The surface in the wake

The moment about a wing’s aerodynamic centre does not change with incidence. That is a useful property when a designer wants a predictable moment to trim out. It is a disastrous one when the same designer wants the aircraft to resist being disturbed, because a body whose moment does not respond to its own attitude has no tendency to return to any particular attitude.

A wing on its own is, to a first approximation, neutrally stable in pitch. Stability comes from a second surface, far enough behind that its own lift acting on a long arm beats the wing’s moment.

The neutral point #

Write the moment about the centre of gravity as the wing’s own contribution plus the tail’s:

$$C_m = C_{m,\text{ac}} + C_{L}(h - h_{\text{ac}}) - \eta V_H\,C_{L,t}.$$

The tail volume ratio $V_H = S_t l_t/(S\bar c)$ carries the tail’s area and arm together, which is the only combination that appears. The tail’s own lift coefficient depends on the incidence it sees, and that is where the wake comes in: the tail is in the wing’s downwash, so its effective incidence is $\alpha(1 - d\varepsilon/d\alpha) + i_t$.

Differentiate, set the derivative to zero, and the centre of gravity position at which it happens is the neutral point:

$$h_n = h_{\text{ac}} + \eta V_H \frac{a_t}{a_w}\left(1 - \frac{d\varepsilon}{d\alpha}\right).$$

The distance between the centre of gravity and that point is the static margin, and it is what a loading sheet exists to keep positive.

Everything in that expression is geometry except one term, and that term is a piece of aerodynamics everybody quotes and nobody computes.

The factor of two #

$d\varepsilon/d\alpha$ is the rate at which the downwash angle at the tail grows with the wing’s incidence. The standard treatment says it is twice the induced angle at the wing itself, or $2 a_w/(\pi\mathrm{AR})$, and moves on.

That number is exactly right and it is the value at infinity.

The reason is worth deriving because it is two lines. Represent the wing by horseshoe vortices. For a trailing leg leaving $(0, y_0, 0)$ and running downstream, the vertical velocity it induces at $(x, 0, z)$ carries the factor

$$1 + \frac{x}{\sqrt{x^2 + a^2}},\qquad a^2 = y_0^2 + z^2,$$

which is 1 at the lifting line — where only half the vortex line is downstream of the point — and 2 far downstream, where the whole of it is. Twice the downwash at the wing, exactly, in the limit.

The computation confirms both ends. At the lifting line the induced angle per unit lift coefficient is $1/\pi\mathrm{AR}$ to within a few per cent — the gate requires five — and four hundred spans downstream it is exactly twice that, required within one per cent by a Biot–Savart sum that was told nothing of the kind.

In between, it is larger than either. The bound vortex — the wing itself — induces downwash behind it too, and close behind, that contribution dominates. So the field overshoots, and comes down to its far-field value from above, monotonically, over several spans.

Where a tailplane actually is #

A tailplane sits two or three chords behind the wing. On a wing of aspect ratio eight, a chord is an eighth of a span, so two and a half chords is about a third of a span.

At that station the computed downwash is 2.46 times the value at the wing, where the standard formula says 2. The gate requires it to exceed 2.2.

That is a quarter more downwash than the sizing formula assumes, which means the tail is a quarter less effective than the formula credits it with, which means the neutral point is further forward than the formula puts it, which means the static margin is smaller than the loading sheet says.

None of those is a large error on its own and the direction of all four is the same: the standard formula is optimistic about stability, and the optimism is largest on wings of low aspect ratio, where a chord is a larger fraction of a span and the tail is proportionally closer.

The near field, which nothing sits in #

The curve close behind the wing is worth a paragraph, because it looks wrong and is not.

Within about a third of a span the computed downwash rises steeply as the station approaches the wing, and at a fifth of a span it is nearly three times the value at the lifting line. That is the bound vortex asserting itself: the wing’s own circulation induces a downwash directly behind it, and that contribution is unbounded as the point approaches the vortex line.

The site’s gate requires the overshoot — a computation that did not have it would have the bound vortex’s sign wrong, which is the error most likely to be made and least likely to be visible. It also requires the field to fall monotonically towards its limit from half a span outwards, which a sign error would break in the other direction.

Nothing is ever put in the near-field region. It is where the flaps are, and a flap is part of the wing rather than a surface in its wake. The useful part of the curve begins at about a quarter of a span and it is already above the far-field value there.

Height, and the reason for a T-tail #

The other variable the formula has no room for is the tail’s height, and it matters at least as much.

Lift the tail a tenth of a span above the wake and the downwash at the same station falls from 2.46 to 1.95 — below the far-field value rather than above it. So a T-tail is not merely a different place to put a surface; it is a surface in a materially different flow, and it is more effective per unit area than a low tail at the same arm.

There is a second benefit that follows from the same figure and is not usually stated in these terms. A tail out of the wake is a tail whose effectiveness does not swing as the wing’s loading changes — flaps down, flaps up, high lift, low lift — because the strong gradients are all near the wake centreline. That is a stability-derivative argument for a T-tail and it is a good one.

It also comes with the corresponding hazard. At high incidence the wing’s wake moves up relative to the aircraft, and a T-tail that was above it in cruise can be inside it at the stall — where the flow is separated, the dynamic pressure is low and the elevator does very little. That is the deep stall, it killed several early T-tail aircraft, and it is the same geometry read at a different attitude.

What ignoring the wake would cost #

The last figure is the reason the term is worth computing rather than assuming.

At a realistic tail volume, taking $d\varepsilon/d\alpha = 0$ — treating the tail as though it were in clean air — moves the computed neutral point aft by thirteen points of chord. An aircraft loaded to a nominal ten per cent static margin against that neutral point would in fact be flying at three per cent aft of the true one: unstable, and unstable by an amount a pilot would notice immediately.

The downwash halves the tail’s effectiveness, near enough. A tailplane is worth about half of what its area and its arm suggest, and every account of tail sizing that omits the term is not making a small error.

What a static margin buys and what it costs #

The number the whole calculation exists to produce is a design compromise, and it is worth saying which way each per cent of it pulls.

A large static margin makes the aircraft strongly self-righting: disturb it and it returns. It also makes it strongly resistant to being deliberately disturbed, so the elevator has to be larger and the stick forces higher, and it makes the aircraft pitch down when it slows and up when it speeds up, which is stability in the axis a pilot most wants it in. And it costs trim drag: a stable aircraft is trimmed with download on the tail, and download is lift the wing has to make and pay for again.

A small static margin is efficient and responsive. Modern transports fly at a few per cent, and fuel is moved between tanks in flight to hold the centre of gravity as far aft as the certification allows, because every point of margin given back is trim drag saved.

A negative one is not a mistake if the aircraft has the authority and the bandwidth to fly it. A relaxed-stability fighter is deliberately unstable and is flown by a computer, which buys manoeuvring performance and removes the trim drag entirely.

So the neutral point is not a limit to stay well away from. It is a boundary whose position has to be known accurately, because the design is deliberately close to it — which is precisely why a thirteen-point error in locating it is not an academic matter.

The canard, which pays the same bill elsewhere #

A surface ahead of the wing is not in anybody’s wake, so it keeps all of its effectiveness. That sounds like a free improvement and it is not, for two reasons that pull in the same direction.

The wing is now in the canard’s wake. The canard’s downwash reduces the wing’s effective incidence over part of its span, which reduces the wing’s lift-curve slope, which moves the whole aircraft’s neutral point forward.

And the stabilising surface is now ahead of the centre of gravity. A surface behind the centre of gravity stabilises because its lift acts on a rearward arm; ahead of it, the same lift is destabilising. What makes a canard configuration stable is that the canard stalls first by design — so at high incidence the canard’s lift stops growing before the wing’s does, and the resulting moment is nose-down.

That is a real and workable arrangement, and it constrains the design severely: the canard must be at a higher local incidence than the wing at every attitude, so it must be loaded harder, so it must be efficient, so it usually has a higher aspect ratio and a more cambered section than the wing does. It also means the wing can never reach its own maximum lift coefficient, because the canard has stalled before it — which is a permanent penalty in wing loading.

Neither arrangement escapes the arithmetic; they pay it in different places.

The other two things the wake does to a tail #

The downwash angle is the term this essay computes, and the same wake acts on the tail in two further ways that the formula carries as a symbol and never explains.

It takes the tail’s dynamic pressure away. The $\eta$ in the neutral-point expression is the ratio of the dynamic pressure at the tail to the free stream’s, and it is below one for the reason a wake is a momentum deficit: the tail is sitting in air the wing and the fuselage have already slowed. Typical values are 0.85 to 0.95 for a conventional low tail, and essentially 1.0 for a tail lifted clear of the wake — so a T-tail gains twice over, once in angle and once in dynamic pressure, and the two gains multiply rather than add.

On a propeller aircraft the same term goes the other way and much further. A tail in the slipstream sees air the propeller has accelerated, so $\eta$ can exceed one substantially at high power and low speed, and fall back towards one at cruise. That is not a small effect on the stability derivatives; it is why such an aircraft trims differently with power, why a go-around produces a strong pitch-up, and why the certification tests for it are flown at several power settings rather than one. A term written as a constant in the formula is a function of throttle position.

And it arrives late. The downwash at the tail is produced by the wing’s circulation, and the information travels downstream at roughly the flight speed, so the tail at any instant is feeling the downwash belonging to the wing’s incidence of $l_t/U$ seconds ago. For a tail arm of fifteen metres at eighty metres a second that is 0.19 seconds — against a short-period pitch oscillation with a period of one to three seconds, so it is a tenth of a cycle and not negligible.

What that lag does is supply damping. When the aircraft pitches nose-up, the wing’s incidence rises at once and the tail’s downwash does not rise until the delay has elapsed; for that interval the tail is at a higher effective incidence than the steady relationship would give, so it makes more download, so the moment opposing the pitch rate is larger. The effect appears in the equations of motion as the derivative $C_{m_{\dot\alpha}}$, it is one of the principal contributions to short-period damping on a conventional aeroplane, and it exists entirely because the wake takes time to travel.

So the tail’s behaviour is decided by three separate consequences of sitting behind something: an angle that this essay computes and the standard formula understates by a quarter, a dynamic pressure that the formula carries as a constant and that a throttle can move, and a delay the static formula has no place for at all. Two of the three are usually written as symbols and left unexamined, and the third is invisible in a static analysis and decides how the aircraft feels to fly.

The unifying observation is the one the whole essay is about. A tailplane’s job is to respond to the aircraft’s attitude, and it is doing so through a medium that carries the wing’s history to it — attenuated, slowed and late.

What the model does not contain #

A rigid flat wake. The trailing sheet here is a set of straight semi-infinite lines. A real wake rolls up within a few spans, which changes the field at exactly the station a tail sits in. The rolled-up field is more concentrated and the downwash at the centreline is somewhat different from the flat-sheet value.

No fuselage. The body between the wing and the tail carries lift, has its own destabilising moment — a slender body’s moment is destabilising for the same reason a canard’s is — and interferes with the flow to both surfaces. A real neutral-point estimate has a fuselage term in it.

Elliptic loading throughout. The downwash field depends on the loading; a bell-loaded wing has a quite different field behind it, and one of the properties of that loading is a smaller downwash outboard.

Lift slopes are given, not computed. Both surfaces’ lift-curve slopes are inputs.

And no dynamics. The static margin says whether the aircraft returns to its trim; it says nothing about how — the short-period frequency and damping, the phugoid, the manoeuvre point — which is the whole subject of flight dynamics and none of it is here.

Who found it, and when #

The static-margin formulation belongs to the 1920s and 1930s and its clearest early statement is in Bryan’s stability work carried forward by the British and American research establishments; the tail volume ratio and the neutral point are in every textbook by the 1940s and have not changed since.

The factor of two for the downwash appears in Prandtl’s own lifting-line papers, correctly labelled as a far-field result. What happened over the following decades is a familiar kind of erosion: the qualification travelled less well than the number, and the number is now quoted in contexts its derivation does not cover.

A limit is the easiest thing in a derivation to lose, because it is usually the tidiest part of the answer. The factor of two is exact, it is beautiful, it is what the geometry gives at infinity, and it is not the number a tail is in.

Where the ladder goes next #

Everything up to this rung has been about how much lift a surface makes and where. The next three essays are about explanations of lift that are repeated everywhere and are wrong, starting with the one that treats a wing as half a nozzle — and which can be refuted with a surface that has no thickness at all.