Prandtl-Meyer expansion waves — isentropic, supersonic turning

3.1.16 · Physics › Compressible Flow & Aerodynamics

Mach wave kya hai, aur fan kyun banta hai?

Jaise flow corner ke around accelerate karti hai, $M$ badhta hai, isliye $μ = arcsin (1/ M)$ ghatta hai. Pehli Mach wave (at $M_{1}$ ) steep hoti hai, aakhri (at $M_{2}$ ) shallow hoti hai → waves ek fan mein spread ho jaati hain.

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

Prandtl–Meyer function ko scratch se derive karna

Step 1 — Ek Mach wave ke across geometry. Mach line ke saath velocity component unchanged rehta hai (sirf normal component badalta hai, bilkul ek bahut weak oblique shock ki tarah). Speed $V$ hai, $d θ$ se turn kar rahi hai. Velocity triangle se standard result ek Mach wave ke across: $d θ = M^{2} - 1 \frac{d V}{V}$

Ye step kyun? Mach wave angle $μ$ par hai. Infinitesimal turn se pehle/baad velocity resolve karo aur $tan μ = 1/ M^{2} - 1$ use karo to exactly yahi milta hai. Factor $M^{2} - 1$ isliye aata hai kyunki sirf supersonic flow ( $M > 1$ ) Mach waves ko support karta hai.

Step 2 — $d V / V$ ko $d M / M$ mein convert karo. Kyunki $V = M a$ hai, log lo aur differentiate karo: $ln V = ln M + ln a \Rightarrow \frac{d V}{V} = \frac{d M}{M} + \frac{d a}{a}$ Perfect gas ke liye, $a = γ R T$ , aur isentropic / adiabatic energy relation deta hai $\frac{T _{0}}{T} = 1 + \frac{γ - 1}{2} M^{2} .$ $a \propto T$ ko $T_{0}$ constant rakh ke differentiate karne par milta hai $\frac{d a}{a} = \frac{- \frac{γ - 1}{2} M d M}{1 + \frac{γ - 1}{2} M ^{2}} .$ Combine karne par: $\frac{d V}{V} = \frac{1}{1 + \frac{γ - 1}{2} M ^{2}} \frac{d M}{M} .$

Ye step kyun? Hum sab kuch $M$ ke terms mein chahte hain, jo natural variable hai, kyunki hum result ko $M$ ke against tabulate karte hain.

Step 3 — Assemble karo aur integrate karo. $d θ = \frac{M ^{2} - 1}{1 + \frac{γ - 1}{2} M ^{2}} \frac{d M}{M} .$ Prandtl–Meyer function $ν (M)$ define karo $M = 1$ se integrate karke (jahan $θ = 0$ hai):

Worked examples

Common mistakes (steel-manned)

Recall Feynman: ek 12-saal ke bache ko samjhao

Socho tum itni tez bhaag rahe ho ki hawa time par raasta nahi de sakti — tum "supersonic" ho. Ab tumhare aage ki wall suddenly door jhuk jaati hai. Hawa ko failne ki jagah mil jaati hai, isliye ye speed up karti hai aur thinner aur cooler ho jaati hai, jaise paani faster beh jaata hai jab pipe chaudi hoti hai. Ye aaram se ek fan of super-thin invisible "ripples" ke through hota hai, isliye kuch bhi mess up ya waste nahi hota (koi heat nahi jaati) — yahi "isentropic" ka matlab hai. Prandtl–Meyer number $ν$ bas ek score hai ki kitna turn kiya hai — wall ko $1 0^{\circ}$ bend karo aur apne score mein $10$ add karo, phir apni nayi speed dekho.

Recall flashcards

Prandtl–Meyer expansion fan supersonic flow ke saath kya karta hai?

Use convex corner ke around turn karta hai, isentropically accelerate karta hai (M↑) jabki p, T, ρ girte hain.

Expansion fan isentropic kyun hai lekin shock nahi?

Fan infinitesimally weak Mach waves ka continuum hai (Δs∝strength³→0); shock ek single finite discontinuity hai jismein finite entropy rise hoti hai.

Mach angle formula batao.

μ = arcsin (1/ M)

Turn angle aur velocity ko link karne wala differential relation kya hai?

d θ = M^{2} - 1 \frac{d V}{V}

Prandtl–Meyer function

ν (M)

likho.

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

Angle

θ

ke turn par

ν

kaise apply karte ho?

ν (M_{2}) = ν (M_{1}) + θ

, phir

M_{2}

ke liye invert karo.

γ = 1.4

ke liye M=1 se start karte hue maximum turn angle kya hai?

ν_{m a x} \approx 130.4 5^{\circ}

(jaise

M \to \infty

Fan ke through Mach angle badhta hai ya ghatta hai?

Ghatta hai, kyunki

M

badhta hai aur

μ = arcsin (1/ M)

Isentropic expansion fan ke across kya constant rehta hai?

Stagnation pressure

p_{0}

, stagnation temperature

T_{0}

, entropy

s

M = 2

γ = 1.4

par

ν

kya hai?

Lagbhag

26.3 8^{\circ}

Connections

Oblique shock waves — compression counterpart; θ–β–M relation vs ν(M).
Mach waves and Mach cone — fan ka building block.
Isentropic flow relations — $M_{2}$ known hone par p, T, ρ get karne ke liye use hota hai.
Method of characteristics — fans characteristics ke saath simple-wave regions hain.
Nozzle design and overexpansion — underexpanded hone par nozzle lips par fans.
Entropy and the second law — kyun weak waves $Δ s \to 0$ deti hain.