The lift at the mean angle

An aeroplane in rough air is not flying at one angle of attack. Every gust changes the direction of the air arriving at the wing, so the angle wanders about its mean by a few degrees, upwards as often as downwards, several times a second. The obvious conclusion is that the effects cancel and the wing averages the lift belonging to its mean angle.

They cancel exactly where the lift curve is straight, and they do not cancel anywhere else.

The arithmetic, which is older than aerodynamics #

For any function $f$ and any distribution of $x$ about a mean $\bar{x}$, expanding about the mean gives

$$\langle f(x)\rangle \approx f(\bar{x}) + \tfrac{1}{2}f''(\bar{x})\,\sigma^2,$$

so the average of the function exceeds the function of the average when the curve bends upwards and falls below it when it bends down. That is Jensen’s inequality in its quantitative form, and the two things it needs are a curvature and a variance.

A wing supplies both. Its lift curve bends downwards as the stall is approached — that is what stalling is — so the mean lift is below the lift at the mean angle. Its drag polar bends upwards, since induced drag goes as the square of the lift, so the mean drag is above the drag at the mean angle. And dynamic pressure is a square, so a fluctuating airspeed delivers more mean load than its mean speed accounts for.

Three nonlinearities, three signs, and all of them ignored by a calculation that uses mean conditions.

How large it is, in numbers #

The averages here are taken by quadrature over a stated distribution rather than by the two-term expansion above, because the expansion is exactly the approximation that fails where the effect is largest. The lift curve is a stated model: linear at the finite wing’s own slope, capped by a quadratic that matches it in value and gradient and peaks at the stall angle.

Two features of that table are worth drawing out.

The deficit is quadratic in the gust size and linear in the curvature. Doubling the gusts quadruples the loss; flying two degrees closer to the stall roughly doubles it. Both are the expansion’s own statement, and both survive the exact quadrature.

The effect is a loss and never a gain. A curve that bends one way bends that way at every angle, so no distribution of gusts recovers the lift the mean angle promised. That asymmetry is the whole content of Jensen’s inequality and it is why the phrase “the gusts average out” is so persistent and so wrong: the gusts average out exactly, and the lift does not, because the map between them is not a straight line.

And nothing has stalled. At a mean of ten degrees with three degrees of gust, the wing spends almost none of its time past the stall angle of sixteen. The loss comes from the curve bending, not from any separation, and a pilot watching a stall-warning system would see nothing at all.

The drag side, where the sign reverses #

In the linear range the lift averages exactly and the drag does not, which makes it the cleanest place to check the arithmetic against the two-term expansion.

The excess is not negligible. For a wing at an aspect ratio of eight with three degrees of gust it is a fifth of the induced drag at that condition, which is a real fuel cost — and it is invisible to any performance calculation done at mean conditions. Aircraft performance in turbulence is measurably worse than in smooth air, and this is one of the reasons.

And the third nonlinearity, which is exact #

The dynamic pressure a wing sees goes as the square of the airspeed, and the mean of a square is the square of the mean plus the variance — exactly, with no expansion and for any distribution:

$$\langle \tfrac{1}{2}\rho U^2\rangle = \tfrac{1}{2}\rho\left(\bar{U}^2 + \sigma_U^2\right) = \tfrac{1}{2}\rho\bar{U}^2\left(1 + \frac{\sigma_U^2}{\bar{U}^2}\right).$$

It is also the reason a turbulent wind delivers more power to a turbine than its mean speed suggests, since power goes as the cube: the mean of $U^3$ exceeds $\bar{U}^3$ by three times the variance ratio, so a site with gusty wind out-performs a smooth one of the same mean. The same arithmetic that costs a wing lift earns a turbine power, and the difference is only the sign of the curvature.

That relation is why a structure’s design load is computed from a gust spectrum rather than from a mean wind, and it is the same statement this collection makes about what an airspeed indicator believes: the instrument reads a dynamic pressure, and a dynamic pressure is a mean of a square.

What is missing: the wing does not see the gust instantly #

Everything above is quasi-steady — the wing is assumed to take up the lift belonging to its instantaneous angle at once. It does not. A wing entering a sharp-edged gust reaches its steady lift only after travelling several chord lengths, because the circulation has to change and the change must be paid for by shedding vorticity into the wake.

The consequence for this essay is a frequency dependence. Slow gusts — those whose wavelength is many chords — are felt fully and the quasi-steady average above applies. Fast ones are attenuated, and the wing’s effective angle fluctuation is smaller than the air’s. So the variance that belongs in Jensen’s formula is not the atmosphere’s but the atmosphere’s filtered through the wing’s own response, and the filter has a name — Sears’ function — and a cut-off at a reduced frequency of order one.

That attenuation reduces the effect and does not remove it, because atmospheric turbulence has most of its energy at long wavelengths, and it is precisely the long ones the wing follows faithfully.

A gust is also a change of speed, and the two do not add #

The essay has treated the gust as a change of direction and the airspeed as a change of magnitude, and a real gust is both. Whether the two effects reinforce or cancel depends on their correlation, and the answer is not obvious.

A vertical gust changes the angle without changing the speed to first order, so it acts through the lift curve alone. A horizontal gust changes the speed without changing the angle, so it acts through the dynamic pressure alone. Atmospheric turbulence has both, they are correlated in a shear layer and nearly independent in the free atmosphere, and the mean load is the average of a product of two fluctuating quantities:

$$\langle L \rangle = \tfrac{1}{2}\rho\,\langle U^2 C_L(\alpha)\rangle,$$

which contains a cross-term $\langle U'\alpha'\rangle$ that neither of the separate calculations has. In a boundary layer near the ground that correlation is not small — it is the Reynolds stress that carries the momentum — so an aeroplane landing in a gusty crosswind is averaging a product whose two factors are correlated, and the sign of the correction depends on the wind’s own structure.

That is the honest general statement of this essay: the mean load is an average of a nonlinear function of several correlated fluctuating quantities, and every simplification of it — mean angle, mean speed, independent gusts — is an approximation with a sign that can be worked out.

Where this shows up in practice #

In every wind-tunnel comparison. A model in a tunnel with one per cent turbulence and an aeroplane in the atmosphere are not doing the same average, which is one more entry on the list of reasons a model cannot be matched to the aeroplane — and one that no Reynolds or Mach correction addresses.

In flight-test data reduction. A lift curve — computed here for a section whose camber fixes where it starts — measured in rough air is a curve of mean lift against mean angle, and its apparent slope and apparent maximum are both reduced. Corrections for this are standard and they are corrections for exactly the arithmetic above.

In stall margins. An aeroplane whose mean angle is comfortably below the stall may be past it for part of every gust cycle, and the mean lift deficit is the first sign of that in the data — appearing before any measurable buffet.

In the stall-warning margin an angle rather than a speed defines, since stalling is an angle and a gust is a change of angle — so a gust of five degrees consumes five degrees of margin however fast the aeroplane is going.

And in wind turbines, where the blade’s angle of attack fluctuates by many degrees every revolution as it passes through the tower’s wake and the atmospheric shear. There the curvature is large — turbine blades operate close to stall by design — and the mean power is measurably below what the mean wind implies, which is one of the reasons a turbine’s measured power curve is a statistical object rather than a calculated one.

The same curvature, protecting the structure #

Everything above is about a mean, and a structure is not designed against a mean. It is designed against the largest load it will ever meet, which lives in the tail of the same distribution — and the curvature that costs the mean its lift is what puts a ceiling on the tail.

The argument is short. A wing’s lift coefficient cannot exceed $C_{L\max}$, whatever the gust does, because past the stall angle the curve turns over. So the load factor a gust can impose is bounded:

$$n_{\max} = \frac{\rho U^2 C_{L\max} S}{2W},$$

which depends on the speed and not at all on the gust. Fly slowly enough and no gust of any size can overstress the aeroplane, because the wing lets go before the structure does.

Setting that bound equal to the structural limit load factor gives a speed, and it is one every pilot knows:

$$V_A = V_S\sqrt{n_{\text{limit}}} ,$$

the manoeuvring speed. For a light aeroplane certificated to 3.8g that is 1.95 times the stall speed — so below about twice the stall speed, full control deflection and any gust the sky can produce are survivable, and above it they are not. The stall is a mechanical fuse. It is the only load-limiting device on the aircraft that requires no sensor, no actuator and no decision, and it works because the lift curve bends.

That is the same bend, in the same place, doing the opposite job. Averaged over a gust distribution it removes five per cent of the mean lift and looks like a nuisance; evaluated at the extreme it removes the entire tail of the load distribution and is the reason the aeroplane is still in one piece. A nonlinearity that costs on the mean pays on the maximum, and which of the two matters depends entirely on whether the question is about performance or about structure.

The V-n diagram is that trade drawn. Its left-hand boundary is a parabola — the stall limit, rising as the square of the speed — and the horizontal line across the top is the structural limit; where they meet is $V_A$. The gust lines are straight lines through the origin whose slope is the lift-curve slope, and their intersections with the stall parabola are the loads a stated gust can actually impose at each speed. Every feature of that diagram is a statement about the two ends of the same curve.

It also explains a real asymmetry between aircraft. A light aeroplane in severe turbulence spends much of its time near the stall boundary and is protected by it; a large transport at cruise flies at many times its stall speed, so the bound above is far above its structural limit and offers no protection at all. For the transport, gust loads are a structural problem and nothing about the aerodynamics limits them — which is why the turbulence penetration speed exists as a procedure, and why it is a speed to slow down to. Slowing down moves the aeroplane back towards the regime in which its own stall is the thing that gives way first.

The same average, on a propeller and a rotor blade #

A helicopter’s advancing and retreating blades meet air at speeds that differ by twice the flight speed once per revolution, and a propeller blade behind a wing passes through its wake. Both are doing this essay’s average deliberately rather than incidentally, and in both the fluctuation is periodic rather than random — which changes the distribution and not the argument.

A periodic fluctuation of the same variance spends more of its time at its extremes than a normal one does, so the deficit is larger: about twice as large for a sinusoid, computed by quadrature over the arcsine distribution a sinusoid actually has. A blade whose angle swings sinusoidally by four degrees loses more mean lift than one whose angle wanders randomly by four, which is a real and slightly counter-intuitive design consequence — and it is why cyclic pitch exists, since undoing the swing recovers the average.

The connection to what a rotor’s wake has to do is direct: the induced velocity at a blade element is itself unsteady, so the angle of attack fluctuates even in perfectly smooth air, and the mean performance of the machine is an average over that.

What the picture cannot show #

The lift curve is a stated model. Its linear part is the finite wing’s own slope, computed from the two-dimensional value and the aspect ratio; its cap is the mildest curvature that joins the line to a peak at the stall angle. A real curve’s shape near the stall depends on the section, the Reynolds number and the surface condition, and the deficit computed here is therefore a lower bound: a sharper stall gives a larger effect.

The distribution is stated too. A normal distribution of gusts is the standard modelling assumption and real atmospheric turbulence is not normal — it is intermittent, with more extreme excursions than a normal law allows, which increases the variance that matters and increases the deficit.

And the response is quasi-steady in every figure. The Sears attenuation is described above and not computed; doing it properly needs a frequency response and a gust spectrum, which this collection has the machinery for in one dimension and has not assembled.

Who found it, and when #

Jensen’s inequality is from 1906 and is a statement about convex functions with no fluid in it. Its application to gust loads is as old as gust-load analysis — the sharp-edged gust formula dates from the 1920s, Küssner’s and Sears’ frequency-domain treatments from the 1930s, and the statistical approach that treats atmospheric turbulence as a spectrum and the aeroplane as a filter from the 1950s.

The surprising connection is with the other essays in this collection about averages, and this is the tidiest case among them. Every essay here computes a quantity defined by an average and finds that operating on the average is not averaging the operation. A wing is the case where the nonlinearity is visible as a curve on a graph, so the whole argument can be seen at once: the chord joining two points on a bending curve lies below it, the average lives on the chord, and the value at the mean lives on the curve. Nothing about fluid mechanics is needed to see the gap — and the size of the gap is set by a piece of aerodynamics, which is where the stall angle came from.

Where the ladder goes next #

Above this rung lies the frequency-domain treatment: the wing as a filter, the atmosphere as a spectrum, and the mean and variance of the load computed from their product. That is the standard apparatus of gust-load analysis and it needs a transfer function this collection has not built. Beside it sits the unsteady lift that supplies the filter, and below it the lift curve whose curvature is the whole of the effect — and whose straightness over most of its range is why the whole question can be ignored in cruise and cannot be ignored anywhere near the stall.