Lift out of a failure

Every other essay on this site treats separation as the end of something. The moment the flow lets go is the moment the lift curve stops being a line, the wake becomes wide, and the wing stops working.

A slender delta wing with sharp leading edges is built to separate. The flow cannot get round an edge that sharp, so it leaves along the whole leading edge, rolls up into a pair of vortices that sit over the upper surface, and the suction under those vortices is where most of the wing’s lift comes from.

Two terms, and only one of them is linear #

The model is

$$C_L = \underbrace{\frac{\pi\,\mathrm{AR}}{2}\sin\alpha\cos^2\alpha}_{\text{potential}} + \underbrace{\pi\sin^2\alpha\cos\alpha}_{\text{vortex}},$$

and the difference between the two is the whole subject.

The first term is slender-wing theory, which is exact in the limit of small aspect ratio: an attached flow past a slender wing behaves in each cross-section as flow past a flat plate of the local span, and integrating along the wing gives a lift-curve slope of πAR/2 per radian. That is a genuine result with a derivation, and this site’s assertion checks that the whole model reduces to it as the incidence goes to zero, at the right rate rather than to a tolerance.

The second term is quadratic in the incidence, and that is why it does nothing at small angles and takes over at large ones. It is also why a delta wing has no linear range worth speaking of: the nonlinearity begins immediately.

Where the second term comes from #

Polhamus’s suction analogy, from 1966, is one of the more elegant pieces of reasoning in the subject, and it is worth stating carefully because it is a hypothesis rather than a derivation.

In an attached flow round a rounded leading edge there is a pressure singularity that produces a force pulling forward along the chord — the leading-edge suction — and it is what keeps an attached aerofoil’s drag down to the induced value rather than the full C_L tan α. A sharp edge cannot support that suction: the flow separates there instead.

Polhamus’s hypothesis is that the force does not disappear. It is rotated through ninety degrees by the separation, reappearing as a normal force on the wing — the suction under the leading-edge vortex — with the same magnitude the attached theory computes for it. For a slender delta that gives a vortex coefficient of π.

It was validated against measurement rather than derived, and it works remarkably well — within a few per cent for slender deltas up to the angle where the vortices break down. This site marks it as what it is everywhere it appears: the shape of the curve is a consequence, the coefficient is somebody’s hypothesis checked in a tunnel.

Why the shape is slender #

That figure is the design argument. A low-aspect-ratio wing is a poor lifting surface in attached flow — the span is the whole story — and it is exactly the shape for which vortex lift is worth having, because the term it adds is independent of the span.

Which is why the shape appears where it does: on supersonic aircraft, whose wings must be slender and sharply swept for wave-drag reasons that have nothing to do with lift, and on missile fins and rocket strakes, where compactness is everything. The delta was chosen for supersonic cruise and turned out to have a way of flying slowly, and that is a fair summary of how the shape survived.

What it costs #

Vortex lift is expensive lift, and the reason is the analogy read backwards: the suction force that would have pulled the wing forward has been rotated into a force that lifts it instead. It cannot do both.

So the resultant force on a sharp-edged delta is normal to its surface, the drag is exactly C_L tan α, and the lift-to-drag ratio is 1/tan α — 2.75 at 20 degrees, before any profile drag at all. A conventional wing of aspect ratio six at its best glide is above twenty.

That is the trade the shape makes and it is not subtle: a delta buys usable lift at high incidence by giving up efficiency at every incidence. Concorde cruised at about four degrees with an L/D near 7, which was the compromise, and the aircraft that followed it did not repeat the compromise.

The vortex itself, and why it stays put #

The two vortices are not shed and swept away; they sit above the wing, growing in strength from the apex to the trailing edge, and they stay there for the whole time the wing is at incidence. That persistence is worth an explanation because it is not obvious.

The mechanism is the same one a vortex pair uses to move each other, balanced against the flow the wing itself imposes. Each vortex sits in the upwash of its partner, which tends to lift it, and in the flow over a wing that is pushing fluid down. At the equilibrium height the two balance, and the vortex is held in place by its own image system in the wing surface — the same image argument the site uses for a vortex near a wall.

What feeds them is the shear layer leaving the leading edge, which spirals inward and adds circulation continuously along the span. A delta’s vortex is therefore not a single filament but a conical spiral sheet, its strength growing with distance from the apex — which is why the lift is concentrated aft and why the centre of pressure moves as the incidence changes.

The suction under them is intense and local. The measured pressure coefficient beneath a delta’s leading-edge vortex reaches −3 or below at moderate incidence, which is deeper than anything on a conventional aerofoil and is concentrated in two narrow bands running from apex to trailing edge. That distribution is the physical content of the “vortex term”; the analogy computes its total and says nothing about where it acts.

What ends it, and why it is a jump #

The vortices hold the wing up until they stop, and how they stop is worth describing, because it is a phenomenon with a name and a criterion rather than a gradual fading.

A leading-edge vortex has two velocity components in it: axial flow running from apex to trailing edge along the core, and swirl around it. Their ratio is what matters. Raise the incidence and the swirl grows faster than the axial flow, and above a ratio of roughly 1.2 the core does something abrupt — a stagnation point appears on the axis, and downstream of it the core expands into a slow, unsteady, disorganised region. That is vortex breakdown, and it comes in two observed forms: an axisymmetric bubble, and a spiral in which the core corkscrews away from the axis.

The most illuminating account of it is Benjamin’s, from 1962, and it is one this collection has already met twice under other names. A swirling flow can carry inertial waves along its own axis, and whether those waves can travel upstream against the axial flow depends on the swirl. A weakly swirling core is supercritical: nothing propagates upstream, so the core cannot know what is ahead of it. A strongly swirling one is subcritical, and standing waves are possible. Breakdown is the transition between the two — a finite jump from a supercritical state to a subcritical one, with energy lost in it.

That is a hydraulic jump, in a vortex. The same structure as the jump in a river and the same structure as a channel choking at its critical depth, with the swirl ratio playing the part the Froude number plays and inertial waves playing the part of gravity waves.

What it does to the wing is decided by geometry. Breakdown first appears downstream of the trailing edge, and as the incidence rises it moves forward along the core. While it is behind the wing the suction is intact and the lift curve keeps climbing. Once it crosses the trailing edge the suction over the aft part collapses, the lift stops rising, and the centre of pressure lurches forward.

And it rarely happens symmetrically. One vortex bursts before the other, the wing rolls, the rolling changes the local incidences, and the aircraft can enter a self-sustained roll oscillation. The burst cores are also violently unsteady, and a vertical tail sitting in that wake is being shaken at high frequency — a fatigue problem that has been designed around on more than one aircraft rather than predicted in advance.

Where else the same trick appears #

The suction analogy’s arithmetic is about a sharp edge, not about a delta, so it applies wherever an edge is sharp enough to fix the separation.

Strakes and leading-edge extensions. A sharp, highly swept root extension ahead of a conventional wing generates its own vortex, which passes over the wing and delays its stall by energising the boundary layer. The lift increment is vortex lift computed exactly as above, on the strake’s own aspect ratio, and it is why almost every fighter since the 1970s has one.

Missile and rocket fins. Low aspect ratio, sharp edges, high incidence — the configuration this model was written for, and the one where the model is most accurate because there is nothing else going on.

Insect and bird wings at high incidence. A leading-edge vortex over a flapping wing is the mechanism usually credited with the extra lift, and the same argument applies with the enormous caveat that the flow is unsteady, the Reynolds number is in the thousands, and the vortex’s stability is maintained by spanwise flow rather than by geometry. This site does not compute any of that.

A sailing yacht’s headsail at high incidence produces the same structure along its luff, and the resulting lift is what allows the sail to work well beyond the angle at which an aerofoil section would stall.

What the wing is like to fly #

The lift curve has a consequence for handling that is worth setting out, because it is the reason delta-winged aircraft look and behave as they do on approach.

A conventional wing has a stall — a definite incidence at which the lift falls — and an aircraft is flown with a margin against it. A slender delta has no stall in that sense: the lift keeps rising until vortex breakdown, which is gradual rather than abrupt, so there is no natural limit and the aircraft can be flown to very high incidence.

What limits it instead is the attitude. To generate a landing-approach lift coefficient of about 0.6 at an aspect ratio of one takes about 17 degrees of incidence, against 8 for a conventional wing, and the nose is 17 degrees up while the flight path is 3 degrees down. That is the whole explanation for Concorde’s droop nose: it was not an aerodynamic device but a way of letting the pilots see the runway.

The drag at that attitude is what set the approach speed, since C_L tan α is a large number, and the engines had to be able to arrest a sink rate rather than merely maintain one. A delta on approach is flown on the back of the drag curve, where more incidence gives more lift and much more drag.

What the model does not contain #

Vortex breakdown, which is what actually limits the wing. The leading-edge vortices are stable only up to a point; past it their cores burst into a slow, unsteady, expanded state, the suction collapses, and the lift falls. That happens at an incidence that depends on the sweep — around 35 degrees for a 70-degree delta — and it means the peak this model computes at 49 degrees is not reached in practice. Nothing in this essay computes breakdown, and the model has no representation of it at all.

No Reynolds number, and no viscosity. The separation point is fixed by the geometry rather than found, which is exactly why a sharp edge is used — and it is also why this model would not survive on a rounded one, where the separation location depends on the flow and the whole calculation becomes a viscous problem.

Slender-wing theory frays above an aspect ratio of about 1.5, where the potential term should be computed by lifting-surface theory rather than by the slender approximation. The two are drawn together in the code and diverge visibly past that point.

No pitching moment. A delta’s centre of pressure moves substantially with incidence, because the vortex lift acts further aft than the potential lift, and that motion is the dominant handling problem of the configuration. It is entirely absent here.

A comparison whose formula has limits. The full-suction drag C_L²/πAR assumes elliptic loading, and at aspect ratios below one it exceeds C_L tan α at high incidence — which says that an elliptically loaded wing of that shape is not a real thing rather than that a sharp edge is free. The assertion in this site’s checks is made only where both formulae mean something, and the caveat is recorded rather than hidden by a wider tolerance.

The one number a designer reads off this #

Almost everything in this essay is a curve, and curves are hard to design with. The exception is the crossover, and it is worth isolating because it is a closed form that a designer can use:

$$\tan\alpha_{\text{cross}} = \frac{\mathrm{AR}}{2}.$$

That is the incidence at which the separation is carrying half the wing. Below it the wing is essentially a poor conventional wing; above it, it is a vortex-lift device. For an aspect ratio of one it is 26.6 degrees, and for two it is 45 — so a stubbier delta spends most of its usable range in the first regime and a slender one in the second.

The consequence for a configuration is direct. A wing that will operate mostly below its crossover should not have sharp edges: it is giving up leading-edge suction, and therefore paying C_L tan α in drag, for a vortex term that is not yet doing anything. That is the argument for the rounded and cambered leading edges on modern slender wings, which recover part of the suction at cruise and still separate cleanly at high incidence.

It is also why the crossover, rather than the peak, is the number this essay’s assertions check in two ways: a closed form and a search agreeing to two hundredths of a degree. The peak is past vortex breakdown on any real wing and is therefore a property of the model rather than of the aircraft, which is a distinction worth keeping in the same sentence as the number.

Who found it, and when #

The leading-edge vortex was seen before it was understood. Wind-tunnel tests of slender deltas in the 1950s produced lift curves that kept rising past the angle at which any conventional wing had stalled, and the flow visualisation showed the vortices, but the extra lift was fitted rather than predicted for a decade.

Edward Polhamus published the suction analogy at NASA Langley in 1966, and its virtue is that it turned a curve-fitting exercise into a calculation with one idea in it. A hypothesis that converts one computable quantity into another is worth more than a correlation with the same accuracy, because it says where it should fail — and this one does fail, exactly where the vortices break down, which is the failure a correlation would have hidden.

The shape’s history is older and stranger. Alexander Lippisch was flying delta gliders in Germany in the early 1930s, for reasons of structure and stability rather than aerodynamics, and the wartime and post-war interest in the planform came from its behaviour at supersonic speed. That it also had a mechanism for flying slowly was discovered afterwards.

Where the ladder goes next #

This is the eleventh anchor in the circulation field against a budget of twenty-eight, and the ones still open are mostly about what a real wing does with the vorticity it sheds: the roll-up of the trailing sheet, the interaction of a wing with its own wake, and the unsteady lift of a wing that is manoeuvring rather than sitting at an angle. Each needs the vortex machinery the essays around this one have been building, and each is a rung rather than a new anchor.