Which speed goes in the number

Somewhere in the first fortnight of every course on the subject a number is handed over: air may be treated as incompressible below Mach 0.3. It is a good number. It is quoted more often than any other threshold in fluid mechanics, it is roughly right, and almost nobody can say what it is a threshold for.

There are three separate claims hiding under it, they give three different numbers, and only one of them is about the body.

The first claim, which is where the number comes from #

Bring air isentropically to rest and its density rises:

$$\frac{\rho_0}{\rho_\infty} = \left(1 + \frac{\gamma-1}{2}M^2\right)^{1/(\gamma-1)}.$$

Set that to $1 + \varepsilon$ and invert. One per cent gives $M = 0.141$; five per cent gives $M = 0.314$; ten per cent gives 0.441.

That is the number. It is a five per cent tolerance on the density at a stagnation point, and it is exact for that purpose. Nothing is wrong with it except that the tolerance has been stripped off in transmission, so it arrives as a boundary of physics rather than as an engineering allowance — and a reader who needs one per cent, or who is happy with ten, has no way to move it.

Nobody who quotes 0.3 says what happens at 0.35. Nothing does; the density change is 6.4 per cent instead of 5.

The second claim, which agrees for a different reason #

The other common justification is about the pressure coefficient rather than the density. An incompressible calculation gives some $c_{p0}$; the Prandtl–Glauert correction scales it by $1/\sqrt{1-M^2}$; so the incompressible answer is wrong by that factor minus one.

Inverting: one per cent at $M = 0.140$, five per cent at $M = 0.305$, ten at $0.417$.

The same two numbers, to two figures, by a completely unrelated route. That coincidence is why the threshold feels robust — two arguments, one answer — and it is a coincidence: one is a thermodynamic relation at a stagnation point and the other is a linearised correction to a potential flow, and their agreement to two figures over the range that matters is arithmetic rather than physics.

The third claim, which is the one about the body #

Neither of the above mentions the body. But every body in a stream has a stagnation point, where the speed is zero, and a suction peak, where it is highest, and the two compete for the title of the worst place on it.

To leading order in $M$, the stagnation point’s density change is $M^2/2$ and the suction peak’s is $-c_{p0}M^2/2$, so the two are equal when $-c_{p0} = 1$. That is exact, and it is worth stating plainly:

The suction peak beats the nose exactly when $c_{p0} < -1$ — which is where the local speed is $\sqrt{2}$ times the free stream.

A thin section at low incidence never gets there. A wing working for its living does: a $c_{p0}$ of $-2$ is an ordinary cruise value, a high-lift section at approach runs to $-4$ or below, and a slat can be at $-8$.

What the numbers do #

Running the three cases through the isentropic relations with Prandtl–Glauert supplying the peak:

The middle row is the crossover, and it lands on 0.3 as well — which is a third coincidence and a tidy one. A section at $c_{p0} = -1$ is exactly the case for which the familiar threshold is correct about the whole body, and it is a fairly lightly loaded one.

The last row is the case that matters in practice. At a free-stream Mach of 0.3 that section’s shoulder is at Mach 0.55 with the local density down by 9.6 per cent — twice the tolerance the threshold was built on, on a wing whose free stream is at the number everybody quotes.

That sweep is over sections at one incidence. The other axis is the incidence itself, and it moves the answer further than the thickness does — because a suction peak deepens roughly with the lift coefficient while the thickness contributes a fixed part of it. Two crossings make the point, drawn for one section at rest and at work.

Where the sequence ends #

The suction peak’s story does not stop at a tolerance. Push the free-stream Mach up and the peak’s local Mach rises faster, until it reaches one — which is the critical Mach number, where a supersonic pocket appears on the upper surface and has to be closed by a shock.

So one section’s incompressible assumption fails at 0.39 and its critical Mach is 0.69; another’s fails at 0.22 and goes sonic at 0.49. Both of them are conventionally described by the same threshold, and the difference between them is the whole of transonic aerodynamic design.

Why the peak wins at exactly minus one #

The crossover deserves its own derivation, because it is the only exact number in the essay and it comes out of two lines.

Expand both density changes for small $M$. At a stagnation point the local speed is zero, so $(V/V_\infty)^2 = 0$ and the density rises; at a point with pressure coefficient $c_p$ the incompressible relation $c_p = 1 - (V/V_\infty)^2$ gives $(V/V_\infty)^2 = 1 - c_p$. To leading order the fractional density change at either place is

$$\frac{\Delta\rho}{\rho} \approx \frac{M^2}{2}\left[1 - \left(\frac{V}{V_\infty}\right)^2\right],$$

which is $+M^2/2$ at the stagnation point and $+c_p M^2/2$ at the peak — negative there, since $c_p$ is. The magnitudes are equal when $|c_p| = 1$.

So the crossover contains no properties of air at all: not $\gamma$, not the speed of sound, not the temperature. It is a statement about a velocity ratio, and $c_p = -1$ is simply the pressure coefficient at which the flow has been accelerated by a factor of $\sqrt{2}$.

That is worth holding onto, because it converts the whole question into one a reader can answer by looking at a pressure plot. If the suction peak is deeper than minus one, the compressibility threshold belongs to the shoulder and not to the nose, and no amount of care with the free-stream Mach number will find it.

And which part of the body, when the parts move at different speeds #

Everything above compares a free-stream speed with a local one produced by the flow. There is a second and much larger discrepancy available whenever the body’s own parts do not all travel at the body’s speed, and every aircraft that has a propeller or a rotor is such a body.

A blade section’s Mach number is formed on the resultant velocity it meets, which is its rotational speed and the forward speed combined. Take a light aeroplane with a two-metre propeller turning at 2,700 rpm and flying at a hundred metres a second. Its free-stream Mach number is 0.29 — comfortably inside the folklore’s incompressible band, and a designer applying the threshold would treat the whole aircraft as incompressible. Its propeller tips are travelling at 283 metres a second rotationally, which combined with the flight speed gives a helical tip speed near 300 and a tip Mach number of 0.88.

So one aeroplane is simultaneously subsonic-and-negligible over most of its surface and firmly transonic at a radius of one metre from its own crankshaft. Nothing about the free-stream number detects that, and the consequences are the ones a pilot meets: propeller efficiency falls away as tip Mach rises past about 0.85, the noise rises sharply, and every certification limit on propeller diameter and engine speed is written to keep that number down rather than to keep any structural stress down.

A helicopter makes the same point with the two ends of one blade. A main rotor with a tip speed of 210 metres a second, flying at 70, has an advancing tip at 280 — Mach 0.82 — and a retreating tip at 140, Mach 0.41. Same blade, same air, same instant, a factor of two in Mach number and a factor of four in dynamic pressure. The advancing side is limited by compressibility and the retreating side by stall, and the gap between the two is what puts a ceiling on a helicopter’s forward speed that no amount of power removes.

The Mach number is a field over the blade rather than a number for the machine, which is the whole reason blade-element methods exist: they cut the blade into strips and give each strip its own local velocity, its own Reynolds number and its own Mach number, precisely because no single value describes it. And the variation is radial as well as azimuthal — a section at the hub of that propeller is at a Mach number a tenth of the tip’s, so the same blade needs a thick high-lift section inboard and a thin transonic one outboard, which is why a propeller blade changes shape along its length as visibly as it changes twist.

The general form of the caution is the same one this essay has already made twice, extended by one step. A dimensionless group is formed on a velocity, and the answer depends on which velocity: the free stream, the local flow, or the local motion of the surface. The first is what gets quoted, the second is what this essay is about, and the third is invisible in any description that treats the aircraft as a single rigid object moving at one speed. A number quoted for a vehicle is not a number for any part of it, and on a machine with something spinning it is not even close.

The same mistake, with a different quantity #

Reading a dimensionless group on the wrong speed is not confined to the Mach number, and the general rule is worth stating: a group formed on a free-stream quantity is a statement about the free stream.

The Reynolds number of a wing is formed on the chord and the free-stream speed, and the flow near the leading edge has a much smaller effective length and a much larger speed — which is why transition happens where it does rather than where a chord Reynolds number would suggest.

The cavitation number is formed on the free-stream pressure, and cavitation begins where the local pressure reaches the vapour pressure — which is why the inception threshold is $\sigma_i = -c_{p,\min}$ and is a property of the body’s shape rather than of the water.

In every case the free-stream group is the right thing to plot against, and the wrong thing to compare with a threshold.

What a wind tunnel does about it #

The practical consequence appears in every tunnel test. A model tested at Mach 0.3 and a full-scale aircraft flying at Mach 0.3 are not doing the same experiment unless their pressure distributions match, and their pressure distributions do not match, because the Reynolds numbers differ and the boundary layer changes the effective shape.

So compressibility corrections applied to tunnel data are applied to a distribution that is itself slightly wrong, and the correction is largest exactly where the distribution is least reliable — at the suction peak, where the boundary layer is thinnest and the Reynolds-number sensitivity is highest.

That is one more entry on the list of reasons a model cannot be matched, and unlike most of them it does not go away by making the model bigger: it is a local mismatch driven by a local speed, and matching the free-stream Mach number does nothing about it.

An audit, for a wing that exists #

It is worth doing the sum once for a case with numbers in it, because the abstraction hides how ordinary the situation is.

A transport wing section at cruise carries a lift coefficient near 0.5 with a peak suction around $c_{p0} = -1.4$, whose critical Mach number is 0.548. Its free-stream Mach number is 0.78, which is far past that and is meant to be: transonic cruise means flying with a supersonic pocket on the upper surface and a shock closing it, deliberately, and the whole of supercritical section design is about making that shock weak.

The same section on approach carries a lift coefficient near 1.8 with flaps and slats deployed, and its slat peak runs to $c_{p0} = -8$. Its free-stream Mach number is 0.2.

At Mach 0.2 with a peak of −8, the local Mach number over the slat is 0.654 and the local density is down by seventeen per cent. A calculation of the slat’s loads done incompressibly — which is a perfectly ordinary thing to do at Mach 0.2 — is being made in a place where the density has moved by a sixth, on the one part of the aircraft where the peak pressure decides whether the flow stays attached. The Prandtl–Glauert factor at Mach 0.2 is 1.02, so the correction a designer would apply is two per cent: it is applied to the free stream, and it is the free stream that did not need it.

That is not an exotic case. It is the second most common flight condition an airliner has, and its free-stream Mach number is well under the threshold everybody quotes.

The general lesson is the one this collection keeps arriving at from different directions. A correction computed from a free-stream quantity is applied uniformly to a field whose departure from the assumption is anything but uniform, so the places where the correction is smallest are the places where it is most needed. That is a statement about where a linearised correction is linearised, and it is why transonic design became a computational subject as soon as computation was available: nothing short of solving the field answers it.

What the picture cannot show #

Prandtl–Glauert is a linearised correction. It assumes small disturbances, which is exactly what a deep suction peak is not, and it over-predicts the peak’s depth at high subsonic Mach numbers. The critical Mach numbers in the table are therefore slightly pessimistic; a full solution gives values a few hundredths higher.

The incompressible $c_{p0}$ is an input. Nothing here computes it; the collection has a panel method that does, and the numbers quoted are round values chosen to span the range real sections occupy.

Nothing here has a boundary layer in it. A real suction peak is shallower than an inviscid one because the displacement thickness changes the effective shape, and the difference grows towards the trailing edge.

And the tolerance is on the density. Five per cent of density is not five per cent of anything anybody measures directly; it becomes about two and a half per cent in a pressure coefficient and rather less in a lift coefficient, because the corrections partly cancel when integrated. Which tolerance on which quantity is the question the folklore leaves out twice over.

Who found it, and when #

The isentropic relations are Euler’s and Bernoulli’s, extended to compressible flow through the nineteenth century. Prandtl gave the $1/\sqrt{1-M^2}$ factor in lectures around 1922 and Glauert published it in 1928. The critical Mach number as a design quantity dates from the late 1930s, when aircraft first got fast enough for it to matter and the phenomenon then called compressibility burble was killing test pilots.

The threshold of 0.3 has no such history. It appears in textbooks from the 1940s onward with no citation, and it is exactly the kind of number that gets into circulation because it is memorable and approximately true.

The surprising connection is that all three routes to it — the stagnation density, the Prandtl–Glauert factor, and the crossover between a body’s nose and its shoulder — land within a few hundredths of each other, for entirely unrelated reasons. A threshold that three independent arguments agree on is very hard to dislodge, even when what the three agree on is a five per cent tolerance rather than a physical boundary, and even when the third of them holds only for a section loaded to exactly $c_{p0} = -1$.

Where the ladder goes next #

Above this rung is the transonic problem proper: the supersonic pocket, the shock that closes it, and the drag rise that follows — which is where the pocket appears and beyond that a nonlinear problem this collection solves only in one dimension.

Beside it sits the same question asked about other groups: which length goes in the Reynolds number, and which pressure goes in the cavitation number. Below it is where air stops being incompressible, which asks the question about the free stream, and the general account of what a group’s own value is worth as a threshold.