The half that carries nothing

Thin-aerofoil theory makes a decomposition that is so useful it is easy to forget it is a claim.

A section is split into a camber line, replaced by a sheet of vortices, and a thickness distribution, replaced by a sheet of sources and sinks. The vortex sheet produces all of the lift; the source sheet produces none — at any incidence, in the linear theory, exactly none. So the whole of a section’s lift is decided by its camber line and its incidence, and the shape everybody actually draws when they draw an aerofoil contributes nothing.

That is very nearly right, which is what makes the exceptions worth measuring.

Nothing is a strong claim #

The first thing to check is the claim itself, and it is worth checking twice because it is true for two independent reasons.

A symmetric Joukowski section at zero incidence has a Kutta circulation of exactly zero, so its lift coefficient is $-U\gamma/\tfrac12U^2c$ with γ = 0: identically zero, and the map returns it to machine precision.

That is the algebra. The pressure field is the physics, and integrating it round the surface is a different check.

The magnitude comes out at parts in a thousand of the dynamic pressure — the residue of a quadrature round a closed curve — for two reasons that do not depend on each other. Symmetry: the pressure is identical above and below, so the vertical components cancel. And d’Alembert: an inviscid flow round a closed body has no drag, so the horizontal components cancel too.

Integrating is the check. Arguing is not, and this collection has been caught by exactly that before: a tangency condition that tested the wrong thing passed every algebraic argument anybody made about it.

And yet the slope moves #

The exact solution is available for this family, so the claim can be tested against something other than the theory that produced it.

Measuring the slope off the Joukowski map at eight thicknesses gives 6.2832 for a plate — 2π, recovered to five digits — rising to 7.098 per radian at 16.9 per cent thickness. Fitted through the origin that is

$$\frac{\mathrm dC_L}{\mathrm d\alpha} = 2\pi\left(1 + 0.766\,\frac{t}{c}\right),$$

which is the 0.77 that appears in the standard texts, measured here rather than quoted.

So thickness carries no lift and changes the lift slope. There is no contradiction: the source sheet produces no force by itself, and it changes the velocity field that the vortex sheet is working in. The thickness makes the section fatter, the flow round it faster, and the circulation required by the Kutta condition at a given incidence larger.

The coefficient in $2\pi(1 + k\,t/c)$ is a property of the map rather than of the thicknesses it was sampled at: fit it to sections at one, three, seven, eleven and sixteen per cent instead and the same $k$ comes back. That is worth one line of checking, because a coefficient recovered from a fit is the kind of number that quietly depends on where the fit was taken.

Three answers, and one of them has the other sign #

The interesting part is the third curve on that figure.

Thin-aerofoil theory has no thickness term, so its slope is 2π for every section — a horizontal line, and the theory is not being sloppy: the linearised problem genuinely separates, and the thickness problem genuinely produces no force.

Exact potential flow says the slope rises, by about 13 per cent at 17 per cent thickness.

Measurement says it falls. A 12 per cent NACA section has a measured slope of about 0.105 per degree against the thin-aerofoil 0.110, and thicker sections are worse. The reason is the boundary layer: it thickens towards the trailing edge, the effective body is fatter at the back than the metal is, and a section made fatter at the back has less effective camber. Decambering, and it is worth more than the inviscid thickness effect at every thickness anybody builds.

Two computed curves and one measured one, and the two computed ones bracket nothing: they are both on the same side. The best of the three, judged against measurement, is the theory with no thickness in it.

What the thickness actually does #

If the thickness produces no force, it is fair to ask what it is for, and the pressure distribution answers immediately.

The distributions are identical above and below, so no lift; and they are not flat, so a great deal of aerodynamics. The flow accelerates round the leading edge, reaches a suction peak, and decelerates to the trailing edge against an adverse gradient.

From −0.0058 for a near-plate to −0.727 at 17.8 per cent thickness. Two consequences follow and both are the practical content of the thickness ratio.

The critical Mach number #

The first is compressible. A section reaches sonic flow when the local speed reaches the local speed of sound, and the standard estimate applies a Prandtl–Glauert-type correction to the incompressible minimum pressure and asks where the corrected value reaches the critical $C_p$.

Deeper suction means an earlier sonic point. Roughly, a section with $C_{p,\min} = -0.4$ goes critical near Mach 0.75 and one with $C_{p,\min} = -0.73$ near Mach 0.68 — which is the whole reason transonic wings are thin, and why the sweep that recovers the loss is worth its structural cost.

The thickness ratio is a proxy for the suction peak, and the proxy is not tight: two sections of the same t/c with the maximum thickness at 30 per cent and at 50 per cent of the chord have different peaks and different critical Mach numbers. That is the residual this essay is about, and the collapse onto one number has no room for it.

The adverse gradient, which is where sections are lost #

The second consequence is viscous, and it is the one that decides whether a section is any good.

Behind the suction peak the pressure rises back to the trailing edge, and a boundary layer climbing a pressure rise is a boundary layer that may separate. The steeper the recovery the sooner it does, and the steepness of the recovery is set by how deep the peak was and how far forward it sat.

That is why the shape of the thickness distribution is a design variable in a way the thickness ratio is not. A laminar-flow section holds its maximum thickness far aft to keep the pressure falling over most of the chord, which delays transition and reduces friction, at the cost of a violent recovery afterwards. A section for high maximum lift puts its thickness forward and shapes the recovery to be long and gentle. Both may be at 15 per cent, and they are different aeroplanes.

What the numbers mean for a wing #

Two consequences of the above are worth turning into design statements, because the thickness ratio is one of the first three numbers anybody chooses about a wing.

Thickness is structure. A wing spar’s bending stiffness goes as the cube of its depth, so a 15 per cent section is nearly four times stiffer than a 9 per cent one of the same chord and weighs far less for the same load. That is why a glider’s wing, which is long and lightly loaded and subsonic, is thick, and why a high aspect ratio and a thin section are a bad combination structurally as well as aerodynamically.

Thickness is speed. The suction peak is what runs into the sound barrier, so a transonic wing is thin — 10 to 12 per cent at the root and 8 to 9 per cent at the tip is typical — and pays for it with a heavier spar and a smaller fuel volume. A supersonic wing is thinner still, at 4 to 6 per cent, because wave drag goes as the square of the thickness ratio.

So the number is doing three jobs and this essay has been about the third: structure, volume, and the peak velocity on the surface. Nothing in it is about lift, which is the quantity the decomposition assigned it none of, and that is the point.

The exception worth naming is the supercritical section, which is thick and transonic at once. It works by flattening the upper surface so that the suction is spread rather than peaked, accepting a shock and placing it far aft, and recovering the lost lift with a heavily cambered lower surface near the trailing edge. It is a redistribution of the same thickness, and it is the clearest possible demonstration that $t/c$ is not the variable.

What the source sheet is, in the linear theory #

The linearised thickness problem is worth writing down, because it makes the “no force” statement transparent.

The body is replaced by a line of sources on the chord with strength

$$q(x) = U\frac{\mathrm dt}{\mathrm dx},$$

so the source sheet emits exactly the volume flux the thickening body displaces. The velocity it induces is symmetric about the chord, which is why it produces no lift; the pressure perturbation it produces is $-2u/U$ where $u$ is its own axial velocity, which is symmetric too.

And the total source strength is $\int q\,\mathrm dx = U[t]_0^c = 0$ for a closed body, which is the statement that a closed body neither creates nor destroys fluid — and is exactly the condition the Sears–Haack body fails when its tail is cut off square, producing a wave drag that is lower than the closed body’s and a design conclusion that is nonsense.

Superposition is complete in the linear theory: camber plus thickness plus incidence, three problems, added. It is the reason a section can be designed as a camber line and a thickness distribution independently, which is how every NACA four- and five-digit section was constructed.

Where the superposition stops #

At what thickness does the linear separation break down? Not where anybody expects.

The exact and linearised lift agree well past any practical thickness: the 13 per cent error at 17 per cent thickness is a small correction, and it is in a quantity everybody measures anyway. Where the linear theory fails badly is the leading edge, and it fails there at every thickness.

The linearised solution has a square-root singularity at a rounded leading edge, and the exact solution does not: the flow round a nose of radius $r$ reaches a finite peak that scales as $\sqrt{c/r}$. So the suction peak — the quantity this essay has been arguing is what thickness is for — is the one quantity the linear theory cannot compute at all. The peak in the figures above comes from the conformal map, which is exact.

That is a familiar shape. The leading-edge suction force is finite and comes from an infinite velocity acting over zero area, and the linear theory gets the force right by an integration through a singularity it does not believe in.

What the exact solution does not contain #

No viscosity, so no decambering — which is the largest of the three effects on the slope and the only one with the other sign.

No compressibility, so the critical Mach argument above is an estimate applied to an incompressible pressure field, and the estimate is the standard one rather than a solution.

And no shape freedom. The Joukowski family has one thickness parameter and its maximum thickness sits at about a quarter of the chord for every member of it, which is exactly the freedom the essay is arguing the collapse discards. A family that could move the maximum would show the effect directly; this one can only argue that it exists.

That is an honest limitation and it is worth stating rather than implying otherwise: what is computed here is that the peak deepens with thickness, and what is asserted from the literature is that it also moves with the position of the maximum.

A third thing the one number quietly decides #

Where the thickness sits is one residual of the collapse. There is another, and it decides something a designer cares about more than the slope: the radius of the leading edge.

In the four-digit family that radius is not free. It is tied to the thickness ratio by a fixed formula and goes as its square, so a six per cent section has about a quarter of the nose radius of a twelve per cent one. Choose $t/c$ in that family and the nose has been chosen too.

What the nose radius decides is the kind of stall rather than the amount of lift. A small radius forces the flow round a tight corner, produces a sharp local peak, and the layer separates right at the nose when the incidence is raised — abruptly, across the whole section at once, with no warning. A generous radius spreads that peak, so separation begins instead at the trailing edge and creeps forward as the incidence rises, giving a gradual loss and buffet before it.

So a section chosen thin for speed has been chosen, in that family, to stall without warning. Breaking that link is exactly what the later families did: the five-digit and six-series sections specify the nose separately, because the one number was deciding two things that wanted to be argued about apart.

Where the decomposition came from #

Munk and Birnbaum had the thin-aerofoil theory by 1922, Glauert gave it its modern form in 1926, and the Joukowski transformation is 1910. The NACA four-digit series, built explicitly as a camber line plus a thickness distribution, is 1933 — and its existence is the strongest evidence that the decomposition is useful, because it produced an entire family of sections whose properties could be predicted before any of them was cut.

What is worth noticing is that the decomposition is a linear statement about a nonlinear problem, and it survives because the nonlinearity is weak at the incidences aeroplanes fly at. At high incidence, or with a deployed flap, or near stall, it does not survive at all — and none of the sections in this essay is at any of those conditions.

Why nobody designs a section this way any more #

The decomposition is exact in the linear theory and it is not how a modern section is produced, and the reason is the one this essay has been circling.

A four-digit NACA section is a camber line plus a thickness distribution, chosen separately, and the pressure distribution is whatever comes out. That is the right order when the two parts really are independent, which is true of the lift and false of everything else: the suction peak, the position of the recovery, and therefore the whole of the boundary layer’s fate depend on the sum, and neither half can be chosen to control them.

So the order was reversed. A section today is designed by asking for the pressure distribution first — a flat rooftop, a specified recovery, a peak below the critical value — and solving for the shape that produces it. The thickness and camber are outputs. They can still be extracted afterwards, and the extraction is still exact, but nothing in the design used them.

That is what a residual looks like when it is taken seriously. The collapse onto $t/c$ discards where the thickness sits; the inverse method treats where it sits as the primary variable and lets $t/c$ fall out. Both descriptions are complete, and only one of them puts the designer’s hands on the quantity that decides whether the section works.

The one place the thickness does carry a force #

There is an exception to the whole argument and it is worth naming, because it is where the entire subject of transonic aerodynamics begins.

In compressible flow the thickness distribution produces a drag. A slender body at supersonic speed has a wave drag proportional to the square of the second derivative of its cross-sectional area — which is a statement about the thickness alone, with no camber and no incidence anywhere in it, and it is what the Sears–Haack body minimises.

So d’Alembert’s cancellation, which is what made the force zero in the figures above, is an incompressible statement. The moment the flow can carry waves away to infinity the pressure distribution stops being fore-and-aft symmetric, the cancellation fails, and the thickness that produced no force at all begins producing the dominant one.

That is a general moral this collection has met from the other direction in the drag that has no viscosity in it: a paradox about zero drag is a statement about what a flow is allowed to radiate, and changing that changes everything.

What this leaves #

A collapse onto one number, $t/c$, which decides no lift, some of the slope and most of the suction peak; and a residual — where along the chord the thickness sits — which decides the rest of the peak and the whole of the recovery.

The next essay is about a collapse of a different kind: a rotor replaced by a disc, and what a disc cannot say because it has no blades.