One wing root feels the rolling gust its lift cannot

A wing averages a gust through its own loading weighted the turbulence across a span by the wing’s loading at uniform incidence, because the reverse-flow theorem says that is exactly how the total lift responds to an upwash that varies along the span. An elliptic wing then filters a gust as a round aperture diffracts light, passes more of the short scales than a uniform strip, and crosses its mean lift a few per cent more often. It ended on a load the structure cares about more than the lift: the bending moment at the root, which weights each station’s lift by its distance from the root, and which ought therefore to feel the outer span more and the middle less.

That expectation is half right, and the half it misses is the interesting one. A root’s moment is not the lift times a fixed arm, and its gust weight is not the lift’s weight times the arm either. It is a different loading altogether, and for a single root it is not even symmetric.

The weight is a loading, and the theorem says which #

The reverse-flow theorem is a statement about symmetry. In lifting-line theory the circulation is a linear response to the incidence along the span, $\Gamma = \mathcal{A}\alpha$, and the operator $\mathcal{A}$ is symmetric: for any two incidence distributions,

$$\int \alpha_1\,(\mathcal{A}\alpha_2)\,\mathrm{d}y = \int \alpha_2\,(\mathcal{A}\alpha_1)\,\mathrm{d}y.$$

The symmetry is the same one a force without the flow that makes it used for Stokes flow, and it comes from the induced-velocity kernel: the downwash a vortex at one station induces at another equals what the second would induce at the first. So any load that is a linear functional of the circulation, $\int a(y)\,\Gamma(y)\,\mathrm{d}y$, can be rewritten with the operator moved across: it is the incidence weighted by $\mathcal{A}a$. The gust weight of a load is the loading the wing would carry at an incidence distribution shaped like that load’s arm. For the lift the arm is one everywhere, and the weight is the loading at uniform incidence — the span-averaging essay’s result, now as one case of a rule. For the bending moment summed over both roots the arm is $|y|$, and the weight is the loading of a wing washed in linearly towards both tips. For the moment at the right root alone the arm is $y$ on the right half and nothing on the left.

The symmetry can be checked rather than assumed. On an elliptic wing of aspect ratio seven, two arbitrary incidences — one with no symmetry about the root at all — give $\int\alpha_1\mathcal{A}\alpha_2$ and $\int\alpha_2\mathcal{A}\alpha_1$ equal to two parts in ten thousand with eighty harmonics, the residual being the series’ convergence on the kinked arm $|y|$. The same eighty harmonics reproduce the lift weight’s filter, the Airy pattern, to three parts in $10^{14}$, because an elliptic wing’s uniform-incidence loading is a single harmonic at every aspect ratio.

The construction says something about the lifting line that its usual statement hides. The span is the whole story for the induced drag of a lifting line, because the drag depends only on the loading and the span; the loading depends on the planform, and the planform enters every load through one operator. A question about any load becomes a question about that operator applied to the load’s arm, and the planform answers it once, for every gust.

Three weights across one span #

The lift’s weight is the ellipse, peaked at the root. The two roots’ moment weight has a hole there instead: the arm vanishes at the root, so an upwash near the root moves the bending moment hardly at all. In the strip limit of infinite aspect ratio this weight is $|\eta|\sqrt{1-\eta^2}$, the arm times the elliptic chord, with a sharp V at the root; at aspect ratio nine the induced flow fills the V in, because a section washed in near the tip induces an upwash all along the span. The dial shows the hole sharpening as the wing grows more slender and filling as it shortens.

The right root’s weight is the one that matters to a spar designer, since each root is a separate joint, and it reaches across the span. Its arm is zero on the left wing, but its weight is not: a wing washed in on its right half only induces a downwash that changes the left half’s circulation too, and the theorem counts that cross-talk. The weight is lopsided, heavier on the right and smaller but nonzero on the left, and that lopsidedness is the whole of what follows.

What each weight lets through #

A weight’s filter is the square of its Fourier transform across the span, and a gust wave that fits into the span an odd number of half-wavelengths is passed or cancelled according to how the weight’s two halves add. The lift’s Airy pattern and the two roots’ moment filter are both transforms of symmetric weights, so both are real, and both fall with the spanwise wavenumber because positive and negative parts of the wave cancel across a weight that sees both halves of the span alike. At $\kappa = 2$ the lift’s passes 0.333 and the two roots’ 0.163: the hole widens the effective span, and the wider average cancels more. The strip limit of the same moment weight passes less still, 0.115, since its sharp hole spreads the weight further out.

One root’s weight is not symmetric, so its transform has an imaginary part — the transform of its antisymmetric half — and that part does not cancel a wave that is positive on one side of the root and negative on the other. It responds to exactly that wave. At $\kappa = 2$ the one-root filter passes 0.658, twice the lift’s.

The arithmetic of the cancellation is worth stating once. A spanwise wave $\cos(k_2 y)$ is symmetric about the root and adds across a symmetric weight; $\sin(k_2 y)$ is antisymmetric and integrates to zero against any symmetric weight whatever. A real gust contains both, in equal measure at each spanwise wavenumber, so the lift and the two roots’ moment discard half of the gust’s spanwise content before the filter’s shape has done anything. One root does not: its antisymmetric half picks up the sine waves, and its filter is the sum of what both halves pass.

The rate at which each load crosses its mean #

The number of times a load crosses its mean per distance flown is the fatigue analyst’s first count, and Rice’s formula gives it from the second moment of the load spectrum: the more of the spectrum at short scales, the more crossings. With the turbulence frozen, the chord’s response from Sears’ function — the gusts that cancel along the chord, the chord’s own version of the span’s averaging — and the span’s from each weight’s filter, the rate comes out in crossings per turbulence scale flown.

The two roots’ moment crosses its mean at the lift’s rate: 0.972 times it at a span of a hundredth of the turbulence scale, 1.000 at one scale. The outward weighting widens the effective span and the hole sharpens the weight’s shape, and the two very nearly cancel — the answer the span-averaging essay suspected. And the uniform strip, at 0.960 and 0.974, is below both: the strip that bounds the lift’s rate from below bounds the symmetric moment’s from below as well.

Why the two effects cancel so closely is visible in the filters. The hole moves the weight outwards, which makes the average wider and cuts the filter at moderate wavenumbers — the two roots’ filter is half the lift’s at $\kappa = 2$. But a weight with a hole has more of its structure at short spanwise scales, its second moment concentrated at two humps rather than one, and at high wavenumbers it passes more: at $\kappa = 5$ the two roots’ filter is 0.138 against the lift’s 0.017. The crossing rate is a ratio of the spectrum’s second moment to its zeroth, and the moment weight loses at the long scales what it gains at the short ones in almost exactly the proportion that leaves the ratio unchanged. It is a property of the elliptic planform rather than a theorem, and whether it survives on a rectangular wing is a calculation this essay has not made.

One root’s moment crosses its mean 1.19 times as often as the lift at a span of a hundredth of the scale, and 1.15 times at one scale. The strip bounds it from neither side; the lift is not even a good estimate of it. Above a few scales the excess goes, because a span many scales wide averages every component of the gust, rolling and lifting alike, down to its long waves.

A sliver of the variance, a fifth of the rate #

The split is exact, because the real and imaginary parts of one transform add in its square: the variance of one root’s moment is the variance of the lifting gust through the symmetric half of its weight plus the variance of the rolling gust through the antisymmetric half, with nothing left over. At a span of a hundredth of the turbulence scale the rolling part is 0.55 per cent of the variance. It crosses its mean 9.6 times as fast as the lifting part, because a gust that differs from one tip to the other is a spanwise gradient, and a gradient has its spectrum weighted towards the short scales. That small, fast component is what raises the total rate by nineteen per cent. At one scale the rolling part carries 16.6 per cent of the variance at 1.7 times the lifting rate; at ten scales, nearly half.

This is the surprising connection these essays on gusts have been circling. The span’s averaging took the infinity out of the crossing rate by cancelling short spanwise waves; the rolling half of a root’s weight is the one part of the wing that does not cancel them, because it measures the difference between the two halves rather than their sum. A structure joined at its root reads the lift and a rolling moment together, and the rolling moment’s rate is set by spanwise wavenumbers the lift never sees.

Slender wings feel it more #

At one turbulence scale of span, the excess rises from 1.12 at aspect ratio four to 1.19 at thirty. A stubby wing’s induced flow is strong: a gust on one wing changes the circulation on the other, and that coupling spreads the one-root weight across the span and makes it more nearly symmetric, which cancels more of the rolling gust. A slender wing’s sections are nearly independent, its one-root weight is nearly the arm times the chord on one side and nothing on the other, and it passes the rolling gust almost undiminished. Sailplanes and long-range airliners, the wings with the most aspect ratio, are the ones whose root-joint fatigue the lift’s rate under-counts most.

An airliner’s root #

On the airliner the crossing-rate essay used — sixty metres of span at 230 metres a second in turbulence of integral scale 760 metres, a span of eight hundredths of the scale — one root’s moment crosses its mean 1.19 times as often as the lift, and the rolling gust carries 2.4 per cent of the root’s variance. For every hundred lift cycles a fatigue count would take from a lift-based gust model, the wing’s root joint sees about 119 moment cycles of the same relative size. The number of excursions past a fixed multiple of each load’s own standard deviation scales with the crossing rate, so the same factor applies to the large cycles that consume most of a fatigue life: a root designed from the lift’s spectrum is counted short by a fifth.

The correction does not apply to every load path. The bending moment that decides how far a wing’s span can grow is a steady, symmetric one, and a wing-box carry-through that adds both roots sees the symmetric moment, whose rate is the lift’s. The single root is the joint that cannot average its own two halves.

The convention: per scale flown, both roots and one #

The crossing rates are per integral scale of frozen von Kármán turbulence flown, from Rice’s formula on the load spectrum; ratios to the lift’s rate are taken at the same span and chord, the chord a ninth of the span throughout, as in the crossing-rate essay. Each weight is normalised to unit integral so its filter is one at zero wavenumber; the one-root filter keeps its antisymmetric part’s square in the same normalisation, so the two parts’ variances add to the whole. “Both roots” is the sum of the two root moments, the symmetric moment; “one root” is the right root’s alone. The wings are elliptic in planform, so the lift’s weight is the same ellipse at every aspect ratio and only the moment weights change with it.

What the lifting line leaves out #

Unsteady aerodynamics across the span. Each section’s response is Sears’ function of its own chord, and the induced flow is quasi-steady; at the high spanwise wavenumbers the rolling part depends on, the wake’s own unsteadiness changes the coupling between the two halves. The aircraft’s response. A real aeroplane rolls in response to a rolling gust, and the roll rate’s own aerodynamic damping cuts the rolling load — a correction that matters most for the long rolling waves, not the short ones that raise the rate; the aeroplane that rises out of its own gust is the same relief for the lifting half. Short wings. The lifting line is a slender-wing theory, and where the line stops being a line its error grows as the aspect ratio falls, to forty per cent at an aspect ratio of one; the stubbiest wings in the figure above are already leaning on it. A lifting surface’s operator is symmetric too, so the rule — the weight is the loading at an incidence shaped like the arm — survives on any wing, while the numbers would have to be computed again. Wing flexibility, which changes the loading as the wing bends. Non-Gaussian turbulence, under which Rice’s formula is an approximation for the count of large excursions. And sweep, which makes a root’s arm depend on the chordwise station as well as the spanwise one.

Fatigue counting itself. Rice’s formula counts crossings of the mean, and fatigue damage depends on the ranges between successive peaks and troughs, which a rainflow count gathers from a load history. A load with more high-wavenumber content has more small cycles for every large one, so the root’s nineteen per cent more crossings would not all be damaging cycles; turning the crossing rate into a fatigue life needs the load’s whole spectrum, which the weights here supply, and a counting rule, which they do not.

Where the weights come from #

Munk stated the reverse-flow theorem for lifting surfaces in the 1920s, and von Kármán and Burgers used it for the same purpose here — to turn an influence into a loading. The gust spectra are von Kármán’s of 1948; Rice’s crossing formula is from his 1944 analysis of random noise; Sears’ function for the chord is from 1941. The lateral, rolling components of gust loads have been part of certification analyses since continuous-turbulence methods were adopted in the 1960s, where they are usually computed for the whole aircraft’s response. The point here is the smaller one the theorem makes exact: the weight a single root’s moment puts on the gust is a loading of its own, and its asymmetry is enough to let the rolling gust in.

Still open: the root of a swept wing #

On a swept wing the root’s moment arm runs diagonally: a station’s distance from the root is measured along the swept spar, and a gust reaches the outer wing later than the inner, because the outer wing is further aft. The next calculation gives the weight a phase as well as a magnitude — each station’s lift arriving with the time delay its sweep imposes — and asks whether sweep, by smearing the gust’s arrival along the span, cancels more of the rolling gust or less, and how much of the one-root excess survives on a wing swept thirty degrees.