Which part of a wing stalls first

A wing has a lift coefficient. It is printed on the polar, it goes into the performance calculation, and it is an average — the total lift divided by the total area and the dynamic pressure. No section of the wing is actually at it.

That gap is where a great deal of aeroplane design lives, because a stall is a local event. The wing does not run out of lift; one station on it runs out of lift, the separation spreads from there, and where that station is decides whether the aircraft drops a wing, whether the ailerons still work while it happens, and whether the pilot gets any warning at all.

What sets the local coefficient #

Prandtl’s lifting line gives the circulation across the span, and the section lift coefficient at a station is

$$C_l(y) = \frac{2\Gamma(y)}{U\,c(y)}.$$

Two things divide there and they are set by different parts of the design. The circulation is decided by the whole wing at once — every station’s downwash depends on every other station’s shed vorticity, which is what makes the lifting line an integral equation rather than a strip theory. The chord is decided locally by the planform.

So a wing that takes chord away faster than it takes circulation away has its highest section coefficients outboard, and one that does the reverse has them inboard. That is the entire argument, and the rest is which planforms do which.

Three planforms, three places to stall #

A rectangular wing stalls at the root. Its chord is constant, so the section coefficient follows the circulation, which is largest in the middle. That is the best possible answer: the separation starts inboard, ahead of the ailerons, and the turbulent wake washes over the tailplane, which is what a pilot feels as pre-stall buffet. A Cub or a Cessna stalls this way and does it gently.

A tapered wing stalls outboard. Taper removes chord fast towards the tip, and the circulation does not fall as fast, so the ratio peaks out at about 0.7 of the semi-span. That is where the ailerons are. A wing that separates there loses roll control at the moment the pilot most wants it, and if one side goes marginally before the other the aircraft drops a wing and starts to spin.

An elliptic wing stalls everywhere at once. Its chord and its circulation are the same shape, so the ratio is a constant: every station has the same section coefficient and reaches its maximum on the same instant. That is the property that makes it optimal — uniform downwash is the condition for minimum induced drag — and it is simultaneously the worst stall behaviour a wing can have. There is no warning, because there is no part of the wing that goes first.

The Spitfire is the famous elliptic wing, and its stall was in fact benign — which is not a contradiction, because it had washout, and washout is the next section.

Why taper is used anyway #

Given that taper puts the stall in the worst place, it is worth being clear about why every fast aeroplane has it.

It is nearly as efficient as an ellipse. A taper ratio of about 0.4 gives a span efficiency around 0.99: the induced drag penalty against elliptic is one per cent. The solver checks this as a comparison rather than by quoting a formula — three wings are brought to the same lift coefficient and the elliptic one is required not to be beaten, which is the theorem stated as the thing a figure could be wrong about.

It is very much lighter. The bending moment at the root is an integral of lift against distance, and taper moves lift inboard, so a tapered wing’s spar carries less moment for the same lift. It also has more depth where the moment is largest.

And it is far easier to build. An elliptical wing is a different rib at every station. A straight-tapered wing is two ribs and a linear interpolation, which in 1935 was the difference between a wing that could be made in numbers and one that could not.

So the design already wants taper for three reasons, and the stall behaviour is a bill that arrives afterwards. The way it is paid is by twisting the wing.

Washout, and what it costs #

Washout is a linear twist that reduces the tip’s incidence relative to the root. Two or three degrees is typical, and it enters the lifting-line solve as a spanwise right-hand side — the monoplane equation is linear in the local incidence, so adding a twist changes the vector without touching the matrix.

It does exactly what it is asked to. Four degrees of washout moves the peak section coefficient from 0.72 of the semi-span to 0.35, which is inboard of the ailerons, and the stall starts where a designer wants it.

The bill is on the next figure.

The logic is airtight and slightly depressing. Elliptic loading is optimal; washout deliberately un-loads exactly the part of the span the optimum wanted loaded; therefore washout costs induced drag, always, in proportion to how much of it there is. A wing with four degrees of washout on a 0.4 taper is several per cent down on span efficiency.

There is no configuration on that figure which is best at both things. The trade is real and it is made every time, and the answer depends on what the aeroplane is for: a glider takes less washout and accepts a sharper stall, an aircraft that will be flown by strangers takes more.

Where the aspect ratio does and does not come in #

It is worth separating two things the aspect ratio does, because they pull in opposite directions and are often run together.

A longer wing has less induced drag, which is the whole content of the span essay and is a statement about the wing as a whole.

A longer wing has a peakier local coefficient, which is a statement about the sections. As the aspect ratio rises the loading approaches the chord distribution more slowly — the induced angle falls, so the sections behave more like isolated two-dimensional sections and the local coefficient tracks the geometry more directly. On a high-aspect-ratio tapered wing the peak is further out and sharper, and the margin between the first section to reach its limit and the rest is smaller.

So a sailplane, which wants the first property as much as any aircraft ever has, gets the second one whether it wants it or not, and pays for it in washout that costs it back some of the drag it went long to save. That is not a small effect on a modern sailplane and it is one of the reasons the washout distribution on one is a designed curve rather than a linear ramp.

The one that changes with lift, and the one that does not #

There is a subtlety worth pulling out, because it separates two fixes that look alike.

Washout is a fixed geometric twist, so its effect on the loading is a fixed increment. At high lift coefficients that increment is a small fraction of the total and the wing behaves nearly like its untwisted self; at low lift coefficients — cruise — the increment is a large fraction, and the tip may be carrying negative lift. That is why a washed-out wing is least efficient exactly where it spends most of its time.

Aerodynamic twist — using a different section outboard, one with a more negative zero-lift angle or a higher maximum — does something different: it changes where the section limit is rather than where the loading is. It costs nothing in span efficiency at all, because it does not move the loading. What it costs is that the outboard section is a different, usually thinner, more awkwardly-behaved shape.

Real wings use both, and the mixture is chosen so that the geometric part is small enough not to hurt cruise and the aerodynamic part does the rest. The two look like the same fix and are charged to different accounts.

The fix that spoils a station on purpose #

There is a third way of moving the stall inboard, and it belongs on the list because its bill is charged to a third account again.

Washout moves the loading and pays in span efficiency. Aerodynamic twist moves the limit and pays in section quality. The third option moves the limit too, and moves it the other way: fit a small sharp strip along the inboard leading edge and that station’s maximum lift coefficient is deliberately lowered, so it reaches its limit before anything outboard of it does.

A stall strip is a triangular section a few millimetres high running along the first thirty or forty per cent of the semi-span. It works by forcing an early leading-edge separation over the stretch it covers, so it does not raise anybody’s margin — it removes some of the root’s. The stall then begins there, spreads outward, buffets the tail and leaves the ailerons working.

Its virtue is that it is nearly free. It changes no loading, so it costs no induced drag at all, and its parasite penalty is a few counts. Set against washout, whose cost is unavoidable and paid at every lift coefficient including cruise, that is a very good bargain — and it is why stall strips are fitted so often, and fitted late. They are the standard remedy when flight test finds a stall that drops a wing or gives no warning, at a stage of a programme where re-twisting a wing is not an option.

The complementary device does the opposite outboard. A leading-edge cuff or drooped outer panel raises the outboard section’s maximum locally, adding margin exactly where the peak sits, and it too leaves the loading alone.

So the taxonomy is four: move the loading, change the section, spoil a station, or extend one. Only the first is charged to the induced drag, and it is the one this essay’s arithmetic can compute.

What a stall station is worth knowing #

Three practical readings, and all three are about the difference between the two curves on the first figure.

A wing-level lift coefficient does not predict a stall. Two wings at the same C_L can be at very different distances from separation, and the one closer to it is the one whose loading is peakier relative to its chord. That is why a polar is not enough and a spanwise loading calculation is part of the design.

A flap changes the stall station. Deploying an inboard flap raises the circulation inboard and lowers the section coefficient outboard for the same total lift — so a flapped wing usually has a more inboard stall than the same wing clean. That is a helpful accident and it is why the approach stall is often gentler than the clean one.

And a damaged or contaminated wing is a wing whose local maximum has moved. Ice on one wing lowers that wing’s section maximum without changing the loading, so the affected wing reaches its limit first, and the aircraft rolls at a lower angle of attack than the pilot expects. The asymmetry is the danger and not the loss of lift.

Reading a wing that has already stalled #

The arithmetic above is a design tool, and there is a matching reading task in front of a real wing — usually a tuft picture or a run of oil, and usually taken in a wind tunnel at the moment somebody is trying to find out why an aeroplane drops a wing.

What to look for is not the extent of the separation but its inboard edge. A tuft picture at one degree past the stall shows a patch of disturbed flow, and the useful information is where the patch starts, because that is the station whose section coefficient reached its limit first and everything else followed. Two wings can have identical stalled areas and completely different handling if one patch starts at 0.3 of the semi-span and the other at 0.75.

The second thing to look for is symmetry. A wing that stalls symmetrically pitches; a wing that stalls asymmetrically rolls, and a roll at the stall is how a spin begins. Manufacturing tolerance, rigging, a repaired leading edge or a film of ice on one side are all ways of moving one wing’s local maximum a little below the other’s, and the arithmetic above says why a small asymmetry is enough: near the stall the section coefficient curve is nearly flat through its peak, so a very small change in the limit moves the station that reaches it a long way.

What the model does not contain #

No stall. The lifting line is inviscid and linear: nothing in it can separate, and the section maximum drawn on these figures is a line put there by hand. What the calculation gives is where the limit is reached first, on the assumption that every section has the same limit — and the reason that is useful is that the ordering is far more robust than the value.

Every section is assumed identical. A real wing changes section along the span, and the maximum lift coefficient changes with it. That is the aerodynamic twist above, and it is deliberately outside this arithmetic.

No sweep. A swept wing’s spanwise drift thickens the outboard boundary layer and moves the stall further out than any planform argument predicts. Sweep and taper push the same way and a swept tapered wing needs more help than either alone.

No Reynolds number variation. A tapered wing’s tip chord is smaller, so its tip runs at a lower Reynolds number and a lower section maximum, which makes the tip stall worse again. Nothing here knows that.

Linear twist only. The solver takes a linear washout because that is what a straight-tapered wing is built with. An arbitrary twist distribution is a design variable and the same machinery would take it; the essay does not, because the argument is about a mechanism.

And the collocation is the usual one. The monoplane equation holds exactly at the ten stations where it is collocated and says nothing between them, so the solver checks the residual off the grid — which is the only version of that check that means anything.

Who found it, and when #

Prandtl’s lifting line is 1918 and the twist term was in it from the start; what took time was the recognition that the section coefficient rather than the wing’s was the quantity that decided the stall. That belongs to the 1930s, and it arrived through accidents rather than through theory: tapered monoplanes were replacing rectangular biplanes, and they were dropping wings on the approach in a way the biplanes had not.

Anderson’s NACA reports of 1936 are the usual reference for the method of adding a twist increment to a basic loading, and they are explicitly a design tool: the object is a table a designer can use, not an account of a mechanism. The mechanism had been available in Prandtl’s equations for eighteen years and nobody had divided one curve by another.

That is the recurring shape of this part of the subject. The theory is old, complete, and phrased in terms of the quantity the theory finds natural — circulation. The design question is phrased in terms of a different quantity — the local section coefficient — and the step between them is a division that nobody takes until an aeroplane makes them.

Where the ladder goes next #

Elliptic loading has been the optimum in every argument so far, and the optimum of a problem is only as interesting as the constraints it was optimised under. The constraint used here is at a given lift on a given span. A wing is not built to a given span; it is built to a structure that has to carry the moment the lift makes about the root — and the loading that minimises drag under that constraint is a different curve, published in 1933, and ignored for eighty years.