The instrument in the answer

Every measured aerodynamic coefficient on this site’s shelves came out of a tunnel, and a tunnel has walls. The model sits in a stream a metre or two across rather than in an unbounded sky, and the walls are close enough to matter: the air squeezing past the model has nowhere else to go, so it speeds up, and the model is therefore sitting in a faster stream than the one the tunnel’s own instruments are reporting from far upstream.

A drag coefficient formed with the reported speed is then too low, by an amount that has nothing to do with the model and everything to do with the room.

This is called blockage, and the word usually arrives with an air of approximation. It should not. For the two-dimensional case it is exact, it is computable in closed form, and the computation is one of the prettiest in the subject.

The walls, made of copies #

A single plane wall is supplied by a single reflection, which is the standard trick: put an identical body on the other side of the plane and the plane becomes a streamline by symmetry, with no boundary condition imposed anywhere.

Two parallel walls cannot be done that way, because reflecting the body in one wall produces an image that is not reflected in the other. Reflect that in the second wall, and the new image is not reflected in the first. The process never terminates: two plane walls are worth an infinite row of images, at spacing 2h, running to infinity in both directions.

Summing the row gives the interference velocity at the model:

$$\frac{\Delta u}{U} = \sum_{n=1}^{\infty}\frac{2\mu}{2\pi (2nh)^2}\cdot\frac{1}{U} = \frac{\pi^2}{12}\left(\frac{a}{h}\right)^2$$

for a cylinder of radius a, using μ = 2πUa² for its doublet strength and ζ(2) = π²/6 for the sum. The Riemann zeta function turning up in a wind-tunnel correction is not a coincidence — a sum over evenly spaced images is a sum over 1/n², and there is only one of those.

The same number, without any summation #

A series is a route to an answer and not a proof of it. The second route shares no arithmetic with the first.

An infinite row of doublets has a closed-form complex potential — the same identity that gives the cotangent as a sum over poles — so the whole flow can be written down in one line:

$$W(z) = Uz + \frac{\pi U a^2}{2h}\coth\frac{\pi z}{2h}$$

Expanding about the origin separates the body’s own doublet from everything else, and what is left is the interference: −μπ/24h², identical to the sum, obtained without adding anything up.

Two independent routes agreeing to five figures is the standard this site holds a result to, and it matters more here than usual, because the answer is a small number multiplying a measured quantity. An error of a factor of two in a five-per-cent correction is invisible in every plot and changes the published coefficient by five per cent.

What it costs a measurement #

The model is sitting in a stream faster than U by the factor (1 + ε). A force coefficient is formed by dividing the measured force by ½ρU²A, so using the nominal speed instead of the true one makes the coefficient too low by (1 + ε)² − 1.

Fifteen per cent is not a refinement. It is larger than the difference between two aerofoil sections, larger than the drag reduction a designer is hired to find, and larger than the scatter that separates two laboratories’ published results.

Which is exactly what happened, historically. Wind tunnels in different countries reported drag coefficients for the same shapes that disagreed by more than either laboratory’s stated uncertainty, through the 1910s and 1920s, and a substantial part of the disagreement was that the tunnels had different working-section sizes and nobody was correcting for it.

The half of the correction that is not exact #

The blockage computed above is solid blockage: the model displaces air, and the air has to get past. There is a second effect and it is not exact.

Behind the model is a wake — a region of slower air — and in an unbounded stream that wake spreads outwards forever. In a tunnel it cannot: the walls confine it, so the flow outside the wake must speed up further to carry the same mass. This is wake blockage, and its size depends on the momentum deficit in the wake, which is the drag being measured.

The correction therefore needs the answer in order to be applied, which makes it iterative in practice and approximate in principle. It is proportional to the drag coefficient rather than to the model’s size, so for a streamlined body it is small and for a bluff one it is not — and it is the part of tunnel correction that genuinely deserves the word empirical.

Nothing on this site computes it. What is computed here is the solid half, exactly.

The body a tunnel actually supplies #

There is a third effect, and it is one the standard corrections do not mention at all.

The image row was assembled to represent a cylinder between walls. It does not quite produce one. The streamline that closes round the body — the ψ = 0 contour, which the streamfunction supplies in closed form — is not a circle: it is a slightly flattened oval, smaller than the circle the doublet was sized for.

The size change is second order in a/h and tracks 1/√(1 + ε) to within a part in a thousand — it is the blockage factor, seen from the geometry rather than from the velocity. The shape change is fourth order: doubling a/h multiplies it by sixteen, which the site measures at a power of 3.92 rather than assuming.

So a tunnel does not merely change the speed the model sees. It changes the model — into a slightly different shape, whose own coefficients differ from the intended one’s by an amount nobody corrects because it is two orders smaller than the effect everybody does correct. That is an honest place to stop rather than a defect: the point of computing the shape distortion is to know that it is negligible, and knowing requires computing it.

A small model, and why everybody uses one #

The quadratic dependence is the practical content of the whole calculation.

And there is the bind. Making the model small makes the correction small and makes the Reynolds number small too, because the Reynolds number is built from the model’s own length. A tenth-scale model in the same air at the same speed runs at a tenth of the Reynolds number, which for a wing means a laminar boundary layer where the full-scale article has a turbulent one, and that changes where the flow separates and therefore the drag it is being tested for.

So a tunnel test is squeezed from both ends: a large model has walls in the answer, and a small model has the wrong physics in it. This is the same trap the similarity essay measures in its general form — a scale model has more constraints than freedoms — arriving here as the specific question of how large a model may be. The escape routes are all expensive: pressurise the tunnel to raise the density, cool it to lower the viscosity, or build the tunnel bigger.

Why this belongs beside a flowmeter #

The previous rung was about a meter that costs pressure to read a flow. This one is about a measurement that is wrong rather than expensive, and putting them in the same ladder is deliberate: both are consequences of the same fact, which is that measuring a flow means being in it.

The two failures are different in kind, though. The orifice plate’s cost is honest — it takes a known fraction of the pressure and returns a correct number. The tunnel’s error is silent: nothing about the reading indicates that it is fifteen per cent low, every gauge is working perfectly, and the only way to know is to compute what the walls are doing.

A measurement that is precise and uncorrected is worse than one that is coarse and honest, because its precision is evidence for a number that is wrong. That is a recurring theme in the site’s misconceptions field and it belongs here too — as does the airspeed indicator’s version of it, where the instrument is faultless and the arithmetic applied to its reading is not.

The pressure coefficient is the quantity that shows the effect most sharply, because it is defined against the dynamic pressure of the oncoming stream. Its exact value at the shoulder of a cylinder is −3 in an unbounded flow, and a measurement in a tunnel at a/h = 0.3 returns something closer to −3.4 — not because the theory is wrong, but because the −3 belongs to a flow whose free stream is the one at infinity, and there is no infinity in a room.

What the picture cannot show #

The flow is inviscid. There is no boundary layer on the tunnel walls in this computation, and a real tunnel’s walls grow one — which reduces the effective cross-section along the working section and produces a slight favourable pressure gradient, corrected for separately by diverging the walls a fraction of a degree.

The body is a cylinder. The blockage of a streamlined body is smaller than a cylinder’s of the same frontal size, because the doublet is a poor model of a slender shape; the correction is usually written in terms of the body’s volume rather than its diameter, and the arithmetic here is the two-dimensional case in its cleanest form rather than the general one.

Nothing here is compressible. At Mach numbers above about 0.3 the blockage correction acquires a Prandtl–Glauert factor and grows sharply, which is why transonic tunnels are built with slotted or perforated walls that partly release the constraint — a solution that this arithmetic explains the need for and does not describe.

The corrections a real tunnel applies, in order of confidence #

It is worth setting the exact result in its practical context, because a tunnel test carries several corrections and they are not equally sound.

Solid blockage is what this essay computes. For a two-dimensional body on the centreline it is exact; for a three-dimensional one it is a body-volume expression of the same kind, slightly less clean and still analytic.

Wake blockage needs the drag being measured, so it is applied iteratively and is proportional to the answer. It is smaller than the solid part for a streamlined body and larger for a bluff one.

Streamline curvature is the correction nobody mentions in a first course, and it is a different kind of thing entirely: the walls not only speed the flow up, they straighten it, because they prevent the streamlines from spreading as far as they would in free air. A cambered aerofoil in a tunnel therefore behaves as though it had slightly more camber than it has, and the correction lands on the lift and the pitching moment rather than on the drag.

Horizontal buoyancy is a small streamwise pressure gradient along the working section, caused by the tunnel’s own wall boundary layers thickening downstream. It produces a spurious drag on any model of finite length, and it is why working-section walls are built with a fraction of a degree of divergence.

Four corrections, and the first is a theorem while the fourth is a measurement of a particular tunnel’s own imperfection. The order matters more than the numbers: knowing which of a measurement’s corrections is exact, which is iterative and which is an artefact of the apparatus is what separates a published coefficient from a reading.

The wall that is deliberately not a wall #

The whole calculation rests on one boundary condition — no flow through the wall — and there is a second kind of boundary a working section can have. Comparing the two produces a design idea rather than a correction.

An open jet has no wall at all: the stream is surrounded by still air at ambient pressure, so the condition on its boundary is a constant pressure rather than a zero normal velocity. Run the image argument with that condition instead and the images change sign. The interference reverses: where a solid-wall tunnel speeds the flow up at the model, an open jet lets it spread and slows it down, and a coefficient measured in one is too low while the same coefficient measured in the other is too high.

Which immediately suggests a wall between the two. Make the boundary partly open — longitudinal slots, or a perforated plate with a chosen open-area ratio — and the effective condition is a combination of the two, weighted by how open it is. There is therefore an openness at which the two interferences cancel and the model behaves as though it were in free air.

That is the ventilated working section, and every transonic tunnel has one. The openness is a few per cent of the wall area, chosen so that the residual interference is small over the Mach-number range the tunnel is built for.

And at transonic speeds it is not an optimisation but an enabling condition. A solid-wall tunnel with a model in it has a reduced minimum flow area, so the working section behaves as a nozzle with a throat in it: the tunnel chokes at a free-stream Mach number below one, and no amount of power raises it further. Before ventilated walls there was simply a band of Mach numbers — roughly 0.9 to 1.1, which is the band every transport aeroplane now cruises in — in which no wind-tunnel data could be taken at all. Slotted walls opened it at the end of the 1940s, and the transonic aerodynamics the type-change essay describes became measurable in the same decade it became computable.

Who found it, and when #

Prandtl’s Göttingen group formulated tunnel-wall corrections in the 1920s, and Glauert’s 1933 monograph gave the two-dimensional cases in the form still used. The image method itself is older than any tunnel: Kelvin used it for electrostatics in 1848, and its transfer to potential flow is the standard route to a wall throughout this site.

The cotangent identity that sums the row was Euler’s, published in 1748 as part of the same body of work that produced the ζ(2) = π²/6 the series needs. That both halves of this correction were available to Euler eighty years before there was a wind tunnel to apply them to is a pleasant fact rather than a useful one, and it is the sort of thing this field keeps turning up: the conservation laws and the classical analysis were finished long before the machines that needed them.

The general shape, which is not confined to tunnels #

The result generalises past wind tunnels, and it is worth stating in the general form because the same arithmetic appears in three other places on this site.

A body near a wall is a body plus one image, which is how a ground effect is computed. A body between two walls is a body plus an infinite row, which is this essay. A body in a pipe is a body plus an infinite two-dimensional lattice of images, which is why a sphere settling down the middle of a tube falls more slowly than one in an unbounded fluid — and the correction there is first order in the size ratio rather than second, because a tube confines in two directions.

In each case the physical statement is the same: a boundary is a constraint on where the displaced fluid may go, and the closer it is the more the flow near the body has to speed up. The image system is a device for computing that, and the reason it works is that a reflection satisfies the same equations and produces the symmetry a wall needs.

Where the ladder goes next #

Metering is finished; both of its rungs are about measurement changing the thing measured. What follows is a different kind of interference, in which the disturbance is not a percentage but a wave: close a valve on a moving column of water and the pressure that appears has nothing to do with the head driving it, and everything to do with how fast a signal can travel in the pipe.