Where the straight line stops

The lift curve is a straight line and the theory that produces it has no way of ending. Feed a Joukowski section forty degrees of incidence and the solver returns a lift coefficient, calmly, with the flow wrapped round the leading edge at eight times the free-stream speed and reattached on the other side.

Nothing in an inviscid model can stall, because stalling is a boundary layer giving up and there is no boundary layer. So the end of the straight line has to be estimated by joining two solvers that share nothing, and the join is the whole content of this essay.

The two halves being joined #

The first half is the inviscid solution, which gives the velocity along the surface exactly. That velocity is what the boundary layer experiences as its external flow — the layer is thin, so the pressure across it is constant, and the pressure just outside it is set by the outer solution.

The second half is the Falkner–Skan family: the exact solutions of the boundary-layer equations under an external flow going as a power of distance, $U_e \propto s^m$. Those solutions are indexed by $\beta = 2m/(m+1)$, and they stop existing at $\beta = -0.198838$ — a number this site computed for itself, by bisecting on the wall slope until the attached branch runs out.

The join is a local similarity assumption. A real aerofoil’s external flow is not a power of distance, but at each station it can be asked what power law it locally resembles, from $m = (s/U_e)(dU_e/ds)$, and the corresponding $\beta$ compared with the separation value.

That is an estimate and it is not a boundary-layer solve. The essay says so three times because it is the kind of claim that gets quoted without its qualification.

What the solver computed, and how it was checked #

The section is 10% thick with 2% camber. At each incidence the Joukowski solution is evaluated at four hundred points along the upper surface, a hair outside it, giving $U_e(s)$ with $s$ measured along the surface from the nose. The derivative is taken by central differences, $m$ and $\beta$ follow, and the first station aft of the suction peak at which $\beta \le -0.198838$ is recorded.

The separation value is recomputed for every figure rather than pasted in as a literal, so it cannot drift from the solver that produced it.

The number that should be alarming #

Read the first row again. At zero degrees of incidence, on an ordinary section, the estimate puts laminar separation at 31% of the chord.

That is not a rounding error or an artefact. It is what a laminar boundary layer under this pressure distribution would actually do, and it is why the earliest aerofoil tests, at low Reynolds numbers, produced results so bad that some early experimenters concluded thick sections were useless.

What saves a real wing is transition. Somewhere along the surface the laminar layer becomes turbulent, and a turbulent layer carries far more momentum near the wall, so it can climb a much steeper pressure hill before separating. Every practical wing operates with a turbulent layer over most of its upper surface for exactly this reason.

This site cannot compute transition. The build-time grid does not resolve it, the local similarity estimate has no representation for it, and nothing on this page should be read as a prediction of where a real wing separates. What the estimate gives is the laminar answer, which is a lower bound in a specific sense: separation cannot be later than the laminar estimate unless something intervenes, and the something is transition.

Why the peak is the villain #

The pattern in the table is that the estimated separation point moves forward as the peak grows, and the two are linked by the recovery between them.

The suction peak is the minimum pressure on the surface. Everything downstream of it is the flow climbing back up to something near ambient at the trailing edge, and how steep that climb is depends on how deep the peak was and how much chord is left to do it in. A peak of 1.24 times the free stream at zero incidence needs a modest recovery; a peak of 2.94 at twelve degrees needs the flow to lose two thirds of its speed, and the layer has to survive all of it.

That is the sense in which stall is a pressure-recovery problem and not a curvature problem. The section’s shape has not changed between the rows of the table. What has changed is where the stagnation point sits — as incidence rises it moves onto the lower surface, so the flow has to come round the leading edge to reach the top, and that journey produces the peak.

What the layer is actually doing at the criterion #

The criterion is a number, and behind it is a picture worth having.

Separation is not the flow “peeling off” a surface, and it is not the layer being blown away. It is the wall shear stress reaching zero and then reversing: the fluid immediately next to the surface, which has the least momentum of anything in the layer, is asked to climb a pressure hill, runs out of energy, and starts moving backwards. Everything above it then has nowhere to go but outward.

That is why the criterion is about the gradient and not about the pressure. A layer can climb an arbitrarily large pressure rise if it is given enough distance; what it cannot do is climb too steeply. And it is why the fluid at the wall is always the first to go, which is the fact that makes a turbulent layer so much more resistant — turbulence transports momentum toward the wall, so the fluid down there has more to spend.

The estimate that does remember its history #

The local-similarity assumption is named above as the second-largest defect, and it is worth saying what the standard repair looks like, because it is one equation rather than a solver and it is available for a laminar layer without any transition model at all.

Thwaites’ method carries the momentum thickness forward along the surface instead of asking each station what power law it resembles. Substituting a correlated shear function into the momentum-integral equation makes it linear in $\theta^2$, and it integrates in closed form:

$$\theta^2(s) = \frac{0.45\,\nu}{U_e^{6}}\int_0^{s} U_e^{5}\,\mathrm ds'.$$

No differential equation is solved. The layer’s thickness at any station is a quadrature over everything that happened upstream — which is exactly the information the local $\beta$ does not have.

The separation criterion is then a parameter of the same shape but a different construction,

$$\lambda = \frac{\theta^2}{\nu}\frac{\mathrm dU_e}{\mathrm ds},$$

with separation at $\lambda \approx -0.09$. Set that beside the local $\beta$ and the difference is the whole point: both are a velocity gradient scaled by a length, and $\beta$’s length is the distance from the nose while $\lambda$’s is the thickness the layer has actually grown to. A layer that has spent its life in a favourable gradient arrives thin and survives a hill that would separate a thicker one; the local criterion cannot tell those apart and Thwaites’ can.

And it is a correlation rather than a derivation, which the site’s usual practice would want marked. The shear and shape functions are fitted to the exact solutions that exist — Falkner–Skan among them, so the family this essay uses is one of the inputs — and the 0.45 and the $-0.09$ are numbers from that fit. Against known solutions it typically places laminar separation within a few per cent of chord, and it degrades where the adverse gradient is strongest, which is unfortunately where it is being asked.

Its second output is more consequential than its first. Thwaites returns the displacement thickness as well, and that is what closes the loop back to the inviscid solve: the outer flow can be recomputed around the section plus its displacement thickness, giving a new $U_e$, giving a new layer, iterated to convergence. That coupling — an inviscid panel solution and an integral boundary layer, talking to each other — is what an aerofoil analysis code is, and it is the rung this essay says it lacks. Half of it is the quadrature above.

Two kinds of stall, and why the difference matters #

The estimate above gives one number, and real sections fail in at least two distinct ways that number cannot distinguish.

Trailing-edge stall happens on thick sections with gentle peaks. The separation point starts at the trailing edge and creeps forward as incidence rises, so the lift curve bends over gradually and the aircraft gives warning — buffet, a soft response, a loss of lift that arrives before it is complete.

Leading-edge stall happens on thin sections with sharp peaks. The layer separates near the nose and either reattaches in a short bubble or does not, and when the bubble bursts the whole upper surface separates at once. The lift curve stops abruptly and the aircraft departs without warning.

The estimate here produces the location and says nothing about the character, because character depends on whether the separated layer reattaches, which depends on transition inside the separated shear layer, which is several models beyond anything this site solves.

What a solved viscous flow does show #

The site does have a viscous solver, and it is worth being clear about which questions it can answer and which it cannot, because “there is a Navier–Stokes solve in the build” invites more confidence than it should.

What that solve resolves is separation on a bluff body at Reynolds numbers in the tens and hundreds: attached flow at 1 and 10, a standing pair of eddies at 40, a longer recirculation at 100. Those are measured from the field rather than asserted, and the circulation in the wake can be checked against the vorticity inside it to a fraction of a per cent.

What it does not resolve is anything about an aerofoil at a useful Reynolds number. A wing operates at $10^6$ or above; this grid is 150 by 76 cells. The boundary layer at those conditions is a small fraction of a cell thick, so the solve would be reporting a layer it has not represented — and the figures on the rest of this page use an inviscid outer flow and an analytic layer precisely because that combination is honest about what it contains.

There is one more limit recorded elsewhere and worth repeating here: this solver does not shed a periodic vortex street at these Reynolds numbers, and nothing on the site may say it does.

What the picture cannot show #

The inviscid field the estimate reads has no separated region in it. That is not a limitation to be apologised for — it is why the estimate is possible at all, since the pressure distribution used is the attached one — but it means the figures show a flow that does not exist past the marked point.

Once separation occurs, the whole outer flow changes: the effective body becomes the section plus its separated region, the pressure distribution flattens, and the peak the estimate was reading disappears. The estimate is therefore self-limiting in a way worth being explicit about. It is a prediction of when the assumptions it rests on stop holding, and past that point it says nothing.

The lift curve, and where this puts its end #

Setting the estimate beside the curve it is supposed to terminate makes the mismatch plain.

Read together, the two figures say something worth saying plainly: the inviscid theory predicts a lift curve that never ends, and the laminar estimate predicts one that ends absurdly early, and the truth is between them because of a process neither contains. Transition is not a detail. It is the single mechanism that makes practical aerodynamics possible, and both of this site’s models manage to omit it in opposite directions.

That is a satisfying place for a rung to stop, because it identifies precisely what the next model would have to add rather than gesturing at complexity. The missing piece is not more resolution or a finer grid. It is a criterion for where a laminar layer becomes turbulent, which is a different kind of physics from anything else on this site.

Where the model stops #

Three limits, in order of severity.

No transition model. The single largest omission, and the reason the numbers are far more pessimistic than a real wing.

Local similarity. Falkner–Skan describes a layer that has grown under a power-law external flow all the way from its origin. A real layer at 30% of chord has a history — the pressure gradients it has already passed through — and the local value of $\beta$ knows nothing about that. Integral methods, which carry a momentum thickness forward along the surface, do better and are the standard engineering answer.

Two dimensions, no sweep, no rotation. Stall on a real wing is a three-dimensional event that begins somewhere along the span and spreads, and where it begins is a design decision made through twist and section changes so that the root stalls before the tip and the ailerons keep working — a decision that interacts with everything the span does.

What a designer does with this #

The estimate is too pessimistic to size a wing with, and the shape of its answer is exactly what aerofoil design is organised around.

Move the peak aft. A section whose minimum pressure occurs at 40% of chord rather than at 5% has a shallower recovery over a longer distance, and it keeps a laminar layer far further back. That is the whole idea behind the laminar-flow sections of the 1940s, and their well-known fragility — they work beautifully until a squashed insect at the leading edge trips the layer early, at which point they are worse than an ordinary section.

Flatten the peak. Load the section over more of its chord rather than concentrating it at the nose. A rooftop pressure distribution, flat for much of the upper surface and recovering gently, is the modern compromise.

Recover as steeply as the layer will bear, and no more. Stratford’s criterion describes the steepest recovery a turbulent layer can just survive, and a section designed to sit exactly on it extracts the maximum lift for a given chord. Nothing here can evaluate that criterion, because it needs the turbulent layer that is not modelled here.

Give the layer help. Vortex generators, slots and blowing all do the same thing: put momentum into the fluid near the wall so it can climb further. A slotted flap is this trick and not a camber trick, which is why an inviscid model gets its geometry right and its value wrong.

Who found it, and when #

The separation criterion belongs to the Falkner–Skan family, published in 1930, and the specific value where the attached branch ends was computed by Hartree in 1937. The local-similarity idea — using it as an estimate on a body whose flow is not self-similar — is a piece of engineering pragmatism from the same decade, most associated with Thwaites and with Stratford’s later separation criterion.

The distinction between leading- and trailing-edge stall was systematised by NACA in the 1940s, in one of the great empirical programmes of the subject: hundreds of sections, tested to stall, and classified by what the lift curve did at the top.

Where the ladder goes next #

Below this rung, when the flow lets go introduces separation and how much uphill a layer can take locates it exactly, in the family of flows where “exactly” is available.

This rung is where that exact result is pointed at a shape it does not strictly apply to, and the honest thing to do with the answer is to state it as an estimate and record what it omits. The rung above would be a boundary-layer march with a transition model in it — which is a solver this site does not have, and would need before it could say anything about the top of a lift curve.