Question bank — Under-expanded nozzle — Prandtl-Meyer expansion, efficiency loss

True. Expansion static pressure ko $p_{\infty }$ ki taraf giraa deta hai, aur isentropic supersonic flow ke liye pressure drop ka matlab hai gas accelerate hoti hai, toh $M_{\text{plume}}>M_e$.

False. Pressure term positive hai, lekin $v_e$ chhota hai (gas ne andar expansion finish nahi ki), aur dono effects cancel nahi hote — perfectly-expanded case $p_e=p_{\infty }$ abhi bhi maximum total thrust deta hai.

True. Turn infinitesimal Mach waves ki ek continuous family ke through hota hai, jinmein se koi bhi shock nahi hai, isliye entropy (aur hence $p_0$) constant rehti hai — over-expanded case ke lossy shocks ke unlike.

False. Expansion waves gas ko thanda karti hain jab woh accelerate hoti hai; loss geometric hai — momentum off-axis ho jaata hai, isliye uska kam hissa thrust axis ke saath peeche ki taraf point karta hai.

True. Badhti altitude $p_{\infty }$ ko kam karti hai jabki $p_e$ (nozzle geometry aur chamber se set) barely change hota hai, isliye $p_e/p_{\infty }$ badhta hai aur plume aur zyada fan out karta hai.

True. Supersonic flow mein disturbances upstream travel nahi kar sakti, isliye pressure gradually andar equalize nahi ho sakta — yeh nozzle lip se fan hoti Mach waves ke through adjust karna padta hai. Subsonic jets bas smoothly spread hoti hain bina kisi fan ke.

True. Ek bada ratio plume ko ek higher $M_{\text{plume}}$ tak expand hone par majboor karta hai, aur kyunki $\nu (M_{\text{plume}})=\nu (M_e)+{\theta }_{\text{turn}}$ aur $\nu$ $M$ mein badhta hai, required turn bhi badhta hai.

False. $\nu (M)$ ek finite ceiling ${\nu }_{\text{max}}$ ke paas pahunchta hai (approximately $130.5^{\circ }$ $\gamma =1.4$ ke liye); ek stream sirf itna hi turn kar sakti hai jis se pehle use infinite Mach ki zaroorat padti.

Inequality ulti hai. Under-expanded ka matlab hai nozzle ne gas ko bahut kam expand kiya, $p_e>p_{\infty }$ chhodkar; $p_e<p_{\infty }$ wala case over-expanded nozzle hai.

Dependency doosri taraf jaati hai. Plume tab tak expand hoti hai jab tak uska static pressure $p_{\infty }$ ke barabar na ho jaaye, isliye tum pehle $M_{\text{plume}}$ paane ke liye isentropic pressure relation use karte ho, phir Prandtl-Meyer functions ${\theta }_{\text{turn}}$ dete hain.

Nahi — yeh physical exit plane par apply hoti hai, wahan $p_e$ aur $A_e$ use karke. Iska poora point yahi hai ki abhi bhi uneven exit pressure ko thrust bookkeeping mein fold kar sake.

Yeh over-expansion describe karta hai. Yahan exhaust zyada pressure par hai, isliye woh baahri taraf hawa mein push karta hai aur fan out hota hai; kuch bhi nozzle mein wapas push nahi karta.

Yeh $\mu =\arcsin (1/M)$ hai, aur yeh $M$ badhne par ghatta hai — faster flow Mach cone ko thinner banati hai, flow direction ke saath aur tightly chipakti hai.

Isentropic ka matlab hai koi entropy loss nahi, lekin ek thrust loss abhi bhi hota hai: expansion baahre hoti hai bina react karne ke liye koi wall ke, isliye momentum axial push add karne ki jagah off-axis redirect ho jaata hai.

Dono losses hain, lekin physics alag hai: over-expanded flow shocks aur yahan tak ki nozzle ke andar flow separation suffer kar sakta hai (potentially damaging), jabki under-expansion ek smoother, aksar gentler external loss hai.

Nozzle physically khatam ho jaata hai gas ke $p_{\infty }$ tak expand hone se pehle; aage expansion guide karne ke liye koi wall nahi hai, isliye bachi-kuchi expansion lip ke baad open air mein hoti hai.

Supersonic flow mein pressure signals sirf Mach waves ke saath travel kar sakti hain aur upstream nahi ja sakti, isliye pressure adjustment lip se nikalne wali kai infinitesimal Mach waves se stitched together hota hai — ek centered fan.

Sirf axial component $v\cos {\theta }_{\text{turn}}$ rocket ko aage push karta hai; sideways hissa real momentum hai jo simply thrust axis ke saath point nahi karta, aur symmetry se sideways bits cancel ho jaate hain aur useful thrust se bas gayab ho jaate hain.

$p_e=p_{\infty }$ par pressure term zero hai lekin gas ne andar walls ke against apna saara expanding kar liya hai, maximum $v_e$ deta hai; mathematically wahan $\partial F/\partial A_e=0$ hai, isliye yeh thrust curve ka peak hai. Dekho 3.3.13-Optimal-expansion-ratio.

Near-vacuum mein maximum thrust ke liye sized nozzle bahut bada hoga; ek chhota nozzle neeche under-expanded rehta hai (chhota sa loss) lekin upar achha perform karta hai, isliye yeh poori trajectory par ek design compromise hai. 3.3.16-Altitude-compensation-nozzles isko directly tackle karta hai.

Turned momentum ka useful axial part total ka $\cos {\theta }_{\text{turn}}$ hai, isliye axial direction se missing fraction $1-\cos {\theta }_{\text{turn}}$ hai — chhote angles ke liye small, turn bada hone par fast badhta hai.

$C_F$ thrust ko ideal momentum flux ke against normalize karta hai, isliye yeh isolate karta hai ki nozzle kitni efficiently flow ko axial push mein convert karta hai, mass flow ya scale se independent. Dekho 2.5.12-Thrust-coefficient-definition.

Lip par koi pressure imbalance nahi, isliye koi expansion fan (aur koi shock bhi) nahi banta — plume seedha aur axial nikal jaata hai. Yeh under- aur over-expanded regimes ke beech perfectly-expanded boundary hai.

Nahi. Subsonic flow pressure signals ko upstream travel karne deta hai, isliye jet bas smoothly adjust ho jaati hai; Prandtl-Meyer expansion strictly supersonic phenomenon hai. (Real rocket nozzles exit par supersonic chalte hain, isliye yeh "kyun yeh choked nozzle ke andar nahi ho sakta" ki yaad dilaane wali baat hai.)

Nahi. $M_{\text{plume}}$ aur hence ${\theta }_{\text{turn}}$ ${\nu }_{\text{max}}$ se cap hain; ek baar jab required turn ${\nu }_{\text{max}}-\nu (M_e)$ se zyada ho jaaye, flow physically aage expand nahi kar sakta aur model ek highly under-expanded, barrel-shaped plume mein break down ho jaata hai.

$\nu (1)=0$ by definition — Prandtl-Meyer function Mach 1 se measure kiya jaata hai, isliye ek sonic exit ne "zero turning use up" kiya hai aur poora ${\nu }_{\text{max}}$ available hai.

Zero — pehli Mach wave $\theta =0$ mark karti hai; turning continuously fan ke across accumulate hoti hai total ${\theta }_{\text{turn}}$ tak aakhri wave par.

$\gamma$ dono isentropic pressure–Mach link aur $\nu (M)$ ki shape set karta hai, isliye ek alag propellant (alag $\gamma$) same $p_e/p_{\infty }$ ke liye alag $M_{\text{plume}}$ aur turning angle reach karta hai — yeh ceiling ${\nu }_{\text{max}}$ bhi change kar deta hai. Iske peeche ka area–Mach relation yahan hai 3.2.7-Isentropic-flow-area-Mach-relation.