson the two-cylinder problem teaches at a smaller scale, and it is worth attaching to this essay rather than to the previous one because it is a lesson about shape and arrangement rather than about turbulence statistics. The Davenport apparatus of the last essay handles a building in a boundary layer very well and says nothing at all about what the building next door is doing to it.

What the picture cannot show #

The amplitude model is empirical. The Strouhal relation and the Scruton number are solid; the coefficient that converts them into a displacement is fitted to measurements, and different codes give noticeably different answers. The shape of the curve is trustworthy and its height is a correlation.

Turbulence disrupts it. Everything here assumes smooth flow. Real atmospheric turbulence breaks up the spanwise correlation of the shedding on its own, so a structure in a rough urban boundary layer responds less than one in open country — and a wind tunnel test in smooth flow overestimates. The worst conditions are steady, smooth, moderate winds, which is another reason the problem is a fine-weather one.

Two cylinders are a different problem. Adjacent stacks, a pair of cables, a chimney beside a building: the wake of one interferes with the other, and the interference produces its own instabilities at speeds neither has alone.

The cylinder can go oval instead of bending #

Everything above treats the chimney as a beam: it has a bending mode, that mode has a frequency, and the shedding locks on to it. A thin-walled cylinder has a second family of modes that has nothing to do with bending, and vortex shedding excites those too.

A cylindrical shell can deform out of round — the section going oval, or into three lobes, or four — without the axis moving at all. These are the shell modes, indexed by how many circumferential waves they have, and the two-lobed one is the lowest and the one that matters. Its frequency is set by the wall thickness against the radius, and for a thin stack it can be a few hertz: the same order as the bending mode, arrived at by a completely different route.

The excitation is the same alternating pressure pattern, and the condition for lock-in is different because the shape is. A vortex pair drives the ovalling mode through two complete shape cycles rather than one, so the mode is excited when the shedding frequency is around half the ovalling frequency, which puts the critical wind speed at roughly

$$U_{\text{crit}} \approx \frac{f_{\text{oval}}\,D}{2\,St}$$

— a different speed from the bending one, at the same site, on the same structure.

Three things follow, and each changes what the remedy is.

It is a shell problem, so the defence is a shell one. Damping does very little for it and mass does not enter the way the Scruton number suggests. What works is a ring stiffener — a circumferential rib that forbids the section going out of round, raising the ovalling frequency out of reach. That is why tall thin stacks carry rings at intervals up their height, a detail that looks like buckling reinforcement and is partly this.

Thinness is the driver, and thinness is what modern fabrication supplies. The ovalling frequency falls as the wall thins, so the same efficiency that lowered the Scruton number and made bending lock-in a problem also brought the ovalling mode down into the range the wind can reach. It is the same story as the previous section, in a mode that story never mentions.

And the two problems share one remedy. Helical strakes spoil the spanwise correlation of the shedding, which starves the ovalling mode exactly as it starves the bending one — so the fix fitted for the visible problem happens to cover the invisible one, which is fortunate and is not a reason to rely on it.

The general point is the one the essay’s own free body should have raised. Drawing a metre of chimney with a lift force on it assumes the metre stays circular, and that assumption is a modelling choice about which degrees of freedom exist. A structure has every mode its geometry permits, and an excitation that is broad in frequency will find the ones that were left out of the picture.

Why it is a modern problem #

Vortex shedding has been going on since the first mast was put up, and it became an engineering problem in the middle of the twentieth century. The reason is entirely in the Scruton number.

A masonry chimney has a mass per metre in the thousands of kilograms and mortar joints that dissipate energy; its Scruton number is in the hundreds and it does not respond. A welded steel chimney of the same diameter has a tenth of the mass and a fortieth of the damping; its Scruton number is around ten and it responds to a breeze. Nothing about the aerodynamics changed — the same vortices leave the same cylinder at the same rate — and the structure stopped being able to ignore them.

The pattern is one this site meets in floor vibration and will meet again: the problems that arrive with better engineering are dynamic ones, because efficiency removes exactly the mass and the accidental friction that used to make dynamics irrelevant. Every kilogram saved is a kilogram not available to the Scruton number.

Where the ladder goes #

Two directions, and they diverge sharply.

One is the fatigue calculation the accepted-motion remedy needs — the same accumulation argument as the load that never came near failing anything, with the cycle count supplied by the wind statistics rather than by a machine: how many cycles, at what stress range, against a detail category — a subject this site has already opened from the materials side.

The other is the family this belongs to. Lock-in is the mild case of a load that depends on the response. There are worse ones, where the feedback is positive without limit and the structure supplies its own excitation from a steady wind — the case where nothing bounds the answer at all.