An angle, not a speed

There is a white arc on an airspeed indicator, a number in a handbook, and a line in every briefing. The wing has no access to any of them.

What the wing knows #

A section’s lift coefficient rises with incidence, bends over, and comes down. The bending over is the boundary layer running out of momentum against the pressure recovery behind the suction peak, and it happens at an angle because the pressure distribution — normalised by the dynamic pressure — depends on the angle and on nothing else.

That is the whole of the wing’s contribution. The stalling angle is a property of the section, the planform and the Reynolds number, and once those are fixed it does not move. In the arithmetic here it is 16.5 degrees, and the solver requires it to be unchanged to machine precision when the weight, the density and the limit load factor are varied.

The speed comes from somewhere else entirely. In steady level flight the lift equals the weight, so

$$V_s = \sqrt{\frac{2W}{\rho S\,C_{L\max}}},$$

and every symbol on the right except $C_{L\max}$ is a fact about the aeroplane and the day rather than about the wing.

Four ways to move it #

Weight. The speed goes as its square root, so twenty-one per cent more weight is exactly ten per cent more speed — the gate requires that identity to nine decimal places, because it is arithmetic rather than aerodynamics. A transport aircraft’s stalling speed at maximum take-off weight and at landing weight differ by fifteen or twenty per cent, which is why the approach speed is computed for the day rather than looked up.

Density. Thinner air is less lift at the same true speed, so the true stalling speed rises with altitude — nineteen per cent at sixteen thousand feet in this arithmetic. The indicated speed does not move, which is the one genuinely useful thing an airspeed indicator does: it reads dynamic pressure, and the stall is at a fixed dynamic pressure for a fixed weight, so the white arc stays where it is.

Configuration. Flaps raise $C_{L\max}$, and the speed follows exactly: the flapped stall speed is the clean one times the square root of the ratio of the two maxima, required here to machine precision. That is why there are two arcs on the indicator.

And the load factor, which is the one that matters most and is the least visible.

The accelerated stall #

Nothing in the lift relation says the lift must equal the weight. It says the lift must equal whatever the aeroplane needs, and in a manoeuvre that is $nW$ — the load factor times the weight.

So

$$V_{\text{stall}} = V_s\sqrt{n},$$

exactly, and the solver checks it at four load factors to machine precision because it is an identity rather than a fit.

In a level turn at bank angle φ the load factor is $1/\cos\varphi$: 1.41 at forty-five degrees, 2 at sixty, 3.86 at seventy-five. So a wing that stalls at 29 metres per second wings-level stalls at 41 in a sixty-degree turn — and a pilot flying a fifteen-knot margin above the handbook figure has, in that turn, no margin at all.

That is the accelerated stall and it is the reason the misconception matters. The base-to-final turn, low, slow, banked and often uncoordinated, is where a very large fraction of light-aircraft stall accidents happen, and every one of them is an aircraft above its published stalling speed.

The whole envelope, on one figure #

The V–n diagram is where all of this becomes one picture, and the shape of it is the argument.

The left boundary is not a speed. It is a parabola, $n = \tfrac12\rho V^2 S C_{L\max}/W$, and every point on it is the wing at the same angle. The stall boundary is the stalling angle drawn in speed coordinates, and it stretches across the whole width of the diagram rather than sitting at one abscissa.

The top and bottom are structural, and they have nothing to do with the air. They are what the spar will take.

And where they cross is the corner speed — the lowest speed at which the aircraft can reach its limit load factor, and therefore the speed at which it turns hardest. Below it the wing stalls before the structure is threatened; above it the structure limits before the wing does. Fifty-seven metres per second here, against a stalling speed of twenty-nine: the corner speed is $V_s\sqrt{n_{\text{lim}}}$ and it is the most useful number on the whole diagram.

The single most telling figure is at the right-hand edge. At the never-exceed speed, this wing could pull 9.6 g before it stalled. The structure gives up at 3.8. So the aircraft can reach its stalling angle at any speed it can fly, and the only thing stopping the pilot is the wing coming off first.

The turn, worked out #

The base-to-final case deserves the arithmetic rather than a warning, because the numbers are more persuasive than the caution is.

An aircraft on final approach is flown at a margin above the stall — typically thirty per cent, so 1.3 V_s. That is a comfortable margin and it is a margin wings level.

Roll into a thirty-degree bank and the load factor is 1.15, so the stalling speed rises by seven per cent. The margin has fallen from thirty per cent to twenty-one. Roll into forty-five and the load factor is 1.41 and the stalling speed rises by nineteen per cent: the margin is now nine per cent. Roll into sixty and the stalling speed has risen by forty-one per cent and the aircraft is below it.

Now add the two things that go with an overshooting turn. The pilot, seeing the aircraft going wide, pulls — which raises the load factor further at the same bank. And, reluctant to bank more steeply close to the ground, uses rudder to tighten the turn — which yaws the aircraft, puts the outer wing at a higher speed and the inner wing at a lower one, and makes the inner wing stall first.

One wing stalls, the aircraft rolls towards the ground, and there is no height to recover in. The airspeed indicator throughout has been showing a number above the one in the handbook.

Why the speed is what gets published #

Given all that, it is fair to ask why the handbook prints a speed at all.

Because the aircraft has an airspeed indicator and most aircraft do not have an angle-of-attack indicator. The one instrument that reads the quantity that actually matters is standard on carrier aircraft — where the approach is flown to an angle rather than a speed, because the weight varies enormously and an angle does not care — and is an option on almost everything else.

Because the published speed is a worst case. $V_{s0}$ and $V_{s1}$ are quoted at maximum weight, which makes them conservative at any lower one. The margin is real if the aircraft is straight and level.

And because a speed is easy to fly. A number on a dial the pilot is already looking at is operationally worth a great deal, and the conditions under which it is misleading — banked, manoeuvring, heavy, iced — are conditions a briefing can name.

The number is a proxy, it is a good proxy under stated conditions, and the stated conditions are exactly the ones that fail in the accidents. That is a fair summary and it is why the misconception is worth attacking rather than tolerating: the belief that a wing stalls at a speed makes the accelerated stall unimaginable, and the accelerated stall is the one that kills people.

What an angle-of-attack indicator changes #

It is worth stating what the instrument that reads the right quantity would do, because it makes the argument concrete.

An angle-of-attack indicator reads a number that is the same at the stall in every configuration and at every weight, altitude, and load factor. The approach can be flown at a fixed indication regardless of what the aeroplane weighs; the manoeuvring margin is visible directly rather than inferred; and the accelerated stall stops being a surprise, because pulling into a turn moves the needle immediately in a way an airspeed indicator does not.

That is why carrier aviation adopted it universally and why it is standard on transports as an input to the stall-warning system, if not always as a display. The stick shaker on a transport is driven by angle of attack, not by speed, and the reason is everything above.

The aeroplane’s own protection system already knows the misconception is wrong. It is only the briefing that does not.

Three other places the same confusion appears #

The pattern — a quantity the physics fixes, reported as a quantity the operator can read — is not peculiar to the stall, and naming the family makes it easier to spot.

The critical Mach number is a Mach number, and it is flown as an airspeed. The maximum operating speed of a transport is a speed at low altitude and a Mach number high up, and the crossover is where the two limits meet. A pilot holding an indicated airspeed through a climb is approaching the Mach limit without the indicator saying so, which is why there are two needles.

A propeller’s limit is a tip Mach number, and it is flown as an engine speed. The thing that matters is the helical tip speed, which combines the rotational speed and the forward speed, and the tachometer knows only the first.

And a structural limit is a load factor, and it is flown as a speed. The V–n diagram’s flat top is the same shape of substitution: the structure cares about n, the pilot has a speed, and the corner speed is where the substitution stops being conservative.

In every case the reported quantity is a usable proxy in the ordinary case and stops being one in exactly the situations that matter. The general remedy is to know which quantity the physics actually fixes, which is a small amount of theory that pays for itself.

The one case where the angle moves too #

The invariant this essay is built on — the speed moves and the angle does not — holds for a wing that is the shape it was designed to be. There is a situation in which the angle moves as well, it is the situation in which the misconception does the most harm, and it deserves stating because it removes the remedy the previous section recommends.

Contamination changes the section. Frost, rime or a ridge of ice behind a de-icing boot alters the suction peak and the pressure recovery behind it, and the layer separates earlier. Both numbers fall: the maximum lift coefficient by something between a tenth and a third, and the stalling angle by several degrees. A wing with a millimetre of frost on it is a different aerofoil, and there is no reason its stalling angle should be the clean one.

That has two consequences and both remove a protection.

The published stalling speed is now wrong in the unsafe direction, because $C_{L\max}$ has fallen and the speed goes as its inverse square root. And — this is the important one — an angle-of-attack indicator is wrong too, because it is calibrated against a clean wing’s stalling angle. The instrument that this essay recommends as the honest one is honest about the angle and not about what the angle means, and a stall-warning system driven from it warns late or not at all. Several loss-of-control accidents on departure after an inadequate de-icing turn on exactly that: the wing stalled below the angle at which anything was set to complain.

So the invariant is a property of a shape rather than of a wing, and anything that changes the shape changes it. That is a narrower claim than the essay has been making and it is the correct one.

The tailplane makes the same point with the signs reversed, and it is worth knowing because the recovery is inverted. A tailplane in a conventional aeroplane carries download, so it operates at a large negative angle of attack, and extending flaps increases the wing’s downwash and drives that angle further negative. A contaminated tailplane can therefore reach its own stall — and when it does, the download disappears and the aircraft pitches abruptly nose down, at the moment of flap extension, in the approach.

Everything a pilot has been trained to do about a stall makes that worse. Lowering the nose increases the tail’s angle further. Adding power increases the downwash on a propeller aircraft. Extending more flap does the same. The recovery is to pull, to retract the flaps and to reduce speed — the exact opposite of the wing-stall drill — and knowing which of the two surfaces has stalled is the whole of the decision, in a situation lasting a few seconds.

Which is the essay’s argument in its sharpest form. A stall is a property of a surface reaching an angle, and an aeroplane has more than one surface. Whose angle, on which surface, in which condition is the question, and a single published speed encodes none of the three.

What the model does not contain #

No stall. The maximum lift coefficient is an input. Nothing here computes where a boundary layer separates, and the whole essay is arithmetic downstream of a number taken from measurement.

The stalling angle is treated as fixed, which it very nearly is and not exactly. It moves a little with Reynolds number, with surface condition, with Mach number and with configuration. None of those movements is remotely as large as the speed variations the essay is about, which is what makes the approximation useful.

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.

Level flight or a level turn. The load factor relation $n = 1/\cos\varphi$ assumes altitude is being held. A descending turn has a lower load factor at the same bank, which is one of the ways a base-to-final turn can be flown safely and one of the ways the arithmetic can mislead.

And no dynamics. A stall entered abruptly behaves differently from one entered slowly: the flow takes time to separate, and a wing pitched up rapidly can exceed its static stalling angle substantially before it lets go. That is dynamic stall, it is a real and useful effect, and it is the mechanism the next essay is about in a different context entirely.

Who said it, and when #

The stalling speed as a published number is as old as certification, and it is in the earliest airworthiness requirements because it determines the field length, the approach speed and the structural design speeds. It has always been defined at a stated weight and configuration, and the qualification has always been in the document.

What the qualification does not survive is retelling. The aircraft stalls at 55 knots is what gets said, and it is what gets remembered, and it is what a pilot has in mind while rolling into a turn at 70.

Accident investigators have been writing about this for as long as there have been accident investigators. The National Transportation Safety Board’s studies of loss-of-control accidents return to it repeatedly, and the phrase that recurs — the aircraft was above its published stalling speed — is not an observation that something unexpected happened. It is a description of the misconception.

A number that is correct under conditions nobody states is more dangerous than a number that is wrong, because nothing about it invites checking.

Where the ladder goes next #

The last of the famous claims is the oldest and the most repeated: that aerodynamics proves a bumblebee cannot fly. It has a traceable origin, the calculation behind it is a real calculation, and what makes it interesting is that doing the arithmetic properly still leaves a gap — which took another sixty years and a new mechanism to close.