The span takes away the infinity, not the gust

The gusts that cancel along the chord filters the atmosphere’s turbulence through a wing section and finds most of it passes straight through: the root-mean-square lift of an airliner’s section in the standard turbulence model is 0.957 of the quasi-steady value. It also finds something that does not pass through so gracefully. The rate at which the load crosses its mean — the statistic fatigue is counted in, and the one that decides how often a structure is loaded and unloaded — has no value. Cut the gust spectrum off at a higher wavenumber and it grows, as the sixth root of the cut-off, without limit.

That essay leaves the reason and the suspected cure on the table. The reason is that Sears’ function falls only as one over wavenumber, and the von Kármán spectrum’s tail, as the minus five-thirds power, is too shallow for the second moment of the product to converge. The suspected cure is the span: a real wing is not two-dimensional, it averages the gust across its width, and an average over a width removes wavelengths shorter than the width. This essay computes that average and finds that the suspicion is right — and that it works in an unexpected way.

The field a wing flies through #

Treat the turbulence as frozen — a pattern carried past the wing at the flight speed, which is Taylor’s hypothesis and is good when the aeroplane is much faster than the gusts — and isotropic, with the von Kármán energy spectrum and an integral scale $L$. The vertical velocity on the horizontal plane the wing sweeps through then has a two-dimensional spectrum, a function of the streamwise wavenumber $k_1$ and the spanwise one $k_2$, and because the field is isotropic it depends only on their combination $q = \sqrt{k_1^2 + k_2^2}$.

It comes from the energy spectrum by integrating the isotropic spectrum tensor over the vertical wavenumber, which the wing never sees. Written with the substitution that makes the integrand smooth,

$$\Phi_2(q) = \frac{1}{2\pi q}\int_0^\infty \frac{E(q\cosh u)}{\cosh^3 u}\,du .$$

That is a new object on the way to the answer, and it is checked against an old one. Integrated over the spanwise wavenumber it must give the spectrum a single probe on a straight flight path records, which is von Kármán’s familiar one-dimensional form, known in closed form. It does, to a part in a hundred thousand over five decades of wavenumber.

The average keeps the gust #

A rectangular wing in strip theory feels the gust averaged uniformly across its span, and averaging over a width $b$ multiplies the spanwise spectrum by $\mathrm{sinc}^2(k_2 b/2)$: one for wavelengths much longer than the span, falling away for shorter ones. The span-averaged gust has the spectrum

$$\Phi_b(k_1) = \int \Phi_2\!\left(\sqrt{k_1^2 + k_2^2}\right)\mathrm{sinc}^2\!\left(\frac{k_2 b}{2}\right) dk_2 ,$$

and its variance is what survives.

Averaging across the span keeps almost all of the gust. An airliner’s span in the standard model’s turbulence is about a tenth of the scale, and the gust it feels is 95 per cent as strong as the gust at a point. That is because the variance of von Kármán turbulence is concentrated at wavelengths comparable to the scale — hundreds of metres — and a wing sixty metres wide sees those as uniform. The span is not a significant load alleviator in the root-mean-square sense, and the certification practice of treating the gust as uniform across the span for the variance is justified by exactly this figure.

The average removes the infinity #

The effect on the short wavelengths is another matter. Above a streamwise wavenumber of about one over the span, the average is over many spanwise wavelengths, and the spectrum loses a factor that grows with $k_1$. Where the point spectrum falls as $k_1^{-5/3}$, the span-averaged one falls as $k_1^{-8/3}$ — one full power steeper. Multiply by Sears’ $1/k_1$ and the load spectrum falls as $k_1^{-11/3}$.

Why exactly one power is worth a paragraph, because it is not a property of the sinc function. A span much wider than a wavelength averages over so many spanwise wavelengths that only one survives the average: $k_2 = 0$, the part of the gust that is uniform across the span. The averaged spectrum is then the plane spectrum at zero spanwise wavenumber, multiplied by the width of the averaging window in wavenumber, which is $2\pi/b$. And the plane spectrum at zero spanwise wavenumber is one power steeper than the spectrum along a line, because the line spectrum is the plane spectrum integrated over every spanwise wavenumber, and the integration adds one power of $k_1$ back. A point probe collects every spanwise wavenumber at once; a wide wing collects only one. Any finite averaging window does the same, whatever its shape, which is why the elliptic loading of a real wing changes the coefficient and not the power.

The crossing rate is Rice’s: the square root of the ratio of the load spectrum’s second moment to its zeroth, divided by $2\pi$. The second moment weights the spectrum by $k_1^2$, so it converges when the spectrum falls faster than $k_1^{-3}$. The two-dimensional load, at $k_1^{-8/3}$, does not. The span-averaged load, at $k_1^{-11/3}$, does.

The figure is the answer to the question as it was posed: a wing whose span is fifty times its chord and a tenth of the turbulence scale. Without the span, the crossing rate climbs from 10.3 per scale at a cut-off of ten thousand per scale to 25 at a million, with a local exponent of 0.19 — the sixth root, as the chordwise calculation found. With the span it goes from 2.25 to 2.29 over the same range, a local exponent of 0.004. The span removes almost none of the gust and all of the infinity. It is the rate at which the load changes, not the size of the load, that the span controls.

Two filters, two lengths #

The wing now filters the gust twice, and the two filters have different lengths in them. The chord filters through Sears’ function, which begins to act at a streamwise wavenumber of about two over the chord — the reduced frequency of order one. The span filters through the average, which begins to act at about one over the span. For any wing with an aspect ratio above one the span acts first, at a wavenumber lower by the aspect ratio, and for an airliner that is a factor of about twelve.

So the ordering of the load spectrum, reading from long waves to short, is: the gust spectrum’s own flat top out to one over the turbulence scale; its minus five-thirds fall out to one over the span; the span’s extra power out to two over the chord; and then Sears’ extra power beyond. The crossing rate is set in the middle of that — by the band between the turbulence scale and the span — which is why it depends on the span as a cube root and hardly on the chord at all. The variance, by contrast, is set at the top, by the turbulence scale, and depends on neither. Each statistic of the load belongs to a different band, and the one that a single number is quoted for is the one whose band is named.

What that means for a real wing #

For the airliner in the chordwise calculation — a five-metre chord in turbulence of the 762-metre scale used above two thousand feet, so a chord of 0.0066 of the scale — with a sixty-metre span, a span of 0.079 of the scale, the settled crossing rate is 1.80 per scale, which is 2.4 load reversals per kilometre flown and 0.54 a second at 230 metres a second. The root-mean-square load, with both the span and the chord’s filter, is 0.93 of the quasi-steady value against 0.957 with the chord alone.

That rate is a property of the wing and the air now, and it depends on both. A wider wing sees fewer reversals, roughly as the cube root of the span, because the span sets where the load spectrum turns steeply down and the rate is decided by the wavenumbers just short of that. A narrower one sees more, and in the limit of no span at all — the two-dimensional wing — the rate is whatever the cut-off makes it.

Which is to say that the infinity was never physical. A two-dimensional wing is an idealisation with no width to average over, and its infinite crossing rate is the idealisation reporting what it left out, in the same way that a vortex sheet’s instability at every wavelength reports a missing thickness. The chordwise and spanwise dimensions of a wing both filter the gust, and the calculation needs both before its statistics mean anything.

What decides fatigue, and what does not #

Fatigue damage is counted in cycles, and the crossing rate is the natural count of them. A structure designed to the two-dimensional rate would be designed to a number that depends on an arbitrary cut-off; one designed to the span-averaged rate is designed to a number set by the wing’s span and the atmosphere’s scale. In practice that distinction is overtaken by a third filter the calculation leaves out: the wing’s own structural response, which in a real aeroplane cuts off everything above its first bending and torsion frequencies. Airliner wings bend at a few hertz, and load reversals faster than that do not reach the structure as bending. What the span average contributes is a proof that the aerodynamic part of the problem is finite before the structure is included — that the structure is refining a finite answer, not rescuing an infinite one.

There is a limit to what the crossing rate can say even once it has a value. Rice’s formula counts the crossings of a Gaussian process, for which a spectrum is a complete description; real turbulence is not Gaussian in its small scales, and a load history with the right spectrum can still have the wrong extremes, because extremes live in the phases a spectrum does not record. The span-averaged rate is the right count of reversals for the Gaussian model of the gust. How often the largest reversals come — which is what fatigue at high stress is most sensitive to — needs the gust’s non-Gaussian structure, and the average over the span, which suppresses the small scales where that structure is strongest, will make the load more Gaussian than the gust it came from. That is a direction, not a number, and it is stated as one.

It is also a warning about measurements. A gust probe on a nose boom measures the gust at a point, and its spectrum has the full $k_1^{-5/3}$ tail. The wing’s load spectrum does not. A load spectrum inferred from a point measurement and a two-dimensional transfer function overstates the load’s short-wavelength content by exactly the factor the span removes, and the overstatement grows without limit at high frequency. What a spectrum records depends on the size of the thing recording it.

The same filter in every finite instrument #

A wing is not the only thing that averages a turbulent field over its own size, and the same arithmetic governs every instrument that does. A hot-wire probe a millimetre long averages the velocity along its wire, and its measured spectrum falls below the true one at wavenumbers above one over its length, by exactly the mechanism above. A pressure transducer set in a wall averages the turbulent pressure over its face and misses the small eddies that carry much of the high-frequency content. A camera with a finite exposure averages in time rather than space, and the shutter is part of the answer for the same reason.

In each case the variance is nearly untouched and the rates are not. A probe that is small against the turbulence scale reports the root-mean-square velocity faithfully and the rate of change of the velocity — the quantity that measures dissipation — too low, and correcting for it is a standard and unglamorous part of turbulence measurement. The wing’s span is a large probe, and its measurement is the load. What averaging costs is the general version: every average is a filter, and what a filter removes is decided by its size, not by the size of the thing it is filtering.

What the picture cannot show #

Every spectrum here is an average over an infinitely long flight through statistically uniform turbulence. A real flight meets patches of turbulence separated by calm air, and the statistics of a patch are not the statistics of the whole; a crossing rate is the long-run average of something that happens in bursts.

The span average is uniform across the span, which is strip theory for a rectangular wing. A real wing’s lift is weighted towards its root, where the chord is largest, and its tips respond less; an elliptic weighting averages slightly less strongly than a uniform one and would raise the settled rate a little. Neither changes the extra power of wavenumber, which comes from the averaging having any finite width at all.

Where the model stops #

Frozen turbulence. The gusts are assumed to be carried past the wing without changing. For an aeroplane flying at two hundred metres a second through gusts of a few metres a second that is excellent; for a slow aeroplane in strong turbulence it is not, and the evolving turbulence adds its own time scale.

Isotropy. Near the ground turbulence is not isotropic — its vertical component is suppressed and its length scales depend on height — and the plane spectrum changes shape. The standard models make it isotropic above a few hundred metres for this reason.

A rigid wing. Nothing here bends, twists or has a structural mode, and the aeroplane is held still; the aeroplane’s own rise alleviates the long wavelengths as the span alleviates the short ones.

The convention the numbers depend on #

Spectra are two-sided in wavenumber and normalised per unit variance; the one-dimensional von Kármán spectrum is its usual one-sided form halved. The turbulence scale $L$ is von Kármán’s, with the constant 1.339 in the spectrum, and 762 metres is the scale the civil standards use above about two thousand feet. Crossing rates are per scale flown unless given per kilometre or per second, and they count zero crossings of the load’s fluctuation in one direction.

Who found it, and when #

Taylor proposed frozen turbulence in 1938. Von Kármán’s spectrum is from 1948. Rice’s formula for the crossing rate of a Gaussian process is from his 1944–45 papers on noise. Liepmann applied Sears’ function to atmospheric turbulence in 1952, and the spanwise averaging of the gust — the “penetration” and “spanwise coherence” effects of the gust-loads literature — was worked out in the 1950s and 1960s, when continuous- turbulence analysis replaced discrete gusts in structural design. The convergence argument above is the question those methods answered in practice by including the structure, asked of the aerodynamics alone.

Still open: what the span does to a gust that is not frozen #

The whole calculation rests on the turbulence being a pattern that the wing flies through unchanged. At the wavelengths the span controls — tens of metres and less for an airliner — the turbulent eddies turn over in a time comparable to the time the aeroplane takes to cross them, for a slow aeroplane in strong turbulence, and then the field seen along the span is not a frozen slice of an isotropic pattern.

The calculation that would follow gives the turbulence a decorrelation time proportional to its eddy turnover time at each wavenumber, so that the span-averaged spectrum becomes a function of frequency and wavenumber together, and asks at what airspeed, in a given strength of turbulence, the settled crossing rate starts to move. Beside it is the question of the weighting: a lift distribution concentrated towards the root averages over an effectively narrower span, and how much that raises the rate for the elliptic wings aircraft actually have is a short calculation with the same spectra and the same averaging.