The street this site cannot draw

Open any book on fluid mechanics at the chapter on wakes and there is a photograph: a cylinder, a stream, and behind it two staggered rows of vortices marching away, alternating in sign, evenly spaced.

This site’s own solver does not produce that picture at any Reynolds number, and the rule this collection draws its figures under forbids any figure here from claiming otherwise. The stepper is two-dimensional and coarse; it produces a steady standing pair at Re = 40 and an unsteady wake at higher Reynolds numbers, and it has never been shown to shed a street with the spacing and regularity a photograph has.

So this essay does the honest alternative: it builds a model of the street, says so in every caption, and extracts from the model the one thing a model of that kind can settle exactly.

What the model is #

Two infinite rows of point vortices, one above the other, staggered by half a spacing, with equal and opposite circulations.

That much can be written down exactly. A row of point vortices at spacing a has a complex potential that sums to a logarithm of a sine, so the velocity field of the pair is

$$w'(z) = \frac{i\Gamma}{2a}\left[\cot\frac{\pi(z - z_1)}{a} - \cot\frac{\pi(z - z_2)}{a}\right]$$

in closed form — no truncated sum of singularities, no numerical convergence question. That matters because it lets the figure carry a measured circulation: the site integrates the velocity round a small loop enclosing one core and gets 1.0000 against the 1 it was built with.

That check earned its place immediately. Written with the opposite sign convention the field is a perfectly good vortex street — smooth, periodic, staggered, indistinguishable by eye — in which every core turns the wrong way. The line integral came back as −1.0000: the magnitude exactly right and the sign exactly wrong, which is the failure mode this whole site was built around and the fourth time on this site that a sign has been caught by a measurement rather than seen.

The one thing it settles #

Kármán’s question in 1911 was not why a street forms. It was: given that one exists, is any arrangement of it stable?

The calculation displaces every vortex slightly, computes the induced velocities, and asks whether the disturbance grows. The answer is that a symmetric double row — vortices directly opposite each other — is unstable at every spacing, and a staggered row is unstable at every spacing but one:

$$\cosh^2\!\left(\frac{\pi h}{a}\right) = 2 \quad\Longrightarrow\quad \frac{h}{a} = \frac{\ln(1 + \sqrt2)}{\pi} = 0.28055\ldots$$

The site bisects the first equation two hundred times and checks the result against the second, and the agreement is to fourteen digits.

A photograph of a real wake measures about 0.28.

That is a remarkable thing for a model containing no viscosity, no body, no shedding mechanism and no fluid in the ordinary sense to have got right, and it is the reason the model is worth drawing at all.

Why a row of vortices is stable at all #

The stability calculation is worth a paragraph of physical reading, because the answer is less arbitrary than a hyperbolic cosine suggests.

Each vortex sits in the velocity field of all the others. Displace one and two things happen at once: the vortices in its own row push it in one sense, because a row of like-signed vortices is a one-dimensional lattice and displacing a member of a lattice produces a restoring or a repelling force depending on the direction; and the vortices in the other row, being of opposite sign, push it the other way.

The two effects have different dependences on h, because the distance to the other row enters through h and the distance along the row through a. There is therefore one ratio at which they balance, and that ratio is what cosh²(πh/a) = 2 encodes.

The symmetric arrangement — vortices directly opposite one another — has no such balance at any spacing, because both effects act in the same sense. That is why an alternating street is what nature produces and a symmetric one is not, and it is the part of the result that survives every qualification below.

Why the agreement is less than it looks, and still worth something #

Three qualifications, because an agreement to two figures between a model this crude and an experiment invites over-reading.

Neutral is not stable. Kármán’s analysis finds the ratio at which the growth rate passes through zero; at that ratio the arrangement is neutrally stable to linear disturbances, not attracting. A real street is not held at 0.28 by this mechanism. What the calculation says is that every other ratio is actively unstable, so an arrangement that persists must be near this one — a selection by elimination rather than by attraction.

Viscosity changes the answer. Real cores are not points; they have finite size and they diffuse. The measured ratio drifts downstream as the cores spread, and the 0.28 is a measurement taken a few diameters behind the body rather than a constant of the wake.

The other spacing is not predicted. The model gives h/a, a ratio. It says nothing about a itself, which is set by the shedding frequency and the convection speed — that is, by the very mechanism the model does not contain.

That last point deserves emphasis because it is where the honest limit of the whole approach sits. The street’s along-row spacing is the convection speed divided by the shedding frequency, and the shedding frequency is the thing an engineer wants: it is what excites a chimney, a bridge deck or a heat-exchanger tube, and it is what the Strouhal number reports.

The number the model does not supply #

The Strouhal number is fD/U, with f the shedding frequency and D the body’s diameter, and for a circular cylinder it sits near 0.2 over an enormous range of Reynolds number — from a few hundred to a few hundred thousand.

That near-constancy is the practical content of the whole subject of vortex shedding. It means a chimney of a given diameter sheds at a frequency proportional to the wind speed, so there is a wind speed at which the shedding matches the structure’s natural frequency, and that speed can be computed in advance and designed around.

It is also, on this site, a measured number rather than a derived one. Nothing here computes it. It does not come out of Kármán’s stability calculation, which contains no frequency at all; it does not come out of the site’s stepper, which does not shed; and the theories that do predict it are global stability analyses of the near-wake that need a resolved computation of the flow past the body.

The site’s practice with such a number is to state where it came from on the figure that carries it, which is what the regime axis above does.

What the solver does produce #

Between those two the real flow passes its own threshold — a cylinder starts shedding somewhere near Reynolds number 47 — so the second picture is already on the wrong side of it and the third is well past. What the stepper produces instead is a steady pair of eddies that grow longer, which is the right answer to a question nobody asked.

The real behaviour behind a circular cylinder, as established by experiment, runs roughly:

Below Re ≈ 5, no separation at all — creeping flow, fore-and-aft nearly symmetric, and the one regime in which the ideal theory’s fore-and-aft symmetry is nearly what a real flow shows, for entirely different reasons.

From Re ≈ 5 to about 47, a steady symmetric pair of attached recirculating eddies, growing longer as the Reynolds number rises. This is what the site’s solver computes, and it computes it as a solution with the conservation checks applied.

Above Re ≈ 47, the wake becomes unstable and begins to shed alternately, by the mechanism the shear layers supply rather than by anything happening at the surface. The transition is a Hopf bifurcation of the near-wake and its threshold is one of the better-established numbers in the subject.

From a few hundred upward, the street is three-dimensional, the cores develop spanwise structure, and the two-dimensional picture is a projection of something more complicated.

The site’s stepper reaches into the third of those and cannot follow it. That is a statement about a 150-by-76 two-dimensional grid rather than about fluid mechanics, and it is why this essay exists in the form it does.

What the street costs the body #

There is one more connection worth making, because it is what makes the wake a subject of this site rather than of a gallery.

Every vortex in the street carries kinetic energy, and that energy came from the body. A street extending behind a cylinder is a continuous record of work done, and the rate at which the energy is being deposited is the drag times the speed.

The same accounting as induced drag, in a different geometry: the drag is not lost to friction, it is deposited, as a structure the body leaves behind. And, as with induced drag, the vortices are the signature rather than the cause. The drag was determined at the body’s surface by the pressure distribution over it — a distribution the wide separated wake produced — and the roll-up happened afterwards.

This is why an argument beginning “the vortex street causes the drag” has the sequence backwards, in exactly the way the tip-vortex version does. What causes the drag is the separation; the street is what the shed vorticity organises itself into once it has left.

What happens when the body can move #

Everything above has the body held rigid, which is the case the Strouhal number is quoted for. Let it move and the wake stops obeying that number, in a way worth knowing because it is the reason anybody outside a laboratory cares about shedding at all.

A cylinder free to vibrate is shaken sideways at the shedding frequency, and its motion in turn changes when the next vortex leaves. As the wind speed brings the shedding frequency near the structure’s own natural frequency, the shedding stops following $fD/U = 0.2$ and captures onto the structure, holding at the structure’s frequency over a band of wind speeds perhaps thirty per cent wide. That is lock-in, and inside the band the two are one oscillator rather than a forcing and a response.

Two consequences follow that ordinary resonance does not have. The band is a range of wind speeds rather than a single one, so the condition cannot be avoided by designing away from one number. And the amplitude is self-limiting — it grows to something of order a diameter and stops, because the motion that large detunes the shedding that drives it. A linear resonance has no such ceiling, which is the tell that this is a feedback and not a resonance, and it is not the mechanism that took down the Tacoma Narrows bridge whatever the textbooks say.

The engineering response follows from the wake being coherent along the span: a street sheds in phase over many diameters, which is what makes the force add up. Break that coherence and the force largely goes away, which is what the helical strakes wrapped round tall chimneys do, and why marine risers and heat-exchanger tubes carry the same devices. They do not stop shedding. They stop it happening everywhere at once.

Where the model stops, which here is most of the essay #

The inventory, stated plainly because the essay’s whole purpose is to be able to draw the picture without implying the claim.

No body. There is nothing in the model for the flow to go around. The vortices exist by stipulation.

No viscosity. The field is ideal flow, so nothing diffuses, nothing decays and no vorticity is created anywhere.

No shedding. Nothing in the model produces a vortex. The street is an initial condition.

No frequency. The along-row spacing is a parameter, so the shedding frequency is an input rather than an output, and the Strouhal number cannot be obtained from it.

No drag. The pressure field of the model integrated over a body would be meaningless, since there is no body; the connection between the street and the drag on a cylinder is real and is not in here.

What is in it: an exact velocity field, a circulation measured by line integral, and a stability result with a closed form. Those three things are the whole of what the figure asserts, and they are asserted at the same standard as everything else on this site.

Why the site does not simply build a better solver #

A fair question, since the essay’s premise is a limitation of the machinery rather than of the subject: why not compute the wake properly?

The arithmetic is more forgiving here than for a genuinely turbulent flow, and that is worth being precise about rather than hiding behind. A two-dimensional shedding wake at Re = 200 needs perhaps 10⁵ to 10⁶ grid points and a few thousand timesteps to establish a periodic state — which is minutes on a laptop, not centuries on a supercomputer.

What makes it awkward is not the flop count but the shape of the build. Every figure on this site is generated inside a static-site build that has to complete in under a minute, from a solver whose current cost is already the build’s cost centre. A shedding calculation would have to run long enough to shed the initial transient, be cached across builds with a key covering every parameter that changes the answer, and be checked — because an unconverged shedding wake is exactly the kind of smooth plausible field that this site refuses.

So it is deferred rather than impossible, and this site records it as owed rather than as out of scope. The distinction matters: a limitation that has been priced is a decision, and a limitation that has not been priced is an excuse.

Who found it, and when #

Strouhal measured the frequency of the tone an air-stream draws from a wire in 1878, which is why the number carries his name — the aeolian tone of telegraph wires being the phenomenon he was after.

Bénard observed the alternating wake in 1908, and Kármán analysed the stability of the double row in 1911 and 1912. The story Kármán told about it is that a doctoral student in Göttingen kept failing to produce a symmetric wake, and that the stability calculation was Kármán’s answer to why.

Roshko’s measurements in 1954 established the Strouhal number’s near-constancy over the Reynolds range and separated the regimes it is constant within.

Williamson’s careful work through the 1980s and 1990s mapped the three-dimensional structure and established that what looks like one phenomenon is several, with distinct spanwise modes appearing at distinct Reynolds numbers.

What a reader should ask of any wake figure #

The general form of this essay’s caution transfers, and it is three questions.

Is there a body in it? A figure of a wake with no body in it is a model of the wake’s structure rather than a solution of the flow that produced it. That is legitimate and it is different.

Where did the spacing come from? If the along-row spacing was an input, the figure cannot be evidence about shedding frequency; if it came out of a computation, the computation had to resolve the near-wake, which is a much more expensive object than a row of vortices.

What is the viscosity? An ideal-flow model has none, so nothing in it decays, nothing diffuses, and no vorticity is created. A real wake spreads and weakens with distance, and a figure showing a street of undiminished strength to the right-hand edge of the frame is drawing an idealisation.

None of those questions is hostile. They are the questions this site’s own model note is designed to answer without being asked, which is why every figure here carries one and why the regime index reads them back out of the rendered SVG rather than out of the source.

Where the field goes next #

This is the last essay of the turbulence field’s first phase, and it ends where the field began: on the difference between what can be solved, what can be modelled, and what can only be reported.

The field’s later work has an obvious first target. The Strouhal number is a measured constant on this site and a derived one in the literature, and deriving it needs a global stability analysis of a solved near-wake — which is within reach of a better solver, and is the kind of thing a later essay here should attempt rather than assume.

Until then the picture stays a model, and stays labelled as one.