Where the lift acts

Every drawing of a wing has an arrow on it labelled lift, and every one of those arrows starts somewhere. The choice is almost never explained, and it cannot be made arbitrarily without consequences.

What is really acting on the wing is a pressure distribution: a different value at every point of the surface, pushing inward everywhere along the local normal. Replacing all of that with one arrow means choosing both a magnitude and a place, and the magnitude is the easy half.

Why a point has to be chosen at all #

A distributed load is equivalent to a single force plus a couple, and the couple depends on where the force is placed. Move the arrow forward and the couple has to become more nose-up to compensate; move it back and the reverse.

So “the lift acts at X” is not a fact about the wing. It is a statement about a bookkeeping choice, and any point at all can be used provided the accompanying moment is carried along with it.

That freedom is why two different points get named, and why they are constantly confused.

The centre of pressure is the point about which the moment is zero. It is the honest answer to “where does the lift act”, in the sense that placing the arrow there needs no couple at all. It has one severe disadvantage: it moves. As incidence changes, the pressure distribution changes shape, and the point where the moment vanishes slides along the chord — forwards as lift increases, and off the back of the section entirely as lift goes to zero.

The aerodynamic centre is the point about which the moment does not change with incidence. It is not where the lift acts in any natural sense, and there is generally a non-zero moment about it. What it has is the property that makes it useful: it stays put.

The distinction is worth an example, because the two are so often used as synonyms. A symmetric section at four degrees has its centre of pressure at the quarter chord; drop it to one degree and the centre of pressure is still at the quarter chord, because a symmetric section is the special case where the two points coincide. Now camber the section. At four degrees the centre of pressure sits somewhat ahead of the quarter chord; at one degree it has moved well forward; at zero lift it is at infinity, because a section making no lift and a non-zero moment has no point at which the moment vanishes. The aerodynamic centre, through all of that, has not moved at all.

What the solver computed, and how it was checked #

The usual demonstration takes the moment about the quarter chord and shows it is constant. That presupposes the answer, and this site’s habit is to refuse demonstrations of things already assumed.

So the chord is swept instead. Forty-one candidate reference points are laid out from 5% to 60% of the chord. About each one, the pitching moment is computed at four angles of attack by integrating the surface pressure — the same integral that gives the lift, with a moment arm in it — and a least-squares slope of moment against incidence is taken. That gives forty-one slopes, one per candidate, and the aerodynamic centre is where the slope crosses zero.

It crosses once, and the crossing is interpolated at 25.6% of the chord. Thin-aerofoil theory predicts exactly 25%.

The difference is not noise. It is thickness: the section drawn has 10% thickness and 8% camber, and thin-aerofoil theory assumes a section of no thickness at all. A thicker section has its centre slightly further aft, which is a known result, and finding it here without having asked for it is the kind of agreement worth more than a match to three decimal places would be.

The assertion that guards this refuses two things. It refuses a result more than 4% of chord away from the quarter chord, and it refuses a sweep in which the slope never changes sign at all — which would mean the search had found nothing and interpolated a number anyway.

Why a quarter #

The thin-aerofoil result is worth an argument rather than a citation, because the number looks arbitrary and is not.

A thin section is modelled as a line of vortices distributed along its chord, with the strength at each station chosen so that the flow follows the camber line and the Kutta condition is met at the back. Working that out produces a distribution whose additional loading — the part that appears when incidence is changed — is a specific shape, and the centroid of that shape sits at a quarter of the chord from the leading edge.

That is the whole content of the number. Extra lift generated by pitching the wing arrives, on average, at the quarter chord. The lift that was already there may be distributed quite differently, depending on the camber, which is why the moment about the quarter chord is generally not zero — only constant.

Camber decides that constant. A symmetric section has zero moment about its quarter chord at every incidence, so for it the aerodynamic centre and the centre of pressure coincide and neither moves. A cambered section has a persistent nose-down moment about the quarter chord, which is the Cm0 that appears in every set of aerofoil data, and which the second figure above shows as the height of the flat line above zero.

There is a design consequence in that constant which is easy to miss. A cambered section’s nose-down moment has to be balanced by something, and on a conventional aircraft the tailplane does it. On a flying wing there is no tailplane, so the sections have to be chosen so that the moment is nearly zero — which means reflexed camber, an aerofoil whose rear portion curves back upwards. That is why flying wings and tailless aircraft look the way they do at the trailing edge, and the reason is one coefficient in this essay.

Why anybody cares #

This is not a filing convention. The aerodynamic centre is the reason aircraft can be made stable.

An aircraft is longitudinally stable if a disturbance that pitches it nose-up produces a moment that pitches it back down. Working out whether it does means adding up the moments from the wing, the tailplane and the fuselage about the centre of gravity — and doing that at every angle of attack the aircraft might encounter.

If the wing’s contribution had to be recomputed from scratch at every incidence, the calculation would be a nightmare. Because the aerodynamic centre exists, it does not: the wing contributes a force at a fixed point plus a constant couple, and the whole incidence-dependence of the problem collapses into the geometry of where the centre of gravity sits relative to it.

The rule that comes out is the one every pilot knows. Centre of gravity ahead of the neutral point — the whole-aircraft equivalent of the aerodynamic centre — and the aircraft is stable. Behind it and it is not. The loading limits printed in every aircraft’s manual are that inequality, converted into kilograms and positions.

The margin between the two positions has a name — the static margin — and it is quoted as a percentage of the mean chord. Fifteen per cent is a comfortable general-aviation figure; five is twitchy; zero is neutral, and the aircraft will hold whatever attitude it is left in and drift away from it at the first disturbance. Modern combat aircraft are deliberately built with a negative static margin, because an unstable aircraft responds faster, and the instability is dealt with by a computer making corrections several dozen times a second. That is a design that could not exist before the computer did, and its whole premise is a number located at 25% of a chord.

The same reasoning explains why a conventional aircraft has a tailplane pushing downwards in cruise. The centre of gravity must be ahead of the wing’s aerodynamic centre for stability, which leaves the wing’s lift acting behind the weight, which produces a nose-down moment, which something has to balance. The tailplane balances it by pushing down, and the wing has to make more lift than the aircraft weighs to compensate. Stability is paid for in drag.

The same integral about a hinge #

Everything above takes the moment about a point on the chord chosen for convenience. There is a point on a real aircraft that is not a matter of convenience at all — the hinge line of a control surface — and the moment about it is what a pilot has to hold.

It is the same integral over the same pressure distribution, restricted to the part of the section aft of the hinge. What makes it a different problem is how it scales. A hinge moment is a pressure times an area times a moment arm, so it goes as the dynamic pressure and as the cube of the control surface’s size. Double the aircraft and the stick force goes up eightfold at the same speed; double the speed and it goes up fourfold again. That single scaling is why a light aircraft has cables and a large fast one has hydraulics, and the boundary between them is a hinge moment rather than a philosophy.

The devices for managing it are all visible on any airfield and all do the same arithmetic. Aerodynamic balance moves the hinge line aft, so that a strip of the surface lies ahead of it and its load acts to reduce the total moment rather than to add to it; a horn balance is the same trick concentrated at the tip, where the arm is longest. A tab at the surface’s own trailing edge, deflected the other way, produces a small force on a long arm and can cancel the moment entirely.

And the balance can be overdone. Move the hinge too far back and the moment reverses sign: the surface now tends to deflect further once it has started, the stick force falls as the deflection grows, and a control that should resist the pilot instead runs away from them. A reversed stick-force gradient is a certification failure and has been an accident cause, and the margin against it is a few per cent of one chord.

What the picture cannot show #

The first figure plots a derivative — how fast the moment changes — and a derivative discards the value. Two reference points with the same slope can have wildly different moments, and the figure cannot distinguish them. The second figure is included precisely because it keeps the value.

Neither figure shows the pressure distribution the moment came from. That is a real loss: the distribution is what is physically there, and the moment is an integral over it that many quite different distributions could produce. A reader could reasonably want to know whether the moment about the quarter chord is constant because the distribution is barely changing or because large changes are cancelling, and these figures cannot say. The answer is the second — the distribution changes a great deal, and the additional loading is what cancels.

The sweep also stops at 60% of the chord, which is a choice. The slope keeps going more negative beyond that, linearly, and there is no second crossing; but the figure does not show that and a reader is entitled to wonder. It stops there because that is where anything of engineering interest happens.

And the whole computation is two-dimensional. A wing has span, and the aerodynamic centre of a three-dimensional wing is not simply the quarter-chord line — sweep, taper and the trailing vorticity all move it. For a swept wing the difference is large rather than technical: the outer panels are a long way behind the inner ones, so the whole-wing centre sits well aft of where any individual section’s does, and the amount depends on how the loading is distributed across the span. That is the same loading distribution the lifting line computes, which means the stability of a swept aircraft cannot be settled without solving the spanwise problem first.

Where the model stops #

Everything here is ideal flow, and the aerodynamic centre survives into real flow better than most ideal-flow results, which is why it is used with such confidence.

It survives because it is a statement about the additional loading, and additional loading is dominated by the outer flow rather than by the boundary layer. Measured centres for real sections at ordinary Reynolds numbers sit within a couple of per cent of the quarter chord, which is remarkable agreement for a theory that has neither thickness nor viscosity in it.

It stops surviving in three places. Near the stall the loading collapses over the rear of the section as the flow lets go, the additional loading moves forward, and the centre moves with it — which is why aircraft pitch abruptly at the stall and why the behaviour is so hard to predict.

In transonic flow a shock forms on the upper surface and moves aft as Mach number rises, taking a large chunk of loading with it. The aerodynamic centre migrates rearwards by 10% of chord or more through the transonic range, which changes the stability of the aircraft in flight and is the reason transonic aircraft need fuel transfer or trim systems that subsonic ones do not.

And at very low Reynolds number, where the boundary layer is thick relative to the section, the effective shape the flow sees is not the shape that was drawn, and everything in this essay is approximate in proportion. A model aircraft’s wing is operating in a regime where the displacement thickness is a noticeable fraction of the section’s own thickness, and the aerofoil the air is following is measurably fatter and blunter than the one on the drawing.

Who found it, and when #

The aerodynamic centre falls out of thin-aerofoil theory, which is Munk’s and Glauert’s in the early 1920s — Max Munk at Göttingen and then at the American NACA, Hermann Glauert at Farnborough and then Cambridge, working the same ground within a couple of years of each other.

Its practical importance was established rather earlier and by more painful means. Early aircraft were frequently unstable in pitch, and the reason was not understood: designers moved the wings and the tail around by trial, and the trials were sometimes fatal. The Wrights deliberately built their first machines unstable, on the reasonable grounds that they intended to control them, which worked for them and for very few others.

There is a connection worth drawing here, and it is this essay’s surprise. The aerodynamic centre exists because the lift curve is straight — the additional loading has a fixed shape whose strength is proportional to incidence, and a fixed shape has a fixed centroid. The two facts are the same fact. Anywhere the lift curve stops being straight, at the stall or through the transonic range, the centre stops staying put, and it stops for exactly the reason it existed.

What thin-aerofoil theory supplied was not a new phenomenon but the ability to calculate before building. The 25% figure meant a designer could locate the neutral point on a drawing, place the centre of gravity ahead of it, and know the aircraft would be stable. That is the transition from aeroplanes that look like guesses to aeroplanes that do not, and it happened in about fifteen years.

Where the ladder goes next #

Next rungs on this anchor: the centre of pressure and its migration, which is what the moment looks like when the other convention is chosen; thin-aerofoil theory in full, where the camber line becomes an integral equation and Cm0 comes out of its first two Fourier coefficients; the neutral point of a whole aircraft, which is this quantity plus the tailplane’s contribution; and the aerodynamic centre in transonic flow, where it moves and the movement has to be designed around.

Then across to the lift curve, which supplies the force this essay is placing, and to the price of having ends, where the two-dimensional section becomes a wing and the centre moves again.