A wing that cannot hold its suction gains little from its span

A finite force from an infinite speed found the leading-edge suction where thin-aerofoil theory puts it: a point force at the sharp front of a flat plate, carried by a velocity that goes to infinity over an area that goes to zero, $2\pi\alpha^2$ along the chord, and the whole difference between a section with no pressure drag and one whose drag is its lift tilted back by the incidence. It noticed that the second drag, $C_L^2/2\pi$, looks like the induced drag of a wing with ends, $C_L^2/\pi A$, with the aspect ratio set to two, and called the resemblance structural. Both are a resultant tilted backwards by an angle proportional to the lift.

On a real wing both are present at once, and that is where the resemblance stops being a curiosity. A finite wing at incidence has an induced drag because its wake carries trailing vortices; if it also loses its suction, it has a second drag due to lift, and the two do not add — one of them is the other with the suction subtracted. The question is how they compare as the wing’s shape changes. The answer turns out to be an argument about what span is for.

Two drags due to lift, and their ratio #

A flat wing at a small incidence $\alpha$ carries a lift $C_L = C_{L\alpha}\alpha$. The pressure on a flat plate acts along the plate’s normal, so without a suction the resultant is normal to the plate and tilted back from the vertical by $\alpha$; its drag is $C_L\tan\alpha \approx C_L^2/C_{L\alpha}$. With the suction collected at the leading edge, the resultant is tilted back only by the downwash the trailing vortices induce, and the drag due to lift is the induced drag $C_L^2/(\pi A e)$, fixed by the spanwise loading in the Trefftz plane. The suction is exactly the difference. Losing it multiplies the drag due to lift by

$$R = \frac{C_L^2/C_{L\alpha}}{C_L^2/\pi A e} = \frac{\pi A e}{C_{L\alpha}}.$$

Everything about the finite wing is in the denominator. The lift slope of a wing is not the section’s $2\pi$; it is smaller, because the trailing vortices’ downwash takes part of the geometric incidence away, and how much smaller depends on the planform. A long wing’s slope approaches $2\pi$, so $R$ approaches $Ae/2$ — half the aspect ratio. A slender wing’s slope is Jones’s $\pi A/2$, the result of treating each cross-section as a two-dimensional flow round a plate, and its loading is elliptic, so $e = 1$ and $R = 2$ exactly. The range from slender to long is a range of $R$ from 2 to roughly $A/2$.

The arithmetic needs $C_{L\alpha}$ and $e$ for real planforms, and lifting-line theory is not good enough for the slender ones. They come here from a vortex lattice: the wing cut into panels, a horseshoe vortex on each with its bound leg at the panel’s quarter chord, and no flow through the plate at each three-quarter-chord point. The induced drag is then computed separately in the Trefftz plane, from the spanwise loading the lattice finds — which is the full-suction drag, whatever happens at the leading edge.

The lift slope, which is the whole of the answer #

A rectangular wing of aspect ratio 6 has a slope of 4.21 per radian, a delta of aspect ratio 1 has 1.29, and both agree with the handbook estimates — Helmbold’s $2\pi A/(2 + \sqrt{A^2 + 4})$ for straight wings, Polhamus’s version with the mid-chord sweep for deltas — to a few per cent. Helmbold’s formula also gives $R$ in closed form for an elliptic wing:

$$R = \frac{\pi A}{C_{L\alpha}} = \frac{2 + \sqrt{A^2+4}}{2} = 1 + \sqrt{1 + \frac{A^2}{4}},$$

which is 2.0 at $A \to 0$, 2.4 at $A = 2$, 3.2 at 4 and 5.1 at 8, and is the dashed curve in the first figure. The rectangles sit a little above it: their span efficiency is a little below one, which lowers $R$, but their slopes are further below the ellipse’s, which raises it more.

The section’s loss, carried unchanged into every span #

Lifting-line theory makes the structure of $R$ transparent, and it is worth doing even though the lattice is more accurate, because it says what the lost suction is on a finite wing. A section of a long wing sees the geometric incidence $\alpha$ less the downwash angle $\alpha_i = C_L/\pi A$ the trailing vortices induce, and lifts at the section’s slope on what is left: $C_L = 2\pi(\alpha - \alpha_i)$. The zero-suction drag is the lift times the whole incidence, and the incidence splits in two:

$$C_L\,\alpha = C_L\,\alpha_i + C_L(\alpha - \alpha_i) = \frac{C_L^2}{\pi A} + \frac{C_L^2}{2\pi}.$$

The first term is the induced drag. The second is the two-dimensional section’s lost suction, $C_L^2/2\pi$, and it does not depend on the span at all. On a finite wing that loses its suction, the drag due to lift is the induced drag plus a section drag that span cannot touch; and

$$R = 1 + \frac{C_L^2/2\pi}{C_L^2/\pi A} = 1 + \frac{A}{2}.$$

That is the long-wing form of the closed expression above, and the “one” in it is the induced drag counting itself. Everything else in this essay follows from those two terms. Span shrinks the first; nothing but the leading edge removes the second. On a long wing the second dominates, which is why the long wing loses the most. On a slender wing lifting-line theory is wrong — the downwash is not uniform along the chord and the slope is Jones’s, not Prandtl’s — and the lattice’s ratio of two replaces the formula’s $1 + A/2$, which would fall towards one; but the split into an induced part and a section part survives, with the section part shrinking as the wing becomes slender enough that its cross-flow, rather than its chordwise flow, carries the lift.

It also explains why the suction is so easy to underrate. In the section it was seven per cent of the lift at four degrees, a small force. On a finite wing it is the whole difference between a drag that span can reduce and one that span cannot, and the difference compounds: the better the wing is in every other respect, the larger the share of its drag due to lift that hangs on one edge. d’Alembert’s paradox holds for a lifting section only because the suction is there to close the force balance; take it away and the section has a drag of exactly the kind the paradox forbids, and the finite wing inherits it whole.

What a slender delta pays #

A slender delta is loaded almost exactly as the ellipse is — its span efficiency at aspect ratio 1 is 0.997 — so its induced drag is very nearly the least any wing of its span can have. Its lift slope is close to $\pi A/2$, the slender value. So the two drags due to lift come out in a ratio of almost exactly two. A slender wing that loses its suction pays its induced drag a second time, and no more. That is the coincidence the plate’s suction suggested, $C_L^2/2\pi$ against $C_L^2/\pi A$ with $A = 2$, turned into a statement about a three-dimensional wing: the slender limit is the one in which the two tilts — by the downwash, and by the lost suction — are equal.

That is the reason sharp-edged slender wings were tolerable at all. A supersonic aeroplane’s wing must be thin and sharp, since a blunt leading edge in supersonic flow costs wave drag out of all proportion to anything a suction could save, so it flies at low speed having lost its suction, and pays for it. A slender delta pays a factor of two; a straight wing of the same aspect ratio as an airliner’s would pay five. Designing a supersonic aeroplane slender was in part a way of losing the suction where it costs least.

The delta has a second way out. Let the flow separate at the sharp edge and roll into a pair of vortices over the upper surface, and their suction acts on the upper surface rather than along the leading edge: lift out of a failure. Polhamus’s analogy says the vortex lift equals the lost leading-edge suction turned through ninety degrees. That buys lift, a great deal of it at high incidence; it buys no drag back, because the resultant is still normal to the plate, and the polar is still the zero-suction one.

What it does to a polar #

The two polars of a rectangular wing of aspect ratio 6 with a zero-lift drag of 0.008 — a typical figure for a clean wing — differ by a factor of three at a lift coefficient of half: 0.0215 with the suction, 0.0673 without. The best lift-to-drag ratio, $1/(2\sqrt{K C_{D0}})$ with $K$ the drag-due-to-lift factor, falls from 24.1 to 11.5. The polar shapes are the same parabola; only the factor $K$ differs, by $R$, and the best ratio falls by $\sqrt R$.

Two wings at the same lift #

Two wings at the same lift coefficient of a half make the comparison concrete. A flat rectangle of aspect ratio 9, the proportions of an airliner’s wing, reaches it at an incidence of 6.1 degrees. With its suction its drag due to lift is 0.0092; without it, 0.0529 — 5.8 times as much, and more than twice the zero-lift drag of a whole airliner, which is about 0.02. A slender delta of aspect ratio 1.8, the proportions of a supersonic transport’s, needs 14 degrees for the same lift and has a drag due to lift of 0.0446 with its suction and 0.1226 without, 2.7 times. The delta is the worse wing either way, by a factor of five with the suction and of two without it. Losing the suction narrows the gap between them from five to about two, which is the arithmetic form of the observation that the slender wing has less to lose.

The incidences matter as well as the drags. The delta’s fourteen degrees are why a delta lands nose-high, and why an aeroplane built round one carries a drooping nose or a tall undercarriage; at that incidence its sharp edge has long since separated into the captive vortices of the section above, and the lift those add moves its polar up without moving it left.

The ceiling a lost suction puts on span #

The best lift-to-drag ratio with the suction grows with the aspect ratio without limit, as $\sqrt{\pi A e/4C_{D0}}$, because the induced drag falls as $1/A$. Without it, the factor is $1/C_{L\alpha}$, and the lift slope stops at $2\pi$. So the best ratio of a wing that cannot hold its suction is bounded, for every aspect ratio, by

$$\left(\frac{L}{D}\right)_{\max} < \frac{1}{2}\sqrt{\frac{2\pi}{C_{D0}}} = \sqrt{\frac{\pi}{2C_{D0}}},$$

which is 14.0 for a zero-lift drag of 0.008. No span carries a wing past what its section alone allows.

From aspect ratio 1 to 12, a wing that keeps its suction improves its best ratio from 9.9 to 33.4. One that loses it improves from 6.75 to 12.5, and by aspect ratio 4 it already has most of what span can give it. The surprising part is which wing benefits from holding its suction: not the slender one, where it is worth a factor of $\sqrt 2$ on the lift-to-drag ratio, but the long, efficient one, where it is worth more than two. The span is the whole story of the induced drag; it is only the whole story of the drag due to lift on a wing whose leading edge is doing its job. A high-aspect-ratio wing with a sharp edge, or one whose edge has stalled, has paid for a span it cannot use.

This is why a sailplane’s wing has a generous, round leading edge and why its designers worry about insects on it: the edge is where the long span is cashed. It is also why a leading-edge stall on a thin, high-aspect-ratio wing is so expensive before it is dangerous. As the suction peak collapses near the edge the drag due to lift moves from the lower polar towards the upper, and on a wing of aspect ratio ten that is a factor of six in the drag due to lift.

Sweep makes the suction worth more #

Sweeping a wing lowers its lift slope — the wind a swept wing feels is the component normal to its leading edge — and since $R$ has the slope in its denominator, sweep raises what losing the suction costs: at aspect ratio 6, from 4.40 unswept to 5.01 at 45° and 6.06 at 60°. That is the opposite of what a designer who sweeps a wing to put its leading edge inside the Mach cone, and sharpens it to keep the wave drag down, would want. A swept, sharp, moderate-aspect-ratio wing is the worst place to lose the suction, and it is exactly the shape a supersonic fighter has. Its long take-off run and its high-power approach are the polar’s upper curve being flown.

The convention, and what the lattice cannot show #

The wings are flat and thin, at small incidence, in incompressible inviscid flow; “full suction” means the whole of the leading-edge singularity’s force is collected and “zero suction” means none of it is. The aspect ratio is span squared over area. The induced drag is the Trefftz-plane drag of the lattice’s own loading, which assumes a flat wake. The zero-lift drag of 0.008 in the polars and the lift-to-drag figures is stated, not computed.

The lattice cannot say how much suction a real edge collects. A rounded edge collects all of it until its suction peak grows too strong for the boundary layer, and then a fraction; a sharp edge collects none, or recovers it in some other form through a captive vortex. The fraction actually achieved — the suction parameter flight-test reports quote — depends on the edge’s radius, the Reynolds number and the incidence, and it is a measurement, not something an inviscid model contains. The figures bracket it between its two ends. Nor does the lattice know about compressibility: a supersonic leading edge loses its singularity altogether, and with it the suction, which is a different argument.

Who found the two polars #

The leading-edge thrust of thin-aerofoil theory is in Munk’s and Glauert’s work of the 1920s; the drag polar with the suction lost, $C_L\tan\alpha$, is the one every textbook gives for a flat plate. Jones’s slender-wing theory of 1946, written to explain why the delta wings of the time lifted at all, gave $\pi A/2$ and an elliptic loading for any slender planform, which makes $R = 2$ immediate. Polhamus’s analogy of 1966 turned the lost suction into vortex lift, and Carlson and others at NASA Langley spent the 1970s and 1980s measuring and estimating the “attainable” suction of supersonic-cruise wings at low speed — which is the fraction this essay cannot compute.

How it was checked #

The lattice loads an elliptic planform of aspect ratio 6 elliptically, with a Trefftz-plane span efficiency of 0.998, and no planform it was given — rectangles, deltas or ellipses at aspect ratios from one to four — beats one. A slender delta of aspect ratio 0.25 has a slope 0.945 of Jones’s $\pi A/2$ and a ratio $R$ of 2.12, both approaching their slender limits as the aspect ratio falls; a rectangle of aspect ratio 40 reaches 0.92 of $2\pi$. Refining the lattice from 8 × 32 to 12 × 48 panels moves the slope of the aspect-ratio-6 rectangle by eight parts in a hundred thousand. The deltas of aspect ratio 1 to 2 agree with Polhamus’s formula within 4.5 per cent. And the lattice refuses a wing with no aspect ratio, a planform it does not know, a span cut into fewer than four strips and a sweep past seventy-five degrees.

Still open: how much of the suction a rounded edge keeps #

The two polars here are bounds, and a real wing lives between them at a fraction set by its leading edge. The next calculation gives the section a nose of stated radius, computes the suction peak there with the incidence and the radius as thin-aerofoil theory with a parabolic nose allows, and lets the edge collect its suction only while the peak’s adverse gradient behind it stays below a laminar separation criterion — the leading-edge stall — so that the suction parameter falls with lift coefficient in a way set by the radius and the Reynolds number. That would say how round an edge a wing of a given aspect ratio needs to keep the upper polar to a given lift, and why the answer is a larger radius on a sailplane than on a fighter.