A canard pays for its stability in induced drag

The surface in the wake computed the downwash a tailplane sits in — the lift curve of each surface slowed by the other’s wake — found the neutral point it gives, and ended with the canard: a stabilising surface in front of the wing, in nobody’s wake. It set out why a canard aircraft can be stable at all — the canard must stall before the wing, so it must be loaded harder — and concluded that neither arrangement escapes the arithmetic, only that they pay it in different places. That is true and it leaves the most common claim about canards untested.

The claim is about efficiency. A conventional aircraft’s tail, it says, pushes down to hold the nose up, so the wing has to lift the aircraft’s weight plus the tail’s download, and all of that extra lift costs induced drag. A canard lifts upwards, so both surfaces work for their living and the wing carries less. It is an appealing argument, and several light aircraft of the 1970s and 80s were sold partly on it. This essay prices it, and finds it backwards.

The theorem that removes the stagger #

The first step is one the argument never takes. Two lifting surfaces one behind the other — a wing and a tail, or a canard and a wing — make a combined wake, and their total induced drag is computed in the Trefftz plane far behind, where the two wakes lie side by side. Munk’s stagger theorem, which the biplane essay proved for two wings one above the other, says that the total does not change if either surface is moved along the stream, provided the lift on each and its spanwise distribution are kept. Whether the smaller surface is in front or behind makes no difference to the pair’s induced drag. What decides it is how much of the lift each surface carries and how that lift is spread.

So the question becomes: at the same stability, how much lift does each arrangement put on its smaller surface? The drag follows.

The least drag at each split #

The first figure is the drag half, computed on a vortex lattice in the Trefftz plane, far downstream, where the wake of two surfaces is a single sheet whose induced drag depends only on how its circulation is shared. The wing has a span of one; the smaller surface, a fifth of the wing’s area on 0.35 of its span, sits a little below it in the wake. For each share of the total lift put on the smaller surface, both loadings are left free to take whatever spanwise shape gives the least drag, and the least drag is expressed as a multiple of the elliptic wing’s on its own.

The curve is a parabola — the minimum of a quadratic form under linear constraints is quadratic in the constraints, and the lattice reproduces an exact parabola to fourteen figures. Its foot is at a share of 1.9 per cent, where the pair pays 0.998: a small surface carrying a little upward lift can fill in the wing’s loading where the elliptic shape would like more, and save a fifth of a per cent. Away from the foot the curve rises steeply, as the square of the share. A surface a third of the wing’s span carrying a fifth of the lift pays a great deal, because induced drag goes as lift squared over span squared, and the span is small.

That is the whole of the drag side, and it already contains the answer. A surface that carries a large share of the lift on a small span is expensive, whichever end of the aircraft it is at.

What stability demands of each #

The other half is balance. An aircraft is statically stable when a nose-up disturbance produces a nose-down moment, which happens when its centre of gravity is ahead of its neutral point — the point about which the moment does not change with incidence. The distance between them, in wing chords, is the static margin, and certification and handling set how large it must be: a tenth of a chord is typical of light aircraft, more of trainers.

The neutral point is the average position of the surfaces, each weighted by how fast its lift grows with incidence. A tail behind the wing sits in the wing’s downwash, which the essay before computed: here, 3.5 chords behind an aspect-ratio-eight wing, the tail’s effective incidence grows 41 per cent more slowly than the wing’s. A canard in front sits in clean air, but puts the wing in its own wake. With both downwash fields taken from the same horseshoe calculation, the neutral point of this tail aircraft is a third of a chord behind the wing’s aerodynamic centre, and that of the canard aircraft more than half a chord ahead of it.

Then the split follows from the moments. The centre of gravity is placed a static margin ahead of the neutral point, and the two lifts must balance about it. For the tail aircraft at a margin of a tenth, the centre of gravity is still behind the wing’s aerodynamic centre, and the tail carries 6.7 per cent of the weight — upwards. For the canard aircraft, the centre of gravity is well ahead of the wing, and the canard carries 19 per cent.

The drag at the same margin #

The second figure joins the halves. The tail aircraft, at a margin of a tenth, pays 1.009 of the elliptic wing’s induced drag: within one per cent of the best that any split could achieve. As the margin grows the tail’s upload shrinks towards the curve’s foot and past it, and the drag barely moves. The canard aircraft at the same margin pays 1.128, twelve per cent more, and every increment of margin makes it worse: more margin moves the centre of gravity further forward, which asks more of the canard, which is further up the parabola. At a margin of three-tenths the tail pays 0.999 and the canard 1.230.

A real wing section is cambered and carries a nose-down moment of its own about its aerodynamic centre — a flap deployed adds more — and that is where the tail’s download comes from. With a moment coefficient of −0.08 at a cruise lift coefficient of a half, the tail’s share falls to two per cent upwards at a margin of a tenth and becomes a small download beyond a margin of about a sixth — the textbook picture, at last. It costs almost nothing: the tail aircraft pays 1.012 at a margin of three-tenths. The same camber moment makes the canard work harder still, since the canard must now also hold the wing’s nose-down moment, and it pays 1.21 at a tenth and 1.33 at three-tenths.

Why so little is the one piece of the calculation that is not obvious. On its own, a wing lifting 103.6 per cent of the weight would pay 7.3 per cent more induced drag than one lifting all of it, since the drag goes as the square of the lift. The pair pays 1.2. The difference is the tail’s position inside the wing’s wake. An elliptically loaded wing leaves a downwash that is the same all across its span, and a surface lifting inside that downwash is charged, in the Trefftz plane, the product of its lift and that downwash — so a download there is paid the same product back. Moving lift from the wing to a surface within the wing’s span is free to first order: the parabola’s slope at zero share is −0.17, where two surfaces that could not see each other’s wakes would have a slope of −2. Only the square of the share is charged, on the smaller surface’s short span, and a share of a few per cent squared is almost nothing.

The download argument has the sign of its effect right and its size wrong. A tail’s download does make the wing lift more than the weight, but the download is a few per cent of the weight, the drag it adds is second order in that share, and the tail’s own contribution to the pair’s loading partly pays it back. A canard’s upload is a fifth of the weight carried on a third of the span, and induced drag is charged on the square of that.

The canard is efficient only when it is unstable #

The parabola has one foot, and each arrangement reaches it at one margin. Solving the trim backwards — which margin puts the smaller surface’s share exactly at the 1.9 per cent the curve prefers — gives the tail aircraft its least drag at a static margin of 0.27 of a chord with sections that carry no moment, and 0.11 with the cambered wing. Both are ordinary margins; the second is almost exactly what light aircraft are flown at. A conventional aircraft is at the bottom of its induced-drag curve when it is comfortably stable, which is a coincidence of geometry rather than a design choice, and a fortunate one.

The canard aircraft reaches the same foot at a margin of −0.50 of a chord, and −0.66 with the cambered wing: its centre of gravity must then be half a chord or more behind its neutral point, which is an aircraft that diverges in pitch unless something holds it. The canard is efficient only when it is unstable.

That is not a curiosity. The canard-delta fighters designed from the late 1970s onwards — the Swedish, French and European ones — are unstable in pitch at subsonic speeds by design, and are flown through computers that move the canard many times a second to hold an attitude no pilot could hold unaided. Relaxed stability lets the canard carry a small share, or even a download, at cruise, and the drag penalty the second figure charges at positive margins disappears. The same computers would let a tail aircraft fly unstable too, and some do; what distinguishes the canard is that it needs the computer to be efficient at all, where the tail aircraft is already efficient without one.

A stable canard is loaded harder than its wing #

The third figure shows the same thing as a loading. A tail’s lift coefficient is a small fraction of the wing’s at every margin — a third of it at a margin of a tenth, and negative with a cambered wing beyond a sixth. A canard’s is larger than the wing’s at every margin: 1.17 times at a tenth, 1.64 at three-tenths, 2.07 with a cambered wing at three-tenths.

That ratio above one is not a defect of this design. It is what the essay before identified as the canard’s necessary property: a canard aircraft is safe at the stall only if the canard stalls first, dropping the nose before the wing reaches its own maximum. A canard loaded more lightly than its wing would let the wing stall first, and then the aircraft pitches up — the direction a stall should never go. Stability and stall safety ask the same thing of a canard, to carry more than its share, and the parabola charges for it.

Why the canard’s wake barely touches the wing #

The fourth figure is the one advantage the canard does have, and it is smaller than it looks. A tail sits in the wing’s downwash across its whole span, which is why it loses 41 per cent of its lift slope. The wing sits in the canard’s wake only partly: the canard’s trailing vortices make downwash between its tips and upwash outboard of them, and a wing three times the canard’s span meets both. Weighted by the wing’s chord, the two nearly cancel, and the wing loses 7 per cent of its lift slope to the canard. That is why the canard aircraft’s neutral point is so far forward of the wing — the canard’s full lift slope acts at the front — and it is why stability costs a canard more: to move the centre of gravity a margin ahead of a neutral point that is already forward, the canard’s share has to grow.

The wing that never reaches its maximum #

The fifth figure is the second bill. If the canard and the wing use sections with the same maximum lift coefficient, the canard reaches its maximum first — by design — and the wing is then at its own lift coefficient divided by the loading ratio. At a margin of a tenth the wing can be flown to 86 per cent of its maximum; at three-tenths, 61 per cent; with a cambered wing, less again. For the same stall speed the wing must be larger by the reciprocal, and a larger wing has more friction drag, which the second of a wing’s two drags charges on its area. Or the canard must be given a section with a higher maximum lift than the wing’s, which is what successful canard designs do — and a high-lift section on a small, heavily loaded surface is exactly what the first figure’s parabola is steepest for.

What was checked #

The sixth figure is the ledger. The least drag at four splits lies on the parabola through three others to $2\cdot10^{-14}$, and the parabola’s minimum equals the optimum found with only the total lift constrained, which is an independent solve. With the two surfaces forty spans apart vertically, where they no longer see each other’s wakes, the lattice returns each surface’s own elliptic drag added, to four parts in a hundred thousand. And every trimmed split, with and without a camber moment, balances the moments about its centre of gravity to machine precision.

What the two halves leave out #

Viscous drag. The comparison is of induced drag only. A canard’s higher loading usually means a higher aspect ratio and a larger section lift, and its profile drag is part of the bill; a tail’s download near the stall adds its own.

The loadings are ideal. Each surface is given the best spanwise loading for its share. A real canard or tail has a planform and a twist fixed for one flight condition, and away from it both pay more; the canard, loaded harder, pays more for being away from its design point.

Downwash from one calculation. Both fields are horseshoe Biot–Savart fields for elliptic loadings, and the canard’s is taken a small height above the wing’s plane. The canard’s net effect on the wing depends on that height and on the span ratio, and a canard in the wing’s plane would put its tip vortices across the wing itself.

Control and trim changes. The comparison is at one lift coefficient. A flap on the wing changes its moment; on a canard aircraft the canard must take up the change, and it is already the harder-worked surface.

The convention the numbers depend on #

Drags are induced drags at a fixed total lift, as multiples of an elliptic wing of the same span carrying all of it. The split is the smaller surface’s share of the total lift, positive upwards. Lengths are in wing chords from the wing’s aerodynamic centre, positive aft; the static margin is the neutral point’s distance behind the centre of gravity. The wing has an aspect ratio of eight, the smaller surface a fifth of its area on 0.35 of its span, 3.5 chords away. Section moments are about each surface’s aerodynamic centre, nose-up positive, and where the lift acts sets out why that point is the one to use.

Who worked it out #

Munk proved the stagger theorem in 1920. The comparison of canard and conventional configurations at equal stability is McGeer and Kroo’s, whose 1983 paper “A fundamental comparison of canard and conventional configurations” reached the conclusion computed here: at the same static margin and with optimal loadings, the conventional arrangement has the lower induced drag, and the canard’s disadvantage grows with the margin. The canard’s stall-first requirement, the pitch version of which part of a wing stalls first, is older than either, and is why the Wright Flyer of 1903, a canard with its centre of gravity too far back, was unstable in pitch.

Still open: a third surface #

The parabola has one free variable because there are two surfaces. A three-surface aircraft — canard, wing and tail — has two, and the extra freedom lets the stability be met by one combination of shares and the least drag sought along the rest. The next calculation adds the third surface to the same lattice and the same trim, and asks whether any three-surface arrangement beats the tail aircraft at the same margin, or whether the tail’s small upload is already so close to the foot of the parabola that a third surface can only add wetted area.