Noesis
3.1.16Compressible Flow & Aerodynamics

Prandtl-Meyer expansion waves — isentropic, supersonic turning

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What is a Mach wave, and why a fan?

As the flow accelerates around the corner, MM increases, so μ=arcsin⁡(1/M)\mu = \arcsin(1/M) decreases. The first Mach wave (at M1M_1) is steep, the last (at M2M_2) is shallow → the waves spread into a fan.

Figure — Prandtl-Meyer expansion waves — isentropic, supersonic turning

Deriving the Prandtl–Meyer function from scratch

Step 1 — Geometry across one Mach wave. The velocity component along the Mach line is unchanged (only the normal component changes, like a very weak oblique shock). Let speed be VV, turning by dθd\theta. Standard result from the velocity triangle across a Mach wave: dθ=M2−1 dVVd\theta = \sqrt{M^2-1}\,\frac{dV}{V}

Why this step? The Mach wave is at angle μ\mu. Resolving the velocity before/after the infinitesimal turn and using tan⁡μ=1/M2−1\tan\mu = 1/\sqrt{M^2-1} gives exactly this. The factor M2−1\sqrt{M^2-1} appears because only supersonic flow (M>1M>1) supports Mach waves.

Step 2 — Convert dV/VdV/V into dM/MdM/M. Since V=MaV = M a, take logs and differentiate: ln⁡V=ln⁡M+ln⁡a  ⇒  dVV=dMM+daa\ln V = \ln M + \ln a \;\Rightarrow\; \frac{dV}{V} = \frac{dM}{M} + \frac{da}{a} For a perfect gas, a=γRTa = \sqrt{\gamma R T}, and the isentropic / adiabatic energy relation gives T0T=1+γ−12M2.\frac{T_0}{T} = 1 + \frac{\gamma-1}{2}M^2. Differentiating a∝Ta \propto \sqrt T with T0T_0 constant yields daa=−γ−12M dM1+γ−12M2.\frac{da}{a} = \frac{-\frac{\gamma-1}{2}M\,dM}{1+\frac{\gamma-1}{2}M^2}. Combining: dVV=11+γ−12M2 dMM.\frac{dV}{V} = \frac{1}{1+\frac{\gamma-1}{2}M^2}\,\frac{dM}{M}.

Why this step? We want everything in terms of MM, the natural variable, because we tabulate the result against MM.

Step 3 — Assemble and integrate. dθ=M2−11+γ−12M2 dMM.d\theta = \frac{\sqrt{M^2-1}}{1+\frac{\gamma-1}{2}M^2}\,\frac{dM}{M}. Define the Prandtl–Meyer function ν(M)\nu(M) by integrating from M=1M=1 (where θ=0\theta=0):

Worked examples

Common mistakes (steel-manned)

Recall Feynman: explain to a 12-year-old

Imagine running so fast that the air can't get out of your way in time — you're "supersonic." Now the wall in front of you suddenly bends away. The air has room to spread out, so it speeds up and gets thinner and cooler, like water rushing faster when a pipe widens. It does this gently through a fan of super-thin invisible "ripples," so nothing gets messed up or wasted (no heat lost) — that's what "isentropic" means. The Prandtl–Meyer number ν\nu is just a score of how much you've turned — bend the wall by 10∘10^\circ and you add 1010 to your score, then look up your new speed.

Recall flashcards

What does a Prandtl–Meyer expansion fan do to a supersonic flow?
Turns it around a convex corner, accelerating it (M↑) isentropically while p, T, ρ drop.
Why is an expansion fan isentropic but a shock is not?
The fan is a continuum of infinitesimally weak Mach waves (Δs∝strength³→0); a shock is a single finite discontinuity with finite entropy rise.
State the Mach angle formula.
μ=arcsin⁡(1/M)\mu=\arcsin(1/M).
What is the differential relation linking turn angle and velocity?
dθ=M2−1 dVVd\theta=\sqrt{M^2-1}\,\dfrac{dV}{V}.
Write the Prandtl–Meyer function ν(M)\nu(M).
ν=γ+1γ−1arctan⁡γ−1γ+1(M2−1)−arctan⁡M2−1\nu=\sqrt{\frac{\gamma+1}{\gamma-1}}\arctan\sqrt{\frac{\gamma-1}{\gamma+1}(M^2-1)}-\arctan\sqrt{M^2-1}.
How do you apply ν\nu to a turn of angle θ\theta?
ν(M2)=ν(M1)+θ\nu(M_2)=\nu(M_1)+\theta, then invert for M2M_2.
Maximum turn angle for γ=1.4\gamma=1.4 starting from M=1?
νmax⁡≈130.45∘\nu_{\max}\approx130.45^\circ (as M→∞M\to\infty).
Through a fan, does the Mach angle increase or decrease?
Decrease, because MM increases and μ=arcsin⁡(1/M)\mu=\arcsin(1/M).
What stays constant across an isentropic expansion fan?
Stagnation pressure p0p_0, stagnation temperature T0T_0, entropy ss.
At M=2M=2, γ=1.4\gamma=1.4, what is ν\nu?
About 26.38∘26.38^\circ.

Connections

  • Oblique shock waves — the compression counterpart; θ–β–M relation vs ν(M).
  • Mach waves and Mach cone — building block of the fan.
  • Isentropic flow relations — used to get p, T, ρ once M2M_2 is known.
  • Method of characteristics — fans are simple-wave regions along characteristics.
  • Nozzle design and overexpansion — fans at nozzle lips when underexpanded.
  • Entropy and the second law — why weak waves give Δs→0\Delta s\to0.

Concept Map

turns around convex corner

composed of

each infinitesimally weak

entropy change to zero

makes angle

from sound circle vs source motion

flow accelerates

mu decreases

one wave geometry

convert dV over V to dM

yields

concave corner instead

contrast, lossy turn into flow

Supersonic flow

Expansion fan

Mach waves

Isentropic process

No shock, lossless

Mach angle mu equals arcsin 1 over M

sin mu equals a over V

M rises, p T rho drop

Waves diverge into fan

d theta equals sqrt M2 minus 1 times dV over V

Integrate over M

Prandtl-Meyer function nu of M

Compression shock

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, jab supersonic flow (M>1) ek convex corner se guzarta hai — yaani deewar flow se door mud jaati hai — tab flow ko phailne ki jagah mil jaati hai. Yeh phailna hota hai bahut saari patli-patli Mach waves ke ek fan ke through. Har wave itni weak hai ki entropy nahi badhti — isliye process isentropic hota hai. Result: Mach number badhta hai, aur pressure, temperature, density girte hain. Yeh shock ka ulta hai: shock me flow andar mudta hai, sab kuch achanak badalta hai aur entropy badh jaati hai; fan me sab smooth aur loss-free.

Mach angle yaad rakho: μ=arcsin⁡(1/M)\mu=\arcsin(1/M). Jaise-jaise M badhta hai, μ\mu chhota hota jaata hai — isliye fan ki pehli wave steep hoti hai aur aakhri shallow, beech me ek pankha ban jaata hai. Core formula hai Prandtl–Meyer function ν(M)\nu(M), jo batata hai ki sonic (M=1M=1) se kitna ghoom kar aap us M tak pahunche. Iska sabse mast trick: agar deewar θ\theta degree mudti hai, to simply ν(M2)=ν(M1)+θ\nu(M_2)=\nu(M_1)+\theta. Phir ν(M2)\nu(M_2) se M2M_2 nikaalo, aur isentropic relations se p aur T ka ratio.

Yaad rakhne ka mantra: "Expand = Away = Add" — flow door mudta hai, θ\theta ko ν\nu me add karo, M upar, p neeche. Aur "Fan, not bang" — fan isentropic hai, shock (bang) me entropy badhti hai. Galti jo sab karte hain: yeh sochna ki pressure girne se flow slow hota hai — nahi! Internal energy kinetic energy me convert hoti hai, flow tez hota hai. Exam aur real nozzle design (overexpanded nozzle ke lip pe yahi fan banta hai) dono me yeh concept bahut kaam aata hai.

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Connections

Oblique shock waves — θ-β-M relationPhysics · 3.1.13Isentropic flow tables — P - P₀, T - T₀, ρ - ρ₀ as functions of MPhysics · 3.1.7Second law — Kelvin-Planck statement, Clausius statementPhysics · 1.7.18Oblique shock waves — θ-β-M relationPhysics · 3.1.13Shock wave angle, deflection anglePhysics · 3.1.14Over - under expanded nozzle flowsPhysics · 3.1.18