What a flap does, and what it does not

A flap is a hinged piece of the back of a wing. Lowering it lets an aircraft fly slower, which is what the whole apparatus exists for, and the reason it works is usually given as “more camber, more lift”. That is true and it is not specific enough to be useful, because there are two quite different things a change to a wing can do to its lift.

It can move the lift curve sideways, so that the section makes lift at an incidence where it made none. Or it can tilt the curve, so that each further degree of incidence buys more than it used to. Only one of those happens.

A flap is a kink in the camber line #

To thin-aerofoil theory a section is not a shape but a line — the camber line — and the thickness distribution round it contributes nothing to the lift. That is a strong claim and it is approximately right for the thin sections aircraft use, which is what makes the theory worth having.

A plain flap, in that language, is a camber line that is straight to the hinge and then bent down. There is nothing else to it: no gap, no slot, no motion, just an angle at one point.

That the whole of the effect follows from a change to the last quarter is worth pausing on, because it is not obvious and it is a claim the solve can be asked about directly.

What the solver computed, and how it was checked #

The camber line is divided into 120 panels. Each carries a point vortex at its quarter point, and at each panel’s three-quarter point the flow is required to be tangent to the camber line. That is a system of 120 equations in 120 unknowns, solved directly; the lift is the sum of the vortex strengths.

Nothing about flaps was built into that solver. It is the same routine that computes ground effect, with a different camber line and no image row.

Its calibration is three results it was not given. The flat plate comes out with a lift-curve slope of 6.2666 per radian against $2\pi = 6.2832$. A circular-arc camber line of 4% camber has its zero-lift angle at −4.522° against the theoretical $-2\varepsilon = -4.584°$. And the flap case itself has a closed form — Glauert’s — which the solve reproduces to within half a per cent.

Glauert’s number, and what it means #

The classical result gives the shift in zero-lift angle per degree of flap deflection as

$$\frac{d\alpha_0}{d\delta} = -\left(1 - \frac{\theta_f - \sin\theta_f}{\pi}\right)$$

where $\cos\theta_f = 1 - 2x_f/c$ locates the hinge. For a hinge at 75% of the chord that evaluates to 0.6090.

The panel solve, asked the same question, gives 6.123° of shift for 10° of deflection — a ratio of 0.6123, which is 0.5% from Glauert. At 20° the shift is 12.602°, a ratio of 0.6301, which is 3.5% away and in a direction worth explaining: the camber line uses the tangent of the deflection angle, so the geometry stops being linear before the theory does. The departure is the model being consistent rather than the theory being wrong.

The number itself says something surprising. A flap occupying the last quarter of the chord is 61% as effective as pitching the whole aeroplane. Deflecting it ten degrees does 61% of what raising the nose ten degrees would do, while moving a quarter of the surface instead of the entire aircraft. Moving the hinge forward raises that: at 70% of chord it is 0.661, at 60% it is 0.748, at 90% it is 0.396.

The reason the small surface punches so far above its area is where lift is generated. The load a change in camber produces is concentrated where the change forces the flow to turn, and the trailing edge is where the Kutta condition acts — so a deflection there resets the circulation of the whole section rather than adding a local force at the back.

Moving the hinge, computed rather than tabulated #

The effectiveness formula says the hinge position matters a great deal, and it is cheap to check that the solve agrees with it at a second hinge as well as at the first.

The pattern the formula describes is worth stating in words, because it is not the one intuition offers. Effectiveness rises as the hinge moves forward, but sub-linearly: doubling the flap’s chord fraction from 10% to 20% does not double the shift. A flap of 25% chord gets 61% of the effect of pitching the whole section; one of 40% chord gets 75%.

The reason is that the load produced by a change of camber is weighted towards the leading edge, so extending the flap forward adds surface in a region that was already contributing. The first few per cent of flap chord are worth far more than the last, which is why control surfaces are the size they are — big enough to be effective, small enough that the hinge moments a pilot or an actuator has to overcome stay manageable.

The same device, doing a different job #

A hinged trailing-edge surface that shifts a lift curve is also, in a different position on the aircraft, the entire mechanism of control and trim.

An elevator is a flap on the tailplane. It shifts the tailplane’s lift curve, which changes the tail’s lift, which changes the pitching moment about the aircraft’s centre of gravity, which changes the attitude the aircraft will settle at. The pilot does not command an attitude directly; the elevator changes where the equilibrium is, and the aircraft goes there.

An aileron is a flap on the outer wing, one up and one down, shifting the two lift curves in opposite directions and producing a rolling moment. A rudder is the same thing on a vertical surface. A trim tab is a flap on a flap, shifting the hinge moment rather than the lift.

All four are the calculation on this page. What differs is which curve is being displaced and what that displacement is wanted for, and the effectiveness number — 0.61 for a quarter-chord surface — applies to all of them.

Shift, not tilt, and why it matters #

The three slopes come out at 0.10944, 0.10890 and 0.10765 per degree for deflections of 0, 10 and 20°. That is a spread of 1.6%, and it is in the direction of slightly less response at large deflection rather than more.

For practical purposes the curves are parallel, and the consequences of that are the whole of what a flap is for.

Approach speed. A section that makes $C_L = 0.44$ at four degrees clean makes 1.10 at four degrees with 10° of flap and 1.79 with 20°. Since lift equals weight, the speed needed falls as the inverse square root of the coefficient: a factor of 1.79 in $C_L$ is a factor of 0.75 in speed.

Attitude. Because the curve moved rather than tilted, the same lift is now available at a much lower angle of attack. An aircraft on approach with flaps down flies nose-down relative to the same approach clean, which is why the runway is visible over the nose at all.

Not manoeuvrability. The response to a pitch input is the slope, and the slope has not changed. A flapped wing is not twitchier, and the belief that it is confuses “more lift” with “more lift per degree”.

What the model leaves out, which is most of a real flap #

A plain hinged flap is the simplest flap there is, and almost no transport aircraft uses one.

Real high-lift systems are slotted: the flap moves back as well as down, opening a gap through which high-energy air from beneath is blown over the flap’s upper surface. That is a boundary-layer device, not a camber device, and its purpose is to keep the flow attached at deflections where a plain flap would have separated long ago. Nothing in an inviscid model can represent it, because the thing being managed is a layer the model does not have.

Real systems also extend the chord, adding area as well as camber, which raises the lift for a reason that has nothing to do with the shift computed here.

And the ultimate limit on any of them is separation. The inviscid model will happily report a lift coefficient of 3 for 40° of flap. What actually happens is that the flow leaves the flap’s upper surface, the extra lift stops arriving, and the drag becomes enormous — which is a viscous story and is where the straight line stops.

Where the extra lift is paid for #

Nothing in an inviscid model costs anything, so the price of a flap has to be found elsewhere — and there are three bills, all of them real and none of them visible in the figures here.

Induced drag. More lift at the same speed is more circulation, and on a finite wing that means more trailing vorticity and more induced drag. The relation is quadratic: induced drag goes as the square of the lift coefficient, so tripling $C_L$ on approach multiplies the induced drag by nine. That is a large part of why an aircraft with flaps down descends steeply, and why the same configuration is unusable for cruising.

Pressure drag. A deflected flap presents a surface at an angle to the flow, and behind it the pressure does not fully recover. In the inviscid model it does, exactly, which is d’Alembert’s paradox appearing again in a place where it is straightforwardly wrong.

Pitching moment. The load added by a flap is concentrated aft of where the section’s existing load sits, so the centre of pressure moves back and the nose-down pitching moment grows. That has to be balanced by the tailplane, which makes downforce to do it, which the wing must then also lift — a chain of consequences that ends with the aircraft carrying more lift than its own weight, and paying induced drag on all of it.

The last of those is the one that makes high-lift design an aircraft problem rather than a section problem. A flap that produced its lift without moving the centre of pressure would be worth a great deal, and where the lift acts is the essay about why that point is so hard to move.

A flap has ends, and the ends do something #

Everything above is a section, and a real flap occupies only part of a span — typically the inboard half or two-thirds, because the outboard part is wanted for ailerons. That partial coverage has consequences that no two-dimensional solve can produce, and they are large.

The shift computed here applies only where the flap is. So the wing’s spanwise loading acquires a step: raised over the flapped portion, unchanged outboard, and falling steeply across the few per cent of span at the flap’s end. Since the vorticity shed into the wake is the rate at which circulation falls along the span, a flap end sheds a concentrated vortex exactly as a wing tip does and for the identical reason. On a humid day those flap-edge vortices are the pair of trails that appear inboard of the tips on an airliner turning onto final, and they are a serious source of airframe noise on approach.

The stepped loading is also a long way from elliptic, so the span efficiency falls and the induced drag rises by more than the lift coefficient alone accounts for. Deploying flaps costs induced drag twice: once for carrying more lift, and again for carrying it in a worse distribution.

Two design consequences follow, and both are deliberate rather than accidental. Loading the inboard wing more heavily means the root reaches its stall first, which leaves the outboard sections and therefore the ailerons still flying — the ordering every designer wants and the reason flaps live inboard. And the arrangement is dangerously asymmetric if it fails asymmetrically: one flap extended and the other not is a large rolling moment applied at the lowest speed and the least height of the flight, which is why the systems that drive them are mechanically interconnected across the aircraft.

Why the model is worth trusting this far #

An inviscid solve of a device whose real virtue is viscous invites the question of what it is good for, and the answer is precise rather than apologetic.

What the model gets right is everything that follows from the circulation being reset. The shift in zero-lift angle, the parallel curves, the effectiveness as a function of hinge position, the load appearing along the whole chord rather than behind the hinge — all of those are consequences of the Kutta condition acting at a trailing edge that has moved, and none of them requires viscosity except in the sense that the Kutta condition is itself a viscous fact wearing an inviscid costume.

What it gets wrong is where the process stops. It offers lift coefficients that no section will deliver, at deflections where the flow has long since left the flap.

That division — right about the mechanism, wrong about the limit — is the same division as everywhere else on this site, and it is the reason the inviscid theory is worth teaching first rather than being replaced by something more complete. The mechanism is what generalises. A slotted Fowler flap on an airliner is a much better device than the hinge modelled here, and it is a better device at doing the same thing: resetting the circulation by moving the trailing edge, and managing the boundary layer well enough that the section can actually carry the result.

What the picture cannot show #

The load distributions are drawn scaled to their own peaks, which is the only way to compare their shapes on one axis, and it hides how much larger the flapped one is in absolute terms. The lift coefficients in the legend carry that information; the curves do not.

The camber-line figure shows the geometry as a mathematical line and therefore shows no hinge mechanism, no gap and no thickness. A real flap at 20° has a visible discontinuity in its upper surface, and the flow over that discontinuity is doing something no camber line describes.

Where the model stops #

The linear theory holds while the deflection is small enough that $\tan\delta \approx \delta$ and the flow follows the surface. The first of those fails gently and visibly — the 3.5% departure at 20° above. The second fails catastrophically and invisibly: an inviscid solve gives no indication whatever that the flow has separated, because it cannot separate.

The practical consequence is that this model predicts the first few degrees of flap accurately, the middle range approximately, and the useful range not at all. The deflections used on approach — 30° or 40° with slots and extension — are outside everything this page computes, and the reason they work at all is machinery the model does not contain.

Who found it, and when #

Hermann Glauert worked out the thin-aerofoil treatment of flaps in the early 1920s at the Royal Aircraft Establishment, and the effectiveness formula above is his. His Elements of Aerofoil and Airscrew Theory (1926) is where the modern presentation of the subject begins.

The device itself is older and was invented several times. Harlan Fowler’s extending, slotted flap — the one that does everything a modern airliner needs — was patented in 1924 and took a decade to be taken seriously. Its advantage over the plain flap analysed here is entirely in the viscous behaviour, which is a reminder of how much of aeronautics is decided outside the theory that predicts the forces.

Where the ladder goes next #

Below this rung, the lift curve establishes the straight line, its slope of $2\pi$ and its displacement by camber. This rung shows that a hinge is a way of moving that displacement at will, in flight.

Above it, the limit is not in the circulation but in the boundary layer, and the next rung is where the straight line stops — the point at which a section can no longer be persuaded to carry the circulation the theory offers.