One twist is chosen for the speed flown per metre

Washout is right at one lift coefficient found that a linearly twisted wing’s induced drag is a quadratic form in its lift coefficient and its washout, so that any given washout is best at exactly one lift coefficient and the best washout grows in proportion to the lift coefficient it is chosen for. A rectangular wing of aspect ratio 8 wants 7.87° of washout for each unit of lift coefficient; four degrees put its design point at 0.509. It ended by asking what a wing should be twisted for when it flies at many lift coefficients in one flight, which every aeroplane does.

Untwisted, the question does not arise: the price of having ends could quote one span efficiency per planform because an untwisted wing has one loading shape, and the span is the whole story for an elliptic wing at any speed. Twist is what makes the answer depend on the flight. The obvious answer is the average: find the lift coefficient the aeroplane spends its time at, and twist for that. The answer computed here is different, and the difference is not a refinement. A flight spends energy, not time, and energy is spent per metre. Weighted that way the mean that comes out is a harmonic mean, and a harmonic mean of lift coefficients turns out to be the lift coefficient of one particular speed — the root-mean-square speed of the flight, averaged over distance rather than over the clock.

Energy is spent per metre, not per minute #

What an aeroplane’s engine replaces, and what a glider’s height pays for, is the work done against drag: drag times speed times time, or drag times distance. For the induced part of the drag at a given weight $W$, a segment flown for a time $t$ at lift coefficient $C_L$ has drag $W C_{D_i}/C_L$ and speed $\sqrt{2W/\rho S C_L}$, so its induced energy is proportional to

$$t\,C_L^{-3/2}\,\big[\,C_L^2(1+\kappa_D) - \kappa_{DL}\,C_L\,\Omega + \kappa_{D\Omega}\,\Omega^2\,\big],$$

the bracket being $\pi A$ times the induced-drag coefficient from Phillips’s form with washout $\Omega$. Summing over the segments and setting the derivative in $\Omega$ to zero gives the mission’s washout in closed form. It is the single-point optimum $\Omega^* = \kappa_{DL}C_L/2\kappa_{D\Omega}$ evaluated at a mission design lift coefficient

$$C_{L,\text{mission}} = \frac{\sum_k w_k}{\sum_k w_k / C_{L,k}}, \qquad w_k = t_k\,C_{L,k}^{-1/2} \propto t_k V_k,$$

which is the harmonic mean of the segments’ lift coefficients, each weighted by the distance flown at it. A segment flown in a turn of load factor $n$ carries an extra $n^{3/2}$ in its weight, since both its drag and its speed rise with the load.

The harmonic mean has a reading that makes it memorable. In level flight $C_L$ is proportional to $1/V^2$, so the distance-weighted harmonic mean of the lift coefficients is the lift coefficient at the speed whose square is the distance-weighted mean of the squared speeds: the wing should be twisted for the root-mean-square speed of the flight, averaged per metre. That is the same kind of mean that sets a gas’s kinetic energy from its molecular speeds, and it weights fast flight twice over — once because a fast segment covers more distance in its minutes, and again because the mean is of the square.

A training sortie, minute by minute and metre by metre #

The aeroplane is the one the single-point calculation used: a rectangular wing of 11.3 metres span and sixteen square metres, aspect ratio 8, carrying a tonne at sea level. The sortie is stated rather than measured, and is typical of a training flight: eight minutes’ climb at 35 metres a second, thirty minutes’ cruise to and from the training area at 55, ten minutes of slow flight and stalls at 28, and twelve minutes of circuits at 33. Their lift coefficients are 0.817, 0.331, 1.276 and 0.919.

By the clock the cruise is half the sortie and the slow flight a sixth. By the measure that spends energy the cruise is 63.3 per cent and the slow flight 10.7 per cent: at 28 metres a second the slow segment covers barely half the distance the cruise covers in the same ten minutes, and the climb and circuits sit between. The time-mean lift coefficient of the sortie is 0.671. The distance-weighted harmonic mean — the one the energy sum asks for — is 0.436, and that corresponds to a speed of 47.9 metres a second, which is exactly the root-mean-square of the four speeds weighted by the distance flown at each.

The twist the sortie asks for #

The average span efficiency over the sortie, defined as the elliptic wing’s induced energy over this wing’s for the same flight, is 0.9367 untwisted, as it must be: an untwisted wing has one loading shape at every lift coefficient. With washout the curve rises to a single peak, 0.9774 at 3.43°, and falls away on both sides. The cruise alone would want 2.60°, so the mission’s slower segments have pulled the optimum up by less than a degree. Twisting for the time-mean lift coefficient instead would mean 5.28°, and the curve has fallen to 0.9651 by then, giving back 1.2 per cent of the efficiency the twist was supposed to buy.

The peak is flat, and that matters as much as where it is. Three degrees of washout gives 0.9767 and four gives 0.9762, both within an eighth of a per cent of the optimum. A light trainer rigged with three or four degrees of washout is therefore at its mission optimum for induced drag, although the washout was put there for the stall — to keep the root stalling first so that the ailerons keep working, which is what which part of a wing stalls first is about, at the angle rather than the speed that a stall is reached at. For a rectangular wing on a flight of this shape the two reasons happen to agree, and nobody had to choose between them.

In energy the agreement is worth something real but small. The sortie’s induced energy is 37.4 megajoules untwisted and 35.8 at the mission optimum: a saving of 1.56 megajoules of work done on the air. At an overall engine-and-propeller efficiency of a quarter that is about six megajoules of fuel, or a little under a fifth of a litre of petrol an hour. The twist was never going to be the largest number on the fuel bill — the cheapest way to stay up puts the profile drag beside the induced drag and the profile drag dominates in the cruise — but it is a free saving, and a stall-safe one.

The slow flight hardly gets a vote #

The stubbornness of the harmonic mean is easiest to see on a mission of just two segments, a cruise and a slow flight, with the slow share swept from nothing to nearly everything. The time mean rises in a straight line between the two lift coefficients. The harmonic mean weighted by time sags below it, as every harmonic mean does. The distance-weighted one sags further, because it discounts the slow segment twice. With half the time spent slow, the time mean is 0.804 and the mission design point is still 0.441 — barely above the cruise’s 0.331. The design point reaches the midpoint of the two lift coefficients only when 88 per cent of the time is slow flight.

That is a general statement about any aircraft whose mission mixes fast and slow: the twist that minimises the energy is set almost entirely by the fast part, and adding slow segments to a mission moves it very little. It is the induced-drag version of a familiar fact about averages of speed. The lift at the mean angle is not the mean lift, because the lift curve bends; here the energy is not spent at the mean lift coefficient, because the weight that converts minutes to metres bends the average the other way.

A tapered wing has nothing to choose #

A wing tapered to 0.4 is within 1.3 per cent of elliptic untwisted at aspect ratio 8, which is why the optimum that does not matter found a broad flat valley around a taper of 0.36. Its best washout for the sortie is 0.09°, worth three thousandths of a per cent over none — there is simply nothing for a twist to correct. Its curve is a parabola falling steeply away from zero. Four degrees, fitted for the stall, cost 5.1 per cent of the sortie’s span efficiency.

The comparison the figure invites is the uncomfortable one. With four degrees of washout each, the rectangular wing averages 0.9762 over the sortie and the tapered wing 0.9358. The planform that is better untwisted is the worse once both carry the twist the stall wants, by four points of efficiency over a whole flight. The bell-shaped loading makes the same point from the structure’s side: a loading is optimal only for the constraint it was chosen under. The single-point calculation had already said that a designer who picks a near-elliptic taper for its drag and then adds washout for its stall has paid for the drag twice; the mission average says how much, in the currency the aeroplane spends.

What a glider’s circling is worth #

A glider’s case looked as though it should be the opposite. A cross-country flight spends a large share of its time circling slowly in thermals at a high lift coefficient, and the obvious guess is that the twist should serve the circling. The height a glider loses is the same integral as the trainer’s fuel — drag times speed times time, over the weight — so the same mean applies, with the circling segment’s weight raised by $n^{3/2} = 1.68$ for the load factor of a 45-degree turn.

It barely moves. The glider here is 300 kilograms on 12.5 square metres, aspect ratio 20; its glide at 40 metres a second is at a lift coefficient of 0.267 and its circle at 1.1 is flown at 23.5 metres a second. At four-tenths of the time circling the mission design point is 0.366. At eight-tenths it is still only 0.639. The best washout for a rectangular wing is 3.08° at four-tenths circling, and for the tapers gliders actually use it is small: 1.91° at a taper of 0.7, 0.85° at 0.5. A glider’s usual washout of a degree or two is set, like the trainer’s, by the stall and spin behaviour it gives at the root, and the circling flight it is often justified by is not what makes it right for drag.

The sink rates show how little is at stake. Circling at 1.1 the 0.5-taper wing’s induced sink is 0.609 metres a second untwisted and 0.603 with two degrees of washout; gliding at 40 metres a second, two degrees raise the induced sink from 0.178 to 0.185. Over a flight that is four-tenths circling, two degrees cost 1.1 per cent of the induced efficiency against the mission optimum — a few millimetres a second of sink either way, against thermals that go up at metres a second.

The part no twist can recover #

Even at the mission optimum a twisted wing does not reach the peak efficiency a single-point design reaches, and the gap has an exact form. Each segment, flown with its own best twist, would sit at the planform’s floor: the departure from elliptic that the best straight twist cannot remove, a fixed fraction of the elliptic drag at every lift coefficient. With one twist for all of them, every segment pays in addition $\kappa_{D\Omega}(\Omega - \Omega_k^*)^2$, its distance from its own optimum squared, and the sum of those terms is a weighted variance of the segments’ optima about the mission’s.

For the rectangular trainer the variance is the larger cost: 1.41 per cent of the elliptic energy against the planform’s 0.90. The spread of speeds in one sortie costs more induced drag than the shape of the wing does, once the wing is twisted. For the 0.4 taper the spread costs a thousandth of a per cent, because a wing whose best twist is a tenth of a degree has no twist for the speeds to disagree about, and its whole excess is the planform’s 1.30 per cent. The split is a way of reading the planform choice: a rectangular wing makes its drag sensitive to the mission and buys low drag at its design point with it; a near-elliptic taper makes the mission irrelevant and accepts a fixed loss everywhere. Each twisted to its own mission optimum, the taper still comes out ahead, 1.30 per cent against the rectangular wing’s 2.31; each given the stall’s four degrees, the order reverses, 6.9 per cent above elliptic against 2.4. Which planform is the low-drag one depends on a twist chosen for a reason that has nothing to do with drag.

How the averages were checked #

The closed form was checked four ways. A golden-section minimisation of the summed mission energy over washout, which uses the form but not its derivative, lands on 3.4272° against the closed form’s 3.4272° — a difference of $10^{-7}$ degrees, the search’s own tolerance — for the rectangular wing and for the 0.4 taper. A mission of one segment gives back the single-point optimum exactly, with zero spread. The form’s mission energy at 3.2° of washout agrees with four direct lifting-line solutions, one per segment, each with its root incidence found for its own lift coefficient, to $10^{-15}$. And the excess at the optimum equals the planform floor plus the spread variance to $10^{-16}$, which is the algebra the split rests on.

The convention: energy, at fixed weight, with profile drag left out #

Every average here is of induced energy — drag times distance — at a fixed weight and air density, over segments of stated length flown at a constant lift coefficient each. The span efficiency of a mission is the elliptic wing’s induced energy over the flight divided by this wing’s, so that it reduces to the ordinary span efficiency for a mission of one segment. Washout is linear, positive with the tip below the root, and the sections have a lift-curve slope of $2\pi$. A turn enters only through its load factor; a turning wing’s loading is not symmetric, and that asymmetry is ignored.

What the average leaves out #

Profile drag, first and largest. Washout changes each section’s lift coefficient and so its profile drag, and on a glider’s laminar section the profile drag can rise sharply outside a low-drag bucket. The mission optimum for total energy is not the induced optimum, and the two drags a wing pays trade against each other here exactly as they do at a single point. The stall is left out except as the reason for the washout: no constraint keeps the root stalling first, and a mission whose optimum is wash-in, as a sharply tapered wing’s would be, would fail that constraint. Weight is held fixed although a powered aeroplane burns fuel, which moves every segment’s lift coefficient down as the flight goes on. The mission itself is a stated sortie and a stated glider flight, not a measured one; the conclusions about which segments vote and which do not hold for any mission of fast and slow segments, but the numbers belong to these two. And lifting-line theory at aspect ratios of 8 and 20 is good to a few per cent in the coefficients and better in their differences.

A harmonic mean that is older than the wing #

Phillips’s quadratic form for twisted wings, which everything here sums, is from 2004, and the observation that a twist is right at one lift coefficient is as old as Glauert’s 1926 Fourier form of the lifting line. Averaging a performance quantity over a mission is the whole method of Breguet’s range equation, from the 1920s, which integrates fuel against distance precisely because distance is what an aeroplane is for; the harmonic mean is what that integral produces whenever the quantity being averaged varies inversely with speed squared. Averages of speed weighted by distance and taken as harmonic means are older still — the average speed of a round trip is the harmonic mean of the two legs’ speeds — and the root-mean-square speed that sets a gas’s energy is Maxwell’s. Nothing new is needed to reach the result here, only the decision to average the thing the aeroplane spends.

Still open: a twist that changes in flight #

Every washout here is fixed, so the spread cost is unavoidable: one twist cannot match a cruise and a circuit at once. A wing whose twist changes with its lift coefficient can. That happens passively on any wing with sweep or with a flexible spar, whose tips bend up and twist nose-down as the load grows — aeroelastic washout, proportional to the dynamic pressure rather than to the lift coefficient — and actively on a wing with trailing-edge surfaces deflected differentially along the span. The next calculation gives the washout a term proportional to the dynamic pressure, as a torsionally flexible wing’s is, and asks whether some stiffness makes the twist track each segment’s own optimum, which would remove the spread cost entirely: a wing stiff enough in bending to carry its load and soft enough in torsion to wash out by exactly the right amount at every speed. Beside it is the constraint the average left out, the root stalling first at the slowest segment’s lift coefficient, and the question of whether a mission’s induced optimum ever falls short of the washout the stall requires.