The condition that can be bought

The single most load-bearing statement in this collection is that lift is circulation, and the second is that the sharp edge decides which circulation. It is worth being precise about what kind of statement the second one is, because it is not the same kind as the first.

Ideal flow round a closed body has a one-parameter family of solutions. Every member of it satisfies the equations everywhere and the wall condition on the body; they differ by a circulation, and nothing in the mathematics prefers one. A panel method meets this as a linear system with one more unknown than it has equations, and the missing row is not a bookkeeping slip.

The Kutta condition supplies it, and its justification is physical rather than mathematical: at a sharp trailing edge every circulation but one produces an infinite velocity, and a real fluid will not do that. The corner is the whole of the argument.

Take the corner away #

An ellipse has no corner. Mapping a circle of radius $a$ with $\zeta = z + c^2/z$ for $c < a$ gives a section with rounded ends, and

$$\frac{\mathrm d\zeta}{\mathrm dz} = 1 - \frac{c^2}{z^2}$$

never vanishes on the circle. So the surface velocity is finite for every circulation there is, and nothing in the ideal problem chooses one.

The circle-plane solution is exact and short. On the circle,

$$|w| = \left|2U\sin(\theta - \alpha) + \frac{\Gamma}{2\pi a}\right|,$$

so the stagnation points sit at $\sin(\theta - \alpha) = -\Gamma/4\pi Ua$ and exist only while $|\Gamma| \le 4\pi Ua$. Everything below follows from those two lines.

The two edges, measured #

The contrast is worth measuring rather than asserting, and the measurement has a subtlety in it.

At the edge itself a cusped map has $\mathrm d\zeta/\mathrm dz = 0$ exactly, so the velocity is 0/0 — infinite in arithmetic and finite in the limit. Sampling just off it evaluates the limit the way a probe would.

At the Kutta circulation the numerator vanishes at the same rate as the Jacobian and the answer is about $U$: 1.29 here, at a sample 0.25° round the circle. At any other circulation the numerator does not vanish, and the answer grows without bound as the sample approaches the edge — 11.3 at the resolution used, and refining the sample fourfold raises it fourfold, which is the signature of a genuine singularity rather than a coarse grid.

The rounded section’s peak surface speed over the same range of circulation is 4.54, and refining the sample does not change it by half a per cent. There is no singularity to find.

That is the whole of the Kutta condition, in two measurements. It needs the corner — and a section without one is not a pathological special case invented for an argument. It is the trailing edge of a helicopter rotor blade, of a circulation-control wing, and of a great many propellers, all of which have a finite trailing-edge thickness for manufacturing reasons alone.

What blowing does #

The engineering consequence is a class of aerofoil this collection has not met before.

A circulation-control section has a rounded trailing edge and a slot just ahead of it that blows a thin sheet of air tangentially. The jet remains attached to the curved surface — the Coanda effect, which is a real phenomenon whatever the misconception essay says about the uses it gets put to — and carries the rear stagnation point round the trailing-edge radius.

Moving the rear stagnation point is changing the circulation, because the two are the same statement: $\sin(\theta_s - \alpha) = -\Gamma/4\pi Ua$. So a blown slot is a circulation control in the literal sense, and the lift follows the stagnation point round the edge.

The ceiling, in closed form #

The stagnation points meet at $\Gamma = 4\pi Ua$, above which there is no stagnation point on the surface at all and the model has run out. That gives a ceiling:

$$C_{L,\max} = \frac{4\pi}{1 + (c/a)^2}.$$

It is 6.35 for a nearly cusped section, 6.94 at ten per cent thickness, and 4π = 12.566 for a circle — checked here to a part in 10⁷. Those are lift coefficients an order above what a plain aerofoil reaches, and measured circulation-control sections do reach 5 to 8, which is the strongest evidence available that this is the right picture of what they do.

Notice which way the ceiling runs. It falls as the section is made more cusped, because the chord grows faster than the merging circulation does — so a fatter trailing edge is worth more, not less, which is the opposite of the intuition a designer brings from ordinary sections.

What the jet is not doing #

There is a momentum argument that looks as though it should explain circulation control and does not, and it is worth disposing of because it is the natural first thought.

A jet leaving the trailing edge and turned through a right angle carries a momentum flux, and turning that momentum produces a force. Non-dimensionalised, the momentum coefficient is

$$C_\mu = \frac{\dot m V_j}{\tfrac12\rho U^2 S},$$

and the lift gained by simply reacting the jet’s own momentum is $\mathrm dC_L/\mathrm dC_\mu = 1$.

Measured sections give 30 to 80. So between thirty and eighty times more lift arrives than the jet’s own momentum could account for, and the jet is therefore not supplying the lift: it is moving the rear stagnation point, and the whole section’s circulation follows. The air doing the lifting is the air going round the aerofoil, and the jet is a control input.

That factor is the reason the technique exists. A jet flap of the reaction kind buys one, and a mechanical high-lift system buys the same lift for far less energy; a technique buying thirty to eighty is worth the plumbing.

Reading it as a boundary-value problem #

There is a way of stating the whole essay in one sentence that is worth having, because it makes the circulation-control section stop being a special case.

Ideal flow round a body is an under-determined boundary-value problem, and every aerofoil theory is a rule for closing it. The Kutta condition is one such rule; the observation that the circulation of an impulsively started wing grows from a seventh of its settled value is the history closing it; the requirement that a body of revolution have no net source is another closure of the same kind. In each case something outside the Laplace problem supplies the missing statement.

That is why the panel method’s singular matrix is instructive rather than embarrassing. The system is telling the truth: there is a one-dimensional null space, its member is a circulatory mode, and the solver has to be told which multiple of it to add. Every panel code in existence contains that decision in one line, and the line is not a numerical detail.

And the rule can be changed. Blowing changes it, viscosity changes it for a rounded edge, and a starting transient changes it in time. A theory that hard-codes one closure and calls it a law will mis-handle every case where a different one applies — which is the situation this essay’s section is in, and is why circulation control looks paradoxical when it is met for the first time.

Super-circulation, and its price #

The name for lift beyond what the Kutta condition would give is super-circulation, and the price is worth naming beside the gain.

The energy is a compressor’s, and it is continuous: a blown flap costs power whenever it is deployed, which is why the systems that reached service — the boundary-layer control on the F-104, the blown flaps on the Buccaneer — bled it from the engines and paid for it in thrust. The Buccaneer’s system blew over the wing, the flaps, the ailerons and the tailplane at once, and the aircraft could not land safely without it.

There is also a stability problem the figures above make visible. The lift is proportional to a circulation set by a slot, so it is not a function of incidence in the usual way: a blown wing’s lift-curve slope is very high, and its lift does not fall off at stall in the ordinary manner because there is nothing conventional about how the circulation was established. That is excellent for maximum lift and poor for handling, and it is why the technique appears on carrier aircraft — where the landing speed matters more than anything — and almost nowhere else.

The other place the corner is doing work #

The essay’s argument generalises, and it is worth pointing at the other place in this collection where a selection rule rests on a geometric singularity.

The leading-edge suction force is finite and arises from an infinite velocity acting over zero area at a sharp leading edge. Round the leading edge off and the singularity goes, the suction becomes a finite peak on a finite area, and the total is nearly unchanged — which is a different outcome from the trailing edge’s, where rounding it removes a condition rather than smoothing an integrand.

The distinction is that the leading-edge singularity is an artefact of the linear theory being asked about a rounded nose, and the trailing-edge one is a genuine feature of the exact solution for a cusped section at the wrong circulation. One goes away when the model is improved; the other is what the model is telling the truth about.

Three other ways the condition gets bought #

Circulation control by blowing is the most literal, and it is not the only device in aviation that works by choosing a circulation the shape would not have given.

A slotted flap. A slot is not a nozzle — the gap does not blow the boundary layer along, it starts a new one on the flap and lets the whole multi-element system carry a circulation a single element could not. The Kutta condition still applies at each element’s own trailing edge; what has changed is how many trailing edges there are and how the elements’ circulations add.

Suction. Removing the boundary layer through a porous surface delays separation, which lets a section reach an incidence at which the Kutta circulation is much larger. That is not buying a different circulation at the same incidence; it is buying a larger incidence — and it is worth separating from what the word usually means here, since nothing in a fluid sucks and the pressure doing the work is always a push from somewhere else.

And a rotating cylinder. Lift with no wing at all is the extreme case: a body with no trailing edge, sharp or round, whose circulation is set entirely by the boundary condition on its surface. The ceiling computed above — 4π for a circle — is precisely that essay’s maximum lift coefficient, arrived at here from the merging of the stagnation points rather than from the Magnus force.

That last coincidence is worth noticing. A rotating cylinder and a blown ellipse are the same section with the circulation established two different ways, and they have the same ceiling for the same reason: the stagnation points have nowhere left to go.

Why the gain has to saturate #

A ratio of thirty to eighty cannot continue indefinitely, and the reason it stops is the same argument as the ceiling.

The gain $\mathrm dC_L/\mathrm dC_\mu$ is large when the jet is being used as a control — a small momentum flux repositioning a stagnation point and letting the whole section’s pressure field do the work. It falls towards one when the jet is being used as a propulsor, at which point the only lift left is the momentum reaction the jet supplies directly. The crossing between the two is where the circulation the section can hold has run out, and the section can hold no more than the ceiling above.

That gives the practical shape of every circulation-control measurement: a steep initial rise in $C_L$ with $C_\mu$, a knee, then a shallow line of slope near unity. The knee is not a viscous phenomenon and it is not the jet detaching. It is the stagnation points arriving at the same point on the surface, and the ideal calculation puts it at a definite lift coefficient for a definite thickness — 6.35 for a nearly cusped section, 6.94 at ten per cent thickness, 4π for a circle.

So the ceiling is a design number rather than a curiosity. It says how much of the wing has to be blown, and it says that a thinner section — the one that is better in every other respect — buys less per unit of compressor power. A blown section is thick for the same reason a high-lift system is complicated: the circulation has to be held by something.

A blunt edge is also a choice #

Finite trailing-edge thickness has been treated here as a case the rule cannot handle, arrived at for manufacturing reasons. It is also designed in, and the reason is worth having because it is the essay’s argument used constructively.

A trailing edge with real thickness separates at its two corners rather than at one point, leaving a small base region at a roughly uniform pressure — and that pressure is higher than the pressure a cusped edge would impose. So the rear of the upper surface is unloaded less than it would be, and the section carries more of its lift aft.

That trade is worth a great deal at transonic speed for a reason peculiar to it. Lift gained at the rear costs nothing in forward suction, and the forward suction peak is what sets the critical Mach number. A blunt or divergent trailing edge therefore buys lift in the one currency a transonic wing has none of, which is why the feature appears on modern transport wings rather than being tolerated on them. The price is base drag, roughly proportional to the thickness, and the optimum is where the two meet.

And the essay’s own warning arrives as a design constraint. Such a section’s circulation is fixed by where the layer leaves the two corners and by the base pressure — by viscosity — so its lift cannot be obtained from an inviscid solve with a Kutta condition applied. The rule the wing was designed around is one the wing does not obey.

What the ideal picture leaves out #

The boundary layer is what actually applies the Kutta condition. The corner argument says an inviscid flow cannot turn a sharp edge at infinite speed; the mechanism is that the boundary layer separates at the edge and sheds vorticity until the stagnation point arrives there. For a rounded edge the layer separates somewhere on the curve and its separation point is what sets the circulation — so a real rounded-edge section is not indeterminate, it is determined by viscosity rather than by geometry. Blowing works by moving that separation point.

The jet is a boundary condition that is not modelled here. There is no slot in the calculation, no jet momentum, and no wall curvature holding a sheet on. The essay computes the circulation’s consequences and quotes the measured response to blowing; it does not compute the blowing.

And the ceiling is inviscid. At a lift coefficient of six the adverse gradient on the upper surface is enormous, and a real section separates well before the stagnation points merge. That is why measured values are 5 to 8 rather than 12.6, and the gap is the whole of the viscous story.

Where the idea came from #

The Kutta condition is Kutta’s, from 1902, and Joukowski’s independently. Both stated it as an observation about sharp edges, and both were clear that it is a selection among solutions rather than a consequence of the equations — a clarity that has eroded in the retelling.

Circulation control by tangential blowing over a rounded trailing edge was developed in Britain in the 1950s and 1960s, principally at the National Gas Turbine Establishment and later at West Virginia University in the United States, and the demonstrator aircraft — the A-6/CCW, the X-Wing rotor — are from the 1970s and 1980s. None reached production, and the reason is the one above: the system is continuously powered and the aircraft cannot fly without it.

What this leaves #

A selection rule that reads as a law, a section for which the rule has nothing to say, and a device that exploits exactly that. The residual the Kutta condition discards is the fact that it was a modelling choice licensed by a corner, and taking the corner away is not a pathological case — it is a wing.

The next essay leaves the section entirely and asks what an aeroplane’s two lifting surfaces do to each other: more lift than weight.