Critical Mach number — onset of local supersonic flow: Question bank

Quick symbol reminders before we start (so nothing is unearned):

$M_{\infty}$ = the free-stream Mach number — the speed of the whole aircraft relative to still air, divided by the speed of sound in that air.
$M_{l oc a l}$ = the Mach number of the air at one point on the skin, which can differ from $M_{\infty}$ because air speeds up or slows over curves.
$p$ = local static pressure at a point; $p_{\infty}$ = the undisturbed pressure far upstream; $ρ_{\infty}$ = far-upstream density; $V_{\infty}$ = free-stream speed.
$C_{p}$ = pressure coefficient, defined precisely as $C_{p} \equiv \frac{p - p _{\infty}}{\frac{1}{2} ρ _{\infty} V _{\infty}^{2}},$ i.e. "how far the local pressure sits below (negative) or above (positive) the undisturbed pressure, measured in units of the free-stream dynamic pressure $\frac{1}{2} ρ_{\infty} V_{\infty}^{2}$ ." See Pressure Coefficient Cp.
$C_{p, cr}$ = the specific value of $C_{p}$ the lowest-pressure point must reach for the local flow there to be exactly sonic ( $M_{l oc a l} = 1$ ).
$C_{p, m i n}$ = the most negative $C_{p}$ anywhere on the airfoil at a given $M_{\infty}$ — the "suction peak." The first point to go sonic is wherever this minimum lives.
$C_{p, 0}$ = the value that suction peak would have in incompressible flow ( $M_{\infty} \to 0$ ) — a single fixed negative number set purely by the airfoil's shape. It is the "seed" that the compressibility correction then scales up.
$γ$ = ratio of specific heats ( $\approx 1.4$ for air).
$Λ$ = wing sweep angle. Here we use the leading-edge sweep (the angle the leading edge makes with the direction perpendicular to flight), since it is the leading edge that first "sees" the flow; the $cos Λ$ rule below uses this convention. See Swept Wings & Transonic Design.

Definition Where the compressibility curve

$C_{p, 0} / 1 - M_{\infty}^{2}$ comes from As $M_{\infty}$ rises, the same wing produces a deeper suction peak because the air is now compressible and crowds together over the curvature. Linearizing the compressible flow equations (small disturbances, thin airfoil) gives the Prandtl–Glauert scaling $C_{p, m i n} (M_{\infty}) = \frac{C _{p, 0}}{1 - M _{\infty}^{2}} .$ Why $1 - M_{\infty}^{2}$ ? The linearized equation for the disturbance potential turns from Laplace's equation (incompressible) into one stretched by the factor $1 - M_{\infty}^{2}$ ; undoing that stretch scales every pressure by its reciprocal. Validity range: it is a first-order result, trustworthy only at moderate subsonic $M_{\infty}$ and thin sections — it under-predicts the peak as $M_{\infty}$ approaches $M_{cr}$ , and (see edge cases) it blows up to $- \infty$ as $M_{\infty} \to 1$ . See Prandtl–Glauert Compressibility Correction.

Figure — Critical Mach number — onset of local supersonic flow

True or false — justify

M_{cr}

is the free-stream Mach at which the whole aircraft becomes supersonic.

False. It is where the flow first reaches sonic at a single point on the surface — the aircraft itself is still subsonic (

M_{\infty} < 1

), only a local patch touches

M_{l oc a l} = 1

M_{\infty} = M_{cr}

there is already a shock wave and drag divergence on the wing.

False. At exactly

M_{cr}

the fastest point just reaches

M_{l oc a l} = 1

; the supersonic pocket and its terminating shock (hence drag divergence) appear only slightly above

M_{cr}

The

C_{p, cr}

versus

M_{cr}

curve is different for every airfoil.

False. It comes purely from isentropic gas dynamics and depends only on

γ

— it is a universal curve. The airfoil enters only through its own minimum-pressure curve.

A more negative

C_{p, cr}

means the flow is harder to drive sonic.

False.

C_{p, cr}

is simply the pressure the peak must reach at a given

M_{\infty}

; more negative just means a lower pressure is required at that

M_{\infty}

. What decides

M_{cr}

is where the two curves cross, not the magnitude alone.

Making an airfoil thinner raises its critical Mach number.

True. A thinner section has a milder suction peak, so

∣ C_{p, 0} ∣

is smaller; its pressure curve crosses the universal

C_{p, cr}

curve at a higher

M_{\infty}

, raising

M_{cr}

. That is why fast jets have thin wings.

Sweeping a wing back lowers

M_{cr}

because the wing is "longer."

False, opposite effect. Sweep reduces the velocity component normal to the leading edge to

M_{\infty} cos Λ

(with

Λ

the leading-edge sweep); the section sees a slower effective flow, so true

M_{\infty}

must be higher to reach sonic —

M_{cr}

rises. See Swept Wings & Transonic Design.

The critical pressure coefficient becomes less negative as

M_{cr}

increases.

True. Higher

M_{\infty}

means less compression is needed to reach

M_{l oc a l} = 1

, so the required minimum

C_{p}

climbs toward zero (a smaller suction is enough). This is exactly the gentle upward slope of the black curve in the figure.

Prandtl–Glauert gives the exact

M_{cr}

for any airfoil.

False. P–G is a linearized estimate; near

M_{cr}

it under-predicts the suction peak, so it tends to over-estimate

M_{cr}

. Karman–Tsien/Laitone give slightly lower, more accurate values — use P–G only for intuition.

Two airfoils with the same incompressible

C_{p, 0}

have the same

M_{cr}

True (to first order, within the P–G approximation). In the intersection method

M_{cr}

depends only on

C_{p, 0}

and the universal curve; equal

C_{p, 0}

⇒ same crossing point. (Higher-order shape effects and better corrections give tiny real differences.)

Spot the error

"Since air is compressible, faster flow over the wing means higher local pressure, so

C_{p}

is positive at the suction peak."

Error: faster flow means lower static pressure (energy trades from pressure to speed), so the suction peak has the most negative

C_{p}

. That is precisely the point that goes sonic first.

"The universal

C_{p, cr}

curve and the airfoil's

C_{p, m i n}

curve are compared at different Mach numbers to find

M_{cr}

Error: both must be read at the same

M_{\infty}

;

M_{cr}

is the single

M_{\infty}

where the two curves have equal value (the crossing in the figure). Comparing at different

M_{\infty}

is meaningless.

"We set

M_{\infty} = 1

in the

C_{p, cr}

formula to get the critical pressure coefficient."

Error: we set the local point to sonic,

M_{l oc a l} = 1

, while the free stream stays at

M_{\infty} = M_{cr} < 1

. Confusing the two Machs is the classic slip.

"Because

p_{0}

is constant, the pressure is the same everywhere on the wing."

Error: the stagnation pressure

p_{0}

is constant along an isentropic streamline, but the static pressure

p

varies as speed varies. Only

p_{0}

is fixed;

p

dips at the suction peak.

"The suction-peak curve

C_{p, 0} / 1 - M_{\infty}^{2}

keeps getting more negative and never crosses the

C_{p, cr}

curve, so

M_{cr}

doesn't exist."

Error: the

C_{p, cr}

curve also moves (toward zero as

M_{\infty}

rises), and the airfoil curve dives faster, so they must cross at some

M_{\infty} < 1

. Every real airfoil has an

M_{cr}

"Above

M_{cr}

the drag jumps immediately and enormously."

Error: there is a small margin between

M_{cr}

and the Drag Divergence Mach Number

M_{dd}

; the sharp drag rise occurs at

M_{dd}

, a bit above

M_{cr}

, once the supersonic pocket and its shock grow strong.

Why questions

Why does the flow reach sonic on the upper surface first, not below?

The upper surface has the greatest curvature, constricting the streamtube most; by continuity the flow speeds up most there, giving the lowest pressure and highest

M_{l oc a l}

— so it hits

M_{l oc a l} = 1

first.

Why do we bother with

M_{cr}

if the plane is still subsonic?

Because the first appearance of a local supersonic pocket seeds the terminating shock that causes drag divergence and transonic buffet — the practical "sound barrier" felt at

M_{\infty} < 1

Why is the

C_{p, cr}

relation independent of the airfoil?

It is derived purely from isentropic flow between free stream and a sonic point (

p_{0}

constant,

M_{l oc a l} = 1

); no geometry enters, only

γ

and

M_{cr}

Why does the isentropic (not just Bernoulli) relation appear here?

Incompressible Bernoulli fails at these Machs because density changes; the flow is adiabatic and reversible, so the isentropic

p_{0} / p

relation correctly links pressure to Mach across compressible speeds.

Why does the prefactor

2/ (γ M_{\infty}^{2})

appear in every

C_{p}

expression here?

It is just the free-stream dynamic pressure

\frac{1}{2} ρ_{\infty} V_{\infty}^{2}

rewritten as

\frac{γ}{2} p_{\infty} M_{\infty}^{2}

, so dividing by it converts a raw pressure ratio into the dimensionless

C_{p}

measured in Mach-based units.

Why does the Area Rule (Transonic) help even though it's about the fuselage, not the wing's

M_{cr}

Just past

M_{cr}

the whole aircraft grows supersonic pockets whose shocks add up; the area rule smooths the total cross-sectional area from nose to tail (see figure s02) so no abrupt jump in area drives a strong shock — this weakens the transonic shock system, delaying and softening drag divergence. It does not raise

M_{cr}

itself but tames what happens above it.

Why does a thicker airfoil have a lower

M_{cr}

More thickness means stronger curvature and a deeper suction peak (larger

∣ C_{p, 0} ∣

); that curve meets the universal

C_{p, cr}

curve at a lower

M_{\infty}

, so sonic flow arrives sooner.

Edge cases

M_{\infty} \to 0

(very slow flight), what is happening in the two-curve picture?

The suction curve equals the incompressible

C_{p, 0}

and the required

C_{p, cr} \to - \infty

; they are nowhere near crossing, so no local sonic flow — consistent with

M_{cr}

being well above zero.

M_{\infty} \to 1

, what does Prandtl–Glauert predict, and why is that unphysical?

The factor

1 - M_{\infty}^{2} \to 0

, so

C_{p, 0} / 1 - M_{\infty}^{2} \to - \infty

— the formula predicts infinite suction. That singularity is a linearization artefact: the flow becomes transonic and nonlinear well before then, which is exactly why P–G must never be trusted near

M_{\infty} = 1

(and one reason it over-estimates

M_{cr}

For a perfectly flat plate at zero angle of attack, what is

M_{cr}

With no curvature there is no suction peak (

C_{p, 0} = 0

), so the flow never accelerates above free stream; local sonic is reached only when

M_{\infty} \to 1

, giving

M_{cr} \to 1

If an airfoil's incompressible suction peak

C_{p, 0}

were extremely mild (nearly zero), where is

M_{cr}

Very close to

1

— the tiny suction curve only meets the fast-rising

C_{p, cr}

curve near

M_{\infty} \approx 1

. Less disturbance to the flow ⇒ later onset of local sonic.

What happens right at

M_{\infty} = M_{cr}

exactly — is there any supersonic region yet?

There is a single sonic point (

M_{l oc a l} = 1

) but no finite supersonic pocket yet; the pocket has zero extent and grows only as

M_{\infty}

increases past

M_{cr}

Can

M_{cr}

exceed the drag-divergence Mach number

M_{dd}

No — sonic flow (at

M_{cr}

) must appear before the shock strengthens enough to diverge drag, so always

M_{cr} < M_{dd}

. See Drag Divergence Mach Number.

γ

increases (a different gas), does the required

C_{p, cr}

at fixed

M_{cr}

get more or less negative?

It shifts because both the isentropic exponent

γ / (γ - 1)

and the prefactor

2/ (γ M_{cr}^{2})

depend on

γ

; the whole universal curve moves, so

M_{cr}

for a given airfoil changes with the working gas — the curve is universal only for a fixed

γ

Recall One-line survival summary

$M_{cr}$ is where the fastest point on the skin first touches $M_{l oc a l} = 1$ while the plane stays subsonic; you find it as the crossing of the universal $C_{p, cr} (M_{\infty})$ curve (formula boxed above, only $γ$ inside) with the airfoil's suction curve $C_{p, 0} / 1 - M_{\infty}^{2}$ — good for intuition but under-predicting near $M_{cr}$ and blowing up at $M_{\infty} \to 1$ , so better corrections nudge $M_{cr}$ slightly lower. Thin + swept ⇒ higher $M_{cr}$ ; drag divergence comes just after.