Slow enough to be steady

A wing that changes its angle of attack does not immediately carry the lift belonging to the new angle. It cannot: lift is circulation, circulation round the wing can change only by shedding an equal and opposite amount into the wake, and the shed vorticity has to be carried downstream before its influence on the wing dies away. The lift that arrives late computes that delay for a step change and finds it takes several chords of travel.

So a wing in oscillatory motion is always behind. How far behind is decided by one number — the reduced frequency

$$k = \frac{\omega b}{U},$$

with $b$ the semi-chord. It is the ratio of the time the air takes to pass the wing to the time the wing takes to complete a radian of its motion, and if it is small the motion is slow compared with the flow and the wing should be able to keep up.

That is the argument, and it is right about what it says. The trouble is that it is used to license a different claim.

The function that measures the assumption #

Theodorsen solved the harmonically oscillating thin aerofoil in 1935 and reduced the whole of it to one complex function of $k$:

$$C(k) = \frac{H_1^{(2)}(k)}{H_1^{(2)}(k) + iH_0^{(2)}(k)},$$

with $H^{(2)} = J - iY$ the Hankel functions of the second kind. It is the factor by which the circulation actually present differs from the circulation a quasi-steady calculation would assign, and its two limits are the whole story: $C(0) = 1$, so a slow enough oscillation is quasi-steady; $C(\infty) = 1/2$, so a fast enough one gets exactly half the circulatory lift and no more, at any frequency whatever.

The Bessel functions here are computed rather than tabulated, and the check that they are right is not a table. It is the Wronskian identity $J_0Y_1 - J_1Y_0 = -2/\pi x$, which holds exactly at every argument and which no independent pair of wrong functions satisfies.

Two thresholds, and only one of them is near one #

For a wing heaving harmonically with amplitude $h_0$, the lift has two parts. The apparent mass term is $\pi\rho b^2\ddot h$ — the inertia of the air the wing is accelerating — and the circulatory term is $2\pi\rho U b\,C(k)\dot h$. Their amplitudes are in the ratio

$$\frac{k}{2|C(k)|},$$

which is one where $k = 2|C(k)|$. Since $|C|$ is near a half up there, the root is close to one: it comes out at $k = 1.086$.

That is a genuine order-one threshold, and it is a comparison of two terms. Above it the air’s inertia dominates and the wing is essentially paddling; below it the circulation dominates and the wing is essentially flying. It is exactly the kind of threshold the folklore promises, and it is correct.

The other threshold is not that at all. The quasi-steady assumption sets $C = 1$, and its amplitude error is $1 - |C(k)|$. That reaches one per cent at $k = 0.0061$.

What a load calculation actually gets wrong #

The amplitude is the smaller half of the problem. A quasi-steady calculation says the lift is whatever the instantaneous angle asks for, which means the lift plotted against the angle over a cycle is a straight line. It is an ellipse at every non-zero reduced frequency, tilted because the amplitude is reduced and open because the lift lags.

The area inside the loop is what matters, because it is energy. A closed loop traversed in one direction is net work done by the wing on the air; traversed in the other, it is net work done by the air on the wing. A quasi-steady calculation, which draws a line of zero area, is unable to represent either — and the second of them is flutter, where the air feeds a structural mode and the structure comes apart.

That is why the phase threshold is the one to quote. A flutter margin is decided by the sign and size of a phase angle, and the phase reaches one degree of lag at $k = 0.003$ — three hundred and sixty times below the value at which the terms are equal.

Which way round the loop is traversed, and why one wing cannot flutter #

The loops in the figure above have area, and area is energy per cycle. The sign of that energy is the whole of aeroelastic stability, and the arc decides it in a way worth following, because it explains why the phenomenon the phase threshold exists for cannot happen to a wing with only one thing to do.

Take pure heaving. The circulatory lift is proportional to $C(k)\dot h$, and the work the air does on the wing over a cycle is the part of that force in phase with $\dot h$ — which is $\operatorname{Re} C(k)$, times a positive constant, with a minus sign in front because the lift opposes the motion. $\operatorname{Re} C$ runs from one down to a half and is positive at every reduced frequency whatever. So the work is negative at every $k$: a heaving wing always does net work on the air, the loop is always traversed in the damping direction, and no amount of tuning makes it otherwise.

A wing free only to move up and down cannot flutter. That is not a statement about this particular aerofoil or this particular structure; it is a property of the function, holding over the whole arc, and it is why aerodynamic damping in bending is something a designer can rely on.

Pure pitching is different, and the difference is where the elastic axis is. A pitching wing’s aerodynamic moment has a contribution from the lift acting at a distance from the axis, and the sign of that contribution depends on which side of the axis the lift acts. Put the axis far enough aft and there is a band of low reduced frequencies in which the moment’s in-phase-with-$\dot\alpha$ component changes sign — the air feeds the motion instead of damping it — which is single-degree-of-freedom torsional flutter, and it is the one case where one mode is enough. It is also the reason control surfaces are mass-balanced and the reason an unpowered control tab is a stability question rather than a convenience.

The classical case needs two. Bending–torsion flutter is an energy transfer between two structural modes with the air as the intermediary, and what makes it possible is precisely that the aerodynamic force is neither in phase with the motion nor a quarter cycle out of it. Bending alone is damped, torsion alone is usually damped, and the pair together can extract energy because the phase of the lift relative to the combined motion can be arranged, by the structure’s own mode shapes, to fall in the quadrant where the loop reverses.

That is what makes flutter not a resonance: a resonance is one mode driven at its own frequency by something outside it, and this is two modes coalescing in frequency and exchanging energy through a force neither of them applies. And it is why the phase threshold rather than the amplitude one is the number to carry. A quasi-steady calculation sets $C = 1$, which is real: it puts the lift exactly in phase with the motion, gives every loop zero area, and therefore predicts neither damping nor flutter. It does not get the flutter speed slightly wrong. It removes the entire mechanism from the problem, and then reports that the wing is neutrally stable at every speed.

Where real machines sit on the axis #

The numbers are not academic, because most things that fly are not at small reduced frequency at all.

An airliner in a phugoid oscillates at about 0.05 Hz at 250 m/s with a semi-chord near 2 m: $k \approx 0.0025$. Genuinely quasi-steady, and the only case in this list that is.

A helicopter blade in forward flight sees a one-per-revolution variation at about 5 Hz with a semi-chord of 0.2 m and a local speed near 150 m/s: $k \approx 0.04$. The amplitude is five per cent low and the lift is eight degrees late — small enough to ignore in a performance estimate and far too large to ignore in a vibration one.

A wing in atmospheric turbulence meets gusts whose wavelengths run from hundreds of metres down to metres, so $k$ runs from $10^{-3}$ to order one across a single spectrum. Averaging the lift curve over a gust distribution treats all of it as quasi-steady, and the correction is exactly this function.

An insect wing has $k$ between 0.2 and 0.4, which is past the point where the lift lags most and approaching the point where the apparent mass takes over. Nothing about such a wing is quasi-steady and nothing about it is well described by a lift curve.

Why the delay exists at all #

The half is worth pausing on. As $k \to \infty$, $C \to 1/2$ exactly, and the exactness is the interesting part: a wing oscillating infinitely fast gets precisely half the circulatory lift its angle asks for, not a tenth and not none.

The reason is that the shed wake immediately behind the wing has not gone anywhere. In the high-frequency limit the vorticity shed during the last half-cycle sits within a fraction of a chord of the trailing edge and induces a downwash that cancels exactly half the circulation the motion would otherwise produce. That factor of two is a property of the flat-plate geometry and the planar wake, and it appears in every unsteady aerofoil theory written since.

The lag that comes back #

One feature of the arc is worth its own paragraph, because it contradicts the intuition that faster means later.

The phase lag does not grow without bound. It rises from nothing, peaks at about fifteen degrees near $k = 0.2$, and then falls again, approaching zero as $k \to \infty$. A wing oscillating infinitely fast is in phase with its own motion — it simply carries half the lift.

The reason is visible in the arc. At small $k$ the function is near one and moving away from it, which means moving into the lower half-plane and acquiring lag. At large $k$ it has arrived near a half and is approaching it along the real axis, so the imaginary part is shrinking again. The maximum lag is the point on the arc furthest from the real axis, and it sits in the middle because the arc has to begin and end on it.

That has a practical consequence which is easy to state and easy to get backwards. The most dangerous reduced frequencies for phase-driven problems are the middling ones, not the high ones. A structure whose flutter mode lands near $k = 0.2$ is at the worst place on the curve; pushing the mode higher in frequency moves it towards less phase error, not more, though it also moves it towards the regime where the apparent mass takes over and a different term dominates the answer.

An audit of the two errors #

It is worth putting numbers on both errors at the reduced frequencies people actually use, because the two behave differently and the difference decides which one to worry about.

The amplitude error rises monotonically towards its limit of one half; the phase error rises, peaks and falls; and the term ratio is still under a tenth at $k = 0.2$, where the amplitude is already a quarter low. At $k = 0.003$ the amplitude is half a per cent low while the phase is a full degree late, which is the general rule stated in numbers: the lag is first order in $k$ and the amplitude deficit is second, so the phase always goes first. Every column crosses its own “small” at a different place, which is the whole difficulty with a single threshold, and no reading of the reduced frequency alone can produce this table.

The same shape, in a pipe #

The structure of this argument is not about wings. Any linear system driven harmonically has a transfer function, its magnitude and phase both depart from the quasi-steady values as the frequency rises, and the frequency at which they depart measurably has nothing to do with the frequency at which the driving and resisting terms are comparable.

A pulsating pipe flow is the same statement with the Womersley number in place of the reduced frequency: the parabolic profile stops being right at $\alpha = 0.28$ and the two terms the number compares are equal at $\alpha = 1$. A factor of three and a half rather than a hundred and seventy-eight, because the observable there is analytic in $\alpha^2$ rather than in $\alpha$ — but the same two questions, and the same answer to the wrong one.

What the picture cannot show #

Theodorsen’s theory is linear and its wake is flat. The wake is assumed to leave the trailing edge along the free-stream direction, convect at the free-stream speed and never roll up. Real wakes roll up, and at the amplitudes an insect or a helicopter blade uses they roll up within a chord. The function is exact within its own theory and the theory is not exact.

The thresholds are amplitude and phase errors in the circulatory lift alone. Adding the apparent-mass term changes the total lift’s error, and above $k \approx 1$ the apparent mass is the larger part, so quoting a “quasi-steady error” up there is quoting an error in a term that is no longer the answer.

Nothing here separates pitch from plunge. They have different apparent-mass terms and different effective angles of attack, and the figures draw the heaving case throughout. The circulatory transfer function is the same for both, which is what makes $C(k)$ worth having.

And there is no stall anywhere in it. Dynamic stall — where a wing pitching rapidly past the static stall angle carries far more lift than the steady curve allows, for a while, before losing it catastrophically — is the dominant unsteady effect on helicopter blades and wind turbines, and it is outside linear theory altogether.

Who found it, and when #

Wagner solved the step-change problem in 1925 and Theodorsen the harmonic one in 1935, both at a moment when aeroplanes had begun to shake themselves apart and nobody could say why. Küssner and Sears added the gust problems shortly afterwards. The functions have been in every aeroelasticity textbook since, always with a plot of $|C|$ against $k$, and almost always with a sentence saying that quasi-steady theory is adequate for small $k$ — without a number.

The surprising connection is that the two thresholds are separated by a factor which is itself a computed quantity of the theory and has never, as far as this collection can tell, been written down. It is $2|C|/\varepsilon$ evaluated where the terms balance, and at one per cent it is 178. That number is not in the literature because nobody has needed the ratio; the two thresholds live in different chapters, one in the section on regimes and the other in the section on accuracy, and the sentence that would compare them belongs to neither.

Where the ladder goes next #

Above this rung is the gust problem, which needs Sears’ function rather than Theodorsen’s and a spectrum rather than a single frequency: the wing is a filter, the atmosphere is an input, and the mean and variance of the load come from their product. That would close the loop with the lift at the mean angle, whose quasi-steady averages are the low-frequency limit of exactly that calculation.

Beside it sits the pipe with the same structure and above both sits the general statement about what a group of order one is worth. Below it is the delay itself, computed in the time domain, and the lift curve whose slope this function multiplies.