The model that cannot be matched

The reason a model tells anything about the full-size article is dynamic similarity: if the dimensionless groups governing the flow are the same, the flows are the same shape, and every coefficient measured on one applies to the other.

That is one of the most powerful ideas in engineering, and it comes with a condition that is usually skated over. All the governing groups have to match. Not the important one — all of them.

What similarity buys #

Non-dimensionalise the equations of motion and the physical parameters disappear into a small number of groups. For an incompressible flow past a body there is exactly one — the Reynolds number — and everything else is geometry.

That single fact is what makes a wind tunnel worth building. A flow at Re = 10⁶ round a shape is the same flow whatever the shape’s size, whatever the fluid, whatever the speed, so long as the product comes to 10⁶. A quarter-scale model in air at four times the speed is not an approximation to the full-scale flow; it is the full-scale flow, at a different size.

The lift coefficient, drag coefficient and pressure coefficient measured on the model are then the full-scale values exactly. Nothing has to be corrected. That is the promise.

It is worth stating how strong that promise is, because the strength is what makes the failure of it matter. Similarity does not say the two flows are approximately alike, or alike in the features anybody cares about. It says they are the same solution of the same equations written in different units. There is no error term to bound and no accuracy to quote.

Where the second number comes from #

The promise holds while there is one group. Air has a second, and it arrives as soon as speeds get high enough for density to change.

The Mach number measures how fast the flow is going compared with the speed at which pressure information travels through it, and above about 0.3 it stops being ignorable. An aircraft cruising at 240 metres a second is at Mach 0.71, which is nowhere near ignorable.

So a model of that aircraft has two conditions to satisfy, and one control — the tunnel speed.

The arithmetic that does not work #

Take a wing of 3.2 metres chord at 240 metres a second in sea-level air. Its Reynolds number is 5.3 × 10⁷ and its Mach number is 0.71.

Now make a quarter-scale model, chord 0.8 metres.

To match the Reynolds number, the speed must go up by four, to 960 metres a second. That is Mach 2.82. The model is now supersonic, has shock waves all over it, and is testing a completely different flow.

To match the Mach number, the speed must stay at 240. The Reynolds number is then 1.3 × 10⁷, four times too small — and four times too small in Reynolds number is not a detail. It changes where transition happens, how thick the boundary layer is, and whether the flow separates.

There is no third option. The two conditions demand different speeds, the difference is exactly the scale factor, and no amount of care with the tunnel resolves it.

What the solver computed #

Both loci in the figure are computed from the definitions rather than sketched. For each of a range of tunnel speeds, the Reynolds number is UL/ν and the Mach number is U/a, both evaluated at sea-level air properties, and the pair is plotted.

The full-scale curve and the quarter-scale curve are parallel on the logarithmic Reynolds axis, displaced by exactly log 4 — which is the whole geometry of the problem in one sentence. Two parallel curves have no intersection, so there is no operating point that lies on both.

The marked points are the two candidates and the dashed line between them is the gap. That gap is 0.60 in log₁₀ Reynolds and 2.11 in Mach, and closing either one opens the other.

What tunnels actually do about it #

Since the conflict cannot be dissolved, it is managed, and the ways of managing it are most of what distinguishes one wind tunnel from another.

Pressurise. Reynolds number is proportional to density, so a tunnel running at five atmospheres gets five times the Reynolds number at the same Mach number. This is why high-Reynolds transonic tunnels are pressure vessels, and why they are so much more expensive than their size suggests: the working section has to be inside a container rated for the pressure, and so does everything a person needs to reach.

Cool it. Viscosity falls with temperature and density rises, and the speed of sound falls too — so cooling the working fluid raises Reynolds number and lowers the speed needed for a given Mach number. Cryogenic tunnels run at around 110 kelvin in nitrogen and achieve full-scale Reynolds numbers on half-metre models. They are enormously expensive to run.

Change the fluid. Some tunnels use heavy gases with a lower speed of sound and higher density, which shifts both numbers at once. The catch is that a different gas has a different ratio of specific heats, and that ratio is itself one of the governing groups in compressible flow — so fixing two mismatches introduces a third.

Accept the mismatch and correct for it. Which is what almost everybody does: test at matched Mach, accept a Reynolds number an order of magnitude low, and apply a correction derived from theory, from tests at several Reynolds numbers, or from previous aircraft. This works, and it works because the dependence on Reynolds number is usually weak and smooth in the range concerned. It stops working exactly where the dependence is not weak, which is near the stall and near the drag rise.

The failure that got a name #

There is a well-known instance of the mismatch being missed and it is worth having, because it is the best available argument for taking any of this seriously.

Wind-tunnel testing of aircraft in the 1930s and 1940s was routinely done at Reynolds numbers ten or twenty times below flight. For most purposes the errors were tolerable. For maximum lift coefficient they were not: a section that stalls gently at full-scale Reynolds number can stall abruptly at model scale, or the reverse, because where the layer separates depends on whether it is laminar or turbulent when it gets there.

Aircraft were consequently delivered with stalling behaviour different from what the tunnel had predicted — sometimes better, sometimes very much worse. The NACA built the Variable Density Tunnel in 1922 specifically to fix this, pressurising to twenty atmospheres to reach flight Reynolds numbers on small models, and the aerofoil catalogues that came out of it are the ones still cited.

The general lesson is the one worth carrying: the mismatch is harmless where the flow depends on the mismatched number weakly, and dangerous precisely where the interesting behaviour is.

That is an uncomfortable shape for an error to have. A mismatch that degraded everything a little would be manageable, because it could be bounded. A mismatch that is invisible across most of the envelope and then decides the answer at the stall is one that gives a false sense of precision right up to the point where it matters, which is a far worse property. It is also why flight test has never been replaced by tunnel test, and why the first flight of a new aircraft still finds things.

The same conflict elsewhere #

This is not an aerodynamic peculiarity. Any flow with two governing groups has the problem, and the resolutions are all the same shape.

A ship has a Froude number and a Reynolds number. Matching Froude requires the model speed to go as the square root of scale; matching Reynolds requires it to go as the inverse of scale. They pull in opposite directions, which is worse than the aerodynamic case, where at least one condition could be met. Froude’s answer — described in the wake essay — was to match Froude, compute the friction separately at both scales from a flat-plate formula, and subtract and re-add it. That is a decomposition rather than a similarity, and it rests on the two contributions being independent, which they very nearly are.

A helicopter rotor has Reynolds, Mach and a tip-speed ratio, and things get worse still.

A flapping wing adds a Strouhal number, which sets the ratio of flapping speed to forward speed and which turns out to be tightly constrained across an extraordinary range of animals — birds, bats, insects and fish nearly all cruise between 0.2 and 0.4. That constancy is the same kind of result as the wake angle: a dimensionless number that refuses to vary across four orders of magnitude of size, and therefore says something structural rather than incidental.

The model is similar and its surroundings are not #

The two numbers are properties of the model. There is a third mismatch that belongs to the room, and it is the one that decides how a tunnel is built.

An aircraft flies in air that extends forever. A model sits in a duct a few metres across, and the walls enforce a condition the sky does not. Their effect is exactly the image system a solid boundary always implies: the model displaces air that cannot move outwards, so the flow past it is faster than the reference speed — solid blockage — and its wake does the same again. Both are why a model is kept to about one per cent of the working section’s area, which is most of the reason a tunnel is so much larger than the thing in it.

The lift interferes too, and instructively the sign depends on the tunnel. Closed walls prevent the wake’s downwash from spreading, so the model is measured at a smaller effective induced angle than it would fly at: the lift-curve slope reads high and the induced drag reads low. An open jet does the opposite. Some laboratories tested the same model in both and took the answer to lie between.

The severe case is transonic, and it stopped the subject for years. As the Mach number rises, the gap between the model and the walls acts as a nozzle throat, and when that gap goes sonic the tunnel chokes: the mass flow is fixed, the working section can no longer be accelerated, and there is a band of Mach numbers either side of one in which no measurement of any kind is possible. That band covered precisely the speeds aircraft were arriving at in the 1940s, and the era’s talk of a barrier owed a good deal to the instruments having a hole in them. The fix was to stop making the walls solid — slotted and perforated walls, opening into a plenum, letting the excess flow escape sideways rather than choking.

What the picture cannot show #

The map plots two numbers and there are more than two. A real test also has to worry about turbulence level in the tunnel, surface roughness relative to the boundary layer, the ratio of specific heats if the fluid has been changed, and the aeroelastic behaviour of a model that is stiffer in proportion than the aircraft.

Turbulence level is the one that most often bites. A tunnel with a disturbed flow will trip the boundary layer to turbulence earlier than the smooth air an aircraft flies in, which partly compensates for the low Reynolds number — and the compensation is uncontrolled. Two tunnels at the same Reynolds number can give different answers for that reason, and reconciling them occupied a great deal of effort in the 1950s.

The figure also treats sea-level air as the only fluid, so the loci are those of a particular tunnel. Pressurising, cooling or changing gas moves the curves, which is the whole point of doing any of those things, and the figure shows none of it.

Where the model stops #

The essay assumes the flow is governed by Reynolds and Mach and nothing else, which is true for a rigid body in a uniform stream of a perfect gas and is a simplification of every real test.

It also assumes the flow is steady, which rules out anything where a frequency matters. Add an oscillation — a flapping wing, a fluttering panel, a shedding wake — and a reduced frequency appears as a further group, and it has to be matched too.

It also assumes geometric similarity, which is quietly demanding. A quarter-scale model must be a quarter-scale everything: every rivet, every panel gap, every surface finish. Roughness that is negligible on an aircraft is proportionally four times larger on a quarter-scale model, and roughness matters through its ratio to the boundary-layer thickness, which is not the same ratio.

And nothing here addresses whether the right numbers have been identified. Dimensional analysis says how many groups there are; deciding which physical quantities belong in the list is a judgement, and a forgotten quantity produces a forgotten group and a similarity that silently does not hold. That is the deepest way this can go wrong and it cannot be checked from inside the calculation.

The surprise: coefficients are not constants #

There is a habit of mind this essay exists to interrupt.

A drag coefficient looks like a property of a shape. It is written as a number, it is tabulated as a number, and it is used as though a sphere simply has a Cd of 0.47.

It does not. The coefficient is a function of the Reynolds number and, above Mach 0.3, of the Mach number too, and for a sphere it varies by a factor of five across the range where people actually use it — collapsing abruptly by about 80% at the drag crisis, where the boundary layer turns turbulent and separation moves aft.

So a coefficient is a reduced quantity rather than a constant, and the whole apparatus of similarity is what makes the reduction possible. Non-dimensionalising removes the dependence on size, speed and fluid, and it leaves the dependence on the groups. Those groups are the axes the answer lives on, and a number quoted without them is a point without coordinates.

Who found it, and when #

Dimensional analysis as a formal method is Buckingham’s, in 1914, and the theorem that says how many independent groups a problem has carries his name. Rayleigh had been doing it informally for decades before and was irritated by the formalisation.

The physical idea is much older, and its practical history runs through ships rather than aircraft. William Froude established in the 1860s that model testing works only under matched similarity, built the first ship tank at Torquay to do it properly, and had to argue the Admiralty into paying for it. Osborne Reynolds’s pipe experiments of 1883 did the same thing for viscous flow, and the number that carries his name was not called that until Sommerfeld named it in 1908.

What is striking is how late the conflict was appreciated. Matching one number was understood by 1870. The impossibility of matching two, and the machinery for living with it, are a twentieth-century development, and they arrived because aircraft made a second number unavoidable.

There is an irony in the order of events. Froude, working on ships, had a two-number problem from the start and solved it by decomposition in the 1870s. The aerodynamicists inherited a one-number problem, built forty years of practice on the assumption that matching Reynolds number was sufficient, and then had to rediscover Froude’s difficulty from scratch when speeds rose past Mach 0.3 in the 1930s. The naval architects had the harder problem first and the aeronautical engineers did not read their papers.

Where the ladder goes next #

Next rungs on this anchor: the Buckingham π theorem worked properly, and how the number of groups is counted; scale effect as a measured quantity, and how corrections are established; the cryogenic tunnel, and what it costs to buy a way out of the conflict; and free-flight and drop-model testing, which sidestep the tunnel entirely by testing at full Reynolds number and giving up on instrumentation.

Then across to the Reynolds number, where the arbitrariness of the length in it turns out to be a related problem, and to the wake angle, where the second number is Froude rather than Mach and the conflict is worse.