Prandtl-Meyer function ν(M) — Hinglish notes

3.1.17 · Physics › Compressible Flow & Aerodynamics

YEH HAI KYA?

Yeh key practical baat hai: turning angle, $ν$ ke change ke barabar hota hai. Koi shock tables nahi, koi iteration on shock relations nahi — bas do lookups.

FAN KYUN BANTA HAI, SHOCK NAHI?

Har infinitesimal Mach wave local flow ke saath Mach angle $μ$ banaati hai, $μ = arcsin (\frac{1}{M}) .$

ν(M) KO SCRATCH SE DERIVE KAISE KAREIN

Hum ek tiny flow deflection $d θ$ aur ek Mach wave ke across tiny speed change ke beech relation banate hain.

Step 1 — Mach wave ke across velocity triangle. Ek Mach wave itni weak hoti hai ki sirf wave ke normal velocity component hi change hota hai; tangential component preserved rehta hai. Velocity $V$ ko wave ke along aur across resolve karke aur ek small turn $d θ$ apply karke, velocity triangle ki geometry deti hai: $d θ = \frac{M ^{2} - 1}{V} d V .$ Yeh step kyun? Normal Mach number $M sin μ = 1$ hai, aur tangential balance iss exact factor $M^{2} - 1 = M^{2} sin^{2} μ \cdot \frac{1}{s i n ^{2} μ} - 1$ ko force karta hai… concretely $cot μ = M^{2} - 1$ triangle se aata hai.

Step 2 — $d V / V$ ko $d M / M$ mein convert karo. $V = M a$ aur isentropic temperature relation $\frac{a _{0}^{2}}{a ^{2}} = 1 + \frac{γ - 1}{2} M^{2}$ se, log lo aur differentiate karo: $\frac{d V}{V} = \frac{1}{1 + \frac{γ - 1}{2} M ^{2}} \frac{d M}{M} .$ Yeh step kyun? $V$ depend karta hai $M$ par directly bhi aur local speed of sound $a$ ke through bhi; accelerating flow mein temperature drop $V$ ki growth ko slow kar deta hai, isliye denominator aata hai.

Step 3 — Combine karo. $d θ = \frac{M ^{2} - 1}{1 + \frac{γ - 1}{2} M ^{2}} \frac{d M}{M} .$

Step 4 — $M = 1$ (jahan $θ = 0$ ) se $M$ tak integrate karo: $ν (M) = \int_{1}^{M} \frac{M ^{2} - 1}{1 + \frac{γ - 1}{2} M ^{2}} \frac{d M}{M} .$

Step 5 — Closed form (standard integral, $M^{2} - 1$ trig substitute karo):

Worked examples

Common mistakes

Recall Feynman: ek 12-saal ke bachche ko explain karo

Socho tum ek curving wall ke andar ke edge ke paas bahut tez daud rahe ho, sound se bhi tez. Jab wall suddenly tumse door mud-ti hai, tumhe thodi extra jagah milti hai, toh tum speed up karte ho aur phail jaate ho — jaise paani ek ramp ke end se rush karta hai. Yeh ek achanak "bump" se nahi hota; yeh bahut saari choti fan-shaped pushes se hota hai. Prandtl–Meyer number bas ek score hai jo count karta hai ki tumhari flow ne kitna "spreading-out turning" kiya hai jab se woh barely supersonic thi. Agar tum wall ko 15 degrees door mod-do, tera score 15 se badh jaata hai. Score se tum padh sakte ho ki tum ab kitni tez ja rahe ho.

Active recall

Prandtl–Meyer function ν(M) physically kya represent karta hai?

Woh angle jis se ek sonic (M=1) flow ko isentropically expand karke Mach number M tak pahunchaya jaata hai; ν(1)=0.

Expansion ke liye turning angle aur ν ka relation?

θ = ν(M₂) − ν(M₁), jisme M₂ > M₁.

Expansion corner smooth fan kyun form karta hai shock ki jagah?

Mach waves alag-alag phailti hain (kabhi coalesce nahi hoti), isliye flow isentropic rehti hai — koi discontinuity nahi.

M ke terms mein local Mach angle μ?

μ = arcsin(1/M).

Derivation ko drive karne waala differential relation?

dθ = [√(M²−1) / (1 + (γ−1)/2·M²)] · dM/M.

ν(M) ka closed form?

ν = √((γ+1)/(γ−1))·arctan√((γ−1)/(γ+1)·(M²−1)) − arctan√(M²−1).

γ=1.4 ke liye ν_max aur uska matlab?

≈130.45°; M→∞ (p→0) ke liye max turn; zyada turn ⇒ vacuum ban jaata hai.

Sharp compression corner ke liye ν(M) use kyun nahi kar sakte?

Woh ek shock produce karta hai, jo non-isentropic hai; ν isentropic flow assume karke derive kiya gaya tha.

Prandtl–Meyer fan ke across static pressure badhta hai ya girta hai?

Girta hai (expansion: flow accelerate, cool, aur rarefy hoti hai); p₀ constant rehta hai.

M₁ aur turn θ diye ho toh M₂ nikalne ka procedure?

M₁→ν₁; ν₂=ν₁+θ; ν₂→M₂ invert karo.

Connections

Oblique Shock Waves — compression counterpart (non-isentropic).
Mach Waves and Mach Angle — fan ke building blocks.
Isentropic Flow Relations — $p / p_{0}$ , $T / T_{0}$ deta hai $M_{2}$ nikalne ke baad use karne ke liye.
Method of Characteristics — $ν (M) \pm θ$ ko Riemann invariants ke roop mein use karta hai.
Expansion Fan / Centered Rarefaction — woh physical structure jise $ν$ describe karta hai.
Nozzle Design (Supersonic) — contour design Prandtl–Meyer expansion par rely karta hai.