Sweep slows a wing root's gust cycles and leaves their excess over the lift

One wing root feels the rolling gust its lift cannot weighted a field of frozen turbulence by the loading the reverse-flow theorem assigns each load. The lift’s weight is the wing’s own loading at uniform incidence; one root’s bending moment weights the gust by the loading of a wing washed in linearly towards that root’s tip, and nothing on the other side. That weight is lopsided, its antisymmetric half lets through the rolling part of the gust the lift never feels, and on an elliptic wing of aspect ratio nine the root’s moment crosses its mean 1.19 times as often as the lift. For a fatigue count based on the lift, a fifth of the root joint’s cycles go uncounted.

That calculation put every station of the wing at the same streamwise position. A real transport wing is swept, and the essay ended on what that changes: a station’s distance from the root is measured along a spar that runs diagonally, and a gust reaches the outer wing later than the inner, because the outer wing is further aft. The weight acquires a phase as well as a magnitude, and the question was what the phase does to the root’s cycles.

It slows them, and the lift’s with them, by nearly the same amount. At 35° of sweep every load crosses its mean about a tenth less often, the root a little more than the lift, and the variance hardly changes. The rolling excess falls from 1.19 to 1.17 and is still there at 45°.

A delay is a phase #

Turbulence moving past a wing at its flight speed is, for loads like these, frozen — the assumption the aeroplane rises out of its own gust also made, with the aeroplane’s own motion added — a pattern of upwash carried bodily past, so that what a station feels at a moment is what a station a little further forward felt a moment before. Sweep the wing back by an angle $\Lambda$ and a station at a fraction $\eta$ of the semi-span sits $\tfrac12 b|\eta|\tan\Lambda$ behind the root. It meets each part of the gust later by that distance over the flight speed.

A delay in time is a phase in wavenumber. A gust component with streamwise wavenumber $k_1$ and spanwise $k_2$ reaches the station at $\eta$ with the phase it would have had at the root plus $k_1$ times the distance aft. The reverse-flow theorem — the reciprocity a force without the flow that makes it uses for a slow flow, here in its lifting-line form — still says each load is the upwash weighted by the right loading, so the load’s response to that component is the weight’s transform with the extra phase in it,

$$W(k_1, k_2) = \int g(\eta)\, e^{\,i\left(\kappa_2 \eta + \kappa_1 \tan\Lambda\,|\eta|\right)}\, d\eta, \qquad \kappa = \tfrac{1}{2}kb .$$

On the straight wing the second term is absent and the weight filters only spanwise wavenumbers; sweep makes it filter streamwise ones as well. And because the phase grows with $|\eta|$ on both sides, the weight can be split at the root into its two halves, on each of which the sweep simply shifts the spanwise wavenumber — by $+\kappa_1\tan\Lambda$ on one half and the opposite on the other. Each half’s transform is then a function of one variable, computed once and read off for every pair of wavenumbers. On the straight wing the split filter equals the earlier essay’s to rounding error, and the crossing rates it gives agree with that essay’s to three parts in ten thousand.

What the sweep does to the aerodynamics is not included. A swept wing’s loading is not a straight wing’s: its root and tips behave differently, as the sweep a root does not have explains, and its sections respond to the component of the flow normal to their own span, as the wind a swept wing feels puts it. Here the weight is the straight elliptic wing’s and only the arrival times change. That is the kinematics of sweep isolated, and it is the part the earlier essay’s question was about.

The rates fall together #

The first figure is the answer. The wing is the earlier essay’s, elliptic with aspect ratio nine and a chord a ninth of the span, at a span of eight hundredths of the turbulence’s integral scale — an airliner in ordinary turbulence. On the left are the lift’s and one root’s crossing rates as fractions of the straight wing’s. Both fall as the sweep grows, slowly at first and then faster, since the delay goes as the tangent: to 0.96 at 20°, to 0.899 and 0.887 at 35°, to 0.83 and 0.81 at 45°.

On the right is the ratio the earlier essay was about, one root’s rate over the lift’s. It falls too, but only from 1.19 to 1.17 at 35° and 1.16 at 45°. The root’s excess over the lift is a property of the shape of its weight, lopsided across the span, and the sweep acts on both weights through the same delay. It takes a little more from the root because the root’s weight sits further out, where the delays are longest: the lift’s weight is the loading, largest at the centre line where nothing is delayed, while the root’s is the loading times the distance from the root, largest beyond mid-semi-span.

The fatigue reading is direct. A lift-based count on a swept wing undercounts its root by 17 per cent rather than 19. Sweep does not close the gap; it shifts both counts down.

Sweep makes a second chord #

The second figure shows the mechanism in its simplest form. Take a gust that does not vary across the span at all — a sheet of upwash with stripes running spanwise — and ask what fraction of it the lift passes. On the straight wing, all of it: every station sees the same upwash at the same time. On a swept wing the stations see it at different times, so for stripes short enough the outer wing is in a trough while the root is on a crest, and the lift cancels. It is the streamwise twin of the cancellation the span takes away the infinity found across the span, where stripes running the other way cancel between one tip and the other.

That is a streamwise average, and it behaves like a chord. The gusts that cancel along the chord found Sears’ function doing the same thing over one chord’s length, cancelling gusts shorter than the chord. The swept span does it over the distance its stations are spread aft, weighted by the loading. The figure puts the two side by side on a wavenumber scaled by half the span: the chord passes half a gust at $\kappa_1 = 1.92$, the 35° span at 4.36 and the 45° span at 3.06. So at 35° the span’s streamwise averaging is a softer filter than the chord’s, cutting in at a little more than twice the wavenumber. It is a second chord, shorter than the first and spread along a diagonal.

The two filters multiply, so the swept span removes some of what the chord has already removed and some it had left. That is why the effect on the crossing rate is a tenth rather than a half: the chord has done most of the work, and the sweep adds a softer cut on top of it.

Where the crossings are counted #

Rice’s formula reads the crossing rate from the ratio of a load spectrum’s second moment to its zeroth: the variance comes from the whole spectrum, the crossings from its short-wave end. The third figure plots the integrand that matters for the rate, one root’s spectrum times the cube of the wavenumber. For the straight wing it peaks at about ninety inverse integral scales, a wavelength comparable to the span; above that the chord and the span’s own averaging cut it off.

The dial moves the sweep, and the whole effect is at that peak. At 35° the peak falls to 0.85 of the straight wing’s and moves to lower wavenumbers, from about ninety to about seventy, and the integrand’s tail is cut harder. Below a few inverse integral scales, where the variance lives, nothing moves at all: a delay of a few metres is nothing to a gust a kilometre long. The swept root keeps 99.7 per cent of its variance and loses eleven per cent of its crossing rate. Sweep changes how often a root’s moment cycles, not how large the cycles are.

The same fall at every small span #

The fourth figure repeats the calculation across spans from a hundredth of the turbulence’s scale to three. Below about a third of a scale the rates’ fall is the same at every span — 0.897 to 0.899 for the lift, 0.884 to 0.887 for the root. Only above a scale does it change, and only a little.

That flatness has a clean reason. For a wing much smaller than the turbulence’s integral scale, the part of the spectrum that sets the crossing rate lies in the turbulence’s inertial range, where the spectrum is a pure power law and has no length of its own. The only lengths left are the wing’s: its span, its chord and the distance its tips sit aft. Their ratios are fixed by the aspect ratio and the sweep, so the rate’s fall is a number that depends on those two alone. A glider in the boundary layer and an airliner at altitude, at the same aspect ratio and sweep, lose the same fraction of their gust cycles.

The variance tells the opposite story. At small spans it changes by a few parts in a thousand, since it lives at the long wavelengths; only when the span reaches a scale or more does the delay reach wavelengths that carry variance, and at three scales the swept wing keeps 94.5 per cent of it. Sweep starts to reduce the size of a wing’s gust loads only when the wing is as large as the gusts — the same threshold at which the span is the whole story for a straight wing’s averaging.

An airliner’s root, swept #

The fifth figure prices the earlier essay’s airliner — sixty metres of span in turbulence of integral scale 760 metres — with its wing straight and swept 35°. Straight, the lift crosses its mean 2.23 times a kilometre and one root 2.64 times. Swept, the lift crosses 2.00 times and the root 2.35.

A fatigue analysis that took the lift’s rate from a straight-wing gust model and applied it to the swept wing’s root would count 2.23 cycles a kilometre against the root’s 2.35 — five per cent short, rather than the nineteen per cent the earlier essay found for a straight wing. The two errors of that shortcut partly cancel: it ignores the sweep, which over-counts the lift by eleven per cent, and it ignores the root’s lopsided weight, which under-counts the root by seventeen. A count that included the sweep but not the root’s weight would be fifteen per cent short, 2.00 against 2.35; one that included the root’s weight but not the sweep would be twelve per cent long, 2.64. Only both together give the root’s number.

As in the earlier essay, the count of large excursions — past a fixed multiple of the load’s own standard deviation — scales with the crossing rate when the load is Gaussian, so the same factors apply to the cycles that consume most of a joint’s fatigue life. And since the variance is nearly unchanged, the cycles are as large as before; there are simply fewer of them.

The phase keeps the two halves apart #

The earlier essay split one root’s load exactly into a symmetric part, which reads the lifting gust, and an antisymmetric part, which reads the rolling gust and carries the excess. The sweep’s phase does not break that split, and the reason is worth a line. The phase grows with $|\eta|$, the same on both sides of the centre line, so it multiplies the weight’s symmetric half by something symmetric and its antisymmetric half by the same symmetric thing: each half keeps its parity. The symmetric half’s transform is then even in the spanwise wavenumber and the antisymmetric half’s odd, so the cross term between them is odd and integrates to nothing over both signs of the wavenumber. The swept root’s variance and second moment are still a lifting part plus a rolling part, with nothing left over.

What the phase does is shrink both parts’ short-wave ends, through the same delay. A delay that grows with distance from the centre line cannot tell a rolling gust from a lifting one; only the weight can, and the weight is unchanged. That is why the ratio in the first figure moves by two hundredths while the rates themselves move by a tenth.

A forward-swept wing would give the same numbers. The delay becomes an advance, the phase changes sign, and the filter’s magnitude does not depend on it.

The convention: rates per scale flown, delays by distance aft #

The crossing rate is the mean number of up-crossings of a load’s mean per integral scale flown, from Rice’s formula; per kilometre for the airliner. The span $b$ is in integral scales of the von Kármán spectrum, the chord is a ninth of the span, and the wing is elliptic with aspect ratio nine throughout. Sweep is the angle of the quarter-chord line, and the delay of a station is its distance aft of the root, $\tfrac12 b|\eta|\tan\Lambda$, over the flight speed. “One root” is the bending moment at the right wing’s root; its weight is the loading of a wing washed in linearly towards the right tip, as in the earlier essay.

What the delay leaves out #

The swept wing’s own loading. The weights here are the straight wing’s. A swept wing carries less lift near its root and more outboard than a straight one of the same planform, and its root bending moment’s weight is correspondingly different; the delay was isolated on purpose, and its effect on a real swept wing’s weights is a separate calculation.

The spar’s own arm. On a swept wing the bending moment about the root is taken about an axis along the root chord — the moment a bending moment never chooses a span sizes a wing’s structure by — and a station’s arm is its spanwise distance, as here. The structural box, however, runs along the swept spar, and the moment the box carries mixes bending and torsion. The torsion’s gust weight is a different load.

The aeroelastic response. The wing here is rigid. A swept wing that bends under a gust also twists nose-down, because its outer sections are behind its root, and that relieves the load — the bending–torsion coupling that makes swept wings respond more softly to gusts. It acts on the same outer stations the delay does.

Unsteady aerodynamics across the span. Sears’ function treats each strip’s chordwise response; a swept strip meets the gust at an angle, which changes its chordwise response, and that is not included.

Who worked it out #

Frozen turbulence, Taylor’s hypothesis, is from 1938; the reverse-flow theorem is Munk’s and Flax’s; Sears’ function, of 1941, is the chord’s response; and Rice’s crossing rate, of 1944, is the fatigue analyst’s first count. The statistical treatment of gust loads on swept wings, with each station delayed by its distance aft, was developed by Houbolt and others in the 1950s and 1960s, and is the basis of continuous-turbulence certification. The result that the delay acts as a second streamwise filter, cutting crossing rates by a sweep-dependent fraction that is the same at every small span and leaving the rolling excess of a single root nearly whole, is this calculation’s, for the elliptic weights and the kinematic delay stated.

How the swept loads were checked #

With no sweep, the two half-transforms added together give the earlier essay’s filter at five spanwise wavenumbers from 0.3 to 30, to rounding error, and the crossing rates of the lift and one root agree with that essay’s to three parts in ten thousand. With 35° of sweep, doubling every quadrature — nodes along the span, wavenumbers streamwise and spanwise — moves the root’s rate by under half a per cent.

Still open: a swept wing’s own weights #

Every number here used the straight wing’s loading and moved only the arrival times. A swept wing’s lifting line, or its vortex lattice, gives a different loading at uniform incidence — less at the root, more outboard — and a different weight for one root’s moment, and the reverse-flow theorem still holds for it. The next calculation would solve a swept planform’s own lattice for the lift’s and one root’s weights, put the delays on those, and ask whether the root’s excess over the lift grows or shrinks when the weights move outboard as well as the delays — and whether a swept wing’s root, whose weight is pushed towards the tips by the sweep itself, crosses its mean more often than a straight wing’s root rather than less.