One coefficient, two errors

The walls are in the answer computes what a wind tunnel’s own boundaries do to a measurement, and its central result is a sign. A closed section constrains the flow and the correction is of one sign; an open jet lets it expand and the correction is of the other, with the same magnitude in the case that essay solves. The two errors bracket the truth.

Anything that brackets the truth invites the obvious move: build a wall between the two, with a coefficient, and set the coefficient so that the error is nothing. That move was made in the late 1940s, the wall was a set of longitudinal slots, and it is why transonic wind tunnels exist.

The reason it is why transonic tunnels exist, rather than merely a refinement, is worth stating at the outset. A solid-walled tunnel with a model in it has a minimum area at the model, and a minimum area chokes: past some Mach number below one the tunnel passes no more mass whatever is done to it, and the flow at the model stops responding to the settings. An open jet does not choke and cannot be run transonically for other reasons — the jet boundary is unstable, and reflected expansions from it return to the model. Between the day the problem was understood and the day the slots were cut, the Mach numbers between about 0.85 and 1.2 were unreachable in a wind tunnel, and aeroplanes were being designed for them from theory and from flight test.

A wall with a coefficient in it #

A wall made of streamwise slots is neither solid nor absent. Averaged over a distance large against the slot spacing, it satisfies a homogeneous mixed condition on the perturbation potential,

$$\varphi + K\,\frac{\partial\varphi}{\partial n} = 0,$$

with $K$ a length that carries the slot geometry — their width, their spacing, and what is behind them. Setting $K = 0$ gives $\varphi = 0$, which is constant perturbation pressure and therefore a free jet. Letting $K \to \infty$ gives $\partial\varphi/\partial n = 0$, which is a solid wall, built out of reflections. Everything between them is a slotted wall, and $K$ is what a designer chooses.

The problem is linear and the walls are straight, so transforming along the tunnel turns it into one algebraic equation per wavenumber. The model’s own singularity is known in closed form in that variable — a doublet for its volume, a vortex for its lift — and the correction field is a combination of cosh and sinh. Nothing here is a numerical field solution; it is one integral over wavenumber per quantity.

The reason a transform is needed at all is the thing that makes a slotted wall different in kind from the two classical ones. A solid wall reflects a singularity as an identical image; an open jet reflects it as an image of opposite sign; in both cases the image system is a row of points and the interference is a sum. A mixed condition reflects differently at every wavelength, so there is no image — or rather there is a continuum of them, which is what a Fourier integral is.

That is not a technicality about method. It is the statement that a slotted wall does something neither extreme does: it treats the long-wavelength part of a disturbance differently from the short-wavelength part, and everything interesting below follows from that one fact.

The two errors #

The first is blockage: the walls put an extra streamwise velocity at the model, so the model is flying faster than the tunnel’s reference instrument says. It runs from $+0.0654$ at an open jet to $-0.1307$ at a solid wall, in units set by the model’s own doublet strength.

The second is streamline curvature: the walls bend the flow through the model, which is indistinguishable from the model having a different camber. In two dimensions this rather than a uniform upwash is the lift interference, because a vortex and its images produce no net vertical velocity at the model and do produce a gradient of one. It runs from $+0.1309$ at an open jet to $-0.0654$ at a solid wall.

That cross-relation is not something anybody arranged, and it is worth a sentence. The two integrands are the same expression with $\cosh$ and $\sinh$ exchanged, because one field is symmetric about the tunnel’s centreline and the other is antisymmetric; exchanging the wall condition between Dirichlet and Neumann exchanges them too. A lifting model in a closed tunnel and a thick one in an open jet see the same wall.

It has a practical use as well as a pleasing one. Any of the four numbers determines the other three, so a tunnel that has been characterised for one kind of model at one kind of wall has been characterised for all four cases — and a measurement of a solid wall’s blockage, which is the easiest of the four to make, predicts an open jet’s curvature, which is the hardest. Whether that transfer is ever used in practice is doubtful; whether it is available is not.

The factor of two, and where it comes from #

Before the slots, the two extremes are worth comparing, because they are not symmetric and the asymmetry is the thing the classical account leaves implicit.

A solid wall’s blockage is exactly twice an open jet’s, with the opposite sign. A solid wall’s curvature is exactly half an open jet’s, again with the opposite sign. So a designer choosing between the two classical walls is not choosing between two equal errors: for a thickness measurement the open jet is twice as good, and for a lift measurement the solid wall is twice as good.

That already explains a division of practice that looks arbitrary from outside. Open-jet tunnels were used for automotive and for propeller testing, where blockage is the dominant concern and the model is large; closed tunnels were used for aerofoil work, where the lift and the moment are what is wanted. Neither choice was a compromise; each was the right end of a two-to-one trade, and the slotted wall arrived because transonic testing needed a third thing that neither end provides.

Two nulls, and they are not the same null #

Both errors change sign between the two extremes, so each has a slot parameter at which it vanishes. Both nulls are computed here by bisection:

$$\left.\frac{K}{H}\right|_{\text{blockage}} = 1.1844, \qquad \left.\frac{K}{H}\right|_{\text{curvature}} = 1.5902.$$

They are thirty-four per cent apart, and a wall has one coefficient.

Neither leftover is negligible and neither can be removed without reintroducing the other. So a slotted tunnel is not a tunnel with no interference. It is a tunnel in which one error has been chosen to be zero and the other has been reduced, and which one is chosen is a decision about what the tunnel is for.

The asymmetry of the two leftovers is worth reading rather than merely noting. Nulling the blockage leaves twenty per cent of the curvature; nulling the curvature leaves seven per cent of the blockage. So nulling the curvature is the better bargain, in the narrow sense that it leaves less of the other error behind — and it is not what is done, because the two errors are not equally damaging.

A residual blockage is a velocity error, and a velocity error is a scale error: every coefficient measured comes out wrong by a known factor and can be corrected afterwards from a measurement of the wall pressures. A residual curvature is a change of effective camber, which alters the shape of the lift curve and the moment about the quarter chord, and cannot be undone by multiplying anything. Transonic tunnels are therefore set to null the blockage — which is also what they were built to do, since a solid-walled tunnel’s blockage is what chokes it before it reaches Mach one.

Why one coefficient cannot do both #

The reason is visible in the spectrum rather than in the totals.

A slotted wall is not transparent at any wavelength. It reflects long waves nearly as a solid wall does and short ones nearly as an open jet does, and the null setting is the one at which the positive and negative contributions happen to balance. That is a cancellation between bands rather than a suppression within each of them.

The two errors weight those bands differently — the blockage integrand carries $\cosh$ in its denominator and the curvature’s carries $\sinh$ — so the setting that balances one does not balance the other. The wall is a filter with one parameter, and it is being asked to flatten two different spectra at once.

Put that way, the thirty-four per cent between the two nulls stops being a disappointing number and becomes a measurement of how different the two spectra are. Had the two errors weighted the wavelengths identically the nulls would have coincided exactly and a slotted wall would be a complete answer; had they weighted them very differently the nulls would be decades apart and the wall would be no use for either. A third of a tunnel height apart is the middle case: close enough that one setting does most of the job for both, and far enough that neither is done.

That is the general shape of what a one-parameter boundary can achieve, and it applies well beyond tunnels. A single adjustable property can null one functional of a spectrum exactly and reduce every other one by whatever the overlap between the spectra happens to be.

What is transferable, which is the design rule #

The null is a fixed multiple of the tunnel’s own height, and it is a property of the wall rather than of the model. That is the whole reason the approach is usable: a slot geometry developed on a small tunnel transfers to a large one by scaling a length, and the same wall serves every model put in it.

It is worth saying what would have happened had that not been true, because the alternative is what a reader might expect. If the null depended on the model as well as on the wall — on its thickness, its lift, its length — then a slotted tunnel would need its walls reset for every test, which is to say it would be a calibration and not a wall. The reason it does not is that the interference is linear in the model’s strength: doubling the doublet doubles the blockage at every slot parameter, including at the null, where twice nothing is nothing.

Linearity is what makes a null a property of the boundary, and it is the same reason a matched transmission line does not care what is being sent down it. It is also the first thing transonic testing loses, because the flow around a model near Mach one is not linear in anything — which is the subject of the essay that measures how badly.

It is also the point at which the idealisation starts to matter. The homogeneous condition is an average over the slots, and the averaging is legitimate only at wavelengths long against the slot spacing. The spectrum above shows that the cancellation depends on the behaviour at wavenumbers of order one on the tunnel’s height, which is safely long — but a model whose own length is comparable with a slot is being measured by a wall that has no boundary condition at all.

What the picture cannot show #

The tunnel is two-dimensional. A real transonic tunnel is a rectangular box with slots in two walls or in four, and the interference is a three-dimensional problem in which the model’s own size against the section brings the aspect ratio in as a second parameter. The structure of the argument survives; the two null values do not.

The wall condition is linear and homogeneous. A real slotted wall has flow through the slots into a plenum, the flow through them is not proportional to the pressure across them when it separates, and the effective $K$ therefore depends on the strength of the disturbance — a slot carrying flow into a plenum being not a nozzle however much it resembles one. Perforated walls, which are the other common arrangement, are worse in this respect and better in others.

There is no viscosity. The boundary layer on the solid parts of a slotted wall is thick, displaces the flow, and changes the effective open-area ratio — which is one of the reasons the measured $K$ of a real wall is not the geometric one.

And the model is a point. Everything above puts the singularities at the tunnel’s centreline and computes the interference there. A model of finite size samples the interference field over its own extent, and the gradient of the interference across it is a third error with a third null.

And the flow is subsonic everywhere. Every expression here comes from Laplace’s equation, which is what a linearised subsonic flow obeys. The tunnels this wall was invented for run at Mach numbers where the flow is not described by a linear equation at all, and the corrections computed from a linear theory are applied there because nothing better exists rather than because they are right.

Who found it, and when #

Slotted walls were proposed and tested at Langley in the late 1940s, by Wright and Ward among others, directly to solve the problem that a solid-walled tunnel chokes before it reaches Mach one and an open jet is unusable at transonic speeds. The first slotted transonic tunnels ran in 1950, and the homogeneous boundary condition used here — with a single parameter standing for the slot geometry — is Davis and Moore’s, of the same period. Baldwin’s and Goethert’s analyses through the 1950s established the null settings for the standard section shapes.

The surprising connection is with an entirely different kind of wall, and the correspondence is exact. An anechoic chamber is the same design problem: a boundary that must reflect nothing, made of a material with one adjustable property, facing a disturbance with a spectrum. A wedge-lined wall absorbs well at short wavelengths and poorly at long ones, for exactly the reason a slotted wall behaves like a solid one at long wavelengths — in both cases the boundary’s characteristic length has to be compared with the wavelength, and one length cannot be compared favourably with a whole spectrum at once. The acoustician’s answer is to make the treatment deep; the tunnel designer’s is to choose which error to null. Neither can make a boundary that is not there.

Still open: what a variable wall would be worth #

The whole difficulty is that a wall has one coefficient, and nothing says it has to.

A wall whose slot parameter varied along the tunnel would have a coefficient at each station, and the interference at the model is an integral over the wall — so in principle a distribution of $K(x)$ could null both errors at once, and probably several more. That is not a fantasy: adaptive-wall tunnels, in which the walls are flexible and are moved until the measured pressures on them match what an unbounded flow would produce, were built in the 1970s and 80s and work.

What has not been done, as far as the record shows, is the corresponding calculation for a slotted wall: given the two error integrals above, both linear functionals of $K(x)$, find the distribution that nulls both, and ask how far from uniform it has to be. If the answer is a few per cent the technique is worth having and is much simpler than a flexible wall; if it is a factor of two the slots would have to be so different at different stations that the homogeneous condition would stop describing them. The calculation is a linear inverse problem needing nothing that is not already in hand, and the interesting possibility is that the required distribution is not smooth — in which case the question becomes which smooth distribution gets closest, and the answer would be a design rather than an impossibility proof.