A wing averages a gust through its own loading

The span takes away the infinity, not the gust settled a question that had been left with an embarrassing answer. Filtered through the chord alone, the gust load’s rate of zero crossings grew without limit as more of the turbulence spectrum was counted, because the spectrum’s tail was too shallow. Averaging the gust across a finite span added one more power of wavenumber to the fall, and the rate converged: 2.29 crossings per turbulence scale flown for the case it posed. The average it used was uniform. Every station across the span counted equally, which is strip theory.

It closed by naming the obvious objection. Real wings do not load their span uniformly; their lift is concentrated towards the root and falls to nothing at the tips. A wing whose lift is mostly in its middle ought to feel the gust mostly in its middle, so it averages over an effectively narrower span and should let more of the short gusts through. This essay computes how much. The weighting turns out to be exact rather than plausible, the filter it makes has a name from a different subject, and the uniform strip turns out to be not an approximation to every wing but a bound on them.

The weight is the loading #

A gust that varies across the span changes the incidence differently at every station. The total lift that results is not simply the average of the local incidences, because each section’s lift changes the downwash on all the others. Lifting-line theory computes it, and it has a theorem that makes the computation unnecessary: the reverse-flow theorem. The total lift produced by an upwash $w(y)$ is

$$L = \frac{1}{U}\int w(y)\,g(y)\,dy,$$

where $g(y)$ is the span loading that the same wing produces at a uniform unit incidence. The influence of an upwash at a station on the total lift equals the lift a uniform incidence puts at that station. The result follows from the symmetry of the induced-velocity operator — the same symmetry a force without the flow that makes it used for Stokes flow — and it can be checked: for a tapered wing of aspect ratio eight and an upwash with no symmetry at all, the lift solved directly and the lift computed as the upwash weighted by the uniform-incidence loading agree to two parts in $10^{15}$.

So the weight is a property of the planform, and it is the most familiar one. An elliptic wing’s weight is elliptic. A rectangular wing’s weight is its own loading, flat across the middle and falling to zero at the tips. Strip theory’s uniform weight is the limit of infinite aspect ratio, where induced effects vanish and every station is independent.

Why the tips count for less #

The reason a tip station weighs less is worth stating physically, because it is not that the tip has less chord. A rectangular wing has as much chord at its tips as at its root and still weights them at nearly zero. An upwash at the tip raises the tip section’s incidence, and the tip section makes more circulation — but circulation that ends at the tip is shed straight into the trailing vortex, which induces a downwash back along the span that cancels much of what the upwash added. A section near the root is far from any end, and what it adds stays bound. The price of having ends is the induced drag that comes from the same trailing vortices; here it is a reduced sensitivity to whatever pushes on the ends.

That is also why the weight depends on the aspect ratio and not on the chord distribution alone. At an aspect ratio of thirty the induced effect is confined to the last few per cent of the span and the weight is nearly the uniform strip. At four it reaches the root, and the weight is nearly elliptic whatever the planform — which is the gust-side version of the span is the whole story: at small aspect ratio every planform loads like an ellipse, and so every planform averages like one. Where the lifting line itself stops being right, where the line stops being a line sets the limit, and below an aspect ratio of about four the weights drawn here are an extrapolation.

Two weights, two apertures #

The weights look as the paragraph above says. Written in $\eta = 2y/b$ and normalised so each encloses unit area, the uniform strip stands at a half across the span. The elliptic weight is $(2/\pi)\sqrt{1-\eta^2}$, 0.637 at the root and zero at the tips. Rectangular wings sit between: at aspect ratio thirty nearly the strip, at four nearly the ellipse, and all of them reaching zero at the tips as a square root, because a lifting line’s loading always does.

What the span average does to a gust component depends on the Fourier transform of the weight. A component varying across the span as $e^{ik_2 y}$ survives in proportion to $|\hat W(\kappa)|^2$, with $\kappa = k_2 b/2$, and for these two weights the transforms are classical. The uniform strip’s is $\sin\kappa/\kappa$. The ellipse’s is

$$\hat W(\kappa) = \frac{2J_1(\kappa)}{\kappa}.$$

Both are formulas from optics. The first is the diffraction pattern of a slit; the second is the Airy pattern, the diffraction of a circular aperture, whose first dark ring sits at $\kappa = 3.83$. A wing averaging a turbulent field across its span is doing the arithmetic of a lens collecting light across its aperture, and an elliptic loading is a round lens.

The Airy filter #

The three filters are drawn on logarithmic axes. The slit’s falls as $\kappa^{-2}$ between its zeros, the Airy pattern’s as $\kappa^{-3}$, and the rectangular wing’s follows the Airy pattern at large $\kappa$ because its loading, too, has square-root tips; its filter is computed by quadrature of its own lifting-line loading. The faster fall at large $\kappa$ invites a wrong conclusion — that the ellipse removes more of the short spanwise scales and so lets less of the gust through. It is the other way, and the reason is at small $\kappa$. The ellipse’s weight is narrower than the strip’s, with a variance of $b^2/16$ against $b^2/12$, so its filter is wider: it stays near one out to larger $\kappa$ before it falls. What the filter passes in total is set by its integral, and by Parseval’s theorem that is $2\pi\int W^2\,d\eta$ — π for the strip, $32/(3\pi)$ for the ellipse. A peaked weight has a larger mean square, and passes more.

More load at every short scale #

Carried through to the load, the difference is a ratio of spectra, plotted against the streamwise wavenumber for a span of a tenth of the turbulence’s integral scale and a chord a ninth of the span. Well below the span’s own wavenumber the two are equal: a gust much wider than the wing loads every station alike and the weighting cannot matter. Around $k_1b = 10$ the ellipse passes twelve per cent more. Far above it the ratio settles at 1.081, and that number is exact: when the streamwise wavenumber is large the gust spectrum is nearly constant across the filter’s width in $k_2$, so only the filter’s integral matters, and the ratio of integrals is $32/(3\pi^2) = 1.0808$.

So the narrower effective span is not a feature of the long gusts, where it might have been expected to show, but of the short ones, which are exactly the ones that set a crossing rate.

The crossing rate #

Rice’s rate of zero crossings of the load is $\sqrt{M_2/M_0}/2\pi$, the square root of the ratio of the load spectrum’s second moment to its zeroth, and the second moment is dominated by the short scales. For an elliptic wing the rate is 4.1 per cent above the uniform strip’s at a span of a hundredth of the turbulence scale, 3.8 at a tenth, 3.0 at a half and 2.2 at one scale. A rectangular wing of aspect ratio nine — an airliner’s planform — sits at about half the ellipse’s excess, 2.0 to 1.0 per cent.

The small-span limit is again exact. When the span is small the crossing rate is set by wavenumbers far above the span’s own, where the load spectra differ by the factor 1.0808, so the second moments do too while the zeroth moments barely differ, and the rate ratio tends to $\sqrt{32/(3\pi^2)} = 1.0396$. At larger spans more of the variance lies near the span’s wavenumber, the zeroth moment grows with the second, and the excess in the rate falls.

Every real wing lies above the uniform strip. Strip theory was a lower bound on the crossing rate, not a central estimate. For the airliner the earlier essay worked through, 0.54 crossings a second at cruise, the correction for an elliptic-like loading is a few hundredths: 0.56.

A narrower strip #

There is a simple way to carry the result into any calculation that uses a uniform average: shrink the span. The uniform strip that passes exactly the gust variance an elliptic wing passes is 0.866 of the ellipse’s span at a span of a fiftieth of the scale, a tenth and a half — the same to three figures. That number is $\sqrt{12/16}$, the ratio of the two weights’ standard deviations, and the reason is that the gust variance is carried by the long scales, where a filter is described by its curvature at the origin and so by the weight’s variance alone. The strip of equal peak, $\pi/4$ of the span, is the wrong equivalent.

The two equivalences do different jobs. For the variance of the load, a uniform strip seven-eighths as wide is exact enough for any purpose. For the crossing rate it is not, because the rate is set by the short scales, where what matters is the weight’s mean square rather than its variance; a strip matched for one is mismatched for the other.

Large loads #

A few per cent in a crossing rate would matter little if it stayed a few per cent, and on a rigid aeroplane that rises out of its own gust the plunging response trims the long scales but leaves the short ones that carry the excess. It does not, because the quantity design uses is the rate of exceeding a large load, $N(y) = N_0\exp(-y^2/2\sigma^2)$, and both factors move. At a span of a tenth of the scale the crossing rate $N_0$ rises 3.8 per cent and the load’s standard deviation $\sigma$ rises 0.39 per cent. The second enters squared in an exponent, so its effect grows with the level: the elliptic wing exceeds a three-sigma load seven per cent more often, a five-sigma load fourteen per cent more often and a six-sigma load nineteen per cent more often. Fatigue counts are driven by the large cycles, and a seventh more of the cycles that matter is not a rounding error.

That is the same arithmetic that made the lift at the mean angle differ from the mean lift: a small change to a distribution’s width matters most far out in its tail.

An airliner’s hour in turbulence #

Put the numbers on one flight. An airliner with a span of sixty metres cruising at 230 metres a second through turbulence of integral scale 760 metres has a span of eight hundredths of the scale. At that span the elliptic weight raises the crossing rate by about 3.8 per cent: the 0.54 crossings a second of the strip estimate become 0.56, or about 2,020 an hour instead of 1,950. That alone changes little. What changes more is the count of large cycles. If the gust load’s standard deviation in a patch of moderate turbulence is a tenth of the wing’s limit-load increment, a five-sigma excursion is half the limit increment, and the strip estimate would count a few such cycles in a long exposure; the loaded wing has fourteen per cent more of them. A fatigue spectrum built from a strip-theory gust model is therefore unconservative in exactly the cycles that consume most of the life.

The same correction has a familiar twin in measurement. A sonic anemometer or a hot wire averages the velocity along a finite path, and its response to the turbulence is the same transform of the same kind of weight. A probe whose sensitivity peaks at its middle — a hot wire is hotter at its centre than at its prongs — filters like the ellipse rather than like the slit, and it too passes more of the short scales than the uniform-path correction assumes.

Checks on the weighting #

The reverse-flow theorem is checked directly, as described above, on a tapered wing with an upwash containing both symmetric and antisymmetric parts, solved with forty harmonics. The elliptic weight’s filter computed by quadrature agrees with $(2J_1(\kappa)/\kappa)^2$ to $10^{-14}$ out to $\kappa = 150$, and $J_1$’s own two branches — its power series below eight and Hankel’s asymptotic expansion above — meet to three parts in ten million, with the first zero found at 3.8317060. A rectangular wing of aspect ratio 400 has a centre weight of 0.504, the strip’s half. The ratio of the elliptic and uniform gust spectra at a streamwise wavenumber of ten thousand is 1.0808, the Parseval value. And the uniform weight with the earlier essay’s chord, a five-hundredth of the scale, gives 2.28 crossings per scale against the 2.29 that essay reported on its own grid. The tests also refuse a span of zero, a negative span and an aspect ratio of zero.

What the lifting line assumes #

Quasi-steady loading across the span. The reverse-flow theorem is applied to the steady lifting line, so each spanwise component of the gust is assumed to load the wing as a steady incidence would. Along the chord the unsteadiness is kept, through Sears’ function — the gust-entry counterpart of the lift that arrives late — as in the earlier essays. Across the span it is not: a short spanwise gust component also changes in time as the wing flies through it, and the wake’s response to it is not the steady wake’s. At the wavenumbers that set the crossing rate this is a real approximation, and the unsteady lifting-surface calculation would replace the steady loading with a frequency-dependent one.

Frozen turbulence. The pattern is still assumed to be flown through unchanged, the assumption the earlier essay named as its own next question.

Rigid wing. The weight is the aerodynamic loading of a wing that does not bend. A flexible wing’s root bending moment, which is often what is being counted, weights the lift by distance from the root and so weights the gust towards the tips — the opposite way.

Gaussian loads. Rice’s formulas assume a Gaussian load. Turbulence is not Gaussian at small scales, and the exceedance ratios far out in the tail are a statement about the Gaussian model rather than about measured gust loads.

The convention: per scale flown, and a chord a ninth of the span #

Rates are in crossings per integral scale of the turbulence flown, as in the earlier essay, and a span is given in integral scales. The chord is a ninth of the span in every figure here, which makes the comparison between weights one at a fixed planform; the earlier essay’s single case used a chord of a five-hundredth of the scale. The gust spectrum is von Kármán’s, frozen and isotropic, with the vertical component’s plane spectrum the earlier calculation derived and checked. The weight is normalised to unit area, so a uniform incidence gives the same lift with every weight and only the distribution differs.

Lens and wing #

The reverse-flow theorem is due to Munk and was generalised by Flax and by Heaslet and Spreiter in the 1950s for wings in steady and unsteady flow; its use for spanwise gust loading appears in the gust-response literature of that period, where the span-averaged gust is weighted by the wing’s loading as a matter of course. The Airy pattern is Airy’s, from 1835. The equivalence of a span average to an aperture is an old observation in the statistics of turbulence — a hot-wire’s finite length filters a velocity field in the same way — and the gusts that cancel along the chord used its chordwise counterpart. That the uniform strip is a lower bound on the crossing rate of any real wing, with an exact small-span limit, is what the calculation adds.

Still open: the bending moment’s own weight #

The load counted here is the total lift. What a wing’s structure feels at its root is the bending moment, which weights each station’s lift by its distance from the root and so turns the elliptic weight into one that rises from zero at the root to a maximum seven-tenths of the way out. By the same reverse-flow argument the gust weight for the root bending moment is the bending-moment influence of a uniform incidence, and its filter is the transform of $|\eta|\sqrt{1-\eta^2}$, a weight with a hole in the middle. The next calculation asks how much the root bending moment’s crossing rate and exceedances differ from the lift’s — whether the outward weighting, which widens the effective span, cancels the loading’s inward one, and so whether the uniform strip that bounds the lift’s rate from below bounds the bending moment’s from above.