A NACA section multiplies its lift by nearly one plus its thickness, and a nose correction cannot supply it

Thickness carries no load of its own and still moves the load forward took a twelve-per-cent Joukowski section, whose flow is exact, and the three exact flows thin-aerofoil theory would add to make it — the symmetric section of the same thickness, the camber line, the flat plate at incidence. Added, they lost a quarter of the suction near the nose and nine per cent of the lift. Thickness, which lifts nothing by itself, turned out to multiply the lift of the other two by $1 + 0.77\,t/c$, a term first order in the lift that no addition of first-order parts can contain.

Joukowski sections were chosen because every part of them is exact, and they are unrepresentative for the same reason: their thickness peaks a quarter of the way back and they end in a cusp. The sections real wings are drawn from — NACA’s four-digit family first among them — peak further aft and close at a finite angle. The essay ended by asking whether such a section follows the same law, how the result depends on where the thickness is, and how much a nose correction of Riegels’s kind recovers. A panel method answers all three, and the third answer is the one that clarifies what the multiplier is.

A section that has no closed form #

A panel method knows only the boundary. The Hess–Smith construction cuts the section into straight panels, each carrying a uniform source strength, with one vortex strength shared by all, and fixes the strengths so that no flow crosses the surface and the flow leaves the trailing edge smoothly. Here it is run on 240 panels clustered towards the nose and the tail, on sections drawn from the four-digit equations with the trailing edge closed, so that the Kutta condition has a definite edge to sit on. Its lift, computed from the circulation, changes by seven parts in ten thousand between 160 and 240 panels and by four between 240 and 320; computed instead by integrating the surface pressure it agrees to half a per cent, the difference being the pressure’s steepness at the nose, which panels resolve more slowly than they resolve circulation.

The parts are taken as thin-aerofoil theory takes them. The thickness form — the symmetric section of the same thickness at no incidence — is solved by the same panels, so its surface speed is exact. The camber line and the flat plate are zero-thickness objects, and their loads from thin-aerofoil theory are exact for zero thickness: Glauert’s series for the camber line, the plate’s $\cot(\theta/2)$ load for the incidence. For the 2412’s camber line the series gives a zero-lift angle of −2.08 degrees, the tables’ −2.1. The three are added as the theory adds them, their disturbance speeds summed, and the pressure taken from the sum either linearly or through Bernoulli.

More than Joukowski #

The first figure is the answer to the earlier essay’s first question. Thickness multiplies the lift of a NACA section, as it does a Joukowski section’s, and it multiplies it more. The symmetric four-digit section’s lift slope over $2\pi$ is 1.050 at six per cent thickness, 1.101 at twelve and 1.153 at eighteen — close to $1 + 0.84\,t/c$, against Joukowski’s $1 + 0.77\,t/c$. The cambered 2412 at four degrees lifts 0.743 against thin theory’s 0.666, a factor of 1.115; at six and eighteen per cent, 1.057 and 1.174. The cambered section’s factor is larger than the symmetric section’s slope factor because the camber’s own lift is multiplied as well, and a little more than the incidence’s.

So the law is the same in kind and different in number. Thickness raises the lift because it raises the surface speed everywhere on the section — the flow has to go round more body — and the circulation the Kutta condition demands is carried on a faster surface: lift is the circulation times the stream speed, but the circulation needed to make the flow leave the trailing edge smoothly is larger round a thick body than round a thin one. That is an effect of the whole section, and so it depends on the section’s whole shape, not on any one dimension of it.

Where the thickness is #

The earlier essay’s second question is the shift’s dependence on where the thickness is concentrated. Moving the crest of a twelve-per-cent thickness form from a fifth of the chord to a half — by stretching the chord coordinate in the four-digit law, so that the shape is the same family and the nose is correspondingly sharper as the crest goes aft — lowers the slope factor from 1.108 to 1.085, about a fifth of the whole excess. Thickness near the nose multiplies more than thickness further back.

That fits the explanation. The circulation is decided at the trailing edge, but the lift is the pressure integrated over the whole surface, and the pressure is decided by the surface speed, which the thickness raises most where the flow is turning hardest — near a blunt nose. A section whose thickness is forward has a blunter nose and a larger region of fast flow ahead of its crest, and the same circulation sits on a faster surface there. Joukowski’s sections, whose crest is at a quarter of the chord but whose nose is relatively sharp and whose tail is a cusp, come out lower than the NACA law because their tail carries almost no thickness: the four-digit section’s finite trailing-edge angle is itself a source of the multiplier, since the flow past the closing tail is slowed less than past a cusp.

The practical reading is that the laminar-flow sections of the 1940s, the six-series with their crests at 40 and 50 per cent of the chord, have slightly lower ideal lift slopes than four-digit sections of the same thickness, by the amounts this figure shows — a difference that in measured lift curves is usually hidden under the viscous effects, which are larger.

What the sum misses #

The pressures show where the sum goes wrong, and it is the same place as on the Joukowski section. Near the nose on the upper surface, where the incidence’s and the camber’s speeds add to the thickness’s, the linear sum is a quarter short: a suction of 0.99 at five per cent of the chord, against the section’s 1.30. Taken through Bernoulli — the summed disturbance speed squared, not just doubled — it is 1.24, most of the way back. That much of the error is the pressure’s nonlinearity in the speed, which the one thing that does not add up named as the cross term between flows: two disturbance speeds that each make a pressure of their own make, together, a pressure that includes their product. The rest is the speed itself, and that is the multiplier: the section’s speed near the nose is higher than the sum of the parts’ speeds, because the circulation the section carries is larger than the sum of the parts’ circulations.

The thicker the section, the worse the linear sum: at six per cent it is a fifth short at five per cent of the chord, at eighteen per cent a quarter. Bernoulli’s version is within four per cent at six per cent thickness and within six at eighteen.

The load moves forward #

As on the Joukowski section, the extra lift is not spread evenly. The section’s load is larger than the sum’s everywhere ahead of mid-chord and almost the same behind it, so the centre of pressure moves forward: to 0.337 of the chord, against the summed parts’ 0.356. Two hundredths of the chord is a small number for a pressure distribution and a large one for an aeroplane’s balance, where the wing’s pitching moment about its centre of gravity is set by exactly that distance times the lift. A design that trims its tail from thin-aerofoil moments is out by a moment coefficient of about 0.014 at this lift, comparable to the camber’s own moment.

The forward shift has the same cause as the multiplier and is consistent with the crest figure: the extra load is made where the thickness speeds up the flow most, near the nose, so it sits forward of the thin theory’s load, which is the incidence’s quarter-chord load plus the camber’s mid-chord one.

Riegels’s correction mends the nose and nothing else #

Thin-aerofoil theory’s surface speed is infinite at the nose: the plate’s load and the thickness’s linear speed both grow as one over the square root of the distance from the leading edge. The classical repair, Riegels’s of 1948, divides the linear surface speed by $\sqrt{1 + (dy/dx)^2}$, the slope of the section’s own surface, which is itself infinite at a round nose in exactly the way that cancels the theory’s infinity. It is the correction designers used before panel methods, and the earlier essay asked how much of the multiplier it recovers.

Near the nose it does what it is for. Linear theory’s suction runs off to six and a half at half a thousandth of the chord and is infinite at the nose; with Riegels’s factor it rises from zero at the stagnation point, peaks at 1.11 two and a half hundredths back and falls away, which is the right shape — the section’s own peak is 1.44 at one and a half hundredths. On the rest of the chord it brings linear theory’s pressures closer to the section’s, within a tenth of the suction where linear theory was within a fifth.

And it adds no lift at all. Integrated over the chord, Riegels’s load is 0.6660, linear theory’s 0.6664 and thin-aerofoil theory’s lift 0.6664: the factor moves load from the region behind the nose, where linear theory overstates the suction, to the nose itself, where the theory had an infinity, and keeps the total. That is the refutation’s point. The section’s eleven per cent of extra lift is not a nose effect that a nose correction can supply. It is the circulation being larger, and a correction applied to the surface speed at a fixed circulation cannot make it larger.

What reaches a finite wing #

A section’s lift slope is not a wing’s. On a wing of finite span the trailing vortices induce a downwash that takes back part of every increase in the section’s lift, and Prandtl’s lifting line puts the wing’s slope at $a_0/(1 + a_0/\pi A)$ for an elliptic loading on a wing of aspect ratio $A$. A larger section slope is therefore partly paid back in a larger induced angle. For a wing of aspect ratio eight, thin-aerofoil theory’s $2\pi$ gives the wing a slope of 5.03 per radian, and the twelve-per-cent section’s 1.101 times $2\pi$ gives 5.42 — an increase of 7.9 per cent where the section had 10.1. At aspect ratio six the wing keeps 7.4 per cent of it, at twelve 8.5: the more slender the wing, the more of the section’s multiplier it inherits, because the less of every increment its own downwash takes back.

That is the same arithmetic the span is the whole story does for the induced drag, applied to the slope, and it says the thickness multiplier is a section effect that a wing passes on at a discount of between a seventh and a quarter. It also says where the discrepancy between thin-aerofoil and panel lifts is most visible in a whole aircraft: on its high-aspect-ratio wing rather than its stubby tail surfaces, which are thinner anyway.

What a designer should take from it #

Two things follow for anyone using thin-aerofoil theory on real sections. The first is that the theory’s lift is low by about $0.9\,t/c$ of itself for four-digit sections, a bit less for sections with their thickness further aft, and that the right correction is a factor on the circulation — the lift slope — not a nose fix; panel methods and measured lift slopes both carry it. The second is that a nose correction is still worth having for what it does do: it puts the suction peak where the boundary layer will meet it, which is what decides where transition and the leading-edge stall begin. The two corrections answer different questions and neither substitutes for the other.

What the panel method cannot add is viscosity. A real twelve-per-cent section’s measured lift slope is lower than the potential-flow 1.10 times $2\pi$, not higher, because the boundary layer thickens over the rear of the section and decambers it — a loss that grows with thickness and nearly cancels the multiplier on a twelve-per-cent section at moderate Reynolds numbers. That is why the multiplier is invisible in most lift-curve tables: the ideal flow’s gain and the viscous loss have the same size and opposite signs. The panel result is still the right reference, because the viscous loss depends on the Reynolds number and the ideal gain does not.

The convention, and the reach of the method #

Coefficients are per unit chord; the section is a four-digit with its trailing edge closed by the −0.1036 coefficient; the cambered section has two per cent camber at forty per cent of the chord, the 2412 shape. “Lift over thin theory’s” is the panel section’s lift from its circulation over $2\pi(\alpha - \alpha_0)$, with $\alpha_0$ the camber line’s zero-lift angle from Glauert’s series; the symmetric slope factor is the change in lift between 2° and 6° over $2\pi$ times that change in angle. The crest is moved by mapping the chord coordinate through a power, which keeps the four-digit family and makes a forward crest go with a blunter nose, as it does in practice. The pressures on the summed side combine the panel thickness form’s exact speed with thin theory’s camber and incidence speeds; on Riegels’s side every part is linear, the thickness speed from its source integral. The flow is ideal, incompressible, and attached to the trailing edge, which is the regime a panel method describes.

Who drew the sections #

The four-digit sections are Jacobs, Ward and Pinkerton’s of 1933 at Langley, and Abbott and von Doenhoff’s book of 1949 collected them with the six-series. Riegels’s nose correction is of 1948, in work on the accurate calculation of section pressures that predates the computer. Hess and Smith’s panel method is of 1967. The thickness–lift multiplier was known to the early section theorists — it is in Theodorsen’s method for arbitrary sections of 1931 — and the four-digit factor, the crest dependence and Riegels’s zero contribution to the lift are computed here.

How it was checked #

The 240-panel lift is within seven parts in ten thousand of the 160-panel one and four of the 320-panel one, so the factors quoted carry three figures. The circulation lift and the integrated surface pressure agree to half a per cent. The camber line’s zero-lift angle from Glauert’s series is −2.08°, the tabulated −2.1. Riegels’s integrated load and linear theory’s agree with each other and with $2\pi(\alpha - \alpha_0)$ to four figures, which is the check that the factor redistributes and does not create load. The model refuses a section of no thickness, a negative camber, and a thickness crest outside the range its mapping can draw.

Still open: whether the multiplier survives a boundary layer #

The ideal multiplier and the viscous loss nearly cancel on a twelve-per-cent section, and that is a coincidence of one thickness and one Reynolds number. The next calculation would couple an integral boundary layer to the panel solution — the displacement thickness added to the section as a source distribution and the panels re-solved until the two agree — and ask, across thickness and Reynolds number, which side of thin-aerofoil theory a real section’s lift slope falls, and whether the thickness crest that maximises the ideal multiplier also maximises the viscous lift, or whether the aft crest that keeps the boundary layer laminar longer wins once the layer is counted.