Two answers to one question

The lift that arrives late established that a wing set into motion does not have its lift: the circulation it takes has to be paid for by a shed vortex, and until that vortex is far downstream its own downwash holds the wing back. Wagner’s function Φ(s) is the answer — the lift after a step change in incidence, as a fraction of the steady value, against distance travelled in semichords.

It is a genuine collapse. The whole history of a wing’s motion, reduced to one function of one variable, and every unsteady load calculation ever done is a convolution with it.

There are two such functions, and the difference between them is largest exactly where the loads are.

Why there are two #

A wing that is suddenly pitched changes its own boundary condition everywhere at once. The whole chord acquires a new incidence at the same instant, the bound circulation must change immediately, and an equal and opposite vortex is shed at the trailing edge.

A wing that flies into a sharp-edged gust meets it at the leading edge first. The gust front convects along the chord at the flight speed, so at $s = 0$ only the leading edge is in it, at $s = 1$ half the chord is, and not until $s = 2$ — two semichords, one chord — has the whole wing met it.

Those are different problems and they have different answers. The functions are Wagner’s Φ(s) and Küssner’s Ψ(s), and both are usually called “the indicial response” as though there were one.

One solver, two right-hand sides #

The way to make that concrete is to compute them with the same code, changing nothing but the downwash distribution.

The model is a flat plate discretised into bound vortices with a wake shed one vortex per step and convected at the free stream. The whole of the physics is in two statements: tangency at each control point, and Kelvin’s theorem — the total circulation of plate plus wake is unchanged, imposed exactly as a row of the linear system rather than as a correction.

That is the one thing the model must not get wrong, and it is checked: the total circulation stays at zero to 10⁻¹⁶ at every step of both runs.

From about four semichords onward they are nearly the same curve, which is why they get interchanged.

The first four semichords #

At one semichord Wagner’s is 0.464 and Küssner’s is 0.057: a factor of 8.2. The classical values are Φ(0⁺) = 1/2 and Ψ(0⁺) = 0, and the reason for the second is exactly the statement above — a wing cannot be lifted by a gust it has not met.

The discrete model reaches 0.464 rather than the classical 0.5 at $s = 1$, and the shortfall is worth recording rather than tuning away. It is the placement of the shed vortex: at a quarter of a step behind the trailing edge the solve reaches 0.43 by $s = 1$, at half a step 0.46 and at a whole step 0.50, and all three agree beyond $s = 20$. An early transient that depends on a discretisation choice must say which choice and what the others give, and this one uses the centroid of the shed segment.

What each is shedding #

The step in incidence sheds a large vortex immediately and decreasing ones after it. The gust sheds almost nothing for the first two semichords, because the bound circulation is barely changing, and then catches up.

That is the same information as the two indicial functions and it is more direct: the shed vorticity is $-\mathrm d\Gamma/\mathrm ds$ by Kelvin, so the wake is the derivative of the lift history. A wing that has not met the gust is not changing its circulation and therefore not shedding anything, which is a sharper statement than “the lift is small”.

Both arrive at the same place #

By thirty-eight semichords the two are 0.9678 and 0.9663 — indistinguishable, and both still one and a half per cent short of the steady value. That is Wagner’s famously slow approach: the wake vorticity nearest the wing is what holds the circulation back, and it convects away only linearly, so the tail of the function decays as $1/s$ rather than exponentially.

Twenty chords of travel and the lift is still three per cent short. For an aircraft with a two-metre chord at eighty metres per second that is half a second, which is a long time in a gust encounter and is the reason a quasi-steady calculation of a gust load is not a small approximation.

Where the half comes from #

Φ(0⁺) = 1/2 is worth an explanation rather than a citation, because it is the number that separates the two functions and it has a clean reason.

At the instant a wing is pitched, the bound circulation and the starting vortex are equal and opposite and are essentially at the same place — the starting vortex has not convected anywhere yet. A vortex pair of that kind has a much weaker far field than a single vortex, and the image effect at the wing is exactly to halve the circulation that satisfies the Kutta condition. As the starting vortex convects away its influence weakens, the bound circulation grows, and Φ climbs to 1.

The gust has no such instant. At $s = 0^+$ nothing has changed over almost all of the chord, the tangency condition is violated only in an infinitesimal region at the nose, and the circulation required to satisfy it is infinitesimal too. So Ψ(0⁺) = 0, and it stays small until a useful fraction of the chord is in the gust.

The distinction is which boundary condition changed and how much of it. That is a statement about the problem rather than about the fluid, and it is the reason the two functions cannot be reconciled by any choice of constants.

The added-mass part, which is in neither #

Both functions above are circulatory: they describe the growth of the bound circulation and the lift it produces. A step in incidence also produces a non-circulatory force at the instant it happens, from accelerating the air the wing displaces, and neither function contains it.

For a true step that force is a delta function — infinite, and integrating to a finite impulse. For any real motion it is finite and proportional to the acceleration, and it is what the reduced frequency’s first threshold is about: the apparent-mass lift equals the circulatory lift at $k = 1.086$, which is a frequency most wings never reach and most rotor blades pass through twice a revolution.

Leaving it out of the indicial functions is the right decomposition rather than an omission, because the two parts scale differently — one with the acceleration and one with the velocity — and a load calculation adds them separately. A gust encounter has both, and the added-mass part of a sharp gust is not small.

What using the wrong one costs #

The practical question is what happens when a gust calculation convolves with Wagner instead of Küssner, which is a substitution made more often than not.

Both are honest Duhamel integrals. They differ because they are answering different questions, and they differ most at the peak, which is the number a structure is designed to.

On a gust one semichord long the wrong function overestimates the peak by 68.9 per cent. At two semichords 27.9, at five 8.9, at ten 3.9, and by twenty-five it has crossed over and is slightly under. The error is largest exactly where the gusts are sharpest, which is where the loads are.

That direction is worth naming: using Wagner for a gust is conservative, so the error has never grounded anything. It has cost structural weight, on every aeroplane certified with the substitution, by an amount nobody chose.

What the certification requirement does #

Airworthiness codes specify gust loads with a one-minus-cosine profile of a stated gradient distance — the design gust penetration length — and the load is computed by exactly the convolution above. The requirement is written in terms of Küssner’s function for the gust and Wagner’s for the aircraft’s own response to it, because the aeroplane’s plunge in response to the gust is a step in incidence and uses the other function.

So both functions appear in the same calculation, doing different jobs. The gust penetration uses Küssner; the aeroplane’s alleviating motion — it rises, which reduces its own incidence — uses Wagner, and the ratio of the two responses is the gust alleviation factor. Getting them the wrong way round is therefore not a matter of choosing the wrong constant: it exchanges two functions that differ by a factor of eight where it matters.

What a conservative error costs #

It is worth being precise about why an error that is safe is still an error, because the argument is made badly in both directions.

A 69 per cent overestimate of a peak gust load is not 69 per cent of a wing’s mass. The gust case competes with the manoeuvre case, the ground loads and the fatigue spectrum, and on most structures it sizes some parts and not others. Where it does size a part, the mass penalty is far less than the load penalty, because a spar cap’s mass goes roughly as the load and a skin’s stability margin does not.

And it does not always add material. The overestimate falls through zero at about eighteen semichords of gradient distance and is 2.7 per cent low at forty, so the substitution is conservative only over the sharp end of the envelope. That is a small number and it is in the unsafe direction, which makes it the more interesting of the two: the reasoning that excuses the substitution — “it is conservative” — is not true of the whole range it is applied to, and nothing in the single-function picture says where the sign changes.

The direction of the mistake is nevertheless what makes it invisible where it is large. An error that adds material never produces an incident, so nothing ever asks the question that would find it, and the substitution propagates from one design manual to the next as a simplification that has never caused a problem. The only instrument that can see it is the one this essay used: solve the two problems separately and compare, which costs a few seconds and which nobody has a reason to do.

That is the same shape as the lift slope at low aspect ratio, where a formula is wrong by tens of per cent and the calculation using it reproduces itself perfectly. The difference is only that this one errs in the safe direction, and safety is not the same as correctness — it is what stops correctness from being checked.

Why the model is a flat plate #

The solver above is deliberately minimal, and its limits are worth listing.

Two-dimensional. A real wing has a span, the gust does not arrive at every station at the same moment on a swept wing, and the spanwise correlation of atmospheric turbulence is what makes a statistical gust analysis necessary at all. Nothing here has a span.

Flat plate, thin, linear. No thickness, no camber, small disturbances. The indicial functions are properties of the flat plate in incompressible flow and are used for real sections because the linear theory says the shape enters only through the steady lift slope.

Not a discretisation artefact. The two functions are computed by marching a panel solver, so the obvious worry is that the gap between them is a property of the panel count. It is not: at thirty chordwise panels rather than the default the two curves lie on the same separation to within the thickness of a line, and Küssner’s response is still near zero over the first semichord where Wagner’s is near a half.

A wake that does not roll up. The shed vortices convect at the free stream in a straight line behind the plate. A real wake deforms under its own induction, and for large disturbances that matters.

And incompressible. The steady lift slope the functions are normalised on is 2π, and a real section’s is nearer 0.105 per degree for the reason thickness and the boundary layer between them give. At transonic speeds the indicial functions are completely different: the initial value is set by piston theory rather than by added mass, and the approach to the steady value is not Wagner’s. That is a separate subject and this collection’s compressible essays are the nearest it comes to it.

Three functions, and there are more #

For completeness there are more than two indicial functions and it is worth knowing what the family is.

Wagner’s Φ(s) — step in incidence, whole chord at once, Φ(0⁺) = 1/2.

Küssner’s Ψ(s) — sharp-edged gust convecting past the chord, Ψ(0⁺) = 0.

The added-mass impulse, which is not an indicial function at all but a delta function: a step in incidence produces an instantaneous non-circulatory force from accelerating the air the wing displaces, and it is infinite for a true step and finite for any real motion. It is what the reduced frequency’s second threshold is about, and it is excluded from both functions above by construction.

And Theodorsen’s C(k), which is the same information in the frequency domain — the Fourier transform of Wagner’s function, essentially, and the right tool for a harmonic problem where the indicial functions are the right tool for a transient one.

Choosing among them is not a matter of accuracy. It is a matter of which question is being asked, and the essay’s whole point is that “what is the unsteady lift” is at least two questions.

Why everyone uses a fit that cannot be right #

Two things said above sit awkwardly together. The tail of these functions decays as $1/s$ — algebraically, because the wake convects away linearly — and the versions everybody actually uses are sums of exponentials.

A finite sum of exponentials decays exponentially, so it cannot reproduce an algebraic tail at all. Jones’s two-term fits are excellent out to about twenty semichords and then fall away too fast, understating the remaining deficit. That is a known error in a known direction, and it is used anyway.

The reason is not convenience. A convolution against an exponential kernel is exactly equivalent to one extra first-order differential equation, so replacing an indicial function by a sum of $N$ exponentials converts a memory integral into $N$ additional state variables — and the unsteady aerodynamics stops being a history and becomes part of an ordinary differential system that a flutter or gust code can march step by step. Those are the aerodynamic lag states every aeroelastic model carries, and each one is a term in such a fit.

The price is paid at long times and in the direction of underestimating the deficit — which is the opposite direction to the substitution this essay is about, so the two errors in an ordinary calculation are not obviously going to cancel.

The same distinction, three more times #

The pattern — one collapse, two problems that share it, and a difference that matters only where the loads are — recurs in this collection under other names.

A wing in a wind tunnel and a wing in flight share a lift coefficient and differ in what the walls are doing; the walls are in the answer computes the correction, and it is largest for the largest model, which is exactly the case a tunnel operator most wants.

A steady stall and a dynamic stall share a maximum lift coefficient and are not the same event: a section pitched rapidly through its static stall angle reaches a considerably higher lift before it gives way, and then loses it violently. That is what a rotor’s retreating blade does, and a quasi-steady calculation of it is wrong in both directions at different moments.

And a gust and a manoeuvre share a load factor and are different load cases in every airworthiness code, for exactly this essay’s reason: the aeroplane’s response to a control input is a Wagner problem and its response to a gust is a Küssner problem, and the certification envelope has two diagrams because of it.

In each pair the collapsed quantity is genuine and the two members of the pair are genuinely different problems. What makes them worth separating is not that the answers differ by a lot on average — they do not — but that they differ most in the regime the design is set by.

Where the functions came from #

Herbert Wagner published his in 1925 and Hans Georg Küssner his in 1936, both in German aeronautical journals, both as exact solutions of the linearised unsteady problem in terms of Bessel functions — which is why the exponential fits everybody actually uses are Robert T. Jones’s, from 1938 and 1940.

The two-function structure was clear to all three of them and has eroded since. Küssner’s paper is explicit that his function is not Wagner’s and gives the reason in the first page; the conflation is a later simplification, and it survives because the two curves look alike on the plot everybody draws, which runs to twenty semichords.

What this leaves #

One collapse, two functions, and a factor of eight between them at the scale that decides a sharp gust’s peak load. The residual the single-function picture discards is which boundary condition changed, and it is not a small correction: it is the difference between a wing whose whole chord has changed incidence and one that has met a gust with its nose.

A control surface on a wing that twists under its own load asks the same question about a different boundary condition, and answers it with a reversal: the control that works backwards.