The gusts that cancel along the chord

Slow enough to be steady put a number on when a wing that moves can be treated as though it carried the lift of its instantaneous angle: Theodorsen’s function, which at a reduced frequency of 0.1 is already fifteen per cent low and which never falls below one half. Two answers to one question showed in the time domain that a wing pitched suddenly and a wing flying into a sharp-edged gust are different problems, with different indicial functions, and that confusing them overstates a short gust’s peak load.

This essay is the frequency-domain half of that distinction, and it has a consequence the time domain hides. A wing that pitches changes its boundary condition along the whole chord at once, so however fast it oscillates, the disturbance it makes is coherent over the chord. A wing that flies through a gust meets the gust one point of the chord at a time. When the gust is many chords long that makes no difference. When it is a few chords long, the chord spans up-gust and down-gust together, and they cancel. The lift that survives has no floor.

A gust is not a moving wing #

The conventions are these. The wing is a thin aerofoil of chord $c = 2b$ flying at speed U, with a planar wake carried away at that speed. The gust is a vertical velocity $w_0\,e^{i(\omega t - kx/b)}$ frozen into the air and carried past the wing, so its reduced frequency $k = \omega b/U$ is also its wavenumber in semichords: a gust of reduced frequency one is $2\pi$ semichords, or about three chords, long. Time goes as $e^{i\omega t}$, so a lag is a negative phase, and the lift is compared with the quasi-steady lift of the lift curve, $\pi\rho U c\,w_0$.

Sears solved the problem in 1941. The lift is $\pi\rho U c\,w_0\,S(k)\,e^{i\omega t}$ with

$$S(k) = \big[J_0(k) - iJ_1(k)\big]\,C(k) + iJ_1(k),$$

where $C(k)$ is Theodorsen’s function and the gust’s phase is taken where it crosses mid-chord. The Bessel functions are the same ones Theodorsen’s function is built from, checked by their Wronskian, so nothing new has to be trusted to compute S.

At low frequency the two are almost the same, since a gust many chords long is indistinguishable from a slow change of incidence: at k = 0.01 they differ by three parts in ten thousand, and at 0.1 by 1.4 per cent. They part above a reduced frequency of about a tenth. Sears’ function falls to one half at k = 0.567, where Theodorsen’s is still at 0.60, and it keeps falling, approaching $1/\sqrt{2\pi k}$ — at k = 10 it is 0.126 and at 50 it is 0.056. Theodorsen’s settles on one half. The ratio of the two is 0.710 at k = 1, 0.252 at 10 and 0.113 at 50, and it has no lower limit.

The difference is the cancellation. A pitching wing’s own motion is in phase along the chord, so the vorticity it sheds is coherent and half the lift survives any frequency. A gust shorter than the chord puts upwash on one part of the wing and downwash on the next, and the bound vorticity each induces cancels in the integral for the lift, more completely the more wavelengths lie along the chord. Liepmann’s fit, $1/\sqrt{1 + 2\pi k}$, captures the whole behaviour to within 7.5 per cent, worst at k = 0.24, which is why it has been used for gust loads for seventy years.

The spiral is where the gust is measured from #

Sears’ function has a notorious picture: in the complex plane it spirals round the origin without end, and a phase that winds through every angle looks like physics.

It is a choice of origin. The formula measures the gust’s phase at mid-chord, but the gust reaches the leading edge a semichord’s travel earlier, and a delay of fixed distance is a phase proportional to frequency — exactly k radians. At high frequency S itself goes as $e^{i(k - \pi/4)}/\sqrt{2\pi k}$, and the winding is that $e^{ik}$. Referred instead to the leading edge, where the gust first arrives, the function becomes $S(k)\,e^{-ik}$, and it stays in one quadrant: its phase is −17.0° at k = 0.1, −38.4° at k = 1 and −44.3° at k = 10, and it settles on a lag of 45°. From mid-chord the same three phases are −11.3°, +18.9° and +168.7°, which describe the same lift.

The leading-edge picture is the physical one — the lift lags the gust’s first arrival by at most an eighth of a cycle, however short the gust — and a plot of mid-chord phase against frequency, which is what a table of Sears’ function usually gives, is mostly a plot of where the reference point was put. The distinction is not academic for anything that acts on the phase. A gust-alleviation system that senses a gust with a probe ahead of the wing and moves a control surface to cancel its lift must time the surface against the lift’s lag from the leading edge; timed from a mid-chord table without the correction, it would be wrong by k radians, which is 57° at k = 1.

Two routes from the time domain agree with it #

The leading-edge form is also what the time domain produces, because a gust in the time domain is defined by when its front reaches the leading edge. Küssner’s function is the lift after a wing enters a sharp-edged gust, and its Fourier transform must be the leading-edge Sears function.

Jones’s two-exponential fit to Küssner’s function, $1 - \tfrac12 e^{-0.13s} - \tfrac12 e^{-s}$ in semichords travelled, transforms in closed form, and up to k = 0.5 it lands within 6.2 per cent of the exact function in magnitude and phase together. Past k = 1 it misses by more than twenty per cent, which is the price of fitting a function with a square-root tail by two exponentials, and the reason it should not be used for short gusts.

The second route is the discrete-vortex plate from the essay on the two indicial functions, flown into a sharp-edged gust for 160 semichords and transformed numerically with its slowly decaying tail added in closed form. Its magnitude agrees with Sears’ to 1.1 per cent up to k = 0.5. Its phase trails, by 7.4° at k = 0.1 and 39.8° at 0.5, and that is a statement about what it records rather than an error: the march follows the circulation bound to the plate, and the unsteady part of the lift is the rate of change of that circulation’s impulse, a quarter of a cycle ahead of it. That shifts the phase in proportion to k and the magnitude only in proportion to $k^2$, which is the pattern the comparison shows.

What the wing lets through of the turbulence it flies in #

A real gust is not a sinusoid. Atmospheric turbulence has a spectrum, and the von Kármán form used for aircraft loads, with scale length L, is flat at long wavelengths and falls as the minus five-thirds power at short ones — the inertial-range slope that the one exact result sits underneath, over a range of scales a real Reynolds number only partly has. Carried past the wing at the flight speed, each wavenumber in it is a sinusoidal gust, and the lift spectrum is the gust spectrum multiplied by the square of Sears’ function.

For a wing of 5 m chord at 230 m/s in turbulence of the 762 m scale used above 2,000 ft, the chord is 0.0066 of the scale. The lift spectrum follows the gust spectrum over the long wavelengths that carry most of the energy and peels away above a wavenumber of a few hundred over the scale, where the gusts are a few chords long, falling thereafter a whole power of wavenumber faster.

Where it peels away is where the structure lives. A 3 Hz wing-bending frequency at 230 m/s is a reduced frequency of 0.205, where $|S|^2$ is 0.511: a quasi-steady load calculation overstates the load spectrum there by a factor of 1.96, and one using Theodorsen’s function by 1.095. An 8 Hz torsion frequency is k = 0.546, where quasi-steady overstates by 3.88 and Theodorsen by 1.42. On approach, at 80 m/s, the same 3 Hz bending mode is k = 0.589 and the quasi-steady factor is 4.13. The filter matters exactly where a structure responds dynamically, and a fatigue or flutter-margin calculation that multiplied the gust spectrum by a quasi-steady lift slope would be doubling to quadrupling the input at its own resonances.

Certification loads are computed differently, with single one-minus-cosine gusts rather than a spectrum, and the same filter sits inside them. The shortest gust the transport-category rules require has a gradient distance of 9.1 m, under two chords of this wing. The wavelength that dominates it has a reduced frequency of 0.86, where Sears’ function is 0.42 and Theodorsen’s 0.56, so the wrong function overstates it by a third. The longest, at 107 m, is twenty-one chords long, and there the two functions are 0.877 and 0.884, one per cent apart. In the time domain that is the difference between Küssner’s indicial function and Wagner’s, and it is why using the pitching wing’s response for a short gust overstates its peak load.

The root-mean-square load barely notices #

The integrated picture is different, and it is the one that explains why quasi-steady gust loads have served for so long.

The root-mean-square lift depends on the chord and the scale only through their ratio. At the airliner’s 0.0066 it is 0.957 of quasi-steady with the von Kármán spectrum — a four per cent reduction — against 0.961 with Theodorsen’s function wrongly used and 0.975 with the Dryden spectrum, whose high-wavenumber tail falls as the minus two power rather than minus five-thirds. The choice of spectrum moves the answer by nearly half as much as the filter does. Only when the chord is a tenth of the scale does the reduction reach twenty per cent (0.796), and at equal chord and scale the lift is 0.490 of quasi-steady while Theodorsen’s function would still claim 0.616. At ten chords to the scale it is 0.218 against Theodorsen’s 0.516, which tends to one half and never lower. Liepmann’s fit, used in place of the exact function, gives 0.943 for the airliner — 1.4 per cent low, a smaller error than the choice of spectrum.

The ratio does not care what is flying. Near the ground the standard turbulence scale shrinks with height, and at 30 m it is taken as 30 m, so a drone with a 20 cm wing at that height has a chord of 0.0066 of the scale — the airliner’s ratio exactly — and the same 4.3 per cent reduction in root-mean-square lift. What differs is where in frequency the filter bites. The lift is halved at k = 0.567, which for the airliner at 230 m/s is a gust arriving at 8.3 Hz and for the drone at 15 m/s is one at 13.5 Hz, and small aircraft are the ones whose structures and controls respond at frequencies like that.

So the filter is nearly invisible in the variance and decisive in the spectrum, and the reason is the one the lift at the mean angle gave: the atmosphere’s energy is at long wavelengths, which the wing follows faithfully. What the long wavelengths do not decide is how often the load changes, and that is the next number.

How often the load crosses its mean #

Fatigue depends on how many load cycles a structure sees, and Rice’s formula gives the rate at which a random load crosses its mean from the second moment of its spectrum: $N_0 = \sqrt{M_2/M_0}/2\pi$, with $M_2$ weighted by the square of wavenumber. For a von Kármán spectrum that weight makes the integral grow without limit, so the answer depends on where the spectrum is cut off.

Without the filter the crossing rate grows as the two-thirds power of the cut-off: from 2.2 crossings per scale length with the spectrum cut at a hundred over the scale to 996 with it cut at a million. It has no value of its own; it is a report of the cut-off. Through Sears’ function the growth drops to the sixth root — between cut-offs of 10⁵ and 10⁶ the computed local exponent is 0.176, still approaching 1/6 — and the same two cut-offs give 1.7 and 14.2. The filter does not make the two-dimensional crossing rate finite, since $|S|^2$ falls only as one over wavenumber, but it turns a number that depends on an arbitrary choice as strongly as the two-thirds power into one that depends on it as weakly as the sixth root.

The quantity is the one a helicopter or wind-turbine blade cares most about, since such a blade meets turbulence at reduced frequencies well above an airliner’s. A rotor blade crossing the wake of the blade before it feels a gust a few chords long by construction, where Sears’ function is at a half and falling.

Sears’ function against what it must reduce to #

Sears’ function reaches one at zero frequency to 1.6 × 10⁻⁷, and $|S|\sqrt{2\pi k}$ reaches one at k = 200 to 7.8 × 10⁻⁷, which is the high-frequency asymptote arriving from the Bessel algebra rather than being assumed. The von Kármán and Dryden spectra integrate to their variance to 1.1 × 10⁻⁵ and 3.2 × 10⁻⁹. The transformed Küssner fit and the transformed marched plate land within 6.2 and 1.1 per cent, the size of their own approximations. The calculation refuses a negative reduced frequency, a wing of no chord, a spectrum with no model behind it and a crossing rate with no cut-off.

The connection that makes the function worth this much attention runs the other way from loads. An aerofoil in turbulence does not only lift in response to it; it radiates sound, and the leading-edge noise of a fan or a rotor is computed from the same response of a thin aerofoil to a convected gust, carried to higher frequency and with the air’s compressibility added. The high-frequency tail that a loads engineer can mostly ignore is the whole of the acoustic problem.

What the picture cannot show #

Span. The wing here is two-dimensional, so a gust is uniform along its span. A real gust field varies across the span as well, and a finite wing averages it there too, attenuating short wavelengths further than Sears’ function alone.

Thickness, camber and incidence. Sears’ function is for a flat plate at zero mean incidence. Thickness and a mean loading both reduce the response to short gusts below it, and the correction grows with the reduced frequency.

Compressibility. At an airliner’s Mach number the gust response is Sears’ function’s compressible counterpart, which differs most at exactly the high reduced frequencies where the filter matters.

The horizontal component and the stall. A gust changes speed as well as incidence, and near the stall the lift curve bends, both effects the frozen, linear picture leaves out and the lift at the mean angle takes up.

Still open: whether the load’s crossing rate has a value once the span is included #

In two dimensions the crossing rate grows without limit however the spectrum is filtered, because $|S|^2$ falls only as one over wavenumber and the von Kármán spectrum’s tail is too shallow for that to be enough. A finite span averages the gust across it, and if that averaging removes another power of wavenumber at scales shorter than the span, the second moment converges and the crossing rate becomes a property of the wing and the air rather than of where a spectrum was cut. Whether it does for a real planform, and what the converged number is for a wing whose span is fifty times its chord and a tenth of the turbulence scale, is the next calculation: the two-dimensional gust spectrum across a lifting line, with Sears’ function applied strip by strip and the spanwise correlation of the von Kármán field between strips. It is also the question what a group of order one is worth asks in general — here with two groups, chord over scale and span over scale, and no reason to expect either to be negligible.