Two lifts at one incidence

When the flow lets go is this collection’s account of separation, and its lift curves are static: each point is a wing held at an incidence until nothing more happens.

Almost no wing works that way. A rotor blade changes its incidence twice a revolution, for the reason the side that cannot keep up sets out. A wind-turbine blade meets a sheared inflow every revolution. A wing in a gust, an aerofoil in a compressor, a control surface being moved — all of them arrive at an incidence with a history, and near the stall the history is what decides the answer.

Why the static curve is not enough #

The mechanism has one moving part. Separation on a stalling aerofoil is a position: the point on the chord at which the flow leaves the surface. As the incidence rises that point moves forward, and the lift falls because less of the surface is producing suction.

The point takes time to move. A boundary layer responds to a change in pressure gradient over the time it takes fluid to travel a few chords, so the separation point at any instant is not the one belonging to the present incidence — it is on its way there, from where it was.

That is a first-order lag, and one lag is all this essay’s model contains. The separation point is taken to relax towards its static value over a fixed number of convective times, the lift is Kirchhoff’s formula in that separation point, and everything below is a consequence.

The loop #

Pitching the wing sinusoidally about twelve degrees, eight degrees either way, at a reduced frequency of 0.1, produces a lift curve that does not close.

Going up, the separation point is behind: at twelve degrees on the upstroke it sits at 0.922 of the chord where the static curve puts it further forward, so more of the surface is attached and the wing carries 1.264 against the static curve’s 1.212. Going down, the same lag works the other way — at the same twelve degrees the separation point is at 0.612, and the wing carries 1.043. Two lifts at one incidence, differing by 0.221, which is 18 per cent of the static value.

At twelve degrees the difference is 0.22 in lift coefficient. The wing’s shape and incidence are identical in the two cases; the only difference is which way it is going.

The loop is traversed anticlockwise, which is the whole of the engineering significance. An anticlockwise loop in a plot of force against displacement is a loop whose enclosed area is work done on the wing by the airstream. A wing that oscillates in this régime is being fed energy, and a structure that can oscillate in pitch will do so.

The area, and what it is largest at #

The loop’s area is not monotone in the pitching rate, and the shape of that dependence is the model’s clearest statement.

Pitched slowly, the separation point keeps up, the wing retraces its static curve, and the loop closes: at a reduced frequency of 0.005 its area is a fortieth of the maximum.

Pitched fast, the separation point cannot respond at all — it sits at wherever it was — so the wing behaves like an attached aerofoil with a fixed effective shape, and the loop closes again: at a reduced frequency of 1.6 the area is 1.254, below the 1.799 at 0.05 and less than a third of the 4.286 at the peak.

The maximum sits in between, at a reduced frequency of 0.2, where the product of the pitching rate and the model’s own lag is 0.8: of order one, which is the standard resonance condition for a first-order lag driven sinusoidally.

That number is the useful part. A wing is at its most hysteretic when it is pitched at the rate its separation forgets at, and both quantities are measurable independently.

The overshoot, which does rise all the way #

The peak lift reached in a cycle behaves differently from the loop’s area: it rises monotonically with the pitching rate and saturates, reaching 1.41 times the largest lift the static curve offers.

Nine pitch rates make the split visible:

The area column peaks at 4.286 at a reduced frequency of 0.2 and has fallen to 1.254 by 1.6, while the overshoot column rises monotonically from 1.005 to 1.405 and never turns over. The peak of the area sits where the reduced frequency times the relaxation time is 0.8 — the rate at which the separation point moves about as fast as the incidence asks it to.

The reason the two columns behave differently is that they are asking different questions. The loop’s area is about the difference between the two directions, which needs the separation point to move some but not all of the way. The overshoot is about how far the wing can get before separation begins at all, and pitching faster always buys more of that.

So a rotor blade can be flown past its static stall angle and carry lift there, which is what makes a helicopter possible on the retreating side — and the price is the loop, and the loop is what shakes the machine apart.

What the solver computed, and how it was checked #

The model is deliberately the smallest one that produces the effect: a separation point relaxing towards its static value over four convective times, and Kirchhoff’s lift formula in that separation point. It is the skeleton of the Leishman-Beddoes model with everything except the lag removed.

That is a model rather than a solution, and the distinction matters more here than in most of this collection. Nothing in it is a solution of the Navier-Stokes equations, no boundary layer is computed, and the four convective times is a number chosen to be plausible rather than derived. What the computation demonstrates is what a lag alone produces, which is the point: the loop, the overshoot and the resonance all follow from one first-order lag and need nothing else.

Three checks. That the loop has an area and that the two lifts at the mean incidence genuinely differ. That the peak exceeds the static peak, since a model that lagged without overshooting would be describing something else — at a reduced frequency of 0.1 it reads 1.479 against 1.295, an overshoot of 14.2 per cent. And that the loop’s area has an interior maximum — the check fails if the largest loop is at either end of the rate sweep, which would mean the resonance was not being captured.

The last of those was written the other way round first, requiring the area to rise monotonically with the rate, and it failed. That failure is the physics: at high enough rates the separation point stops moving and the loop closes. The check was wrong and the model was right, which is the useful direction for a check to fail in.

The same curve, averaged rather than traversed #

There is a neighbouring effect that is easy to confuse with this one and is completely different, and separating them is worth a section because both are about a wing whose incidence varies.

The lift at the mean angle is about a wing meeting a random gust field and asks what its average lift is. The answer is not the lift at the average angle, because the lift curve is bent: averaging over a bent curve and evaluating at the average are different operations, and near the stall they differ by five per cent at three degrees of gust.

That is a statement about the curve’s shape and it holds even if the wing responds instantly. It has no memory in it at all: freeze the aerodynamics, make the response quasi-steady, and the effect is unchanged.

This essay’s effect is the opposite. It survives a perfectly straight lift curve — put a lag on a linear response and the loop still opens — and it vanishes if the response is instantaneous, however bent the curve is.

So one is a nonlinearity without a memory and the other is a memory without a nonlinearity, and a real wing near the stall has both, acting on the same signal, at the same time. Any measurement of one of them in the presence of the other has to separate them by their rate dependence, since only the second has any.

What a real dynamic stall adds #

The model above is a lag and a lift formula, and a real dynamic stall has more in it. The additions are worth naming because they make the effect larger rather than smaller.

A leading-edge vortex. Past the static stall angle a vortex forms at the leading edge, grows, and convects back over the surface. While it is over the surface it adds a large suction, so the lift overshoots far beyond anything an attached model gives; when it passes the trailing edge the lift collapses. The peak lift on a real pitching aerofoil can be twice the static maximum rather than 1.4 times.

A moment that collapses with it. The vortex’s suction moves aft as it convects, so the pitching moment goes sharply nose-down as it leaves — the “moment stall”, which typically happens after the lift peak and is what actually breaks things.

And a slow reattachment. The flow does not reattach at the incidence it separated at; it reattaches several degrees lower — the same species of effect as the transition hysteresis in where the straight line stops, where a boundary layer that has already given way is not in the state a boundary layer at that condition would be in. That is a second hysteresis, on top of the one modelled here, and it is why the loop in a measurement is fatter at the bottom than this one.

Where this decides an engineering answer #

The mechanism being traversed is the one lift out of a failure computes statically — a separation that is useful while it is forming and ruinous once it has formed.

Retreating-blade stall. A helicopter’s retreating blade meets low relative speed and needs high incidence, and it gets there by pitching. The overshoot is what lets it work at all; the loop is what puts a large oscillatory torsion load into the blade and control system, and it is the limit on the forward speed of a conventional helicopter.

Stall flutter. A wing oscillating in pitch inside the loop is being given energy every cycle. Unlike classical flutter it needs no coupling between two modes — one mode suffices, because the aerodynamic work is already positive. That makes it possible at speeds far below the flutter boundary the lag that makes flutter possible computes, and it is the failure mode of compressor blades and of stalled wings.

Wind-turbine fatigue. A blade passing through the tower’s wake and the atmospheric shear changes incidence once a revolution, at reduced frequencies squarely in the range where the loop is largest. The resulting load cycles are what set the blade’s fatigue life.

And a measured lift curve. A curve obtained by sweeping incidence continuously rather than in steps has a hysteresis in it that belongs to the sweep rate, not to the aerofoil. The instrument in the answer is the general form of that warning.

What kind of memory this is, beside the others #

Placed against the other memories this collection measures, this one is unusual in three ways, and each of them limits what can be done with it.

It is nonlinear, so there is no kernel. The response to a general motion is not the motion convolved with anything, because the separation point moves on a curve whose shape depends on where it is. That is the sharpest contrast with the lag that makes flutter possible, where the wake’s memory is exactly one complex number per frequency and superposition holds.

It is bounded in extent rather than in time. The separation point can only be somewhere on the chord, so however violent the history the state is confined — which is why the overshoot saturates rather than growing without limit.

And it is stored on the body rather than in the fluid. The wake memories in this collection travel away and eventually stop mattering; this one sits on the surface and is reset only when the flow reattaches.

That last property is what makes stall flutter possible with a single structural mode. A memory that convects away supplies a phase lag; a memory that stays supplies a phase lag and a state the structure can drive, and the loop is the energy that comes out of driving it.

Why the memory is in a position rather than in the flow #

The other memories in this collection are stored in the fluid — in the wake, in a vorticity distribution, in a displacement. This one is stored in the location of a point on the surface, which is an unusual place for a fluid to keep anything, and it is worth asking what it really means.

The separation point is not itself a physical object; it is where a computed quantity changes sign. Its lag is really the lag of the boundary layer’s own profile, which is an integral of the pressure gradient it has met — and that integral is how far before a duct forgets what was fed into it applied to a wing rather than to a pipe.

So the memory is the boundary layer’s, and the separation point is a one-number summary of it. That is why the model works at all with a single state variable, and it is exactly why it stops working when the leading-edge vortex forms: at that point the flow has a second state — the vortex’s position — and one number is no longer enough.

How a measurement separates the two #

Since a real experiment produces both effects at once, it is worth saying how the separation is actually done, because the method is a good example of using a rate dependence as a fingerprint.

Sweep the rate. The hysteretic part depends on the reduced frequency and the curvature part does not, so measuring at three or four rates and extrapolating to zero rate gives the quasi-steady answer with the memory removed. That extrapolation is the standard technique and it is why measured static curves are quoted with a sweep rate beside them.

Watch the direction. The curvature effect is symmetric between upstroke and downstroke; the hysteretic one is antisymmetric. Adding the two directions removes the memory and subtracting them isolates it, which is a cleaner separation than a rate sweep and needs only one run.

And check the loop’s sign. An anticlockwise loop is energy going into the wing; a clockwise one is energy coming out. A measurement that produces a clockwise loop near the stall has almost certainly got a phase error in its instrumentation rather than a wing that damps itself, and the sign is the first thing to check because it decides whether the structure is being driven or damped.

What the picture cannot show #

The loop is drawn in lift against incidence, which is the plane the work is done in, and the drawing therefore hides the time. Two points on the loop at the same incidence are separated by half a cycle, and nothing in the picture says so. A reader tracing the curve is tracing a trajectory, not a function.

Nothing here draws the flow either. There is no boundary layer in the figures, no separation line and no vortex, because the model contains none of them: it contains a number between zero and one that stands for all of it.

Who found it, and when #

Dynamic stall was found by helicopter engineers in the 1960s, in the sense that it was named then; the phenomenon had been limiting rotor speeds since rotors existed. The systematic measurements are McCroskey’s, from the 1970s and 1980s, and they are still the reference data.

The reduced-order models came afterwards and for a practical reason: a rotor code has to evaluate the loads at every blade element at every azimuth, and cannot solve a boundary layer to do it. The Leishman-Beddoes model of 1989 is the standard one, and the lagged separation point used here is its central idea with the rest removed.

Limits recorded rather than smoothed over #

This is a model, and a deliberately minimal one. No boundary layer, no leading-edge vortex, no compressibility, no moment. Its purpose is to show that a lag alone produces a loop, an overshoot and a resonance, and it does not predict any real aerofoil’s numbers.

The lag is a chosen number. Four convective times is plausible and is not derived. The resonance condition — that the largest loop sits where the pitching rate times the lag is of order one — is the robust statement; the reduced frequency of 0.2 at which it happens moves with the lag.

The static curve is a fit. The separation-point curve behind the static lift is an algebraic expression chosen to have a stall in about the right place, not a measurement.

No compressibility. A retreating blade is at low Mach number and an advancing one is not, and a real rotor’s dynamic stall interacts with shock-induced separation on the other side of the disc. The shock that lies on the body is the other half of that problem and nothing here contains it.

And the loop’s area is quoted in degrees times lift coefficient, which is a work per unit span divided by the dynamic pressure, the chord and a radian conversion. It is a comparison between the rates, not a number of joules.