Prandtl-Meyer function ν(M): Question bank — Physics deep dive

Symbols and conventions used on this page

Before the traps, we pin down every symbol so no item leans on something undefined. Look at the figure below while reading this list — it shows the corner, the fan, and the velocity triangle that all these symbols live on.

Definition Every symbol on this page

$M$ — the Mach number: the flow speed divided by the local speed of sound. $M > 1$ means supersonic (faster than sound). The whole topic needs $M \geq 1$ .
$V$ — the flow speed itself (metres per second), the actual velocity magnitude of the gas. In the picture it is the long arrow; $M = V / a$ where $a$ is the local speed of sound.
$a$ — the local speed of sound, how fast a tiny pressure ripple travels in the gas right there. It shrinks as the gas cools.
$γ$ — the ratio of specific heats (a fixed property of the gas: $γ \approx 1.4$ for air). It measures how "springy" the gas is when compressed; it is a pure number with no units and sets the numbers like $ν_{m a x}$ .
$μ$ — the Mach angle, $μ = arcsin (1/ M)$ : the angle a single Mach wave makes with the local flow direction (the thin blue line in the figure).
$θ$ — the turning angle of the wall/flow.
$ν (M)$ — the Prandtl–Meyer function, the total angle a sonic flow has "spent" expanding to reach $M$ , with $ν (1) = 0$ .
$p, T$ — static pressure and temperature (what a thermometer riding along with the gas reads).
$p_{0}, T_{0}$ — stagnation (total) pressure and temperature: the values the gas would reach if brought smoothly to rest. In an isentropic flow $p_{0}$ and $T_{0}$ stay constant even as $p$ and $T$ change; a shock lowers $p_{0}$ .

True or false — justify

A flow can only use

ν (M)

if it is supersonic.

True — the whole construction starts at

M = 1

and the Mach angle

μ = arcsin (1/ M)

only exists for

M \geq 1

; a subsonic flow has no Mach waves to build a fan from.

The Prandtl–Meyer function depends only on Mach number, not on the actual velocity or temperature.

True —

ν (M)

is a pure function of

M

(and

γ

); the derivation converted every

d V / V

into

d M / M

, so all dimensional quantities cancelled.

Turning a flow by

θ

always increases its Mach number.

False — only an expansion (

θ > 0

, convex, turning away) increases

M

; a compression turn would decrease it, but a sharp compression makes a shock and

ν

no longer applies.

Static pressure and static temperature both drop across an expansion fan.

True — the flow accelerates isentropically, so both

p

and

T

fall while stagnation quantities

p_{0}, T_{0}

stay constant.

Stagnation pressure

p_{0}

is conserved across a Prandtl–Meyer fan but not across a shock.

True — the fan is isentropic so entropy (and hence

p_{0}

) is unchanged; a shock raises entropy, dropping

p_{0}

ν (M)

grows without bound as

M \to \infty

False — it approaches the finite ceiling

ν_{m a x} \approx 130.4 5^{\circ}

(for

γ = 1.4

); the

arctan

terms saturate at

π /2

For a fixed turn angle

θ

, the same

θ

produces the same downstream Mach number regardless of the incoming Mach number.

False —

θ = ν (M_{2}) - ν (M_{1})

fixes the change in

ν

, but because

ν (M)

is nonlinear, the resulting

M_{2}

depends strongly on where

M_{1}

sits.

If a corner demands a turn larger than

ν_{m a x} - ν (M_{1})

, the flow simply reaches

M = \infty

and stops turning.

False — the ideal inviscid limit predicts

p \to 0

and a vacuum void, but in real flow the boundary layer separates first (or the flow finds another structure); the wall keeps turning yet the flow physically cannot follow it that far.

The Prandtl–Meyer relation

θ = ν (M_{2}) - ν (M_{1})

requires you to know the fan's angular width

μ

False —

θ

comes purely from the

ν

difference; the Mach angles

μ_{1}, μ_{2}

describe the fan's shape (front and back edges) but are not needed to find

M_{2}

Spot the error

"Since a compression is just a negative expansion, I'll use

θ = ν (M_{2}) - ν (M_{1})

with

M_{2} < M_{1}

for a sharp concave corner."

The error: a sharp compression corner produces a shock, which is non-isentropic, and

ν (M)

was derived assuming isentropic flow — you must use oblique-shock relations instead.

"To get the downstream Mach number, add the turn angle directly to the Mach number:

M_{2} = M_{1} + θ

The error: you add

θ

ν

, not to

M

;

ν

and

M

are nonlinearly related, so you must go

M_{1} \to ν_{1} \to ν_{2} = ν_{1} + θ \to M_{2}

"I looked up

ν (2) = 26.38

in degrees and added a turn of

θ = 0.26

radians to get

ν_{2} = 26.64

The error: mixing units —

26.38

is degrees and

0.26

is radians; convert to a common unit (

0.26 rad \approx 14. 9^{\circ}

) before adding, per the house rule.

"The expansion fan is a discontinuity in the flow, just a smooth one."

The error: a fan is not a discontinuity at all — flow properties vary continuously through it; that continuity is exactly why it stays isentropic (a shock is the discontinuity).

"Because the flow speeds up, the density must increase across the fan."

The error: expansion rarefies the gas — density falls along with pressure and temperature; the speed rises but the fluid spreads out.

"The Mach waves in the fan all make the same angle with the wall."

The error: each wave makes the Mach angle

μ = arcsin (1/ M)

with the local flow, and since

M

rises through the fan,

μ

shrinks — the waves fan out to different angles.

ν (1) = 0

means a sonic flow has done zero turning, so it can't turn at all."

The error:

ν (1) = 0

is just the chosen zero reference; a sonic flow can still expand — it simply starts its "turning budget" from zero and climbs upward.

Why questions

Why is the factor

M^{2} - 1

rather than

M

in the differential relation

d θ = \frac{M ^{2} - 1}{V} d V

It is

cot μ = M^{2} - 1

coming from the velocity triangle (in the figure) across a single Mach wave — resolving velocity along and across a wave inclined at

μ

produces exactly this cotangent.

Why does the denominator

1 + \frac{γ - 1}{2} M^{2}

appear when converting

d V / V

d M / M

Because

V = M a

depends on

M

both directly and through the local sound speed

a

; the accelerating flow cools, so

a

falls and partly offsets the rise in

M

, giving that damping denominator.

Why does an expansion stay isentropic while a compression does not?

In an expansion the Mach waves spread apart and never cross, so no discontinuity forms; in a compression they converge and coalesce into a shock, which jumps entropy.

Why is

ν_{m a x}

finite even though

M

can grow without limit?

Both

arctan

terms approach the finite value

π /2

M \to \infty

, so their weighted combination converges to a fixed ceiling rather than diverging.

Why can we integrate a differential relation to get

ν (M)

here, but not across a shock?

Integration requires the process be smooth and reversible (isentropic); a shock is a finite discontinuity with entropy jump, so there is no smooth path to integrate along.

Why do tables give

ν

in degrees while the raw integral yields radians?

The integral of the geometric relation naturally produces an angle in radians; tables convert to degrees purely for engineering convenience — you must keep

θ

in the same unit.

Why is

ν (M)

guaranteed to be invertible (one

ν

↔ one

M

Its integrand

\frac{M ^{2} - 1}{M ( 1 + \frac{γ - 1}{2} M ^{2} )}

is positive for all

M > 1

, so

ν

is strictly increasing (monotonic) — every

ν

value maps back to a unique

M

Edge cases

What is

ν

exactly at

M = 1

, and why can't the fan begin below it?

ν (1) = 0

; below

M = 1

the term

M^{2} - 1

becomes imaginary and

μ = arcsin (1/ M)

is undefined — there is no Mach wave to expand through.

M = 1

the Mach angle is

μ = 9 0^{\circ}

— what does that mean for the leading edge of the fan?

The first (leading) Mach wave stands perpendicular to the flow; as the flow accelerates through the fan, later waves lie at ever-shallower angles.

What happens to the fan's angular spread if the incoming flow is only just supersonic, say

M_{1} = 1.01

The leading wave is nearly normal (

μ \approx 8 2^{\circ}

) and

ν \approx 0

, so almost the entire

ν_{m a x}

budget is still available — such a flow can turn through a very large total angle.

For an incoming flow already near

M_{1}

giving

ν (M_{1})

close to

ν_{m a x}

, what is special about further expansion?

Very little turning budget remains, so even a small extra

θ

drives

M

toward infinity and

p

toward zero — the flow is on the verge of the ideal vacuum limit (in practice the boundary layer separates first).

θ = 0

(no wall bend), what does the relation say?

ν (M_{2}) = ν (M_{1})

, so

M_{2} = M_{1}

— no turn means no expansion and the Mach number is unchanged, as expected.

Can

ν (M)

ever be negative for a real supersonic flow?

No — the integrand is positive for all

M > 1

starting from

ν (1) = 0

, so

ν

is always

\geq 0

; a "negative

ν

" would require

M < 1

, outside the domain.

What determines

ν_{m a x}

physically, and does it change with the gas?

It is set by

γ

through

\frac{π}{2} (\frac{γ + 1}{γ - 1} - 1)

; a different gas (different

γ

) has a different ceiling —

130.4 5^{\circ}

is specifically for

γ = 1.4

What happens physically if a corner demands more turn than the flow can supply — vacuum, shock, or separation?

The idealized inviscid answer is a vacuum void (

p \to 0

), but real viscous flow almost always separates at the boundary layer well before that, and confined geometries can instead terminate the expansion in a shock — the "pure vacuum" is only the frictionless limiting case.

Connections

Oblique Shock Waves — the non-isentropic compression counterpart these traps keep contrasting against.
Mach Waves and Mach Angle — where $μ = arcsin (1/ M)$ and the $M = 1 \Rightarrow μ = 9 0^{\circ}$ edge case come from.
Isentropic Flow Relations — supplies the $p / p_{0}$ , $T / T_{0}$ behind the "pressure falls / density falls" items.
Expansion Fan / Centered Rarefaction — the physical structure whose continuity the True/False set probes.
Method of Characteristics — uses $ν (M) \pm θ$ invariants, relying on the monotonicity established here.
Nozzle Design (Supersonic) — practical use of these limits and traps.