How much uphill a layer can take

Separation is the moment the flow next to a surface stops following it. It causes most of the drag on most objects, it is what stalls a wing, and the usual account of it is qualitative: the layer runs into rising pressure, loses momentum, and eventually gives up.

That account is correct and it is not a number. This essay produces the number.

One parameter for the whole question #

Blasius describes a plate in a stream that neither speeds up nor slows down, which is the one case where the pressure does nothing at all. Every real surface has a pressure gradient along it, so the useful generalisation is the one that adds exactly one parameter.

Falkner and Skan took the external flow to vary as a power of distance, U ∝ xᵐ, and found that the similarity structure survives. The equation becomes

$$f''' + f f'' + \beta\,(1 - f'^2) = 0$$

with β = 2m/(m+1), and the whole of what a pressure gradient does to a laminar layer is in that single number.

β positive means the outer flow is accelerating, which by Bernoulli means the pressure is falling — a favourable gradient. β negative means the outer flow is decelerating and the pressure is rising: adverse. β = 0 recovers the flat plate.

What each sign does to the shape #

The figure above is the answer in one picture, and it is worth reading carefully.

At β = 1, the profile is full: it rises steeply from the wall and reaches the free stream quickly. The falling pressure is pushing the fluid along, and every parcel near the surface has been helped.

At β = 0 it is the Blasius shape.

At β = −0.1 and −0.15 the profile is progressively hollowed. The fluid furthest from the wall barely notices — it has plenty of momentum and the pressure rise costs it little — but the fluid closest to the wall has almost none to begin with, having been slowed by friction, and the rising pressure eats what is left.

At β = −0.198838 the slope at the wall reaches zero. The fluid immediately against the surface has been brought to a standstill, and one step further would have it moving backwards.

That is separation, and this is what it is: not a dramatic event but the vanishing of a derivative. Everything downstream of it — the wake, the pressure that never recovers, the drag, the stall — follows from one number reaching zero.

Why the wall is where it fails #

The mechanism deserves stating plainly because it explains why separation always begins at the surface rather than in the middle of the layer.

Every parcel in the layer is being decelerated by the same adverse pressure gradient, which acts equally at every height. What differs is what each has to spend. A parcel at the top of the layer is moving at nearly the free-stream speed and can afford a great deal of deceleration. A parcel a hundredth of a millimetre from the wall is barely moving at all, because friction has taken its momentum, and it can afford almost none.

So the pressure rise is a flat tax on a very unequal distribution of wealth, and the poorest fluid fails first. The reverse flow that follows starts as a film against the surface and thickens outwards.

What the solver computed, and how it was checked #

Falkner–Skan is a harder shot than Blasius and the difficulty is worth describing, because it produced a bug that converged confidently on nothing.

The obvious criterion for a shooting method is the sign of f′(far) − 1: overshoot means the wall slope was too big, undershoot means too small. That is correct for Blasius and wrong here whenever β is positive. With a favourable gradient a wall slope that is too small does not leave f′ short of the free stream: the profile sags, f′² drops below one, the β(1 − f′²) term turns strongly positive, and the solution runs away upwards. Both an overshoot and an undershoot end at f′ = +4710. The test cannot tell them apart, every shot looks like an overshoot, and the bisection walks the wall slope smoothly down to zero and reports it.

What separates the two cases is the curvature. The real profile rises to the free stream with f″ ≥ 0 throughout, decaying to zero. A shot that is too large crosses f′ = 1 while still curving upwards; one that is too small turns over first, and f″ goes negative before f′ ever gets there. Classifying on that instead fixes it.

With the criterion right, the solve reproduces the tabulated wall slopes to six decimal places: 1.232 588 at β = 1, 0.927 680 at β = 0.5, 0.774 755 at β = 0.3, 0.469 601 at β = 0, 0.319 271 at β = −0.1.

The check that ties two solvers together #

There is an independent test available, and it is the best kind: two different equations that must describe one layer.

The Falkner–Skan equation carries f f″ where the Blasius equation carries ½ f f″. That is not a different physical problem — it is the same problem with η scaled by √2 — so the two wall slopes must stand in exactly that ratio.

Blasius gives 0.332 057. Times √2 that is 0.469 600. Falkner–Skan at β = 0, solved by a different root finder with a different criterion and a different step count, gives 0.469 601.

Nothing in either solve knows about the other. The gate asserts the agreement and refuses a wall slope that does not stand in the ratio.

Finding separation by asking where the solutions stop #

The separation value is located by bisecting on β, and the predicate is not what it first appears.

The obvious approach is to look for where the wall slope goes negative. It never does, because the shooting brackets f″(0) from zero upwards and cannot return a negative value. Bisecting on that condition simply walks to the bracket’s lower end and reports it — which is the second way this computation can converge confidently on nothing.

What actually happens below the separation value is that the attached branch stops existing. The curve of wall slope against β reaches zero with infinite steepness there, dβ/df″(0) going to zero, and below it the shooting has nothing to find. So the predicate is whether an attached profile exists at all, and the bisection is on that.

It converges to β = −0.198838, in 287 milliseconds, against a tabulated value of −0.198838.

The assertion beside it refuses three things: a value more than 0.002 away from the tabulated one, a value at which the wall slope has not actually reached zero, and a value below which a slightly less adverse gradient is not still attached — which would mean the search had found the wrong side of something.

What the number means in practice #

β = −0.198838 corresponds to m = −0.0904: an external flow decelerating as x⁻⁰·⁰⁹.

That is an extraordinarily gentle deceleration. Over a chord, it is a free-stream speed falling by about nine per cent. A laminar boundary layer cannot survive a nine per cent slowdown of the flow above it without separating, and that single fact explains a great deal of what aerofoils look like.

It explains why the pressure recovery over the rear of a wing has to be so gradual, and why a section designed for laminar flow has its maximum thickness well aft — to keep the pressure falling for as long as possible and make the recovery that follows as gentle as it can be.

It explains why a sphere separates just past its widest point, at around 82 degrees, when the ideal theory says the pressure does not begin to rise until 90. The layer does not merely fail to complete the recovery; it fails almost immediately after the recovery begins.

It also explains something about diffusers, which are the same problem in a duct. A diffuser slows a flow down to recover pressure, and its half-angle cannot exceed about seven degrees before the flow separates from the walls and the recovery is lost. Seven degrees is a famous number in internal-flow engineering, it is the reason wind tunnels have such long expansion sections, and it is the same constraint as this essay’s β wearing different units.

And it explains why turbulent layers are so often desirable. A turbulent layer mixes fast fluid down towards the wall and therefore arrives at the pressure rise with far more momentum near the surface. It tolerates gradients several times steeper. The dimples on a golf ball exist to trip the layer into turbulence early, accepting more friction in exchange for a much later separation and a much smaller wake.

The shape factor as a warning #

The shape factor tracks the whole approach to separation, and it is measurable in a way the wall slope is not.

For the flat plate it is 2.59. At β = −0.1 it is 2.80, at −0.15 it is 3.02, and at separation it reaches about 4.0. Those are the numbers in the figure’s legend, computed by integrating each solved profile rather than quoted.

The usefulness is that H can be obtained from a traverse across a real layer — measure the profile, integrate it twice, take the ratio — without knowing the pressure gradient or anything about the layer’s history. An experimentalist watching H climb past 3.5 knows separation is close. It is a one-number state of health, and this is where the scale on it comes from.

What the picture cannot show #

The profiles are drawn in similarity units, so nothing on the vertical axis is a length, and the layers at different β cannot be compared for thickness. They can only be compared for shape.

More importantly, the family is a set of separate similarity solutions rather than a picture of one layer developing. Each curve is the profile of a flow whose external velocity has followed a power law with that exponent all the way from the leading edge. A real layer on a real aerofoil experiences a gradient that changes along the chord, and it carries a history — its profile at any station depends on where it has been, not only on the local β.

That is a genuine limitation of the whole similarity approach, and it means the separation value is a guide rather than a criterion for a real surface. Predicting separation on an actual aerofoil requires marching a layer along it, which is a different calculation.

The figures also stop at separation and show nothing beyond. There is something beyond — profiles with reverse flow near the wall exist on a second branch — but the boundary-layer equations themselves become ill-posed once the flow reverses, and marching downstream through a separation point is not a well-defined operation. Where the curve stops is where the theory stops.

Why the branch stops, and how the stopping was got round #

Where the curve stops is where the theory stops is the right description and it is not the end of the story, because the reason it stops turned out to be a defect in how the question was asked.

Marching a boundary layer along a surface with the pressure prescribed — the outer flow computed once, inviscidly, and handed to the layer as data — produces a singularity exactly at separation. The wall stress approaches zero like the square root of the distance remaining, and the displacement thickness’s slope becomes infinite there. Goldstein established that in 1948, and for twenty years it was taken as a statement that a marching calculation simply could not pass a separation point.

It is not a property of the flow. It is a property of the splitting. Prescribing the pressure asserts that the outer flow does not care what the layer is doing, and near separation that is false: the layer is thickening rapidly, its displacement pushes the outer streamlines aside, and the pressure distribution the layer is being handed is not the one it would produce. The problem has been over-determined, and a singularity is what an over-determined problem produces.

Two repairs work and both are used. Run the calculation inverse: prescribe the displacement thickness, or the wall stress, and compute the external velocity that goes with it. Catherall and Mangler showed in 1966 that the solution then passes through separation without any singularity at all and continues smoothly into a region of reversed flow. Or couple the two problems and solve them simultaneously, so that the layer and the outer flow negotiate the pressure between them rather than one dictating to the other.

That second repair is what made computational aerofoil analysis practical. Every viscous–inviscid code in ordinary use solves the panel method and the integral boundary layer as one system, which is why such codes can compute a laminar separation bubble — separation, transition in the free shear layer, and reattachment — on a section that a direct march would have refused to continue past.

So the boundary-layer equations are not ill-posed at separation. The habit of solving them one way and then the other is.

Where the model stops #

Everything here is laminar, steady, two-dimensional and incompressible.

Laminar is the biggest restriction, and it means this essay describes the case that separates most easily. Turbulent separation is a different calculation with a different criterion and it happens much later, which is why almost every practical surface is deliberately turbulent.

Similarity is the second, discussed above: real gradients are not power laws.

Two-dimensional excludes the three-dimensional separation that dominates on swept wings and slender bodies, where the flow can escape sideways instead of reversing and the whole topology of the separated region is different.

Incompressible holds only to about Mach 0.3; a shock wave imposes a pressure rise that is essentially a step, which no layer of any kind survives, and shock-induced separation is a category of its own that the compressible essay opens.

And this site’s own grid solver, which produces the separated pictures elsewhere on the site, is coarse enough to under-resolve the bubble it draws. The number in this essay comes from an ordinary differential equation solved to six figures; the pictures of separation elsewhere come from a grid that gets the phenomenon right and the dimensions wrong. It is worth keeping the two apart.

Who found it, and when #

V. M. Falkner and Sylvia Skan published the family in 1930, at the National Physical Laboratory. Skan did the numerical work, by hand, and the tabulated values in that paper are still the ones quoted — including the separation value this essay’s bisection converges on.

The context was the same one that motivated most British aerodynamics of the period: aerofoil design, and specifically the question of how far the pressure recovery could be pushed before the flow let go. Falkner and Skan gave the laminar answer, and the answer was discouraging enough that the field turned its attention to turbulent layers and stayed there.

Skan’s part in it is worth naming properly, because the paper is usually cited as though it were one person. She was a mathematician at the NPL who spent a career doing the numerical work behind other people’s aerodynamics, on mechanical calculators, and the tables in this paper are hers. The computation that this essay reproduces in 287 milliseconds on a laptop took her a great deal longer, and the values agree to every digit she published.

The equation’s later career is mostly mathematical. It has been studied intensively as an example of a nonlinear boundary-value problem with multiple solution branches, non-uniqueness below the separation value, and behaviour at the turning point that is a small industry on its own. That the physically interesting value of β is exactly where the branch turns is not a coincidence — it is the same fact stated twice.

Where the ladder goes next #

Next rungs on this anchor: Thwaites’s method, which predicts separation on a real gradient by tracking momentum thickness alone and is accurate to within a few per cent despite being one correlation; turbulent separation and why it survives so much more; the separation bubble, where a laminar layer separates, becomes turbulent in the free shear layer, and reattaches — which is what actually happens on most model-aircraft wings; and three-dimensional separation, where the topology changes and the two-dimensional criterion says nothing.

Then across to when the flow lets go, which is the phenomenon this essay has put a number on, and to how thick is thin, whose solve this one generalises.