A wing that twists with its speed washes out the wrong way

One twist is chosen for the speed flown per metre took a trainer’s sortie — climb, cruise to the training area, slow flight and stalls, circuits — and chose one washout for the whole of it. The best was the single-point optimum at the distance-weighted harmonic mean of the sortie’s lift coefficients, 3.43° on a rectangular wing of aspect ratio eight. And whatever twist was chosen, the sortie paid for flying at more than one lift coefficient: 1.41 per cent of its induced energy on top of the 0.90 its planform cost at any single speed, because a fixed twist can be right at only one of them. It ended on the obvious escape. A wing whose twist changed with its speed could be right at all of them, and real wings do twist in flight.

They twist the wrong way. The calculation that shows it is short, and the direction is the whole result.

What each speed wants #

Washout is right at one lift coefficient wrote the induced drag of a linearly washed-out wing as Phillips’s quadratic form in the lift coefficient and the washout, and found the best washout at a lift coefficient $C_L$ to be in proportion to it: $\Omega^* = \beta C_L$, with $\beta = 7.87°$ per unit of $C_L$ for this planform. The four segments of the sortie fly at lift coefficients of 0.33 in cruise, 0.82 in the climb, 0.92 in the circuits and 1.28 in slow flight, and want 2.6°, 6.4°, 7.2° and 10.0° of washout. Slow flight wants the most. A wing at high lift coefficient is carrying a loading whose shape the twist has most to correct, and the correction is in proportion to the load per unit dynamic pressure.

Why the optimum goes with the lift coefficient is worth a sentence, because the rest follows from it. Induced drag comes from the ends of the wing: the circulation that cannot be carried off a tip is shed into the wake, and the drag is least when the shedding is spread in the elliptic way. A rectangular wing loads its tips too heavily for that, and washout unloads them by lowering their incidence. The excess tip load is in proportion to the wing’s load per unit dynamic pressure, which is the lift coefficient, so the incidence that removes it is in proportion too. At a low lift coefficient there is little tip load to remove and a large washout would push the tips into carrying negative lift; at a high one there is a great deal. The washout that is right depends on how hard the wing is working, not on how fast it is going.

What a real wing’s twist follows #

A wing’s elastic twist answers to the moments on it, and in flight there are two kinds. A cambered section has a pitching moment at zero lift, $C_{m0}\,q\,c$ per unit span, nose-down for the usual camber, and it twists the wing nose-down towards the tips by an amount in proportion to the dynamic pressure $q$. In level flight $q = W/(SC_L)$, so this twist grows as the lift coefficient falls: it adds washout at cruise and takes it away in slow flight, the reverse of the line the optimum follows. The second kind is the lift itself, acting at a chordwise point other than the elastic axis, and on a swept wing the bending that couples into twist: both go with the load the wing carries, $nW$, which in level flight is the same at every speed. A wing twisted by its load washes out more in a turn and not at all differently between cruise and slow flight.

Neither is the law the optimum asks for, a twist in proportion to $C_L$ — to the incidence, which at a fixed weight is the lift coefficient by another name. That is a twist that answers to the angle the air meets the wing at, not to the force it puts on it, and no stiffness gives it: only a moving surface scheduled with the lift coefficient, or a tip that pivots about its own aerodynamic centre against a spring, so that what twists it is the angle and not the pressure.

Three laws, each at its best #

Each law has two parameters — a base washout and a gain on its driver — and the sortie’s energy is quadratic in the washout, so the best pair for each law is one small linear solve, checked against a direct search that never finds a lower value.

The fixed law is the earlier essay’s answer: 3.43° and 2.31 per cent above the elliptic wing. The load law can do nothing in level flight, because the load is the same in every segment, and returns the same 2.31. The incidence law reaches the planform’s floor exactly, 0.902 per cent, with no base washout at all and a gain equal to $\beta$: every segment flies at its own optimum, and the 1.41 per cent spread cost is gone.

The dynamic-pressure law, given its best parameters, comes within a twentieth of a point of the floor: 0.958 per cent. But it gets there by washing in as the aeroplane speeds up — a base of 10.4° less 2.59° divided by the lift coefficient, which is 8.4° of washout in slow flight and 2.6° in cruise, a wash-in of nearly six degrees across the speed range. A law of the form $a + b/C_L$ can bend to fit a straight line over a factor of four in $C_L$ only by making $b$ negative, which in a real wing means a pitching moment of the opposite sign to the one camber gives.

The natural sign costs energy #

So the twist a light wing actually gets makes things worse. The reason is visible segment by segment. The natural twist adds washout in cruise, which already had too much of the fixed 3.43° against its own 2.6°, and removes it in slow flight, which already had too little against its own 10°. Every degree of it widens the gap between what each segment is given and what it wants, and the base washout can only move the whole line up or down. With one degree of extra washout at cruise, growing with dynamic pressure as a nose-down pitching moment gives, and the base washout re-chosen to suit, the sortie’s excess rises from 2.31 to 2.68 per cent; with two degrees, to 3.09. The re-chosen base washout falls to compensate — 2.53° and 1.64° at the root of the law — but no base can make a twist that grows at the wrong end of the speed range point the right way. Two degrees of wash-in with dynamic pressure, the opposite sign, would lower the excess to 1.71 per cent.

The surprising connection is that the drag a flexible wing pays for its flexibility is set by the sign of its camber’s pitching moment. A section with a nose-down moment, as nearly every cambered aerofoil has, twists the wrong way; a reflexed section, whose pitching moment is nose-up — the section tailless aircraft use for their pitch balance — twists the right way, washing in as it speeds up. A tailless wing’s reflex, chosen for trim, is therefore also the one camber whose aeroelastic twist helps its induced drag across a mission.

The tailless designer’s constraint is the one no elevon reaches past a quarter chord priced: a wing with no tail trims itself with the back of its own section, and a reflexed camber line is how it carries the nose-up moment it needs without deflecting a surface. That moment twists a flexible wing’s tips nose-up as dynamic pressure rises, which is to say it washes in at speed and out at low speed. The same reflex that the trim calculation imposes is the camber the mission calculation asks for, and a flying wing with a little torsional flexibility gets for nothing the variable twist that a conventional wing would need flaps to approach. Whether the amount is right is a matter of stiffness, which a designer can choose, but the direction comes free.

How much twist a light wing gets #

The size of the effect is the size of a light wing’s torsional flexibility. A uniform cantilever of the trainer’s chord, 1.41 metres, and semi-span, 5.65 metres, carrying a pitching moment of $C_{m0} = -0.05$ at its cruise dynamic pressure of 1,850 pascals, twists at the tip by $|C_{m0}|qc^2(b/2)^2/(2GJ)$: 1.7° for a torsional stiffness of $10^5$ newton-square-metres and 0.85° for twice that. That is the range of a light aircraft’s wing, and so the third of a point of the figure above is a real cost, paid in cruise fuel, by any trainer whose wing is not much stiffer in torsion than it needs to be for strength. It is small beside the profile drag, and it is in the opposite direction from the folklore that a flexible wing adapts.

What the swept wing’s load-twist is for #

The load law, useless in level flight, is not useless everywhere. A swept-back wing bending under load twists its tips nose-down in proportion to the load, so it washes out more in a turn than in straight flight, and the turn is flown at a higher lift coefficient: for once the twist and the optimum move together. A glider’s circling is the case that gains, since a glider spends much of its flight in turns at a load factor of $\sqrt2$ and a lift coefficient three times its gliding one; with two segments, any law with two parameters can fit both exactly, and the load law reaches the planform’s floor with a gain of 4.7° of washout per unit of load factor. A swept-back glider’s bending, if it twisted by that much, would be doing the right thing for its circling. The trainer’s sortie, flown level, gives the load law nothing to work with.

Where twist with load does pay: a gust #

There is one place a twist that follows the load is unambiguously wanted, and it is not a mission average. In a gust the load rises in a fraction of a second while the flight condition stays the same, and a wing that washes out under load sheds part of the gust’s lift from its tips, where the bending arm is longest. That is passive gust-load alleviation, and it is the reason aeroelastic tailoring of composite wings is usually aimed at the load rather than at the induced drag. One wing root feels the rolling gust its lift cannot found the root’s bending moment responding to more of a gust than the lift does; a load-twist relieves exactly that root, so the structure’s preference and the mission’s indifference to the load law point the same way.

What the stiff wing is worth #

The comparison puts a number on an old design choice. Making a wing stiff in torsion costs structure weight; the calculation here says a stiff wing also keeps the induced-energy penalty at the fixed twist’s 2.31 per cent rather than letting it rise. And it says what an active twist would be worth: a scheduled change of washout with the lift coefficient — trailing-edge surfaces deflected differentially along the span, as some sailplanes do with their flaperons and as morphing-wing studies propose — recovers the whole 1.41 per cent of the spread cost on this sortie, which is about a twentieth of the cruise’s induced drag and much less of its total drag. Whether that is worth a mechanism is a question about profile drag and weight, which the span is the whole story reminds is the larger account, and which this calculation does not price.

What a sailplane’s flaps already do #

Racing sailplanes carry trailing-edge flaps along the whole span and set them by speed: down for circling, neutral or reflexed for the fast glide between thermals. That schedule is a camber change rather than a twist, and it is chosen for profile drag — keeping each section in the low-drag bucket of its polar at the lift coefficient it is flying — but when the outer flaps are set differently from the inner ones it is also a scheduled twist, and its schedule runs with the lift coefficient: more trailing edge down, more lift, at the slow end. Pilots set it by airspeed, but airspeed at a fixed weight is the lift coefficient by another name, so the schedule is the incidence law in practice. The calculation here says that the induced-drag share of what it buys, on a sortie with the trainer’s spread, is the whole of the spread cost, a little under one and a half per cent of the induced energy; the profile share, which is why the flaps are there, is larger.

The convention: energy per mission, at fixed weight #

The figure of merit is the sortie’s induced energy — induced drag times speed times time, summed over segments — divided by an elliptic wing’s of the same span over the same sortie, as in the earlier essay. Washout is the tip’s incidence below the root’s, linear along the span; positive gains add washout as their driver grows. Dynamic pressure and load are in level flight unless a segment is a turn. The base and gain of each law are chosen for this sortie and this planform; another mission weights the segments differently and moves them.

What the quadratic form leaves out #

Profile drag, which the twist also changes, more at the extremes of lift coefficient than in between. Stall, which washout delays at the tips and which is the reason slow flight wants washout for handling as well as drag. Aeroelastic divergence and control reversal, the limits a wing flexible in torsion has to stay away from, which bound how much twist with dynamic pressure is allowed at all. A twist that is not linear along the span, as a real wing’s elastic twist is not: a cantilever twists most near its tip. The lifting line itself, adequate at an aspect ratio of eight but, as where the line stops being a line measured, increasingly in error as wings get shorter. Compressibility, irrelevant to a trainer. And the time a twist takes, since a gust changes the load faster than the flight condition does.

And the structure’s own price. A wing made stiff enough in torsion to suppress its natural washout weighs more, and the weight is paid for in lift and so in induced drag. The calculation prices one side of that trade — a degree of natural washout at cruise costs about a third of a point of the sortie’s induced energy — and a designer would set it against the mass a stiffer torsion box adds, which is a structural calculation this one does not make.

Where the laws come from #

Phillips’s quadratic form for washed-out wings is from 2004, and the mission average that the earlier essay derived from it is a direct consequence. The aeroelastic twist of wings under pitching moment and bending is the content of every aeroelasticity text since Bisplinghoff, Ashley and Halfman’s of 1955, where it is studied for its effect on control and stability rather than on induced drag. Active span-load control to follow the lift coefficient has been flight-tested on several research aircraft. The finding here is the elementary one that the passive twist’s driver, dynamic pressure, rises where the optimum falls.

Still open: a twist with a pivot #

The incidence law needs a twist that answers to angle rather than to pressure, and the one passive device that gives it is a tip panel pivoted near its own aerodynamic centre and held by a spring: the aerodynamic moment about the pivot then depends on the panel’s incidence, not on the dynamic pressure, and the panel settles at an angle set by the wing’s lift coefficient. The next calculation gives the trainer’s wing a pivoted outer panel of stated span and spring stiffness, finds the twist the panel takes at each segment of the sortie, and asks how much of the 1.41 per cent spread cost a passive device of that kind recovers — and what it gives away in a gust, where the panel’s freedom is also a freedom to flutter.