A roll that feeds itself

The rung below this one settles what a stall is: a wing reaches an angle, and the speed at which an aeroplane arrives at that angle is an answer with four other variables in it. The essay’s own list of what it does not contain ends with an exclusion:

A rigid, symmetric, coordinated aeroplane. No sideslip, no yaw, no asymmetry, and therefore no spin — which is the thing that happens when one wing stalls before the other and is the reason the accelerated stall is dangerous rather than merely startling.

A spin is the second most misdescribed event in flight after the stall itself, and the usual description — a stall that got worse — is wrong in the same way the stall’s own description is: it names a real thing and gives it a job it does not do.

What actually holds a wing level #

Start with the thing that stops working, because it is invisible while it works.

An aeroplane rolling at rate $p$ meets the air differently along its span. The down-going wing’s local incidence is increased by $py/V$ — it is descending into the air — and the up-going wing’s is reduced by the same. Below the stall the down-going wing therefore makes more lift, and the difference is a rolling moment opposing the roll.

That is roll damping, and it is why an aeroplane disturbed in roll returns to level rather than continuing over. It is not the ailerons, it is not the dihedral, and it is not the pilot: it is a property of a lift curve with a positive slope, integrated across a span. And the lift curve’s slope is the one number a thin aerofoil theory gets exactly right, which is why the damping is so reliable while the curve is straight.

And the sign that removes it #

Past the peak the slope of the lift curve is negative. The down-going wing is at the larger incidence and therefore makes less lift than the up-going one, and the rolling moment that was opposing the roll now reinforces it.

A wing in that condition does not merely fail to damp a roll. It amplifies one. Any disturbance — a gust, a control input, an asymmetry in when the two wings stall — grows, and the roll rate rises until something else limits it.

That is autorotation, and it is a spin’s whole engine. The rotation is not the aircraft falling untidily out of a stall; it is a self-sustaining motion with a positive feedback in it, and it will continue as long as the wing is held past the peak.

Where it begins, which is not where a reader would put it #

The obvious guess is that autorotation begins at the stalling angle, since that is where the slope changes sign. The calculation says otherwise and the margin is worth knowing.

The stalling angle of the prescribed curve is 16.46°, and at a modest roll rate the damping vanishes at 17.07° — a little over half a degree past. At a fast roll it is further still.

The reason is that a rolling wing is not at one incidence. The down-going tip is at a larger angle and the up-going tip at a smaller one, and at a base incidence just past the peak most of the up-going wing is still on the rising part of the curve and still damping. The integral over the span is a competition between the two, and it takes a little more base incidence for the losing side to win.

Which produces a small, useful conclusion. There is a band of incidence — narrow, and a real one — in which the wing is stalled and the aircraft still damps a roll. An aeroplane entering a stall symmetrically and gently is in that band, which is why a well-behaved training aircraft can be stalled straight ahead all afternoon without anything dramatic happening. Where the flow has actually let go when that is happening is a separation the pressure gradient decides, and it starts at the root on a wing designed to have it start there. What takes it out of the band is depth, and what takes it out fastest is yaw.

Why a spin needs yaw and a stall does not #

Autorotation is a roll, and a spin is a roll and a yaw together. The connection between them is where the misdescription does its damage.

A rolling aircraft has more drag on the down-going wing, because that wing is at a higher incidence and past the peak its drag is still rising while its lift falls. The drag difference is a yawing moment towards the down-going wing, which sideslips the aircraft, which increases the incidence of that wing further — and the motion becomes a coupled roll and yaw rather than either one.

That coupling is what makes a spin stable rather than merely transient. The aircraft settles into a steady rotation with a definite rate, a definite incidence and a definite rate of descent, and it will hold that state indefinitely.

And it is why the recovery is a yaw control rather than a roll control. Standard spin recovery is full opposite rudder, then the elevator forward — stop the yaw first, then unstall the wing. Using ailerons to pick up the down-going wing does the opposite of what a pilot expects: deflecting an aileron down on a stalled wing raises its incidence further and drives it deeper past the peak, which increases the autorotation rather than opposing it.

A control that reverses its effect is the most dangerous kind of control, and this collection has met the pitch-axis version of it in the tailplane that stalls on flap extension, and this one does so at exactly the moment its use is most instinctive.

The design variables, all of which are the same variable #

Everything an aircraft designer does about spin resistance is an attempt to control where and how sharply the wing’s lift falls, and the figures above make each one legible.

Washout — a twist that gives the tip a lower incidence than the root — means the root stalls first and the tip is still on the rising part of the curve when it does. The span-integrated damping therefore survives past the point where the root’s own curve has turned over, and a wing with washout has a wider band of stalled-but-damping incidence.

A stall strip at the root does the same thing by making the root stall earlier and more definitely, which sounds perverse and is the point: a wing that stalls at the root first drops its nose and keeps its ailerons.

And a sharper aerofoil is worse. A thin, sharp nose gives a deep suction peak and a steeper pressure rise behind it, which is what makes the separation abrupt. The dropDeg parameter in the figures is exactly this — how many degrees the lift takes to fall from the peak to its plateau — and a thin, sharp-nosed section with an abrupt leading-edge stall has a small one. That is why the aerofoils on training aircraft are thick, docile and aerodynamically unexciting.

The rate a spin settles at, and why it settles at all #

An instability that grows without limit is not what a spin is, and the difference is worth spelling out because it is the second half of what makes a spin steady rather than divergent.

The autorotation figure shows the moment becoming negative and then, further past the peak, returning towards zero as the lift curve flattens onto its plateau. On the plateau the slope is zero, the down-going and up-going wings make the same lift, and the driving moment vanishes.

So the negative region is bounded on both sides, and a roll that starts growing accelerates until the wing has been carried far enough past the peak that the moment has come back to zero. That is an equilibrium roll rate, and it is stable: a faster roll finds a restoring moment and a slower one finds a driving one.

A spin’s rotation rate is therefore a property of the wing’s post-stall lift curve rather than of how the spin was entered — which is why a given aircraft spins at a characteristic rate, why that rate is quoted in its flight manual, and why two entries that looked completely different converge onto the same motion within a turn or two.

And it explains the one thing about spins that surprises everybody who watches one. The rotation is not violent. It is a steady, almost leisurely turn at a rate a person can count, in an aircraft that is descending fast and doing nothing sudden at all. The drama is entirely in the fact that it will not stop.

Why the misdescription persists #

Three reasons, and only the last is a genuine confusion.

The two events are adjacent. A spin requires a stall, so every spin is preceded by one, and a sequence is easily read as a cause. But a stall requires nothing of a spin, and the overwhelming majority of stalls do not become one — which is the asymmetry the phrase “stall-spin” conceals.

The recovery drills share a step. Both end with reducing the incidence, so a pilot’s actions in the two cases look similar from outside. They are not the same drill: a stall recovery is one action and a spin recovery is a sequence in which getting the order wrong makes things worse.

And the word “stall” is doing two jobs. In the stall of the rung below, the wing has reached its peak and lift has stopped rising. In a spin, the wing is far past that, on the plateau, in a condition the lift curve of ordinary flight says nothing about. Calling both “stalled” is accurate and it hides that they are dozens of degrees apart, in a region where the aerodynamics is entirely different — and that is the same failure of vocabulary the rung below identifies in “stalling speed”.

What a stall warning cannot warn about #

The rung below argues that an angle-of-attack indicator is the honest instrument, since the wing stalls at an angle and the speed moves. This rung adds a limit to that recommendation, and it is not the contamination case the previous essay already names.

An angle indicator reports the aircraft’s incidence — one number, measured by one vane, usually on the forward fuselage. What decides whether the wing autorotates is the incidence at each station along the span, and those differ from the fuselage’s by the wing’s twist, by any asymmetry between the two wings, and — the important one — by the roll rate itself.

So an aircraft that is already rolling has one wing well past the indicator’s reading and the other well below it, and the instrument reports the average. At a roll rate that would be unremarkable in ordinary flight, the tips differ from the mean by a degree or two, which is the whole width of the band this essay has been measuring.

That is not an argument against the instrument, which remains far better than an airspeed. It is an argument about what a single scalar can report: the quantity that decides the outcome is a distribution across a span, and every instrument that reduces it to one number has thrown away the asymmetry that does the damage.

Which is the same objection this collection makes to a site average of anything. A mean over a distribution is silent about the tail, and the tail is where the events are.

What a wing does about it, which is a distribution #

The design section above listed washout, stall strips and a docile aerofoil, and all three are doing one thing: arranging where along the span the wing stalls first. It is worth making that explicit, because it turns three apparently unrelated devices into one variable.

The roll damping is an integral over the span of the local lift difference, weighted by the distance from the centreline. Two things follow immediately.

The tips dominate. The weighting is by $y$, so a station at the tip contributes several times what the same station’s worth of area contributes near the root — and on an elliptic planform the chord is falling there, which offsets it only partly. What the tips are doing is most of what the damping is.

So the design goal is that the tips stall last. Washout gives them a lower incidence than the root; a stall strip makes the root give way earlier and definitely; a thicker root section with a gentler stall does the same by a third route. All three buy the same thing: a band of incidence in which the root has let go and the tips have not, so the integral is still stabilising and the aircraft still damps a roll while its wing is partly stalled.

That band is what a training aircraft’s docility consists of, and it is why the lift curve’s straight part matters less than what happens at its end. A wing designed for a high maximum lift coefficient and nothing else has no such band, and a wing designed with one gives up a little lift for it.

The tips also decide something else that follows from the same integral. Ailerons are at the tips, because that is where a control surface has the most leverage — and it is where the wing must not stall, because a stalled aileron reverses. The two requirements point the same way, which is a rare and welcome coincidence and is why the design rule is as durable as it is.

What the picture cannot show #

The post-stall lift curve is prescribed. Nothing in this collection computes the lift of a stalled wing. What is prescribed is a curve that rises, peaks and falls, with the fall’s steepness and depth named as parameters in every figure — and every conclusion above depends only on the sign of the slope rather than on the particular shape.

No yaw is computed. The spin is described and not modelled. What the figures compute is the roll damping, which is autorotation’s engine and is only one of the three axes a spin involves.

The strip integration is a strip integration. Each spanwise station is treated as a two-dimensional aerofoil at its local incidence, with no account of the induced flow the rest of the wing produces — which is a serious approximation on a rolling wing, since the roll rate itself alters the wake — the same objection a lifting line answers for an unrolled wing and that nothing here answers for a rolling one.

And the wing is elliptic and untwisted. Washout is discussed above and is not in the calculation; adding it would move the crossing and would not change its existence.

The assertion behind the figures is the one that could reject: the damping must be positive at four degrees and negative past the stall, with the crossing found by bisection rather than assumed to be at the peak. A model that placed the crossing at the peak by construction would have hidden the essay’s one genuinely new number.

Who found it, and when #

Autorotation was identified in the 1910s, at a time when spins were killing pilots at a rate that made the question urgent and nobody knew what a spin was. Lindemann’s spin experiments at Farnborough in 1917 — in which he learned to fly in order to make the measurements himself, then deliberately spun an aircraft while recording the motion — established that a spin is a steady state with a definite rate rather than a loss of control, which is the observation everything else rests on.

The theoretical side is Glauert’s and Lanchester’s, and it is exactly the argument above: a wing past the peak of its lift curve has negative roll damping, and a negative damping is an instability.

The vocabulary never caught up. “Stall-spin accident” is a phrase that names two things as though one caused the other, and while a stall is necessary for a spin it is nowhere near sufficient — what is also needed is yaw, and yaw is very often the pilot’s own rudder input trying to correct a wing that has dropped. That sequence, and not the stall, is what the phrase should describe.

Where the ladder goes next #

Autorotation is a roll that will not stop, and the aircraft in it is in a rotation it can be flown out of by a pilot who knows the drill. The rung above is the case that cannot be, and its mechanism is neither roll nor yaw but pitch.

A tailplane mounted high — on top of the fin, in the T-tail arrangement most rear-engined transports use — is swept into the wing’s wake at large incidence. The wake is thick, slow and turbulent, so the tail’s download collapses; and with the download gone, the wing’s own nose-up moment carries the aircraft to a second stable trim point far past the stall, from which the elevator cannot recover it because the elevator is inside the wake too. That is a stall which is a place rather than an event, and the arithmetic that produces it is a search for the zeros of a pitching moment.

The one beside it is the flat spin, which is where this rung’s motion and that one’s meet: a spin at high incidence in which the aerodynamic damping in all three axes has collapsed at once and the aircraft is descending nearly level, rotating, with every control surface in a separated wake.