The pocket on top of the wing

An aircraft at Mach 0.85 is comfortably subsonic. The flow over the top of its wing is not.

The flow accelerates over the upper surface — that is where the lift comes from — and if the free stream is fast enough, the local speed over the shoulder passes Mach one. There is then a pocket of supersonic flow embedded in a subsonic field, and getting out of it is the whole of the transonic problem.

The correction, and its remarkable honesty #

The Prandtl–Glauert rule is the simplest useful statement in compressible aerodynamics:

$$c_p = \frac{c_{p,0}}{\sqrt{1 - M^2}}$$

Take the incompressible pressure coefficient at a point and divide by $\sqrt{1-M^2}$. That is the whole rule. It comes from linearising the compressible potential equation, noticing that it becomes Laplace’s equation under a stretch of the $x$ coordinate, and mapping the result back.

What makes it worth an essay rather than a footnote is that it announces its own failure. The denominator goes to zero at Mach one, so the prediction diverges — and it diverges precisely where the linearisation it rests on stops being valid.

That is unusually well-behaved for an approximation. Most break down quietly, giving plausible wrong answers with nothing to indicate the trouble. This one produces an infinity, and the infinity is at the right place.

The solver refuses to evaluate it at or above Mach one, with a message saying so: the factor being undefined there is the theory speaking rather than a numerical accident, and returning a large finite number instead would misrepresent it.

The critical Mach number, as a crossing rather than a rule of thumb #

The free-stream Mach number at which the peak suction first reaches sonic conditions is the section’s critical Mach number, and it is found by intersecting two curves that come from different places.

The first is the section’s own suction peak, corrected. This site does not assume a value for it: it solves the exact incompressible flow round a Joukowski section, walks the surface, computes $c_p = 1 - |v|^2/U^2$ at every point, and takes the minimum. Then it applies the correction, which deepens it as the free stream speeds up.

The second is the sonic condition. The pressure coefficient at which the local flow reaches Mach one is a function of the free-stream Mach number alone,

$$c_p^* = \frac{2}{\gamma M^2}\left[\left(\frac{1 + \frac{\gamma-1}{2}M^2}{1 + \frac{\gamma-1}{2}} \right)^{\frac{\gamma}{\gamma-1}} - 1\right]$$

and it comes from the isentropic relations — nothing to do with the section at all. As the free stream speeds up, less suction is required to reach sonic, so this curve rises.

One curve falls, the other rises, and their crossing is a genuine prediction from two independent relations rather than a fitted rule.

Thicker sections give up sooner #

The dependence is the practical content, and it explains a great deal of aircraft shape.

A thicker section makes the flow accelerate more over its shoulder, so it has a deeper suction peak, so the corrected peak reaches the sonic value at a lower free-stream Mach number.

assertCriticalMachFallsWithSuction requires exactly that ordering across a family, and also requires the two curves genuinely to have been intersected — at each returned crossing it checks that the corrected peak and the sonic value agree to $10^{-6}$. The rejection test hands it a family in which the deeper suction has the higher critical Mach, which is what a sign slip in the correction would produce, and requires the refusal.

What happens above it #

Passing the critical Mach number is not itself a problem. A small pocket of supersonic flow over the shoulder does very little.

The trouble arrives when the pocket has to be closed. The flow inside it is supersonic and the flow downstream of it is subsonic, and there is no smooth route between them in a flow that is still accelerating. So the pocket terminates in a shock, standing on the wing.

That shock does two things, and the second is much worse than the first.

It costs total pressure, which is a drag. At the Mach numbers involved — the shock is typically at Mach 1.2 to 1.4 — the loss is a few per cent and would be tolerable on its own.

It imposes a violent adverse pressure gradient on the boundary layer beneath it. A shock at Mach 1.4 raises the static pressure by a factor of 2.12 over a distance of essentially nothing, and a boundary layer can only climb so much before it separates.

Where the suction peak comes from in the first place #

The critical Mach number is a property of a pressure distribution, so it is worth looking at the distribution that produces it rather than only at its minimum.

This is the tension that makes transonic design hard rather than merely fiddly. The suction peak is the lift. A section with no suction peak has no lift, so the critical Mach number cannot simply be raised by flattening the pressure distribution — it has to be raised by redistributing it, holding the area under the curve while lowering its maximum.

That is what the supercritical section does, and it is why the fix took forty years after the problem was identified: the obvious moves all trade lift for Mach number one-for-one, and the useful move is the one that does not.

Drag divergence, buffet, and the rest #

The consequences of shock-induced separation are the ones that killed test pilots in the 1940s, and they are all downstream of one thing: a separated wing behaves nothing like an attached one.

Drag divergence. Drag rises sharply above a Mach number a little past critical, as the shock strengthens and the separation grows. The rise is steep enough that it is usually characterised by where the slope reaches a threshold, and it sets cruise speed for every transport aircraft in service.

Buffet. The shock position is unstable when the layer beneath it separates: separation changes the effective shape, which moves the shock, which changes the separation. The oscillation shakes the airframe and limits the usable envelope from above at the same time stall limits it from below.

Control problems. A shock ahead of a control surface changes the pressure the surface can develop and can render it ineffective or reverse it, and a shock moving as the surface deflects makes the whole system nonlinear.

Nose-down pitch. The centre of lift moves aft as the wing goes transonic, since the shock kills the forward suction and leaves the rear loading, and the resulting pitching moment is nose-down and large.

Two ends of one envelope #

An aircraft in cruise is squeezed from both sides, and the two limits have entirely different causes while producing the same symptom.

From below, the limit is stall: too little speed, too much incidence, and the boundary layer separates because it cannot climb the pressure recovery.

From above, the limit is buffet: too much speed, a shock on the wing, and the boundary layer separates because it cannot climb the shock.

Both are separation, and at high altitude the two limits converge — the aircraft has to fly fast enough for the thin air to support it and slow enough to stay below the shock, and the gap between them narrows with height until there is very little of it left. The name for that gap is not this site’s to give, but the physics of both its edges is: one is a gradual adverse gradient the layer cannot survive, and the other is a discontinuous one.

The fix, which is to move the curvature #

Everything above says that the problem is the suction peak, so the fix is to arrange for a shallower one.

The supercritical aerofoil — Whitcomb, in the 1960s — flattens the upper surface and moves the curvature aft, which spreads the suction over a longer stretch instead of concentrating it. That raises the critical Mach number for the same thickness, and, more importantly, gives a weaker shock when the pocket does form.

The section then loses lift at the front, so it is given a strongly cambered aft underside to recover it — the distinctive rear loading of a modern transport wing. It also allows the section to be thicker at the same Mach number, which buys structure and fuel volume.

Sweep is the other answer, and it is a different argument. What matters to the section is the component of the free stream normal to the leading edge, and sweeping the wing back by $\Lambda$ reduces that component by $\cos\Lambda$. A wing swept 35° sees a normal component of 0.82 of the flight Mach number, so an aircraft can fly at Mach 0.85 with a wing that thinks it is at Mach 0.70.

Both are on every large transport built since the 1970s, and both are answers to a curve crossing another curve.

What the numbers actually come to #

Worth putting the values in one place, because the range is narrower than the discussion suggests and that narrowness is the design difficulty.

For the Joukowski family solved here at two degrees of incidence:

Doubling the thickness from 6 to 12 per cent costs nearly three points of Mach number, and tripling it costs six and a half. There is no range in which the penalty disappears, and — worth noticing — none in which it is dramatic either. A designer wanting to cruise at Mach 0.85 with a straight wing of plain section cannot get there by thinning it: even the six per cent section goes critical at 0.645, which is a long way short. That is the corner sweep and supercritical sections were invented to escape, and it explains why both were needed rather than either alone.

Note also what the table does not depend on: any property of the air. Critical Mach number is a function of the section shape and the incidence, and the same wing has the same critical Mach at sea level and at altitude. What altitude changes is the true airspeed that Mach number corresponds to, which is the temperature argument from the beginning of this field.

What the solver computes, and how it is checked #

The peak suction is not a supplied number. peakSuction builds the exact Joukowski flow at the given camber, thickness and incidence, walks 600 points round the outline stepping slightly outside the surface, computes the pressure coefficient from the solved velocity at each, and returns the minimum.

That means the critical Mach numbers here are properties of a solved section rather than of a plausible value, and the family in the second figure is a family of genuinely different flow solutions.

criticalMach then bisects for the crossing of the two curves and returns both values at the crossing so the caller can verify they meet. prandtlGlauert refuses at or above Mach one, and criticalMach refuses a section whose peak is not a suction at all — a positive $c_p$ has no place on the section for the flow to go sonic first.

The number that is computed, and the number that is used #

There is a gap between the quantity this essay computes and the quantity an aircraft is specified by, and it is worth naming because the two are constantly conflated.

The critical Mach number is where the first point on the section reaches sonic conditions. It is a crossing of two curves, it is computable from a solved incompressible field and a linear correction, and — this is the awkward part — nothing happens there. A pocket of supersonic flow a few per cent of chord long, closed by a shock at Mach 1.05, costs a loss of total pressure too small to measure.

What matters is the drag-divergence Mach number: the speed at which the drag rise stops being negligible. It sits some way above critical — typically five to eight points of Mach number for a conventional section — and every cruise speed in service is set just below it.

And it is a convention rather than a measurement, in exactly the way a Reynolds number’s length is. One manufacturer defines it as the Mach number at which the drag coefficient has risen twenty counts above its low-speed value; another as where the slope of drag against Mach reaches a tenth. The two definitions give different numbers for the same wing, and a divergence Mach quoted without its definition is not comparable with anything.

The gap between the two is also where the real design gain hides. A supercritical section’s advantage is not only that it goes critical later; it is that it can be flown further past critical before the shock becomes strong enough to matter, because the pressure distribution keeps the pocket weak. So the useful quantity is a margin, and nothing in a linear correction can compute a margin — the correction has already failed by then, which is the section above.

Where the model stops, and this one stops early #

This is the most heavily qualified essay in the phase, and the qualifications are not incidental.

The correction is first order. Prandtl–Glauert linearises the potential equation, so it is valid for thin sections at small incidence and modest Mach numbers, and every one of those conditions is strained near the critical Mach number. Better corrections exist — Kármán–Tsien, Laitone — and they differ from this one by several per cent in exactly the range that matters.

The onset is computed and nothing above it is. The critical Mach number is a crossing of two curves and is honest. The shock’s position, the pocket’s extent, the separation and the drag divergence Mach number are all not computed here, and no figure on this site claims them. The transonic flow field is a mixed elliptic-hyperbolic problem with a free boundary, and it needs a solver of a different kind entirely.

The boundary-layer half is out of reach. This site’s separation estimate has no transition model and puts laminar separation at 31 per cent of chord at zero incidence, which is why real wings work and this estimate cannot say when they stop. Applying it under a shock would be worse than useless.

Two-dimensional. Sweep, taper, and the three-dimensional relief that makes a real wing’s behaviour differ from its sections’ are absent.

What the picture cannot show #

The critical-Mach figure is a plot of two curves crossing, and everything interesting happens on the far side of the crossing where neither curve means anything any more.

The corrected suction curve continues past the crossing in the drawing, and it should be read as extrapolation rather than prediction: once there is a supersonic pocket on the section, the flow is not the incompressible flow scaled, and the correction has no claim on it.

Nor is there any picture on this site of the pocket itself — no sonic line, no shock standing on a wing, no separated region behind it. That is a deliberate absence. Drawing one would require solving a transonic flow field, and drawing a plausible one instead would be exactly the failure this site exists to avoid: a smooth, convincing, captioned picture of a flow nobody computed.

Who found it, and when #

Prandtl had the correction by 1922 and Glauert published it in 1928, both working from the linearised potential equation. Glauert’s paper is short and the result is one line, and it was the standard tool for two decades.

The transonic difficulties arrived in the field before the theory did. Propeller tips reached critical Mach numbers in the 1920s and lost efficiency for reasons nobody could explain; dive recoveries in the early 1940s met control problems that were attributed to a “sound barrier” that was described in the press as a wall. It is not a wall — there is nothing discontinuous about the drag rise, and the aircraft that came apart did so from shock-induced separation and control reversal rather than from anything at Mach one.

Whitcomb’s area rule in 1952 and his supercritical aerofoil in the 1960s came from wind tunnels rather than from theory, and between them they are why a modern transport cruises where it does.

Where the phase goes next #

This phase has taken the compressible story from the speed of a signal to the drag it produces, and the transonic range is where it stops — not because there is nothing further, but because the honest computation stops here and the site does not draw what it has not solved.

What is left over from the phase is an instrument. If the pressure a pitot tube reads is not ½ρU² above the static pressure, then the airspeed every aircraft’s most basic instrument displays is wrong in a way that grows as $M^2/4$, and what it actually believes is the last essay of the phase.