The lift that arrives late

A wing is at rest. At $t = 0$ it starts forward at incidence, and from that instant the flow round it satisfies the Kutta condition, has a stagnation point at the trailing edge, and looks in every photograph like a wing that is flying.

It has almost none of its lift.

The circulation the steady theory promises — $\Gamma = \pi c U \alpha$, the number behind the lift curve — takes tens of chord lengths to arrive, and the reason is a conservation law rather than any sluggishness of the fluid.

The debt, and who holds it #

Kelvin’s theorem says the circulation round a circuit of fluid particles cannot change. Draw a circuit enclosing the wing and all of its wake, far enough out that it was in still air before anything moved, and its circulation was zero and must stay zero:

$$\Gamma_{\text{bound}} + \Gamma_{\text{wake}} = 0 \quad\text{at every instant}$$

So a wing that takes circulation must leave the opposite behind it. That much is the starting-vortex argument and it is usually where the story stops.

The part that is not usually told is what the shed vorticity does while it is still nearby. A vortex of the opposite sign, sitting a fraction of a chord behind the trailing edge, induces a strong upward velocity at the wing — a negative downwash — which is indistinguishable, as far as the wing is concerned, from a reduction of its own incidence. The wing is therefore flying at an effective angle much smaller than the one it was set to, and it can only take more circulation as the old wake moves away and its influence weakens.

The lag is not inertia and it is not viscosity. It is the wing being shadowed by its own debt.

The model, stated in full #

Everything below comes from an unsteady vortex lattice, and since the essay’s numbers are the model’s, its assumptions are worth listing rather than implying.

• The wing is a flat plate at small incidence, represented by sixteen bound vortices at the quarter point of each panel, with a collocation point at the three-quarter point of each.
• The wake is a row of point vortices along the wing’s own line, convected downstream at the free stream speed and never rolled up.
• At every step there are two conditions and two kinds of unknown: no flow through the plate at every collocation point, and Kelvin’s theorem. The bound strengths and the newly shed vortex come out of one linear solve.
• The fluid is inviscid, the flow is incompressible, and the plate has no thickness.

The quarter-chord and three-quarter-chord arrangement is not a fudge: it is the placement for which this model’s steady answer is $2\pi\alpha$, which is checked rather than assumed.

What the solver computed, and how it was checked #

Three numbers, and the third is the one nobody put in.

Kelvin’s theorem holds to machine precision. It is imposed as one row of the linear system, so what is being checked is that the solve did what it was told: the worst total circulation anywhere in a four-hundred-step run is $6\times10^{-17}$.

The steady limit is $2\pi\alpha$. After two hundred semichords the circulation is 99.34 per cent of $\pi c U\alpha$, and it approaches from below, monotonically, at every step. The assertion refuses a run that overshoots, one that falls anywhere, and one whose first step already has most of the lift — the last being the case where there would be nothing to draw.

The arrival is slow in a way that is worth quoting. Half the circulation after 1.5 semichords, three quarters after 5.2, nine tenths after 14.3, and ninety-nine per cent only after 125 — which is sixty chord lengths. The tail is the striking part: the last per cent takes longer than the first ninety-nine.

The starting vortex is not one of many #

The wake is not a uniform sheet. The vorticity shed in the first instants is far stronger than anything that follows, because that is when the wing’s circulation is changing fastest.

That is why the starting vortex is visible in the classic laboratory photograph and the rest of the sheet is not, and why the wake behind a wing in steady flight carries no circulation at all along its length — only the trailing pair from the ends of the span, which is a different vortex system entirely.

The wake as an incidence the wing did not ask for #

There is a way of reading the growth curve that makes it obvious, and it is the way to keep.

At any moment the wing is flying at an effective incidence: the geometric one, minus whatever upwash or downwash the wake is inducing at it. Since the circulation is proportional to that effective angle, the curve above can be read directly as the fraction of the incidence that has survived the wake.

At the first step the wake is cancelling 86 per cent of the angle. After one semichord it is cancelling 57 per cent, after ten 15 per cent, and after a hundred about two. The wing was never at the wrong angle; it was at the right angle in air the wake had already turned.

That reading also explains why the tail is so long. The influence of a vortex falls off as $1/r$, which is a slow decay — the same slow decay that makes an incompressible pressure field global — so a starting vortex fifty chords behind still induces a percent-level upwash at the wing. Nothing dissipates it, and the only thing weakening it is distance.

Where the model is wrong, and by how much #

Wagner solved this problem exactly in 1925, and his answer starts at half the steady value rather than at a seventh. The model here is slower everywhere: the gap peaks at 0.20 of the settled circulation at about 0.6 of a semichord, and closes to a few hundredths by ten.

The reason is identifiable and is not a coding error. Each step’s shed vorticity is concentrated at a point a quarter of a step behind the trailing edge, while the exact solution spreads it along a sheet whose strength is singular there. A point cancels more of the incidence than the sheet it stands for, so the wing takes up its circulation more slowly. Refining the time step does not remove it — the model converges, as the step is refined, to a curve that is still low, because the error is in what a point vortex represents rather than in how many of them there are.

This is recorded rather than repaired for the reason this site records such things: tuning the shed position until the curve matched Wagner’s would produce a figure that agrees with a result it was fitted to, and would prove nothing at all. What the model gets right without tuning is the shape, the two ends, and the extraordinary length of the tail.

Why any of it matters #

Three places, and they are not exotic.

Gusts. An aircraft flying into a sharp-edged gust does not receive the load the steady curve predicts, at once; it receives it over the same kind of distance. The whole apparatus of gust load alleviation is built on functions of exactly this shape — Küssner’s, for a gust, beside Wagner’s, for a change of incidence.

Anything that flaps. A bird’s wing, an insect’s, a helicopter blade in forward flight and a fluttering aerofoil all change their incidence continuously, so they never reach the steady value at all. The governing parameter is the reduced frequency $k = \omega c / 2U$ — the number of oscillations per semichord of travel — and when $k$ is not small the steady lift curve is simply the wrong tool.

Wind-tunnel work. A model started impulsively in a tunnel takes the same tens of chords to settle, which at a metre chord and thirty metres a second is a second or two — long enough to matter for a short-duration facility and short enough to be missed.

There is a fourth place worth naming because it is where the effect is largest and least expected. A wing that has just been tripped — by a control surface deflecting, a gust arriving, or a change of incidence at the start of a manoeuvre — is in exactly this transient, and the load it carries during it is smaller than the steady curve says. Every load calculation that uses a steady coefficient for a rapidly changing condition is therefore conservative in one direction and optimistic in the other, depending on whether the load or the damping is what matters. That is why gust response is computed with indicial functions rather than with a lift-curve slope, and it is why the shape of the curve at the top of this essay, rather than its endpoint, is the thing an aeroelastician needs.

The same result, asked as a frequency #

A step change in incidence is one way to interrogate a system; a sinusoidal one is the other, and for this problem the two are the same result wearing different clothes.

An aerofoil oscillating at angular frequency $\omega$ in a stream $U$ has a reduced frequency

$$k = \frac{\omega c}{2U}$$

which counts radians of oscillation per semichord of travel — the same clock the growth curve above is plotted against. At $k \ll 1$ the wing travels many chords per cycle, the wake gets clear between changes, and the steady lift curve applies. At $k$ of order one it does not: the wing is always surrounded by the vorticity it shed a moment ago.

Theodorsen’s 1935 result gives the factor exactly — the circulatory lift is multiplied by a complex function $C(k)$ whose magnitude falls from one at $k = 0$ towards a half at large $k$, with a phase lag in between. It is borrowed here, like Wagner’s curve, and nothing on this site computes it; what the model above does supply is the physical content behind it, which is that half the lift is lost to the wake’s own induced angle and the other half is not.

The practical numbers follow at once. A helicopter blade at 4 Hz with a 0.5 m chord at 100 m/s has $k = 0.06$ and is quasi-steady. An insect wing at 200 Hz with a 3 mm chord at 3 m/s has $k = 0.6$ and is not remotely quasi-steady — which is why insect flight cannot be understood with a lift curve, and why the whole subject is written in terms of unsteady circulation instead.

What the picture cannot show #

Added mass is missing. The lift on a suddenly-started plate has a second part that the circulation curve does not include: an impulsive force from accelerating the fluid around the plate, which is infinite at the instant of the start and gone immediately after. The force of getting going is that term, and the two are usually plotted together with the impulsive part drawn as a spike.

The wake does not roll up. A real shed sheet rolls into a starting vortex within a chord or two, and the model convects its points in a straight line forever. This is a first-order effect on the picture and a small one on the circulation, because what the wing feels is mostly the total shed circulation nearby rather than its arrangement.

Nothing here separates. At incidences where a real wing would stall, the lattice continues to produce a linear answer. The unsteady case is worse than the steady one in this respect: a wing pitched rapidly can exceed its static stall angle by ten degrees before separating, and none of that is in these equations.

The case where being late is an advantage, briefly #

The note above that a rapidly pitched wing exceeds its static stall angle deserves more than a parenthesis, because it is the most consequential unsteady effect in the subject and it runs the opposite way to everything else in this essay.

Pitch an aerofoil up quickly — at a reduced frequency of a few hundredths or more — and the boundary layer does not have time to respond to the pressure distribution it is being given. It stays attached past the angle at which it would have separated in steady flow, so the lift keeps climbing along the linear curve well beyond $C_{L,\max}$. Then the leading edge does give way, and what forms there is not a separation but a vortex: a concentrated core of the shed leading-edge vorticity, sitting on the upper surface.

While that vortex remains over the aerofoil it is doing what a delta wing’s leading-edge vortices do — inducing an intense local suction — so the lift overshoots the static maximum substantially, sometimes by half again. It is a real force and it is available for as long as the vortex stays put.

It does not stay put. It convects downstream at some fraction of the free stream, and as it passes aft of the quarter chord the centre of pressure goes with it. The pitching moment breaks nose-down, hard, before the lift has fallen at all — moment stall precedes lift stall — and when the vortex finally leaves the trailing edge the suction goes with it and the lift collapses.

Reattachment on the way back down happens at a much lower angle than separation happened on the way up, so a cycle traces a hysteresis loop rather than a curve, and the area inside that loop is work done on or by the aerofoil each cycle — which is the mechanism of stall flutter.

The place this decides something is a helicopter’s retreating blade, which sees a low relative speed and must therefore take a high incidence once per revolution. Its forward-speed limit is dynamic stall, and what limits it is not the lift — the overshoot is helpful — but the nose-down moment excursion, which twists the blade and loads the pitch links once per revolution at whatever forward speed the machine is flown to.

None of it is in the lattice above, which has no boundary layer to be late and no leading-edge vortex to shed. What the lattice supplies is the other half of the same clock: the same reduced frequency governs both, and both say that a wing changing its incidence is not a wing at an incidence.

Where the model stops #

The lattice is a small-disturbance theory: flat plate, small angle, planar wake. It is at its best where the steady thin-aerofoil theory is at its best, and it inherits every one of that theory’s limits — no thickness effects, no viscosity, no separation, no compressibility.

It is also two-dimensional, which for the unsteady problem is a real restriction. On a finite wing the shed vorticity is a sheet with structure across the span as well as along it, and the settling is correspondingly different: a low-aspect-ratio wing settles faster, because its wake escapes sideways as well as downstream.

And the model is inviscid, so nothing in it decays. Wagner’s function approaches one and never quite gets there; a real wake diffuses, and after a few hundred chords its influence is gone for reasons this model has no mechanism for.

Who found it, and when #

Wagner published the indicial function in 1925 — the response of a two-dimensional aerofoil’s circulation to a step change in incidence, as an integral of Bessel functions with no elementary closed form. Küssner did the corresponding problem for a sharp-edged gust in 1936. Theodorsen’s 1935 report gave the frequency-domain version: an aerofoil oscillating at reduced frequency $k$ has its circulatory lift multiplied by a complex function $C(k)$ whose magnitude falls to a half as $k$ grows, which is the same physics with a different question asked of it.

The discrete-vortex method used here is much later and comes from the computational literature of the 1970s and 1980s; Katz and Plotkin’s textbook treatment is the standard statement of it. R. T. Jones’ two-exponential fit to Wagner’s function, drawn on the first figure, is from 1940 and is the form almost every flight dynamics code still uses.

Where the ladder goes next #

The wing’s circulation has now been followed from where it comes from through what fixes its value to how long it takes to arrive. What this whole field has taken for granted throughout is that the pressures acting on the wing are pushes — and the word “suction”, which appears in every account of a wing’s upper surface, suggests otherwise. It is worth settling.