The wind a swept wing feels

Every large aeroplane built since about 1950 has its wings swept back, and the reason given is almost always the same sentence: the swept wing meets a slower wind, so it can fly faster before the air over it goes supersonic.

The conclusion is right and the sentence is not. A wing does not meet a slower wind. It meets exactly the wind it would meet unswept, and the reason it behaves as though the wind were slower is a theorem — one of the very few exact decompositions in this subject, and considerably stronger than the hand-waving version that stands in for it.

The split #

Take a wing of infinite span, yawed at an angle Λ to a steady stream, in an incompressible laminar flow. Resolve the free stream into two components: one in the plane normal to the leading edge, of size U cos Λ, and one along the span, of size U sin Λ.

Write out the Navier–Stokes equations in wing-fixed axes and something remarkable happens. The two equations governing the flow in the normal plane contain no spanwise velocity at all. They are the ordinary two-dimensional equations, driven by U cos Λ, and they can be solved with no knowledge of the sweep whatever.

The spanwise equation is then

$$u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} = \nu\frac{\partial^2 w}{\partial y^2},$$

with w the spanwise velocity and u, v the normal-plane flow the first calculation produced. There is no pressure gradient in it — the wing is infinite, so nothing varies along the span. The spanwise velocity is a passive scalar: it is advected by a flow it does not influence, diffuses, and exerts no force on anything.

That is the independence principle, and it is exact. Not asymptotic, not a small-angle approximation: an exact statement about an infinite yawed wing in a laminar incompressible flow.

Checking it where it cannot hide #

An assertion this strong deserves a test that could fail, and there is one available.

On a flat plate with no pressure gradient, the normal-plane flow is Blasius’s, which this site already solves by shooting the nonlinear third-order equation. Substituting the similarity variable into the spanwise equation gives

$$g'' + \tfrac{1}{2} f g' = 0, \qquad g(0) = 0,\ g(\infty) = 1,$$

a linear second-order equation with no eigenvalue in it — which is itself the computational shadow of the flow being passive: nothing has to be shot for, the equation is integrated once from an arbitrary wall slope and scaled to meet the outer condition.

Divide through: $g'' / g' = -\tfrac12 f$, so $g' \propto \exp(-\tfrac12\!\int\! f)$. The Blasius equation says the same thing about $f''$. So $g' \propto f''$, and integrating with both boundary conditions gives $g = f'$ exactly.

The two profiles are not similar. They are the same function.

The solver integrates the spanwise equation independently and requires the two curves to agree to five parts in ten thousand; they agree to two parts in a hundred million. And it then asks the consequence: a swept flat plate must have no crossflow at any sweep, which is required at four angles and comes out below 10⁻³ at every one.

That is the independence principle in the form that has no wriggle room in it. A boundary layer on a yawed plate at sixty degrees of sweep is the boundary layer on an unswept plate, with a spanwise flow superimposed that the layer does not feel.

What the split buys #

Once the normal-plane problem is the whole of the section’s problem, two design numbers fall out by division.

The critical Mach number rises as 1/cos Λ. The pocket of supersonic flow on top of a wing forms when the local Mach number reaches one, and the local Mach number is a property of the normal-plane flow — so it is $M\cos\Lambda$ that has to stay below the section’s own critical value. A section good to Mach 0.72 unswept is good to 0.88 at thirty-five degrees of sweep, and a designer wanting Mach 0.85 out of that section needs 32.1 degrees.

The lift-curve slope falls as cos²Λ. The dynamic pressure in the normal plane is $q\cos^2\Lambda$, and the chord measured streamwise is longer by $1/\cos\Lambda$, so the lift coefficient referred to the streamwise chord carries two factors of the cosine. At thirty-five degrees the wing has 67 per cent of the lift slope it had, which is a very large amount to give away.

That is where every high-lift device on a swept wing comes from. The slats, the triple-slotted flaps, the enormous wing area of a swept-wing airliner relative to its weight: all of it is buying back the lift slope that the sweep spent. A swept wing is a machine for converting low-speed performance into high-speed performance, and the exchange rate is the cosine.

Where the split breaks #

Put a pressure gradient on the normal-plane flow and the coincidence ends.

The chordwise profile now solves Falkner–Skan, which has an extra term $m(1 - f'^2)$ in it. The spanwise profile still solves the passive equation, which does not. The two are no longer the same function, and the resultant velocity in the layer therefore points a different way at every height.

That difference is the crossflow, and it is the thing a swept wing has that an unswept one cannot. It is easiest to see as a curve rather than as two profiles.

The crossflow profile has a property that decides everything downstream of it: it is zero at the wall and zero at the edge, so it has a maximum in between and therefore an inflection point. And an inflected profile is inviscidly unstable — Rayleigh’s criterion, which this site owns and which says that a velocity profile with an inflection has a growing mode without needing viscosity to supply one.

So a swept wing’s boundary layer carries, everywhere it has a pressure gradient, a small velocity profile that is unstable for the strongest reason a profile can be. That is crossflow instability, and it is why a swept wing transitions far earlier than an unswept one at the same Reynolds number, and why laminar flow on a swept wing is an enormously harder engineering problem than laminar flow on a straight one.

The angle where it is worst #

The crossflow needs a sweep and a gradient, and it vanishes at both ends of the sweep range: at no sweep there is no spanwise flow for the gradient to act on, at ninety degrees there is no chordwise flow to act with. So the maximum is interior, and it is at exactly forty-five degrees — the product $\cos\Lambda\sin\Lambda$, and the solver requires it at that angle rather than near it.

That is a geometric fact about resolving a velocity, not a fact about boundary layers, and it is useful precisely because it is so blunt: an airliner wing at thirty-five degrees is close to the worst possible place to be for this particular mechanism, and it is there because the compressibility benefit is worth more than the transition penalty costs.

The attachment line, which the principle makes into a problem #

There is one place on a swept wing where the independence principle stops being a convenience and becomes a hazard, and it follows directly from the passivity of the spanwise flow.

On an unswept wing the leading edge is a stagnation point in every section, and each section’s boundary layer starts there from nothing. On a swept wing the leading edge is an attachment line, and the spanwise flow runs along it. There is nowhere for that flow to start from: it arrives from further inboard, carrying whatever boundary layer it has already grown.

So a swept wing has a boundary layer that runs the length of its own leading edge, thickening as it goes, with its own Reynolds number based on the spanwise velocity and the leading-edge scale. Above a critical value that layer becomes turbulent on its own account, and everything downstream of it is turbulent too — the wing has lost its laminar flow before any section’s own transition has been considered.

Worse, it can be contaminated. If turbulence arrives at the wing root — from the fuselage boundary layer, or from a bracket, or from an insect — it travels outboard along the attachment line and turns the whole leading edge turbulent, at a Reynolds number well below the one at which the line would have gone turbulent by itself. That is attachment-line contamination, it was found in flight rather than in a wind tunnel, and the standard remedy is a small bump near the root — a Gaster bump — that forces a fresh stagnation point and cuts the line.

A mechanism that exists only because the spanwise flow is there, on a wing whose whole justification is that the spanwise flow does not matter. The principle is exact and it is exact about a section; the attachment line is not a section.

The S the outer flow makes, and the reversal it causes #

The similarity solutions above are shapes at a station. They cannot show the thing a swept wing’s layer is most famous for, which is that the crossflow reverses direction along the chord — and the reason for that is in the inviscid flow, with no boundary layer needed at all.

The spanwise component of the edge velocity is constant along the chord, because the wing is infinite and nothing varies along the span. The chordwise component is zero at the attachment line, rises to a maximum at the suction peak, and falls again. So the edge streamline starts out running almost along the leading edge, swings forward towards the chordwise direction as the flow accelerates, and swings back as it decelerates.

The outer streamline is an S, and it is exact — that figure is computed from the section’s own inviscid pressure distribution, with no layer anywhere in it.

Now put a boundary layer under it. A curving streamline is a balance between a pressure gradient normal to itself and the centrifugal force of the fluid going round the bend. The fluid near the wall is slower, so its centrifugal force is smaller, and the same normal pressure gradient pushes it to the inside of the bend. Ahead of the suction peak the bend goes one way and behind it the other, so the near-wall flow is pushed outboard first and inboard afterwards.

That reversal is the real crossflow pattern on a wing, and it is why the two most dangerous places on a swept wing’s surface are the attachment line and the pressure recovery — different mechanisms, different remedies, and both invisible to a similarity solution that has one fixed pressure gradient.

Two instabilities pulling opposite ways #

The crossflow is not the only way a layer goes turbulent, and the relation between the two ways is what makes laminar flow on a swept wing so much harder than on a straight one.

An unswept wing transitions through Tollmien–Schlichting waves, which grow in the chordwise profile and are suppressed by a favourable pressure gradient: accelerating the flow fills the profile out, removes any inflection, and keeps the waves damped. That is the whole of natural laminar flow design — put the maximum thickness well aft, hold the flow accelerating over the front half, and the layer stays laminar because it has no unstable mode.

Now read the crossflow figures above with that in mind. The crossflow’s size is set by the chordwise pressure gradient, and it grows with it. The same acceleration that quietens the first mechanism feeds the second.

So on a swept wing the two are in opposition. A strongly favourable gradient gives a quiet chordwise profile and a violent crossflow; a flat one gives a small crossflow and a chordwise profile with nothing holding it. The design is a compromise between two instabilities rather than a defence against one, and the compromise narrows as the sweep rises — past twenty-odd degrees there is no gradient that keeps both subcritical, which is why laminar flow beyond that sweep needs the layer removed by suction rather than merely shaped.

What sweep does to the rest of the wing #

Three consequences that follow from the same split and that a design has to answer:

The spanwise flow accumulates. On a real wing the layer thickens as it drifts outboard, so the tip region is running a thicker layer than a strip theory would predict, and it separates earlier. That is the origin of tip stall on a swept wing, and it is worse than the taper effect this site treats separately, because it adds to it.

Sweep and taper stall the same part of the wing. Both push the trouble outboard, and outboard is where the ailerons are. Fences, notches, vortilons and sawtooth leading edges are all attempts to stop the spanwise drift, and every one of them is a device for interrupting a flow the independence principle says should not matter — which it does not, until the wing is finite.

And the root and the tip are not swept. The independence principle is about an infinite wing. At the root the fuselage forces the flow to be symmetric and the isobars unsweep themselves; at the tip the flow can escape round the end. Both regions lose the compressibility benefit, and the fix is to give them extra sweep, extra area or a different section — which is why a swept wing’s isobars, plotted, are a design objective rather than a consequence.

What the model does not contain #

Infinite span. Every exact statement above assumes it. A real wing is finite, has a root and a tip, and has a spanwise pressure gradient that the independence principle explicitly excludes.

Laminar and incompressible. The split is exact for a laminar incompressible flow. In a turbulent layer it is approximately true and the approximation is not sharp; in a compressible flow the energy equation couples the two directions through the temperature, and the split is again approximate.

No transition prediction. The crossflow Reynolds number is computed here and calibrated nowhere. The value at which a swept wing actually goes turbulent is somebody else’s measurement, and quoting one would be putting a number on a mechanism nothing here models.

Simple sweep theory is simple. The 1/cos Λ on the critical Mach number treats the section’s critical value as unchanged, which it is not: the streamwise thickness ratio changes with sweep, and a real wing is designed with the two effects together.

And the crossflow profiles here are similarity solutions, so each is a shape at one fixed pressure gradient. The reversal along the chord is argued from the inviscid streamline and is not computed as a marching boundary-layer solution, which is a different and much larger piece of machinery.

Who found it, and when #

The independence principle is Prandtl’s, from around 1945, and Rudolf Sears and Wilhelm Jones each published careful statements of it shortly afterwards. Adolf Busemann had proposed sweep for supersonic aircraft at the Volta conference in 1935; Albert Betz worked out the subsonic argument in 1939 and it became a German wartime research programme.

What is striking is how completely the idea was missed elsewhere. The Volta paper was published, openly, and nobody outside Germany acted on it for nearly a decade — the swept wing arrived in Britain and America in 1945 with the captured research, and every jet airliner since is descended from it. It is one of the clearest cases in this subject of an idea sitting in print for ten years while the people who needed it did not read it.

The crossflow instability came later and from measurement: Gray at Farnborough in 1952 found transition on swept wings occurring far earlier than any two-dimensional criterion allowed, and Owen and Randall identified the inflected crossflow profile as the cause the following year. The theorem that makes a swept wing possible and the instability that makes it difficult were found eight years apart, and the second one was found by somebody noticing that an aeroplane did not behave like the theory.

Where the ladder goes next #

The independence principle is about an infinite wing; every difficulty above came from the wing being finite. The next question is what a finite wing’s span does to the lift it carries at each station, and which station runs out first — which is the question a taper ratio and a twist distribution exist to answer.