The sweep a root does not have

The wind a swept wing feels is one of the cleanest arguments in this subject and it is worth restating before it is taken apart, because nothing below contradicts it.

An infinite wing yawed at Λ to a stream cannot know about the velocity component along its own span: that component produces no gradient in any direction the wing has, so it convects along and does nothing. Only $U\cos\Lambda$ is felt. Every pressure coefficient therefore scales by $\cos^2\Lambda$, and the free-stream Mach number at which the flow first goes sonic rises by $1/\cos\Lambda$.

A forty-five degree wing is promised a forty-one per cent higher critical Mach number, which is the whole reason transonic aircraft are swept, and it is exactly right for a wing with no ends.

A wing has three ends #

A real wing has a root where the two halves meet and two tips where the pressure relieves round an edge, and at both the argument fails — because the spanwise component is no longer producing no gradient.

That locus is the isobar the simple theory says should run parallel to the leading edge. Over the middle of the span it does. Near the root it does not, and near the tip it does not.

At a geometric sweep of 35° the computed local sweep is 35.2° at mid-span — the theory being right, and the check that the lattice is not inventing anything — 31.6° at the tip, and 14.3° at the root.

Twenty-one degrees of unsweep, at the station where the wing is thickest, most heavily loaded and joined to a fuselage.

Why the root unsweeps #

The mechanism is not subtle once it is drawn.

Consider the port and starboard halves of a swept-back wing as two yawed wings meeting at the centre. Each half’s own upwash field acts on the other, and at the centreline the two contributions add. On a swept-back wing the root’s own bound vortex runs aft as it goes outboard, so the influence of the far half arrives from behind the root section — and the effect is to reduce the loading at the front of the root chord and increase it at the back.

The pressure distribution at the root is therefore flatter and shifted aft than the yawed-wing solution says, and the isobar — the locus of a given fraction of the load — moves aft near the centreline. A line that has moved aft at one end and not at the other is a line with less sweep.

At the tip the opposite happens for a different reason: the pressure relieves round the edge, the loading collapses over the last few per cent of span, and the isobar bends forward.

What that does to a transonic wing #

The consequence is the whole of transonic wing design.

The benefit of sweep goes as the local isobar sweep, not as the geometric one, because it is the component of velocity normal to the isobars that decides whether the flow goes sonic. So a wing with a geometric sweep of 35° and a root isobar sweep of 14° is getting a critical Mach number benefit at the root of $1/\cos 14° = 1.03$ instead of $1/\cos 35° = 1.22$.

The shock forms at the root first. That is observed on every swept wing ever built, and it is why the fixes exist:

Fuselage waisting, the area rule’s cousin — narrowing the body where the wing joins it so that the combined cross-sectional area varies smoothly. It works partly by the area rule and partly by removing the root’s own overspeed.

Root fairings and glove sections — a locally different aerofoil at the root, usually with less camber forward and more aft, chosen so that its own pressure distribution compensates for the unsweep.

And a “yehudi” — the inboard trailing-edge extension on almost every Boeing transport, which increases the root chord and reduces the local lift coefficient without changing the outboard wing.

All three exist because the collapse onto one sweep angle is wrong at the root, and none of them appears in a calculation that has only Λ in it.

What sweep does to the loading #

There is a second consequence, and it is what makes a swept wing stall differently.

Sweepback unloads the root and loads the tip. The load centroid moves from 0.435 of the semi-span at zero sweep to 0.464 at 45°, and the outboard sections are working harder than their unswept equivalents.

That is a stability problem rather than a drag one. A wing that is most heavily loaded outboard stalls outboard first, and a swept wing that stalls outboard loses lift behind its own centre of gravity — which produces a nose-up pitching moment, which raises the incidence, which deepens the stall. That is pitch-up, and it is the reason swept wings carry washout, wing fences, vortilons, leading-edge notches and every other device for making the root give way first. This collection computes the same trade for a tapered wing, and sweep pushes it in the dangerous direction.

The lift-curve slope falls too — from 4.82 per radian unswept to 3.77 at 45° — and by more than the simple theory’s $\cos\Lambda$ would give, because the three-dimensional relief at the tips is stronger on a swept wing.

What a designer does about it #

Küchemann’s statement of the problem is the useful one and it is worth writing as a design rule, because it inverts the usual presentation.

The wing is not swept; the isobars are. A wing whose leading edge is swept at 35° and whose isobars are swept at 14° over the inner third is a 14°-swept wing over that third, whatever the drawing says. So the design variable is the isobar pattern, and the planform is one of several ways of controlling it.

The others are section shape and twist. Giving the root a section with less forward camber moves its own suction peak aft, which pushes the isobar aft at the root and increases the local sweep; giving the tip a more forward-loaded section does the same at the other end. A modern transonic wing therefore has a different aerofoil at every station, chosen so that the isobars come out straight and swept rather than so that the sections are individually good.

That is why a transonic wing’s sections look strange in isolation. The root section of a Boeing wing has a pressure distribution nobody would design for a two-dimensional test, and it is right for the wing it is part of.

And the same argument runs the other way for the tip. An outboard section that unsweeps its isobars forward is running at an effectively lower sweep as well, which is one reason the outer wing is thinner — thinness and sweep are substitutes for each other in raising the critical Mach number, and where the sweep is not available the thickness has to give.

The tip, which fails for a different reason #

The root has had most of this essay and the tip is worth its own paragraph, because the mechanism is not the same one and neither is the fix.

At the root the isobars unsweep because the two halves of the wing are in each other’s upwash. At the tip they unsweep because the pressure relieves round an edge: the loading collapses over the last few per cent of span, the suction peak weakens, and the locus of a given fraction of the load bends forward as it approaches the tip. The computed sweep falls from 34.9° at η = 0.78 to 31.6° at η = 0.92 for a 35° wing.

The consequences are milder than the root’s and are not negligible. A tip whose isobars are unswept has a lower local critical Mach number in the same way the root does, and it is carrying more load than an unswept wing’s tip would — which is the outboard shift the load figure shows — so it is the more likely place for the wing to reach its section maximum lift.

The fix is a different one: tip washout and a tip section chosen for a lower design lift coefficient, rather than a change to the planform. A raked or swept tip helps too, by moving the sharp relief further outboard and away from the loaded span, and that is what most modern transports have instead of a winglet or in addition to one — the same reasoning a winglet is designed by, applied to the isobars rather than to the wake.

What the lattice is, and what it is worth #

The solve is a vortex lattice: forty spanwise strips by eight chordwise panels, horseshoe vortices on each panel’s quarter-chord line, control points at three-quarter chord, tangency enforced with Biot–Savart.

Its check is the unswept case. At aspect ratio 8 with a rectangular planform it gives a lift slope of 4.65 against the monoplane equation’s 4.84 — four per cent low, and the sign is right: a rectangular wing’s lifting line has a singular tip and the lattice does not, so the lattice carries less. That discrepancy is reported rather than tuned away.

The isobar is computed from the load rather than from a pressure field, because a lattice has no continuous one. What is tracked is the chordwise station carrying half the section’s cumulative load, which is the same curve integrated and is what an isobar’s position is a proxy for. The sweep is then a central difference in the spanwise direction, so it is a local angle rather than a fit through the whole span.

The compressible half, which is not here #

Everything above is incompressible, and the argument it is about is a compressible one. That is worth being explicit about.

The isobar sweep computed here is the incompressible one. In a transonic flow the loading redistributes — the shock’s own position feeds back on the pressure distribution, the root’s overspeed is worse than the incompressible calculation suggests, and the isobar pattern near a shock is not a smooth curve at all.

So the numbers are an indication of the mechanism rather than a transonic design tool. What survives compressibility is the geometry of why the root unsweeps — two halves meeting, each in the other’s upwash — and that is present at every Mach number.

The pocket on top of the wing is where this collection computes what a supercritical region actually does, and it is at a section rather than on a wing for exactly this reason.

What else is missing #

No fuselage. The root condition modelled here is a plane of symmetry, which is a wing joined to a mirror image of itself. A real root is joined to a body, and the body’s own flow field adds an interference that is the same order as the effect computed above.

Flat plates. No thickness and no camber, so the loading is the lattice’s and the pressure distribution over a chord is the flat-plate one.

And no viscosity, so no boundary layer running outboard along a swept isobar — which is a real effect, is what the crossflow instability is about, and is the reason a swept wing transitions further forward than an unswept one at the same Reynolds number.

The forward-swept case #

One consequence is worth stating because it is counter-intuitive and follows immediately.

For a wing swept forward, everything above reverses in sign: the load moves inboard, the root isobars unsweep in the other direction, and the wing stalls at the root first — which is exactly the benign behaviour a designer wants, with the lift lost ahead of the centre of gravity and a nose-down moment resulting.

Forward sweep is aerodynamically superior on that count and is almost never used, for a structural reason: a forward-swept wing’s bending under load increases the incidence at the tip, which increases the load, which increases the bending. That is divergence, and it is why the X-29 needed composite aeroelastic tailoring to exist at all.

The loading is not the one the lattice computed #

The forward-swept case reverses the sign of a structural coupling, and the aft-swept case has that coupling too. It is worth stating, because it qualifies this essay’s own centroid figure.

An aft-swept wing’s outboard sections sit behind the root as well as beside it, so bending the wing upwards rotates them leading-edge-down. Sweepback converts bending into washout, automatically and in proportion to the load. Where forward sweep’s version of this runs away — more load, more incidence, more load, which is the divergence the X-29 had to be tailored against — aft sweep’s version is self-correcting, and it is one of the quiet reasons sweepback is structurally the easy direction.

The consequence for everything computed above is that the span load has an extra argument. The lattice solved a rigid wing, and a real one washes out more at high load factor than at cruise, so the tip unloads and the centroid moves back inboard — partly undoing the outboard shift that made the stall argument dangerous, by an amount that depends on dynamic pressure and on how much the wing is carrying. A swept wing does not have a span load. It has a family of them.

Which forces a decision visible on any airliner at a gate. The aerodynamicist specifies the twist the wing should have in flight at cruise, the structures group computes how much the wing will wash out under that load, and the wing is manufactured with a jig twist that is neither — deliberately wrong on the ground so that it is right at altitude. The shape drawn in the wind tunnel exists at one flight condition and nowhere else.

The same torsional flexibility has a hostile face. A deflected aileron twists the wing against its own lift, and above some speed the twist wins outright — the control reverses, and the roll goes the other way.

What “the sweep” of a wing even means #

One more consequence follows, and it is a definitional one that catches people out.

A tapered swept wing has at least four candidate sweep angles — of the leading edge, the quarter-chord line, the half-chord line and the trailing edge — and on a typical transport they differ by ten degrees or more. Which one is “the” sweep depends on what the number is being used for: the quarter-chord sweep is the one that enters the lift-slope corrections, the half-chord sweep is nearer the right one for the critical Mach number, and the leading-edge sweep is what decides whether a leading-edge vortex forms.

The isobar argument settles it. None of them is the right number, because the quantity the physics depends on is the local sweep of the isobars, which is a function of spanwise station and is not any of the four constants. The four are proxies of varying quality, and the half-chord line is the best of them because it is nearest to where the isobars actually run over the middle of the span.

That is a small point and it is the practical residue of the whole essay: a wing’s sweep is not one number, and the number that would be worth having is a curve.

Where the argument came from #

Adolf Busemann proposed sweep for reducing wave drag at the Volta Conference in 1935, in a paper that was published and ignored for a decade. Robert T. Jones arrived at the same idea independently in 1945 and — this is the part that matters here — his memorandum is explicit that the argument is for an infinite yawed wing and that the ends are a separate problem.

The root and tip corrections were understood by the early 1950s from measurement, and the fixes listed above date from then. Küchemann’s work on isobar shaping in the 1950s and 1960s is where the design method comes from: shape the wing so the isobars are swept, rather than so that the leading edge is.

That is the right statement of the principle, and it is not the one in the textbooks.

What this leaves #

Sweep collapses a three-dimensional wing onto one angle, and the residual is the two places a wing has ends. Twenty-one degrees of it, at the station where the wing is thickest.

The next essay leaves the geometry for the history — what a wing’s lift does when the flow changes, and why entering a gust is not the same as being pitched.