The walls are in the answer

A wind tunnel is a device for putting a wing in a flow whose properties are known. What it actually does is put a wing in a box, and a box is a set of boundary conditions the aeroplane will never meet again. The walls are close, they are solid, and every one of them insists that no air passes through it — which is a statement about the flow round the model, and therefore a statement about the numbers the model produces.

This site has already dealt with one half of that: the blockage a body’s own displacement causes, where a cylinder in a channel reads a speed that is not the tunnel’s. The other half is what the walls do to a lifting model, and it is a different mechanism with a cleaner answer.

The tunnel makes the model look better than it is #

The mechanism is the same one as ground effect, run in every direction at once. A wing’s trailing vortices induce a downwash on the wing itself, and that downwash is the whole of its induced drag. Put a wall nearby and the wall’s condition adds an upwash; the net downwash falls; the induced drag falls with it.

So a closed tunnel flatters. The model appears to have a steeper lift-curve slope than it has, because it is reaching its lift coefficient at a smaller geometric incidence, and it appears to have less induced drag, because it does.

For a model of half the tunnel’s diameter in a working section, at a lift coefficient of 0.8, the computed corrections are an incidence error of 0.365 degrees and a drag error of 15 per cent of the induced drag. The incidence error is 3.75 per cent of the incidence itself, which is 3.75 per cent of the lift-curve slope — a quantity aircraft are designed against to a per cent or two.

An eighth, exactly #

The correction factor is defined by

$$\Delta\alpha = \delta\,\frac{S}{C}\,C_L,$$

with $S$ the model’s area and $C$ the tunnel’s cross-section, so that $\delta$ is a property of the tunnel’s shape alone. For a closed circular working section it is exactly one eighth.

That is exact rather than fitted, and it comes from the one image system in this subject that is finite: Milne-Thomson’s circle theorem puts the image of a vortex at the inverse point, $R^2/\bar{\zeta}$, with reversed circulation for a solid wall. Two trailing vortices at $\pm s$ have their images at $\pm R^2/s$, the upwash they induce at the centre is $\Gamma b/2\pi R^2$, and every step from there to $\delta = 1/8$ is arithmetic.

The factor of two in it is the one worth watching. The upwash at the wing is half the upwash far downstream, because the trailing vortices are semi-infinite at the lifting line and doubly infinite in the Trefftz plane. The incidence correction uses the first and the drag correction uses the second, which is why the same $\delta$ appears once in each with different powers of $C_L$ beside it. Getting that factor wrong halves or doubles every corrected measurement a tunnel has ever made and looks like nothing on a graph.

The images do not work for a rectangle, and the wall does #

The textbook route to a rectangular tunnel’s correction is a doubly infinite array of images, built by reflecting in each wall in turn. It was tried here first, and it does not converge in any useful sense: summed over a disc of growing radius, the correction factor for a 1.5-to-1 section came out 0.1197 at one truncation, 0.0214 at twice that and 0.2650 at twice again, while the wall condition the images exist to impose was violated by more than the interference velocity itself.

The images cancel to a high multipole only when whole groups of them are present, and no truncation of an infinite plane keeps every group whole.

So the boundary condition is solved for instead, which is what a boundary condition is for. A sheet of sources on a closed wall, with the strength chosen so that the wall passes nothing; a sheet of vortices on a free boundary, chosen so that the boundary carries no tangential perturbation. Both are one linear system, and the closed one has the defect that a Neumann problem always has: a uniform source strength on a closed boundary has nowhere to send its flux, so the matrix is singular and the system has to be closed by requiring the total strength to vanish. Which it does, to $5\times10^{-18}$.

The check that licenses all of it: the same solver pointed at a circle, whose answer is known in closed form, gives 0.1236 at a hundred and twenty panels, 0.1243 at two hundred and forty and 0.1246 at four hundred and eighty — converging on an eighth from below, at first order, as a panelled curved boundary must.

The shape a tunnel should be #

With the wall solved rather than reflected, the correction can be computed for any section at all, and the natural question is which section is best.

The minimum is at a breadth of about one and a half heights, with $\delta = 0.119$ against a square section’s 0.136. Wind tunnels are built at close to that proportion, and the usual explanation for it is that a wing is wider than it is tall. That is true and it is not the reason: the reason is on the axis of this figure, and it is a property of the interference rather than of the model.

The square section’s computed 0.136 is worth one more sentence. The classical value for a closed square tunnel, tabulated by Glauert in the 1930s from a hand-summed image series, is 0.137. That is not a check of the physics — it is the same image system done by somebody else — but it is a check of the arithmetic, and the arithmetic here reached it by a route that does not use images at all.

The drag correction, and why it is the other factor of two #

The incidence correction moves a measurement along the $\alpha$ axis. The drag correction moves it along the $C_D$ axis, and it uses the far upwash rather than the near one.

The reasoning is the same as for a wing’s own induced drag. Drag is lift multiplied by the angle through which the lift vector is tilted, and that angle is the induced one at the lifting line — but the interference induced drag is what the wall’s upwash does over the whole wake, so it collects the Trefftz-plane value. What comes out is

$$\Delta C_{D_i} = \delta\,\frac{S}{C}\,C_L^2,$$

the same $\delta$, the same area ratio, and one more power of $C_L$. For the model above it is 0.0051 against an induced drag of 0.0339 — fifteen per cent, at a lift coefficient a transport aircraft cruises at.

Fifteen per cent of the induced drag is between three and six per cent of the whole drag of a subsonic aeroplane, which is the difference between a design that meets its range guarantee and one that does not. Tunnel corrections are not a refinement applied to a measurement; they are a part of the measurement whose size is comparable with the thing being measured.

The sign, which is the whole argument #

Replace the solid wall with a free boundary — an open jet, where the flow discharges into still air at atmospheric pressure — and the condition changes from no normal velocity to no pressure perturbation. In the crossflow plane that is a condition on the potential rather than on its gradient, the image reverses sign, and every consequence reverses with it.

So two tunnels, two answers, one aeroplane. A closed section reports a lift-curve slope that is too steep by 3.75 per cent for the model above; an open jet reports one that is too shallow by the same amount. The difference between the two measurements is 7.5 per cent, and nothing about the air, the model or the instrumentation differs between them. What differs is what the boundary was told.

What the picture cannot show #

Four limits, and the last one is the one that keeps tunnel engineers up at night.

The correction computed here is the value at the model’s centre. The interference upwash varies across the span, so a full correction weights it by the loading and picks up a dependence on the ratio of model span to tunnel breadth that the centre-line value does not have. For a circular section the centre value is exactly an eighth at every span, which is a property of the image being a single vortex and is not true of the loading-weighted average that a tabulated $\delta$ actually is.

Only lift interference is here. A real correction has three more terms: solid blockage from the model’s volume, wake blockage from its drag, and — in a closed section — a streamline-curvature term that changes the effective camber as well as the incidence. This site has computed blockage separately and the two are combined in practice rather than solved together.

The walls are perfect. A real working section has a boundary layer on it, slots or perforations in half the tunnels ever built, and a wall that is neither wholly solid nor wholly free — for which the condition is a linear combination of the two, with a coefficient that is measured rather than derived. The whole point of a slotted wall is to land between $+1/8$ and $-1/8$ and choose zero.

And none of it addresses the Reynolds number, which is the tunnel’s real difficulty and which no amount of correction fixes: a model is small, so its Reynolds number is low, and the transition point sits somewhere the aeroplane’s will not.

Correcting from the wall instead of from the model #

Every correction in this essay starts by assuming what the model is — a horseshoe vortex of known strength at a known position — and computing what the walls do to it. That is the classical method and it has an obvious weakness: the answer depends on a representation of the model, and a model that is not well represented by a single horseshoe is not well corrected.

The modern method inverts it. Instead of modelling the model, measure the walls. A working section instrumented with static tappings along its length gives a pressure signature that the model has written on the boundary, and that signature contains everything the outside world knows about the model’s disturbance. Fit an equivalent singularity distribution to it — or use it directly as boundary data — and the interference velocity at the model follows without anybody having decided in advance what the model looks like.

The gain is that the assumption moves from the model to the linearity of the outer flow, which is a far safer place for it. A model with a separated wake, a high-lift configuration with several elements, a propeller with a slipstream: each of them is badly described by a horseshoe vortex, and each writes an honest signature on the wall.

Or move the wall until there is nothing to correct #

The other route is to stop correcting and change the boundary instead, and it follows directly from the sign result of this essay. A closed wall gives $\delta = +1/8$ and a free boundary gives $-1/8$, so somewhere between them is a boundary that gives zero — and rather than trying to build a fixed wall with that property, an adaptive-wall tunnel adjusts its own.

The condition to be met is worth stating because it explains why the method needs an iteration rather than a setting. At a boundary in free air, two quantities are determined: the normal velocity and the pressure. A wall can be given one of them — a solid wall sets the normal velocity, a free jet sets the pressure — and free air requires both to be right at once, which no single fixed condition can deliver.

So the tunnel measures one and imposes the other. Flexible top and bottom walls are moved to some shape; the flow near them is measured; the measurement is compared with what free air would give outside that shape; the walls are moved again. A few iterations bring the residual interference down to something below the measurement’s own uncertainty, and the correction is then not calculated because there is nothing to calculate.

Which sets out the three positions a tunnel can take, in increasing order of cost and decreasing order of assumption. Compute the interference from a model of the model, which is cheap and rests on the model being what it was assumed to be. Compute it from the measured wall signature, which rests only on the outer flow being linear. Or deform the wall until the interference is not there, which rests on nothing and requires a tunnel built for it. The classical eighth is the first of the three, and its value is that it says how large the thing being avoided is.

How a correction is applied, and why the order matters #

The corrections do not commute, which is the practical detail most easily got wrong.

A tunnel measurement produces a set of forces at a set of geometric incidences, referred to a dynamic pressure the tunnel believes it has. Three things are then wrong with it: the dynamic pressure is too low because the model blocks the section, the incidence is too small because the walls induce upwash, and the drag is too small for the same reason. Correcting them in the wrong order mixes the errors.

The order used in practice is: blockage first, because it changes the dynamic pressure that every coefficient is divided by and therefore changes the lift coefficient that the interference correction is proportional to; then lift interference, which moves the incidence and the induced drag; then any streamline-curvature term, which is a change in effective camber and therefore a change in the zero-lift angle rather than in the slope.

Each of those is a small number and their product is not. For a model at six per cent of the section’s area at a lift coefficient of 0.8, blockage is a per cent or two on the dynamic pressure, the incidence correction is a third of a degree, and the drag correction is fifteen per cent of the induced drag. Applied in the right order they combine to a lift-curve slope correction of a few per cent; applied in the wrong one they can differ by as much again. The corrections are a calculation, not a set of allowances, which is why every tunnel keeps its own documented sequence and why comparing data between tunnels means comparing their sequences first.

Who worked it out, and the connection worth keeping #

Prandtl — whose lifting line this is an application of — had the closed-tunnel correction by 1919 and Glauert tabulated the rectangular cases through the 1930s, hand-summing image series that this site could not persuade to converge. What has changed is not the answer but what is cheap: the boundary condition can now be solved for directly on any section, including the ones with fillets and slots that no image system will accept.

The connection worth keeping is with the essay before this one. A box wing’s wake is closed on itself and a tunnel closes the boundary round it, and both closures produce the same kind of mathematical object — an operator with a null space, a solvability condition, and an answer that is fixed only once somebody says which member of a family is wanted. In the box wing’s case the freedom is a circulation that costs nothing; here it is a source strength that emits nothing. Both are harmless physically and fatal numerically, and both were found by the arithmetic refusing rather than by anyone thinking about it in advance.

Where the ladder goes next #

The rung above is the slotted wall — a mixed boundary condition with a measured coefficient, chosen so that the two errors cancel — which is where tunnel design became an optimisation rather than a correction.

The one beside it is the ground plane, which is a tunnel with one wall and no others, and where the same image system produces a saving rather than an error. That is what a wing near the ground actually gets, and the answer is not the one the phrase ground effect suggests.