Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Gust — where it appears

A change in the wind a wing meets, treated as an input to the wing's unsteady response rather than as a new steady condition. Its load is computed by convolving the gust's shape with Küssner's indicial function, which is not the function used for a change of incidence and differs from it by a factor of eight at one semichord.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

The mean lift is not the lift at the mean angle. A finite wing's lift curve, with a gust distribution of standard deviation 3° about a mean angle of 10° drawn along the foot. The wing spends its time spread across that distribution, so what it averages is the average of the curve: 0.7951 against the 0.8368 the mean angle promises, a deficit of 5.0 per cent. Nothing has stalled, and no gust has taken the wing past the stall angle: the deficit comes entirely from the curve bending over, and it is there at every angle where the curve is not straight.

The lift at the mean angle

A wing in rough air flies at every angle in turn, so what it averages is the average of its lift curve rather than the lift at its average angle. Where the curve bends over near the stall the two differ by five per cent at three degrees of gust and by fourteen at five, and the drag goes the other way.

circulation · Gust Wagner's function and Küssner's, from one solver and two inputs. Lift as a fraction of its steady value, against distance travelled in semichords. The step in incidence and the sharp-edged gust are the same unsteady problem with two different right-hand sides, and Jones's exponential fits to both are drawn over the solve. From four semichords on they are nearly the same curve — which is why they get interchanged.

Two answers to one question

Unsteady aerofoil theory collapses a wing's whole history onto one function of one variable, and every quasi-steady gust calculation convolves something with it. There are two such functions, not one: a wing that is pitched changes its boundary condition everywhere at once, and a wing flying into a gust has not met most of the gust yet.

circulation · Unsteady lift A gust's lift keeps falling where a pitching wing's stops at a half. The magnitude of Sears' function, the lift a wing gets flying through a sinusoidal gust as a fraction of the quasi-steady value, against the reduced frequency on a logarithmic axis, beside Theodorsen's function for a wing that pitches or heaves. The two agree at low frequency. Above a reduced frequency of about a tenth they part: Theodorsen's levels off at one half, because a moving wing changes its whole boundary condition at once, while Sears' keeps falling as one over the square root of 2πk, because several wavelengths of gust lie along the chord and cancel.

The gusts that cancel along the chord

A wing that pitches keeps half its circulatory lift however fast it moves. A wing flying through a gust does not: once the gust is a few chords long, its ups and downs lie along the chord together and cancel, and the lift falls without limit. For an airliner that barely touches the root-mean-square gust load, and cuts the load spectrum at the wing's own torsion frequency to a quarter of the quasi-steady value.

regimes · Reduced frequency A gust, the aeroplane's response, and the load it is left with. A one-minus-cosine gust twenty-five chords long, as an angle of attack, against distance flown, and the lift it produces on two aeroplanes free to rise — a light single with a mass parameter of 16 and an airliner at eleven kilometres with 149. Both loads lag the gust, because lift takes several chords to build. The light aeroplane's also peaks lower and falls away early: it has started to rise with the gust, which lowers its own angle of attack.

The aeroplane rises out of its own gust

A gust pushes a wing up, and the wing, being attached to an aeroplane that is free to move, starts rising — which lowers its angle of attack and takes away part of the load the gust was delivering. How much depends on one number, and the formula every airworthiness code used for forty years is a fit to the calculation this essay does again.

circulation · Gust A gust spectrum at a point, across a span, and as a load. The spectrum of vertical gust velocity at a single point — von Kármán's, falling as the minus five-thirds power — and the same gust averaged across a wing whose span is a tenth of the turbulence scale and across one as wide as the scale. Averaging removes the wavenumbers shorter than the span, one power steeper. Through Sears' filter for a chord of a five-hundredth of the scale, the tenth-span load falls as the minus eleven-thirds power, steep enough to converge.

The span takes away the infinity, not the gust

A two-dimensional wing in turbulence reverses its load infinitely often, because the gust spectrum never falls fast enough. A real wing averages the gust across its span, and the average keeps ninety-five per cent of the gust's strength and removes the infinity. The load's crossing rate settles at about two per turbulence scale flown and stops depending on where anyone cut the spectrum off.

circulation · Gust An elliptic wing filters the gust like a round aperture. How much of a gust component with spanwise wavenumber k₂ survives the span average, against κ = k₂b/2, for three weights. A uniform strip is a slit and its filter is sinc², falling as κ⁻² between its zeros. An elliptic loading is a circular aperture and its filter is the Airy pattern (2J₁(κ)/κ)², whose first zero is at 3.83 and whose envelope falls as κ⁻³, but which stays near one to larger κ: the ellipse's weight is narrower, so its filter is wider. A rectangular wing of aspect ratio nine weights the span by its own loading and lies between, with the ellipse's κ⁻³ fall because its loading too goes to zero at the tips as a square root.

A wing averages a gust through its own loading

A finite span averages a turbulent gust and so tames the load it causes, but it does not average uniformly. The reverse-flow theorem says exactly how it weights the span: by the loading the wing makes at a uniform incidence. An elliptic wing therefore filters the gust the way a round aperture diffracts light, passes eight per cent more of its short scales than a uniform strip, and crosses its mean load up to four per cent more often.

circulation · Gust A root's bending moment weighs the gust by the loading of a wing twisted like its moment arm. Across the span of an elliptic wing of aspect ratio 9, each normalised to unit area: the lift's weight, the loading at uniform incidence; the two roots' bending moment's, the loading of a wing washed in as |y|, with a hole at the root; and the right root's alone, the loading of a wing washed in as y on the right half and not at all on the left. The one-root weight reaches across to the left wing through the induced flow, and it is not symmetric.

One wing root feels the rolling gust its lift cannot

The reverse-flow theorem says the gust weight of any load is the loading the wing would carry at an incidence shaped like that load's arm. For the lift that is the familiar uniform-incidence loading. For the bending moment at one root it is the loading of a wing washed in towards one tip only — a weight that is not symmetric, and so passes the rolling half of a gust that the lift and the two roots together cancel. That half is a sliver of the variance and adds up to a fifth to how often the root's moment crosses its mean.

circulation · Gust A symmetric panel with no spring stiffness washes out exactly as each lift coefficient wants. The outer panel's washout against the wing's lift coefficient over the trainer's sortie, with a spring stiffness of 0 cruise dynamic pressures: each lift coefficient's own best, 5.31° per unit of it; the best fixed washout, 2.31°; a symmetric panel at its best hinge and preload, 1.34% above an elliptic wing; and a panel with a cambered trainer section's moment of −0.05, 9.31%. The fixed twist spends 2.65% and the floor is 1.34%.

A pivoted wingtip follows the lift only with no spring stiffness and no camber

The washout a trainer's sortie wants grows in proportion to the lift coefficient, and the twist a wing gets for free follows the dynamic pressure instead. A tip panel free to pivot on a hinge ahead of its own aerodynamic centre can do better, and on one condition it does exactly what is wanted: held by a constant torque and carrying no camber moment, its own lift coefficient stays in proportion to the wing's, and every segment of the sortie flies at its own best twist. Every unit of spring stiffness gives part of that back, a cambered section's moment turns the panel into the wrong law altogether, and the free panel sheds a quarter of a gust's lift for nothing.

circulation · Taper Sweep slows every load's crossings by about a tenth at 35°, and leaves one root's excess over the lift standing. Against the sweep, for an elliptic wing of aspect ratio nine whose span is eight hundredths of the turbulence's integral scale: left, the lift's and one root's crossing rates as fractions of the straight wing's; right, one root's rate over the lift's. At 35° the lift crosses its mean 0.899 times as often as on the straight wing and the root 0.887; the root's excess over the lift goes from 1.19 to 1.17, and at 45° to 1.16.

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

On a straight wing every station meets a gust at the same instant, and one wing root's bending moment crosses its mean nearly a fifth more often than the lift, because it reads the rolling half of the gust. Sweep the wing back and the outer stations meet the gust later, by their distance aft. That delay averages the gust along the flight path as a second, softer chord would: at 35° every load crosses its mean about a tenth less often, the root slightly more than the lift since its weight sits further out, and the variance hardly changes at all. The rolling excess survives almost whole.

circulation · Gust

Named alongside it

The objects these essays reach for when they reach for this one.

Model limitSpectrumFrequency responseLoad factorSpan loadingUnsteady liftAveragingFatigueLifting lineReciprocityTurbulenceBending moment

All concepts