Where lift starts

The site has an exact aerofoil. flow.js maps a circle onto a wing section, applies the Kutta condition, and hands back a closed-form velocity everywhere in the plane — no approximation, no grid, no residual. It is one of the few genuinely exact solutions in the subject and it has been the backbone of a dozen figures here.

It answers the wrong question.

A conformal map takes a circle and reports what section came out. A designer has a camber line in mind and needs to know what it will do — and there is no circle that produces an arbitrary camber line, because the map has three parameters and a camber line has infinitely many. The exact theory can only answer about the shapes it happens to produce.

Glauert’s thin-aerofoil theory runs the other way and answers about any camber line at all. The price is that it throws the thickness away entirely, and this essay ends by measuring what that costs — it turns out to be first order in the thickness ratio, which puts a number on an apology that is usually made without one.

The trick, and why it is not a numerical method #

Replace the section by a sheet of vorticity laid along its camber line, and require the flow to be tangent to that line rather than to a surface. The unknown is the sheet’s strength γ(x), and the tangency condition is an integral equation for it: the downwash the whole sheet induces at each station must cancel the component of the free stream across the line there.

Glauert’s substitution is what makes it tractable. Write the chordwise coordinate as

$$x = \frac{c}{2}\left(1 - \cos\theta\right),$$

so that θ runs from 0 at the leading edge to π at the trailing edge, and expand the sheet as

$$\frac{\gamma(\theta)}{2U} = A_0\,\frac{1 + \cos\theta}{\sin\theta} + \sum_{n\ge 1} A_n \sin n\theta.$$

Every term of that series is chosen so its induced downwash is a single cosine, and the integral equation collapses to one algebraic relation per harmonic. It is not a discretisation and it does not converge to something — it is an exact solution of the linearised problem, and the only approximation anywhere is that the boundary condition was applied to a line instead of to a surface.

That figure carries the structural point of the method. A₀ is the only coefficient that depends on the incidence. Every other one is an integral of the camber line’s slope and does not change when the wing is pitched. So the shape’s contribution and the attitude’s contribution are separated before any number is computed, and everything worth knowing about the section falls out of that separation.

The zero-lift angle, and what it is a property of #

The lift coefficient is $\pi(2A_0 + A_1)$, and substituting the definitions gives

$$C_l = 2\pi\left[\alpha - \frac{1}{\pi}\int_0^{\pi}\frac{dz}{dx}\,(1 - \cos\theta)\,d\theta\right].$$

The bracket is the incidence measured from an angle that belongs to the shape. Call it α₀:

$$\alpha_0 = \frac{1}{\pi}\int_0^{\pi}\frac{dz}{dx}\,(1 - \cos\theta)\,d\theta,$$

and the whole of thin-aerofoil theory’s answer about lift is $C_l = 2\pi(\alpha - \alpha_0)$.

Two things about that. The slope is 2π per radian for every camber line — camber shifts the curve sideways and never tilts it, which is why the lift curve is the same straight line for every section a designer might draw. And α₀ contains no speed, no density, no viscosity and no thickness. It is a weighted average of the camber line’s slope, and it can be computed for a shape that has never been near a wind tunnel.

This site computes it by quadrature rather than by substituting into a table, and then checks the quadrature against the cases that have closed forms. A parabolic arc of maximum camber $m$ has $z = 4mx(1-x)$, so $dz/dx = 4m\cos\theta$, and the integral is elementary:

$$\alpha_0 = \frac{1}{\pi}\int_0^\pi 4m\cos\theta\,(1-\cos\theta)\,d\theta = -2m.$$

The solver’s quadrature reproduces −2m to ten decimal places, and refuses to run at all for a camber it considers too large to call thin.

The angle where the leading edge stops screaming #

Look again at the sheet strength. The first term carries a factor $(1+\cos\theta)/\sin\theta$, which is unbounded as θ → 0. That is the leading-edge singularity, and it is the reason thin-aerofoil theory is excellent about lift and useless about the suction peak: lift is an integral and survives the singularity, while the local velocity at the nose does not exist.

There is exactly one incidence at which it goes away, and it is the one where A₀ = 0:

$$\alpha_{\text{ideal}} = \frac{1}{\pi}\int_0^\pi \frac{dz}{dx}\,d\theta.$$

At that angle the flow arrives at the leading edge along the camber line’s own tangent and comes onto the section smoothly. For a circular arc it is exactly zero incidence, because the mean slope of a symmetric parabola over the chord is zero — a small, pleasing fact that falls out of the arithmetic and that nothing about the shape makes obvious. For an unsymmetric mean line it is somewhere else, and the difference between α₀ and α_ideal is the whole design tension of a laminar-flow section: the shape wants to be cambered so that it lifts at low incidence, and it wants its ideal angle to be where it will actually be flown so that the nose is not overloaded there.

The lift the section makes at its ideal angle is its design lift coefficient, and it is the number stamped on the second digit of a six-series aerofoil designation. It is computed here rather than looked up.

The moment that does not move, and the centre it implies #

The moment about the leading edge does depend on the incidence — it has A₀ in it. The moment about the quarter chord does not:

$$C_{m,c/4} = \frac{\pi}{4}\left(A_2 - A_1\right).$$

There is no α anywhere in that expression. For a parabolic arc it comes out at exactly −πm, and the solver checks it against that closed form.

That flatness is what an aerodynamic centre is. The essay that located it found the quarter chord numerically on the exact section by searching for the point about which the moment stopped changing. Here it is not found at all — it falls out, for every camber line, because the α-dependence lives entirely in A₀ and A₀ is exactly the term whose moment about the quarter chord vanishes.

The centre of pressure is a different point and it moves, running off to infinity as the lift goes through zero. The two are constantly confused, and the reason the aerodynamic centre is the useful one is on this figure: a designer needs a point about which the moment is a constant, because a constant can be trimmed out once and a moving point cannot.

Turning the tail up, and the price of it #

A tailless aircraft has nothing to trim against, so its sections have to carry no moment of their own: $C_{m,c/4} = 0$. The way to get it is to turn the trailing edge up, and the question is how far.

The solver answers it by bisection on the station at which the camber crosses back through zero, and the answer is worth the figure it gets.

Two rationals come out of that search and neither was put in.

The crossing station is exactly 7/8 of the chord, and it does not depend on how much camber the line had. That is not a coincidence: the camber is a multiplicative constant in $dz/dx$ and every Glauert coefficient is linear in it, so the ratio $A_2/A_1$ that decides the moment is a function of the reflex station alone. Bisection finds 0.875000000010; the solver requires it to be within a millionth of 7/8.

And the lift it costs is exactly 6/7. The reflexed line at zero incidence makes one seventh of the lift the plain arc made, again independent of camber. That is a startlingly steep price and it is the reason flying wings are hard: the camber that was there to provide lift at low incidence has been almost entirely cancelled by the camber that was added to remove the moment, and the aircraft has to get its lift from incidence instead — which it pays for in drag, and which puts it closer to its stalling angle at every point in the flight.

Both numbers are asserted in flowcheck against the rationals, by arithmetic the bisection knows nothing about.

What the thinness costs, measured #

Everything above is exact for a camber line and says nothing about a real section, which has thickness. Textbooks introduce the theory with an apology — valid for thin sections at small incidence — and the apology never comes with a number. Here it can, because this site has the exact answer for a family of shapes.

The measurement runs like this. Take a Kármán–Trefftz section, whose lift is known in closed form. Read its camber line off its own outline — split the surface at the leading and trailing edges, interpolate both arcs at common stations, average them — and put that camber line through the approximate theory. The only thing the theory is now missing is the thickness. Then thin the section and watch.

The exponent is 1.012, measured, and the site’s gate requires it to lie between 0.9 and 1.15. So the apology has a number: thin-aerofoil theory’s lift error is proportional to the thickness ratio. A one per cent section is right to under a per cent; a twelve per cent section — which is most aeroplanes — is out by about eight.

The other half of the measurement is more interesting. The zero-lift angle extracted from that same camber line agrees with the exact map’s to within a fiftieth of a degree at every thickness, including the eighteen per cent one where the lift is wrong by twelve per cent. The theory is not uniformly approximate. It is wrong about what it threw away — the thickness’s own contribution to the circulation — and right about what it kept, which is the camber line’s.

That is a much more useful thing to know than a blanket caution, and it is why thin-aerofoil theory is still the tool for the first question a designer asks. Where does this shape start lifting? is answered to a fiftieth of a degree. How much does it lift at four degrees? is answered to eight per cent, and for that the answer has to come from an exact solution or a panel method.

Reading a camber line, once the arithmetic is in place #

Three practical readings follow, and they are the reason this theory outlived the exact maps that came before it.

Camber buys lift at low incidence and is paid for in moment. The two integrals move together: anything that makes α₀ more negative also makes $C_{m,c/4}$ more negative, because both are dominated by A₁. There is no camber line that lifts at zero incidence and carries no moment — except a reflexed one, which pays six sevenths of the lift for the privilege.

Where the camber is matters more than how much. The weighting $(1-\cos\theta)$ in the zero-lift integral is small near the leading edge and large near the trailing edge, so camber near the tail is worth several times as much zero-lift angle as the same camber near the nose. That is why a flap is at the back, and it is a stronger statement than the usual one about flaps: the reason a trailing-edge device is effective is visible in the weight function.

And a mean line is a design variable in its own right. The four-digit sections chose a mean line and a thickness distribution independently, which was a genuine invention: it made camber and thickness into two knobs instead of one shape, and thin-aerofoil theory is what says the first knob is the one that decides where lift starts.

Camber lines add, and that is the whole design method #

Every Glauert coefficient is an integral of $dz/dx$ against a fixed weight, so each is a linear functional of the camber line. Add two mean lines and their coefficients add; scale one and its coefficients scale. Which means zero-lift angles add, quarter-chord moments add, and design lift coefficients add.

That is not a curiosity, it is how sections are actually built. A flap deflection is a camber line — a straight line with a kink in it — so its contribution superposes on whatever camber the section already had, and its effectiveness therefore depends on where the hinge is and not at all on the aerofoil it is hinged to. The flap essay’s number is a property of the weight function above, evaluated over the last quarter of the chord.

The stronger consequence is that the map runs backwards. The theory takes a camber line and produces a loading; because it is linear and invertible, a designer can specify the loading and integrate to get the camber line. Ask for a particular distribution of lift along the chord, and the shape that delivers it is a quadrature rather than a search.

That is where the six-series sections came from, and it explains their designation. A flat loading over the forward fraction of the chord means a flat pressure distribution there, which means the flow is still accelerating — a favourable gradient, which holds the boundary layer laminar. So the specification was written as uniform loading to a stated fraction of the chord, tapering thereafter, that fraction is the number after the letter in the section’s name, and the mean line was computed from it rather than drawn.

The catch is the one measured above. What is inverted is the linearised problem, so the section built to the specification delivers a pressure distribution differing from the requested one by something first order in its thickness.

What the model does not contain #

No thickness at all. The section is a line. Every conclusion about the suction peak, the pressure distribution near the nose, the critical Mach number and the stall is outside the theory, because all of them depend on a curvature the model does not have.

No viscosity, so no stall and no limit. $C_l = 2\pi(\alpha-\alpha_0)$ has no maximum. A real section’s curve bends over and comes back down, and where that happens is a boundary-layer question that nothing here can be asked.

Small angles. The boundary condition was applied on the chord line rather than on the surface, which is a linearisation in the incidence as well as in the thickness. At twelve degrees the linearisation is worth about as much as the thickness approximation; at twenty it is worthless regardless of how thin the section is.

The leading-edge singularity is real in the model and absent in the world. A real nose has a radius and a finite suction peak. The theory’s infinity is the price of the linearisation, and the ideal angle is the one place the price is not paid.

And the sheet is not a wake. Everything here is steady. A section changing incidence sheds vorticity and gets its lift late; none of that is in a Glauert series.

Who found it, and when #

Munk gave the essential integral in 1922 and Glauert put it in the form used here in 1926, in The Elements of Aerofoil and Airscrew Theory — a book whose first half is still the clearest account of this material anybody has written. The context matters: the exact conformal maps were already twenty years old and were regarded as the serious theory, and Glauert’s method looked like a convenience.

It won because of the direction it runs. Joukowski’s map answers what does this circle become, and in 1910 that was miraculous. By 1925 the industry had a different question — given a section with this much camber at this station, what will it do? — and only the approximate theory could answer it. The four-digit and five-digit families were designed against exactly this arithmetic, and the six-series sections were designed against the ideal angle it produces.

There is a general moral in that, and this site keeps meeting it. An exact solution answers the question it was built around, and an approximate one can be pointed anywhere. The right response is not to prefer one but to have both and measure the gap, which is what the convergence figure above is for.

Where the ladder goes next #

The theory here throws the thickness away because a line is all it can handle. The obvious repair is to stop insisting on an analytic solution: cut the actual surface into panels, put a singularity on each, and solve for the strengths. That gets the thickness back and works for any shape — and it turns out to run straight into the thing thin-aerofoil theory quietly assumed, because a panel method’s matrix is one row short and the row that is missing is the Kutta condition.