The wing that is flat, and flies

This collection refutes the equal-transit story by computing the parcels and showing that they do not meet. That refutation is correct and it leaves something standing.

What it leaves standing is the story’s premise, and the premise is the part most readers actually believe: that a wing has to be curved on top, that the shape is what makes the lift, and that a flat or symmetric section could not fly.

That is a claim about shapes. It is therefore testable with shapes, which is the easier half.

A plate #

A flat plate at incidence has upper and lower surfaces of identically equal length. Not nearly equal: they are the same segment, so the path-length excess is zero as an identity rather than as a small number.

Its lift-curve slope is exactly $2\pi$ per radian, with no arithmetic in it at all, and it flies. Model aircraft have been built from flat plates for a century; a sheet of card at an angle to a stream produces lift, and every wind tunnel in every teaching laboratory has demonstrated it.

Whatever makes a wing fly, it is not a difference between two path lengths, because here there is not one.

A symmetric section #

A symmetric section is thick — it has a nose, a maximum thickness, a rounded upper surface — and its two surfaces are the same curve reflected. So its path-length excess is exactly zero at every thickness, and its lift curve is the flat plate’s.

At six, twelve and twenty-one per cent thick it is the same line, and a lift coefficient of 0.4 comes at 3.65 degrees.

That disposes of a natural retreat from the plate case. Somebody might grant that a thin plate is a special case and hold that a shaped section works by its shape. A symmetric section is fully shaped and has no excess, and it flies exactly as well.

A section flown upside down #

Turn a four per cent cambered section over. Its longer surface is now on the bottom, so the story predicts a downward force at every attitude.

It makes a lift coefficient of 0.4 at 7.80 degrees, against minus 0.51 upright. The camber’s contribution has changed sign and the incidence pays for it, which is why an aerobatic aircraft can fly inverted and why doing so costs some incidence and some drag rather than being impossible.

Three shapes, three refutations, none of which requires computing a single streamline.

There is a fourth case that costs nothing and is worth adding, because it is the one a reader can check without any apparatus at all. A paper aeroplane is a flat plate. So is a kite, so is a hand held out of a car window, so is a sheet of plywood on a trailer. All of them make lift, all of them have upper and lower surfaces of exactly equal length, and none of them is a marginal case.

The shape story has to say something about each of those, and what it says is that they should not work. That is a considerable amount of everyday evidence against a premise, and it is available before any calculation.

The premise’s usual defence at this point is that a flat plate is inefficient, which is true and is a different claim. A plate at incidence has a sharp leading edge, so it separates early and has a lower maximum lift and a higher drag than a shaped section. Efficiency is what a shape buys. Lift is not.

The premise as a correlation #

The three cases above are each a counter-example. A stronger test is whether the two quantities are related at all across a family, and they are not.

Take one camber line at six and at twenty-one per cent thickness. The upper surface’s excess goes from 1.32 to 3.77 per cent, nearly a factor of three. The lift coefficient does not move by one part in 10¹³, because thin-aerofoil theory reads the mean line and the thickness is not in it.

And the family contains a pair the wrong way round. A section with a wavy upper skin has an excess of 2.52 per cent — twice a four per cent cambered section’s — and makes 0.437 against its 0.833. Twice the path-length difference, half the lift. If the story described a mechanism, that pair could not exist.

Why thickness is invisible and camber is not #

The thickness result deserves an explanation rather than only a number, because it is the sharpest part of the refutation and it looks like a coincidence.

Thin-aerofoil theory represents a section by a vortex sheet along its mean line — the average of the upper and lower surfaces — and computes the lift from the sheet’s strength. The thickness distribution is the difference of the two surfaces, and it is represented by a source distribution that produces no circulation and therefore no lift.

So the split is exact rather than approximate: mean line to lift, thickness to displacement. Adding thickness to a section changes its pressure distribution, its suction peak, its drag and its stall, and changes its lift by nothing at all in this theory and by very little in reality.

The path-length excess, meanwhile, comes almost entirely from the interaction between camber and thickness. A symmetric section has zero excess at any thickness and a flat camber line has zero excess at any camber; put both together and the upper surface, being the sum of a rising camber and a thickness, is longer than the lower, which is the difference.

That is why the excess and the lift are unrelated. One is a property of the mean line; the other is a product of the mean line and the thickness. They share a factor and neither determines the other, which is exactly the pair the family above demonstrates.

The story’s own arithmetic, priced #

It is worth putting a number on the story rather than only refuting its premise, because the size is part of the answer.

Taken at its word — the parcels meet, so the upper flow is faster by the path-length ratio, so Bernoulli gives a pressure difference — the story produces lift coefficients between 0.013 and 0.077 for sections that make between 0.35 and 0.83. It is out by factors between nine and forty-six.

And inverting its own arithmetic: for the story to produce a cruise lift coefficient of 0.4, the upper surface would have to be about eighteen per cent longer than the lower. Real sections are between half a per cent and four. There is no aerofoil shaped like that and there could not be one.

The trend is the wrong way round too. The story’s factor is worst on the thinnest section — forty-six on a six per cent one against eleven on a twenty-one per cent one — which is backwards for a story in which the thickness is doing the work.

Why the premise is so durable #

It is worth asking why a claim with this much evidence against it survives, because the answer is not that people are careless.

The correlation is real in the sample most people see. Nearly every aeroplane wing is cambered, so nearly every wing a reader has looked at does have a longer upper surface, and the premise is consistent with all of them. It fails on the cases people do not look at — the flat plate, the symmetric section, the aerobatic aeroplane inverted — and those are not part of the everyday sample.

The story explains something. It offers a chain from a visible feature to a familiar equation to the answer, and a chain like that is satisfying in a way that “circulation” is not. Being wrong is a much smaller defect in an explanation than being unavailable.

And the true account requires a quantity that cannot be seen. Circulation is a line integral of the velocity round a closed curve. There is nothing in a photograph of a wing that is the circulation, and there is something in a photograph that is the shape.

That is the general shape of a durable misconception, and it is worth recognising because it predicts where the next one will be. A visible cause, a familiar mechanism, and a correlation that holds in the ordinary sample — three properties that between them make a claim comfortable, and none of which is evidence.

What the premise and the mechanism are, separately #

The equal-transit story has two parts and they fail independently.

The mechanism — that the parcels meet at the trailing edge — is false, and computing the parcels shows it. The upper parcel arrives well ahead of the lower one, by a margin far larger than the path lengths would give.

The premise — that the shape is what makes the lift — is also false, and shapes show it. A plate with no shape flies, a symmetric section flies, and a cambered section flies inverted.

Refuting one leaves the other standing, which is why both are worth doing. A reader who has been shown that the parcels do not meet may conclude that the story is nearly right and needs a better account of how much faster the upper flow is. That reader still believes the premise, and the premise is the more consequential belief, because it is the one that makes a wing’s shape seem like the explanation.

The anchor, tested for closure #

This is the second essay under the equal-transit claim, and it was written to ask deliberately whether the claim is finished. It is worth recording the answer.

The claim has two parts and both have now been answered from the direction that suits them: the mechanism by computing the parcels, the premise by building the shapes. What remains — that the upper flow is faster, that the pressure there is lower, that air is pushed down, that the shape matters — is a set of true statements, each of which belongs to an anchor that already holds it and none of which is a further refutation.

So a third essay here would restate one of the two. That is what closure means in this collection: not that nothing more could be written, but that every remaining candidate argument would repeat one already made.

The same test applied to the neighbouring claims in this field gives the same answer for some and not for others. The bee that cannot fly is answered once and the remaining arguments about unsteady insect aerodynamics belong to a different anchor; the siphon that does not need the air is a single mechanism with a single refutation. Against that, what a photograph of a flow shows has more to say, because there are several distinct ways for an image to mislead and they do not reduce to one another.

The general shape of it is worth stating, since it is not obvious in advance. A false claim has a bounded number of distinct refutations and a mechanism has an unbounded number of distinct depths. Which is why the misconceptions in this collection are the anchors that close first, and why closing them is a finding rather than a shortage.

What is left of the claim #

What survives from the story is a set of true statements that belong elsewhere.

The flow over the top is faster. True, and it is the circulation that makes it so rather than the path length.

The pressure there is lower. True, and it is fast meaning low pressure along a streamline, which holds for the reason that essay gives.

Air is pushed down. True, and the usual sum done with it is wrong for a reason that has nothing to do with this page.

And the shape matters. Also true, and not in the way the story says: the camber line’s first three Fourier coefficients set the lift and the moment, and everything above them changes the shape and neither force. Shape decides the pressure distribution, the stall and the drag. It does not decide whether there is lift.

What would have made the premise testable earlier #

There is a lesson about how to phrase a claim that this page illustrates, and it is worth extracting because it applies to the other stories in this field.

The equal-transit story as usually told is a mechanism — the parcels meet — and mechanisms are hard to test without computing the flow. Its premise is a correlation between two measurable properties of a shape, and correlations are easy to test: build a family, measure both, and look.

So the useful move when meeting a plausible account is to find the correlation it implies and test that first, because it is cheaper. Here the correlation was “more path-length excess, more lift”, and it fails on a family of seven sections that took an afternoon to define.

This collection has used the same move before. The bee that cannot fly implies that quasi-steady aerodynamics should predict insect lift, which is a testable correlation and fails. The cushion that is not there implies that ground effect should be a pressure build-up under the wing, which is testable and is not what a solution shows. In both, the mechanism took a computation and the implied correlation took a comparison.

Find the implied correlation. It is nearly always cheaper to test than the mechanism, and where it fails, the mechanism cannot be right whatever else is true of it.

What actually makes the flow over the top faster #

Having refuted the shape account, it is worth stating the true one compactly, because a refutation that leaves nothing in its place is not much use.

The section has circulation round it. That circulation is not a free choice: of the infinitely many potential flows past a given shape at a given incidence, all but one turn the flow round the sharp trailing edge at infinite speed, and the one that does not is the one nature picks. Fixing that condition fixes the circulation.

Circulation added to a uniform stream adds to the speed on one side and subtracts on the other. That is why the flow over the top is faster: not because it has further to go, but because the circulation is superposed on it in the same direction there and in the opposite direction below. The speed difference is then whatever the circulation requires, which is far more than any path-length difference would give — and it depends on incidence, which a path-length difference does not.

The lift follows from the circulation by a formula that does not ask what the shape is, and the shape enters only through what circulation the Kutta condition demands of it. That is the whole chain, and the shape appears once, near the end, as an input to a condition rather than as a cause of a speed.

Which explains every case on this page in one sentence. A flat plate has a sharp trailing edge, so it has a Kutta condition, so it has circulation, so it flies. A symmetric section has the same at incidence and none at zero. A cambered section inverted has circulation of the opposite sign at zero incidence and needs enough angle to overcome it.

What a reader should take from the shape family #

The family of sections used here is worth keeping as a mental tool, because it settles the question faster than any argument does.

Four shapes, four sentences. A flat plate has no shape difference and flies. A symmetric section has none either and flies. A cambered section flown upside down has the difference the wrong way and flies. A section with a bumpy upper skin has twice the difference and half the lift.

Any one of those is enough. Together they close the question, and none of them requires computing a flow, measuring a pressure, or knowing what circulation is — which is what makes them worth carrying into a conversation where the story is being told.

The positive statement takes one more sentence and is worth having beside them. The lift is set by the circulation, the circulation is set by the condition at the sharp trailing edge, and the shape enters only by deciding how much circulation that condition demands. Shape matters enormously to how well a wing works and not at all to whether it works, and keeping those two apart is the whole of what this essay is for.

A closing note on where the story does apply. A surface with a longer path on one side does produce a small force by the mechanism the story describes, and it is the size computed here — a lift coefficient of a few hundredths. That is not nothing; it is simply two orders below what a wing at incidence makes. The story is not describing a mechanism that does not exist. It is describing one that exists and is negligible, which is a more precise complaint and a harder one to argue with.

What is not claimed #

Camber is not useless. A cambered section makes lift at zero incidence and reaches a given lift coefficient at a lower angle, which is worth having: less incidence means a smaller suction peak, a gentler adverse gradient and a higher maximum lift. The claim refuted is that camber is necessary, not that it is pointless.

Inverted flight is not free. The section needs 7.80 degrees rather than minus 0.51 for the same lift, which is more induced drag and less margin to stall, and a real aerobatic aircraft uses a symmetric section for exactly that reason.

The lifts are thin-aerofoil theory’s. A real section’s lift-curve slope is a little different from $2\pi$ and its zero-lift angle a fraction of a degree from the theory’s. The statements refuted here are qualitative — that a flat plate cannot fly, that a symmetric section cannot — and no accuracy in the theory is needed to refute them.

And the wavy-skinned section is a construction. It is built to have a long upper surface for a reason unrelated to lift, which is exactly what the test needs, and it is not a section anybody would manufacture. The counter-example establishes that the correlation fails; it does not claim that real aerofoils are distributed that way.