A right total from a wrong picture

This collection refutes the equal-transit story by computing the parcels and showing that they do not meet, and it refutes the Newtonian picture by computing what that theory says a wing can do.

Both refutations leave a question unasked, and it is the one a sceptical reader should ask: what would it have taken for a wrong picture to look right? Because a wrong mechanism with a free constant in it can be tuned to give the correct answer at one condition, and once it has been, no measurement of that condition tells it from the truth.

The story with no constant to tune #

Start with the one that has no freedom, because it sets the scale.

The equal-transit story is fully specified: the parcels meet at the trailing edge, so the upper flow is faster by the ratio of the path lengths, and Bernoulli converts the speed difference into a pressure difference. Every step is stated and nothing is free.

A NACA 2412’s upper surface is 1.215 per cent longer than its lower. That gives a lift coefficient of 0.0244. The section at five degrees makes 0.776.

A factor of thirty-two. The story is not a rough version of the answer; it is out by more than an order of magnitude, and this collection has said so before.

It is worth noticing how the arithmetic fails, because the failure is instructive about the story’s structure. A path-length excess of one and a bit per cent gives a speed excess of the same size, and Bernoulli turns a speed excess into a pressure coefficient of roughly twice it — so the story’s lift is fixed at about two and a half per cent of a dynamic head, whatever the incidence.

That is the second problem with it, and it is worse than the first: the story predicts a lift that does not depend on the angle of attack at all. The path lengths of a rigid section do not change when it is tilted, so the story gives the same lift at zero incidence, five degrees and twelve. A model that cannot produce a lift curve is not being slightly inaccurate about the lift; it has nothing to say about the quantity that matters most.

Inverting the story’s own arithmetic makes the size of the gap plainer. For it to produce a cruise lift coefficient of 0.4, the upper surface would have to be about eighteen per cent longer than the lower. There is no aerofoil shaped like that and there could not be.

The story with a constant #

Newtonian impact theory is different: the air is a stream of particles that strike the surface and bounce, so the force is proportional to $\sin^2\alpha$, and the constant in front of it depends on what is assumed about the bounce.

Choose that constant so the model passes exactly through the section’s lift at five degrees. It takes 102.6, which is a large number and is a number, and the model is then exact at five degrees — to the last bit of double precision, on a mechanism that is false.

Any measurement at five degrees now agrees with it. A balance, a pressure integration, a wake survey: all of them return the number the tuned model predicts.

What separates them #

Not the lift. The derivative of the lift.

A model tuned to a lift coefficient at an angle $\alpha$ has, at that angle, a slope of very nearly $2C_l/\alpha$, because $\sin^2$ is quadratic there and a quadratic through the origin has twice the slope of the chord. That is 17.67 per radian. The section’s slope is $2\pi$, which is 6.28.

A factor of 2.8, at a point where the lifts agree to sixteen decimal places.

That factor is not subtle to measure. Two runs a degree apart in a wind tunnel would show it. A single run at five degrees would not, however carefully it was made, because the disagreement is not in the quantity being measured.

There is a second discriminator that costs nothing: the zero-lift angle. A cambered section makes a lift coefficient of 0.23 at zero incidence and reaches zero lift at minus two degrees. A model built on $\sin^2$ of the incidence gives exactly nothing at zero incidence, whatever its constant, because the constant multiplies a function that vanishes there.

So the tuned model fails two tests that a single lift measurement does not perform, and both of them are cheap.

The constant is the diagnosis #

There is a third tell, and it is the one that generalises furthest.

Tune the model at two degrees and the constant is 610; at five, 103; at eight, 44; at twelve, 22. The constant a mechanism needs moves by a factor of thirty across the ordinary operating range.

A mechanism whose coefficient depends on the condition it was measured at is not a mechanism. A constant that has to be re-fitted for every operating point is not a property of anything; it is a record of the model’s error at that point, wearing the name of a physical parameter.

That is worth stating as a general test because it costs nothing to apply and it does not require knowing the right answer. Fit the model at several conditions; if the fitted parameter is stable, the model may be capturing something; if it drifts systematically, it is absorbing the discrepancy.

The general test, and where else it applies #

The pattern that has emerged is a test that can be applied to any model with a fitted parameter, and it is worth setting out in general because it needs no knowledge of the particular subject.

Fit at one condition and check at another. A model with $n$ free parameters can be made to agree at $n$ conditions and says nothing until the $(n+1)$-th. Reporting agreement at the fitting points is reporting the fit.

Check a derivative, not only a value. Derivatives are cheap — two nearby runs — and they are the first thing a wrong mechanism gets wrong, because getting a value right requires one number and getting a slope right requires the functional form.

Watch the fitted parameter across conditions. A drifting constant is the model absorbing its own error, and the drift is a measurement of how much error there is.

And prefer a test the model cannot have passed by construction. The zero-lift angle is such a test here: no choice of the constant can produce one, so the test discriminates before any fitting happens at all.

Those four are worth having because the situation recurs everywhere in this subject. A turbulence model with a calibrated constant, a drag correlation with a fitted exponent, a wall model with an additive constant, a discharge coefficient: all of them agree with the data they were fitted to, and all of them have to be tested somewhere else.

Why one agreement is worth so little #

The general statement is this collection’s own, turned round.

An exact constraint on a profile leaves an enormous family free, and the residual freedom is measurable. Agreeing with one measurement is the same statement about models: one measurement is one linear functional of the model’s predictions, the space of wrong models that satisfy it is as large as the space of profiles a wake survey cannot distinguish, and satisfying it is worth exactly as much as a single constraint ever is.

Two measurements cut the family down. A whole lift curve is a function rather than a number, and it is what the tuned model fails against. A pressure distribution is a function on the surface, and it would fail against that more comprehensively still — the tuned Newtonian model puts all of its load on the lower surface, and a real aerofoil carries most of it as suction on the upper.

Why the two stories fail differently #

The equal-transit story and the tuned Newtonian model are both false and they are false in different ways, and the difference is worth naming because it decides how each should be argued against.

The equal-transit story is falsified by its own numbers. It has no free parameter, it predicts a definite lift, and the lift is out by a factor of thirty-two. Nothing more is needed: the refutation is arithmetic, and it does not require any knowledge of what the right mechanism is. That is the strongest kind of refutation available and it is why this collection reaches for it first.

The Newtonian model is not falsified by any single number, because it has a parameter that absorbs one. Refuting it needs a second measurement, and choosing which second measurement is the skill: the slope and the zero-lift angle are the right choices because the model’s functional form cannot be right about them for any parameter value.

That distinction is the useful one for a reader meeting a new claim. A story with no adjustable constants can be tested by a single computation; a story with one has to be tested by a relation between two quantities. And a story with three or four adjustable constants — which is what most plausible-sounding accounts turn out to have once they are written down carefully — cannot be tested by a handful of measurements at all, and must be tested by its predictions of shape: which way things move, what is proportional to what, and what happens in a limit.

That last is why the limits matter so much in this subject. A wing at zero incidence, a flat plate, a section flown inverted — each is a case where the alternatives make qualitatively different predictions, and qualitative differences survive fitting.

What this does not excuse #

It would be easy to read all of this as a defence of a wrong model on the grounds that it can be made to fit. It is the opposite.

The point is that fitting is cheap and therefore proves little, which cuts against the model rather than for it. A mechanism that has to be tuned at every condition has been reduced to a curve fit with a physical story attached, and the story is doing no work — it is not predicting the slope, not predicting the zero-lift angle, not predicting the pressure distribution, and not predicting how any of them change with the section’s shape.

The true account predicts all four with no free constants at all. Lift is circulation, the circulation follows from the condition at the sharp edge, and the formula does not ask what the shape is. There is nothing to tune.

What a wrong picture costs when it is not tested #

There is a practical reason to care beyond tidiness, and it is worth stating because the tuned model is not obviously harmful.

A model that is right at one condition and wrong about the derivative is wrong about every increment, which is what most engineering questions actually ask. How much does the lift change if the incidence is raised a degree? What happens to the trim when the flap moves? How does the load redistribute in a gust? Every one of those is a derivative, and the tuned model is out by a factor of 2.8 on all of them while agreeing exactly on the lift itself.

It is also wrong about the shape of the dependence, which decides stability. A lift curve that steepens with incidence, as $\sin^2$ does, and one that is straight, as a real section’s is, give different answers to whether a disturbance grows — and a vehicle’s static stability is a statement about slopes rather than about forces.

And it is wrong about the pressure distribution, which decides everything structural and everything about separation. A model that puts the load on the lower surface predicts the wrong bending moment distribution, the wrong skin thicknesses, and no suction peak to worry about — while agreeing with the balance.

So a wrong picture that has been tuned is not harmlessly wrong. It is wrong in exactly the quantities that are asked of it after the measurement it was tuned against.

What the anchor looks like after this #

This is the second essay under the claim that air must be pushed down, and it is worth saying what the two together have established and what is left.

The first established that the momentum statement is true and that the usual sum done with it is wrong — the air is pushed down, and the volume of air involved is not the volume swept by the wing, so the arithmetic that produces a plausible-looking answer from a plausible-looking control volume is not the right arithmetic.

This one establishes something about the class of argument rather than about the claim: that a momentum picture, or any picture, can be arranged to produce the right total and that doing so is worth almost nothing. The two are complementary — the first says the sum was done wrongly, the second says that doing it rightly at one condition would not have settled anything either.

What is left under the anchor is not obviously another essay. The remaining candidates all restate one of the two: that the downwash is real, which the first essay computes; that the reaction is at the ground, which the far field already answers; that the momentum flux and the circulation are the same statement, which Kutta and Joukowski’s formula is. The claim has been answered from both directions, and a third rung would be a third telling.

The same test, applied to the models this collection uses #

It would be unfair to apply this test only to stories nobody defends, so it is worth turning it on the machinery this collection relies on.

Thin-aerofoil theory has no free constant at all. It predicts a lift-curve slope of $2\pi$, a zero-lift angle from the camber line, and a moment about the quarter chord, and none of the three can be adjusted. That is why it can be wrong in a useful way: a measured slope of 5.9 rather than 6.28 is a discrepancy that has to be explained rather than absorbed.

A turbulence model has several. The constants of a two-equation model are fitted to decaying grid turbulence, a log layer and a shear layer, and this collection has an essay on the constant that makes a variance negative — which is a case of a fitted constant being pushed outside the range its fit supports. The test above applies directly: a model whose constants have to be re-tuned per flow is absorbing its error.

A wall model has one, and the essay that measured what it is worth: six tenths in the additive constant is worth 6.6 per cent in the friction, which is more than the entire shape of the layer the model was written to describe.

The general position is not that fitted constants are illegitimate. It is that a constant’s stability across conditions is a measurement of the model, and reporting it is as much a part of a model’s validation as any comparison of values. A stable constant is evidence; a drifting one is a diagnosis.

What a good model looks like by this test #

It is worth ending constructively, because everything above has been about how to fail a model rather than what passing looks like.

A model passes when it predicts quantities it was not fitted to, in the right direction and by roughly the right amount, with parameters that do not move. Thin-aerofoil theory passes: it was fitted to nothing, and it predicts a lift-curve slope, a zero-lift angle, a moment about the quarter chord, an aerodynamic centre and a load distribution, all from the camber line and none adjustable.

That is a very high bar and most working models do not clear it. A turbulence model, a wall model, a drag correlation, a discharge coefficient: each has fitted constants and each is used far outside where it was fitted. The response is not to reject them but to be explicit about which of their outputs are interpolations and which are extrapolations, and to check the constants’ stability when the model moves to a new class of flow.

The test is not “does it agree” but “what did it predict that it could have got wrong”. A model agreeing with the data it was fitted to has predicted nothing; one agreeing with a derivative, a limit, or a qualitatively different case has predicted something, and the amount it has predicted is the amount it is worth.

A last note on why the tuning constant was 102.6. A number that large is itself a diagnosis. The Newtonian model’s natural constant is two — the momentum flux of a stream turned through the surface angle — so a factor of fifty is needed to reach a real aerofoil’s lift at a small angle. That gap is the ratio between what a stream of independent particles delivers and what a continuous fluid with circulation round the body delivers, and it is the size of the effect the model has no representation of at all.

What is not claimed #

Newtonian impact theory is not useless. At hypersonic speeds it is a good approximation, for a reason: the shock lies close to the body, the flow behind it turns abruptly, and the momentum picture is nearly what happens. The theory’s constant then has a derivation rather than a fit. What is being tested here is its use at low speed, where it was proposed and where it fails.

The tuned model is a construction. Nobody proposes Newtonian impact theory with a constant of 102.6. It is built here to make a point about what a single measurement can distinguish, and the point does not depend on anybody having made that error.

Thin-aerofoil theory is the reference and is itself a model. Its lift-curve slope is $2\pi$ per radian; a real section’s is a little less, because of the boundary layer’s displacement, and a little more, because of thickness. The factor of 2.8 is against $2\pi$ and would be against 5.7 or 6.6 just as well.

And the equal-transit factor of thirty-two is for one section at one incidence. A thinner section has a smaller path-length excess and a larger factor; a thicker one at a lower incidence has a smaller factor. What is general is that the two quantities are unrelated, which a family of sections shows directly.