A load carried to the trailing edge needs a hook

Thin-aerofoil theory is usually run one way. A camber line goes in, its slope is expanded in Glauert’s cosine series, and a loading, a lift and a pitching moment come out. Where lift starts pointed out that the theory is linear and invertible, so it can be run the other way: ask for a loading and integrate for the camber line that delivers it. Three numbers out of a camber line showed why that is the useful direction — a force balance reads only three numbers of a camber line, so force requirements leave the shape almost free, while a loading fixes every coefficient at once. That is how the NACA six-series mean lines were made, and this essay makes them, and asks what happens at the simplest request of all.

The loads asked for #

The six-series family asks for a load that is uniform over the front of the chord, to a fraction $a$ of it, and then falls linearly to nothing at the trailing edge. The load is the difference between the lower- and upper-surface pressure coefficients, $\Delta c_p$, and its area is the section’s design lift coefficient $c_{l_i}$. A uniform load is the case $a = 1$; $a = 0$ is a load that falls linearly from the nose. Every member has the same lift and differs in where along the chord that lift is put.

The reason to want a flat load is the boundary layer, not the lift. A flat load means a flat pressure difference, and over the forward part of a section a flat pressure on the upper surface means a flow that is still accelerating or level — a favourable or neutral gradient that keeps the layer laminar for longer.

The lines that carry them #

Running the theory backwards gives the lines in closed form, a combination of $x\ln x$ and $(c - x)^2\ln|c - x|$ terms that Abbott and von Doenhoff tabulate. For the uniform load it collapses to one line:

$$\frac{z}{c} = -\frac{c_{l_i}}{4\pi}\Big[(1-x)\ln(1-x) + x\ln x\Big].$$

That line is symmetric about mid-chord and 5.52 per cent of the chord high at a design lift coefficient of one. Tapering the load over the last fifth of the chord, $a = 0.8$, moves the highest point aft to 0.515 of the chord and raises it to 6.79 per cent; tapering it over the whole chord, $a = 0$, brings the highest point forward to 0.323 and the height to 6.29 per cent. The shapes look like ordinary camber lines. Their slopes do not.

Both ends turn vertical #

Differentiate the uniform line and the slope is $(c_{l_i}/4\pi)\ln\!\big((1-x)/x\big)$, which grows without bound at both ends: positive and infinite at the nose, negative and infinite at the tail. The line meets the chord vertically at both of its ends.

The two infinities have different meanings. At the leading edge every line in the family has the same $-\ln x$ behaviour, whatever $a$ is. That is the price of asking for a finite load at the nose. In ordinary thin-aerofoil theory the load at the leading edge is infinite — the $(1+\cos\theta)/\sin\theta$ term that carries the leading-edge suction — unless the incidence is exactly the camber line’s ideal angle, at which the stagnation point sits on the nose and the flow enters smoothly. A line designed to carry a finite load at its nose is designed to be at its ideal angle, and the theory makes the line meet the oncoming stream head on.

At the trailing edge only the uniform load does it. Any taper, however short, brings the load to zero before the edge and the slope arrives finite: at a hundred-millionth of a chord from the tail the uniform line’s slope is −1.47 and still falling as the logarithm, while the $a = 0.8$ line’s is −0.204 and settled. The trailing-edge infinity belongs to the load being finite there, and that is what thin-aerofoil theory’s version of the Kutta condition forbids.

The hook #

The sharp edge decides is the account of why a real wing’s circulation is the one that makes the flow leave a sharp trailing edge smoothly. In the linearised theory that statement becomes a condition on the load: the vortex sheet’s strength, which is the load, must be zero at the trailing edge. Glauert’s series builds it in — every term of it vanishes there — which is why a forward calculation never has to impose it.

Asking for a uniform load is asking the theory to violate it, and the theory complies in the only way it can: by making the edge something else. The magnified tail shows it. The tapered lines come down to the trailing edge at a slope, as a cambered section’s mean line ordinarily does. The uniform line drops the last 0.446 per cent of a chord in the final one per cent of it and arrives vertically. It is a hook.

That shape has a name in practice. A Gurney flap is a small tab, one or two per cent of the chord, fixed perpendicular to the lower surface at the trailing edge. It carries load to the edge: the pressure difference across the tab is finite, a pair of small counter-rotating vortices sits behind it, and the section’s lift rises at the same incidence. The racing driver it is named after found it by experiment, and Liebeck’s measurements in 1978 showed a tab of 1.25 per cent of the chord raising a section’s lift and slightly lowering its drag at a given lift. The uniform-load line is the inviscid theory’s version: the same violation of a smooth Kutta condition, paid for with the same turned-down edge. The theory cannot represent the vortices behind the tab, so it gives a hook whose last part has no thickness and infinite slope instead.

A flap of no chord #

There is a second way to see the hook, from the forward direction. What a flap does and the elevon essay both used Glauert’s closed form for a hinged flap of chord fraction $E$ deflected by $\delta$: the load it adds has a logarithmic spike at the hinge and spreads forward along the whole chord. Shrink the flap towards the trailing edge while raising its deflection to keep its lift, and the hinge spike crowds ever closer to the edge. The uniform-load line’s tail looks like the end of that sequence: a turned-down edge spread as a logarithm over the last per cent of the chord, with no hinge left to point at. A Gurney flap is a member of that sequence that stopped early, with a chord of one or two per cent and a deflection of ninety degrees, which is why the analogy is closer than it first looks.

Why the exact inverse refuses what the linear one grants #

The same request made of the exact theory gets a different answer. Ask for the pressure, and see what shape that is inverted the full potential-flow problem by a conformal map and found that a pressure distribution cannot be asked for freely: three closure conditions must hold, or the shape that comes back does not close or does not leave the flow undisturbed at infinity. A distribution that loads the trailing edge violates the one that encodes the Kutta condition, and the exact inverse refuses it. The linearised inverse has no closure conditions — any integrable load returns a camber line — so it grants the request and pays in the only currency it has, a slope. The singular hook is what a refused request looks like in a theory too linear to refuse.

The two answers are consistent. The linear theory’s camber line is finite in height everywhere, so it satisfies the closure that keeps the section closed; what it cannot satisfy is a smooth trailing edge, and it does not pretend to. The condition that can be bought found the same freedom from the other side: blow a jet over a rounded edge and the Kutta condition becomes a quantity to be chosen, at a price paid in jet momentum rather than in shape. A hook, a jet and a tab are three ways of putting load where a sharp edge will not carry it.

The theory’s own series cannot carry it #

Running the theory forwards on the designed lines is the natural check, and for the uniform line it shows the limitation in another form. Expand each line’s slope in Glauert’s series to sixty harmonics and rebuild the load. The tapered load comes back everywhere except at its kink at $x = 0.8$, where a sine series of sixty terms rounds a corner. The uniform load comes back along the chord and is then dragged to zero at the trailing edge, with an overshoot just before it — 1.14 a thousandth of a chord from the edge, against the 1 asked for.

That is not a numerical accident. Each term $\sin n\theta$ vanishes at $\theta = \pi$, the trailing edge, so any finite sum of them carries no load there. The uniform load is reached only in the limit of infinitely many terms, and it is reached the way a square wave is reached by its Fourier series, with an overshoot that moves towards the edge as terms are added and never shrinks below about eighteen per cent of the load — the Gibbs overshoot of a sine series asked to jump from the load to its mirror image. A camber line whose load breaks the Kutta condition is a camber line that the theory’s own description of a camber line can approach and not contain.

What moving the load aft costs #

The family’s other numbers follow from where the load sits. The quarter-chord moment is just the load’s first moment about the quarter chord: $-0.0833$ for $a = 0$, whose load is concentrated forward, falling to $-0.202$ at $a = 0.8$ and $-0.25$ exactly for the uniform load, whose centre is at mid-chord, a quarter of a chord behind the reference point. A uniform load is the most nose-down of the family, and every unit of design lift carried this way costs a quarter of a unit of moment — which a tailless wing has to trim at a poor exchange rate.

The ideal angle runs the other way: 4.56° for $a = 0$, 1.54° for $a = 0.8$, zero for the uniform load. A load concentrated forward needs the section pointed up into the stream for the flow to enter smoothly, because the nose is carrying lift; a uniform load, symmetric about mid-chord, needs no incidence at all. Both numbers scale with the design lift coefficient, since everything here is linear in it.

How much camber the request needs #

The height of the line needed to deliver one unit of design lift is not monotonic in $a$. It rises from 6.29 per cent at $a = 0$ to a peak of 7.5 per cent near $a = 0.47$ and falls to 5.52 per cent for the uniform load, while the highest point moves steadily aft from 0.323 of the chord to mid-chord. The uniform line is the flattest of the family for the same lift because its hooked tail does part of the work: the steep last few per cent of the chord turn the flow down hard, as a flap would, and the rest of the line need not.

That is the trade a designer of the six-series faced, and the answer was not the one the hook suggests. The uniform line is the series’ default — a section named without an $a$, such as a 65-415, carries it — because a real section has thickness and a finite trailing-edge angle, and the hook lives in the last fraction of a per cent of the chord, inside the region a real trailing edge rounds or cuts off. What the infinity costs in practice is a pressure recovery at the very back that the boundary layer handles badly. The 6A-series of the 1950s used a modified $a = 0.8$ line instead, whose taper keeps the load off the trailing edge and whose straight-sided tail was easier to build; it buys that for an extra 1.3 per cent of camber and a degree and a half of ideal angle.

How the lines were checked #

The closed form was checked by not trusting it. Each line’s slope was expanded in Glauert’s series by quadrature in the angle variable, which never evaluates the logarithmic end singularities and integrates them correctly, and the theory was run forwards. At the line’s own ideal angle the leading-edge coefficient $A_0$ comes out at $10^{-6}$ — no nose spike, as designed — for $a = 0.8$ and $a = 0$; the lift is the design value, 1, to seven figures; and the moment matches the load’s first moment to $10^{-12}$. The rebuilt load matches the asked-for load to three parts in a thousand away from the kinks, which is the sixty-term series’ own truncation. And the general formula at $a = 1 - 10^{-7}$ agrees with the uniform line to $6\cdot10^{-8}$ of a chord everywhere, so the special case is the limit of the family rather than a separate formula. The forward run also fixed a sign: written with Abbott and von Doenhoff’s constant $h$, the ideal angle is $-c_{l_i}h/2\pi(a+1)$ measured nose-up from the chord, which is the angle at which the forward theory’s $A_0$ vanishes.

The convention: load as a pressure difference, incidence from the chord #

The load is the lower-surface minus the upper-surface pressure coefficient, positive for lift. Heights are in chords, $x$ runs from the leading edge, and the incidence is measured nose-up from the chord line, as thin-aerofoil theory measures it. The design lift coefficient $c_{l_i}$ is one throughout; every height, angle and moment scales in proportion to it.

What the picture cannot show #

The theory is linear in the camber and has no thickness, so what it designs is the mean line of a section, not the section. Wrapped round a real thickness distribution, the a-series lines deliver a pressure distribution that differs from the one asked for at first order in thickness, which is why six-series sections were designed with corrections and checked in a tunnel. The figures cannot show the boundary layer, which is the reason any of this was wanted, or the vortices behind a Gurney flap, which are what a real hooked edge has instead of an infinite slope. And the claim that the hook is a Gurney flap is an analogy between an inviscid limit and a viscous device: both carry a finite load to the edge, and only the second has been measured doing it.

Who found it, and when #

Glauert’s form of thin-aerofoil theory dates from 1926 and Munk’s from 1922. The mean lines designed from a specified load were worked out at the NACA through the late 1930s and published by Jacobs, Abbott and others as the six-series in the early 1940s, and Abbott and von Doenhoff’s Theory of Wing Sections of 1949 is where the closed form and its tables are collected. Liebeck described the Gurney flap’s effect in 1978, from a device Dan Gurney had fitted to racing-car wings by trial. The observation that the uniform-load line’s trailing-edge singularity is the Kutta condition’s violation follows from the theory’s structure.

Still open: a load the edge is allowed to carry #

The hook is the inviscid theory’s answer to a request it should refuse. A real Gurney flap carries its trailing-edge load because viscosity makes a small separated region behind the tab, and the load it can carry depends on the tab’s height relative to the boundary layer. The next calculation gives the trailing edge a finite load as a boundary condition rather than a violation — a thin-aerofoil theory in which the Kutta condition is replaced by a prescribed trailing-edge pressure difference, as the tab imposes — and asks how much extra lift each unit of that load buys, how far the zero-lift angle moves, and how the moment changes, so that a measured Gurney flap’s lift increment can be read as a trailing-edge load and compared with the hook drawn here.