Where the line stops being a line

Prandtl’s lifting line is the result this collection’s circulation field is built on. Replace a wing with a single bound vortex whose strength varies along the span, let the variation shed a trailing sheet, and the price of having ends falls out: induced drag, the elliptic optimum, and

$$a = \frac{a_0}{1 + a_0/\pi\!\;\mathrm{AR}} \;\longrightarrow\; \frac{2\pi}{1 + 2/\mathrm{AR}}$$

for an elliptic wing. One number, the aspect ratio, decides how much of the section’s slope survives — and it does so through a formula whose derivation and whose small parameter are the same quantity.

The formula is exact as the aspect ratio goes to infinity and is used at every aspect ratio there is. This essay is about where that stops being reasonable, and about the fact that nothing in the calculation says so.

The assumption that is a word #

The derivation assumes the wing is a line: that its chordwise extent is negligible beside its span, and that the induced angle is uniform along the chord so a section can be given one effective incidence.

At aspect ratio 20 that is excellent. At aspect ratio 1 the chord is as long as the span, “the trailing vortices leave from the tips” describes nothing in particular, and the induced angle at the leading edge and at the trailing edge are different numbers.

The correct limit at the other end is a different theory altogether. R. T. Jones’s slender-wing theory gives

$$a = \frac{\pi\,\mathrm{AR}}{2},$$

with no section slope in it at all — a slender wing’s lift does not care what its aerofoil is, because the flow is essentially two-dimensional in the cross-flow plane rather than in the streamwise one.

Two theories, each exact in its own limit and each generous outside it.

They cross at exactly two #

$2\pi/(1+2/\mathrm{AR}) = \pi\mathrm{AR}/2$ has one root, and bisection puts it at $\mathrm{AR} = 2.0000000$ with both theories returning $a = \pi$ exactly.

Two approximations agreeing is not evidence about the answer, and here it is demonstrably not. Helmbold’s formula,

$$a = \frac{2\pi\,\mathrm{AR}}{2 + \sqrt{\mathrm{AR}^2 + 4}},$$

is the composite of the two in exactly the sense matched asymptotics means: it reduces to the lifting line as AR → ∞ and to Jones as AR → 0, with no free constant. At AR = 2 it gives $4\pi/(2 + \sqrt8) = 2\pi(\sqrt2 - 1) = 2.6026$, and the two limiting theories are above it by

$$\frac{\pi}{2\pi(\sqrt2 - 1)} - 1 = \frac{\sqrt2 - 1}{2} = 0.2071,$$

exactly. Twenty-one per cent, in closed form, at the one aspect ratio where the two theories agree with each other.

Where the line stops #

The lifting line is within one per cent above aspect ratio 20 and within ten per cent down to aspect ratio 3.42. At aspect ratio 1 it is 41 per cent high.

Aspect ratio 3.4 is not an obscure corner of the design space. A missile fin, a delta wing, a control surface treated as a wing in its own right, a fighter’s wing — most of those are between 1 and 4, and that is precisely the range in which the formula everybody quotes is wrong by tens of per cent.

The theory does not warn anybody #

Here is the part worth sitting with, and it is about numerical work rather than about wings.

Solving Prandtl’s integral equation numerically — twelve odd harmonics, collocated on the half-span, the standard method — returns the closed form to 2.4 × 10⁻¹⁶ at every aspect ratio from 0.5 to 20. At AR = 0.5 the solve gives 1.2566370614359175 and the formula gives 1.2566370614359172, and the true answer is nearer 1.14.

Nothing in the solve gets larger. No residual grows. No matrix becomes ill-conditioned. Nothing fails to converge. The Fourier coefficients are well behaved and the collocation is stable. A person who had only the numerical solution and no external reference would have no indication whatever that they were computing an answer that is forty per cent wrong.

That is the general point. A model outside its range does not report that it is outside its range, and a numerical solution of a wrong equation is a wrong answer computed carefully. The refinement studies, the residual checks and the convergence tests all pass, because every one of them is a question about the numerics rather than about the equations.

This collection has met the same shape before, in a lift coefficient computed from an unconverged Poisson solve and in a spurious root of a shooting problem — but those were numerical failures with numerical symptoms. This one has none.

What actually goes wrong at low aspect ratio #

Three things, and they are worth separating because they fail in different directions.

The chordwise variation of downwash. With a chord comparable with the span, the induced angle is not uniform along the chord; the wing is effectively cambered by its own wake, and the effect is a reduction in lift the lifting line does not have.

The tips are not points. A lifting line sheds its wake from a line of zero chordwise extent. A low-aspect-ratio wing sheds it from an edge with real length, and the wake leaves at an angle the line model cannot represent.

And the leading edge stops being a leading edge. On a slender wing the flow round the side edges is what matters, and at any appreciable incidence it separates into a pair of leading-edge vortices that add a nonlinear lift the linear theory of either kind has no term for. That is a delta wing’s whole aerodynamic proposition, and it means that below about aspect ratio 2 the linear answer is not merely inaccurate — it is describing a different flow.

Why the slender answer has no aerofoil in it #

The formula $\pi\,\mathrm{AR}/2$ has been quoted twice above and its most striking property — that no property of the section appears in it — has been asserted rather than explained. The derivation is three lines and it makes the whole low-aspect-ratio end of the axis intelligible.

Take a slender wing at incidence and watch one cross-flow plane fixed in space as the wing passes through it. What that plane sees is a flat plate, oriented across the stream, whose span grows as the wing sweeps by, sitting in a fluid moving across it at $U\alpha$. That is a two-dimensional unsteady problem, and the two-dimensional problem of a plate accelerating a fluid sideways is an added-mass problem: a plate of semispan $s$ drags along a mass of fluid $\rho\pi s^2$ per unit length, whatever it is made of and whatever its thickness.

The momentum given to that plane per unit length is therefore $\rho\pi s^2 U\alpha$, and the lift is the rate at which the wing hands it over as it flies through:

$$L = \rho U^2 \alpha\, \pi s_{\max}^2 \quad\Longrightarrow\quad \frac{\mathrm{d}C_L}{\mathrm{d}\alpha} = \frac{\pi\,\mathrm{AR}}{2}$$

Two of the theory’s oddities follow immediately. The section cannot appear, because added mass depends on the plate’s width and on nothing else — camber, thickness and nose radius are invisible to a cross-flow plane. And only the largest span enters, because the result is a total momentum handed over rather than an integral along the body: everything downstream of the widest station has nothing left to add, which is why a slender wing’s lift is set at its trailing edge and why cropping or notching a delta behind that station costs almost nothing.

What the induced drag does at the same time #

The lift slope is the quantity this essay has been tracking and it is not the only thing the collapse gets wrong, so it is worth saying what happens to the drag.

Lifting-line theory gives $C_{D_i} = C_L^2/\pi\,\mathrm{AR}\,e$ with the span efficiency $e = 1$ for elliptic loading, and that result survives to much lower aspect ratio than the lift slope does — because it is a statement about the wake rather than about the wing. The induced drag can be computed in the Trefftz plane from the trailing sheet alone, and the trailing sheet of a low-aspect-ratio wing is still a sheet of the right total strength and the right lateral extent.

What changes is the loading that produces it. A slender wing’s optimum span loading is not elliptic in the lifting-line sense, and the roll-up is far more advanced by the time the wake reaches any measurement plane. So the formula is nearly right and the theory that produced it is not, which is a combination worth recognising: a result can survive the failure of its own derivation when it depends on less of it than the derivation used.

That is why the span is the whole story generalises further down the aspect-ratio range than the lift slope does, and it is why a low-aspect-ratio wing’s problem is described in terms of lift rather than drag.

What the composite is worth #

Helmbold’s formula is used constantly and it is worth being clear about its status.

It is not a theory. It is a construction with the two correct limits, and it has no derivation beyond that: the square root under it is chosen because it produces the right behaviour at both ends and nothing else. Its accuracy in the middle is an empirical matter, and against wind-tunnel data it is good to a few per cent over aspect ratios from about 1 to 8 — which is the range in which neither limiting theory is any use.

And it is a lower bound on both. At every aspect ratio where the two limits disagree, the composite runs under the pair, because each is exact in its own limit and too generous outside it. That is the usual behaviour of a composite of two over-predictions and it is the reason it is worth having: it says which direction the two theories are wrong in, which neither of them can say about itself.

The bracket it puts on the answer at AR = 2 — 21 per cent below where both theories agree — is the most useful single number in this essay, because it is exactly the case a designer would be most confident about and most wrong.

What the collapse discarded #

Prandtl’s collapse replaces a wing with a line, and the residual is the chord.

That is a different kind of residual from the ones the regimes essays found. There the discarded quantity was a second dimensionless group — a viscosity ratio, a frequency parameter, a flow type. Here it is a geometric assumption buried in the derivation, and the aspect ratio is exactly the parameter that says whether it holds.

So the aspect ratio is doing two jobs at once, which is where the confusion comes from. It is the physical variable in the answer — induced drag really does go as $1/\mathrm{AR}$ — and it is the small parameter of the expansion, and a formula that is a function of it does not distinguish the two.

How anybody would have known #

If the calculation gives no warning, the question is what does — and the answers are worth listing, because they are the standing methods for finding this class of error.

A second theory with a different assumption. That is the whole content of this essay: the slender-wing answer is not a refinement of the lifting line, it is an independent statement, and the disagreement between them is the diagnostic. Neither could have found the problem alone.

A refinement in the direction the assumption points. The lifting line assumes the chord is negligible; solving with a lattice of horseshoes distributed along the chord as well as the span relaxes exactly that, and it returns a slope four per cent below the lifting line at aspect ratio 8 and far below it at aspect ratio 1. A convergence study in the number of spanwise harmonics finds nothing, because the harmonics are not the approximation.

A measurement. Wind-tunnel data for low-aspect-ratio wings has been available since the 1930s and disagrees with the formula by exactly the amounts above. That is the reason the problem was known long before Helmbold wrote his formula down.

And an argument about limits. As AR → 0 the lifting line gives a slope going as $\pi\mathrm{AR}$ and Jones gives $\pi\mathrm{AR}/2$ — a factor of two, in a limit where the physics is simple enough to settle by hand. A theory that gets a clean limit wrong by a factor of two is wrong somewhere.

Each of those is a check against something outside the calculation, and none of them is a check the calculation could have performed on itself. That is the whole lesson, and it is why this collection’s figures carry a stated model and a stated regime: the hypothesis is the thing a numerical result cannot report.

The other end, and what it is good for #

It would be wrong to leave the impression that low aspect ratio is a problem to be endured.

A slender wing’s lift slope is small, so it needs a large incidence for a given lift coefficient, and at large incidence the leading-edge vortices form and add lift nonlinearly. The result is a wing that does not stall in the ordinary sense: its lift keeps rising to incidences of thirty degrees or more, which is exactly what a delta wing is for.

It is also the configuration for which the Prandtl–Glauert transformation is kindest, because compressibility acts on the aspect ratio and a small one is barely changed. A slender wing’s lift slope is nearly independent of Mach number where a high-aspect-ratio wing’s is not, which is a large part of why supersonic aircraft have the planforms they do.

So the two ends of this axis are two different aerodynamic worlds, and the interesting thing about the middle is that the theories for both ends are simultaneously available and simultaneously wrong there.

Where the theories came from #

Prandtl’s lifting line is 1918 to 1921 and is the foundation of the subject. Jones’s slender-wing theory is 1946, written for the supersonic configurations then being designed, and its result — that the lift depends only on the span at the trailing edge — is a genuinely different physical statement rather than a limit of Prandtl’s.

Helmbold’s formula is 1942, from a paper on the effect of aspect ratio at high speed, and it has been quoted ever since without anybody calling it a matched asymptotic expansion. It is one, in the strict sense that it is a uniformly valid composite of two limits with no adjustable constant, and noticing that is what puts this essay next to the composite essay rather than beside the other finite-wing rungs.

What to do with a slope that cannot be trusted #

The practical residue of all this is short, and it is worth stating because the temptation is to treat the disagreement as a curiosity rather than as an instruction.

Quote the composite, not either limit. Above aspect ratio 20 it makes no difference; below aspect ratio 8 it is the only one of the three that is anywhere near the measurements, and it costs a square root.

Carry the bracket. The gap between the two limiting theories is a computable number at every aspect ratio, and it is the honest error bar on a lift slope obtained from any of them. At aspect ratio 2 it is 21 per cent; at aspect ratio 8 it is under 4. A stability derivative quoted without it is quoted to a precision the theory does not have.

And check what the number is being used for. A lift slope that is 40 per cent high produces a neutral point in the wrong place, because the tail’s contribution scales with its own slope and a tail is usually the low-aspect-ratio surface on the aircraft. The error does not stay in the aerodynamics; it arrives in the handling qualities, one derivative later, with nothing on it to say where it came from.

What this leaves #

A theory whose small parameter is also its answer’s variable, a numerical solve that reproduces its own formula everywhere including where the formula is wrong, and a composite that says by how much.

The next essay keeps the wing and adds a sweep, and finds a residual of a different kind: the sweep the root does not have, where the isobars unsweep.