Over-expanded nozzle — oblique shocks in plume, efficiency loss: Question bank

3.3.14 · D5 · Physics › Rocket Propulsion › Over-expanded nozzle — oblique shocks in plume, efficiency l

Yeh parent topic ke liye ek rapid-fire trap bank hai. Har item ek question ::: answer reveal hai — answer cover karo, zor se bol ke commit karo, phir check karo. Point arithmetic nahi hai (uske liye D3/D4 dekho) balki exam se pehle wrong-but-tempting picture ko pakadna hai.

Koi bhi trap judge karne se pehle, usmein use hone wale har symbol ka ek seedha-saadha matlab aur ek picture honi chahiye. Isliye hum pehle poori vocabulary banate hain, phir misconceptions dhundhte hain.

Symbols, ek ek karke

Thrust equation almost har trap mein aati hai, isliye use karne se pehle uske teen symbols ko unpack karo.

Definition Thrust equation aur uske parts

$F = \overset{m}{˙} v_{e} + (p_{e} - p_{a}) A_{e}$

$\overset{m}{˙}$ = mass flow rate — exhaust ke kilograms jo har second nozzle se nikalte hain. Dot Newton ka shorthand hai time derivative ke liye, toh $\overset{m}{˙} = \frac{d m}{d t}$ : wo rate jis par mass $m$ per unit time $t$ nikalti hai. Zyada gas per second → zyada push.
$v_{e}$ = exit velocity — kitni tez (m/s mein) woh gas nikalti hai. Tez exhaust → zyada push.
$A_{e}$ = exit area — nozzle ke munh ka physical area m² mein. Pressure mismatch $(p_{e} - p_{a})$ is poori disc par act karta hai, isliye bada munh dono gain (agar $p_{e} > p_{a}$ ) aur penalty (agar $p_{e} < p_{a}$ ) ko multiply karta hai.
$\overset{m}{˙} v_{e}$ momentum thrust hai (jet ka recoil); $(p_{e} - p_{a}) A_{e}$ pressure thrust hai (exit disc par net push). Poori picture Thrust Equation note mein hai: $F$ force hai, "cheezein phenki gayi" aur "munh par pressure imbalance" mein split.

Ab flow-geometry symbols. Har definition padhte waqt neeche ki figure dekho.

Figure — Over-expanded nozzle — oblique shocks in plume, efficiency loss

Definition Mach number, shock angle, deflection angle

$M$ = Mach number — flow speed divided by local sound speed. $M > 1$ supersonic hai; nozzle exit $M_{e} > 1$ hota hai. $M_{1}$ shock ke bilkul upstream ka Mach number hai.
$β$ (beta) = shock angle — incoming flow ke relative shock line ka tilt (figure mein slanted violet line).
$θ$ (theta) = deflection angle — us shock ko cross karte waqt streamlines kitni bend hoti hain (orange kink). Sign convention: over-expanded plume mein hum $θ > 0$ lete hain centerline ki taraf inward turn ke liye, jo direction ambient squeeze hamesha force karta hai; compression shock flow ko sirf apne andar turn kar sakta hai, kabhi bahar nahi, isliye $θ$ is poore page par positive hai. Attached oblique shock ke liye $β > θ$ hamesha: shock line turn se zyada steep hoti hai jo woh produce karta hai.
$M_{1 n} = M_{1} sin β$ = normal Mach component, flow ki speed ka woh hissa jo shock ke seedha andar point karta hai. Sirf yahi component compress hota hai; sideways wala hissa unchanged slide karta hai. Isliye oblique shock head-on (normal) wale se gentle hota hai — woh flow ki speed ka sirf ek fraction process karta hai.

Aakhir mein, stagnation pressure — woh single quantity jo loss measure karta hai.

Definition Stagnation pressure

$p_{0}$ aur yeh "quality gauge" kyun hai Socho ek moving gas stream ko bina friction ke dead stop par lao. Jo pressure woh tab reach karega woh uska stagnation pressure $p_{0}$ hai — yeh static pressure plus saari kinetic energy ko wapas pressure mein convert karta hai. Ise flow ka total "usable pressure budget" samjho.

Jab hum ek shock cross karte hain toh hum value ko bilkul upstream $p_{0, 1}$ aur bilkul downstream $p_{0, 2}$ label karte hain; ratio $p_{0, 2} / p_{0, 1}$ woh hai jo bachta hai.
Smooth (isentropic) flow mein $p_{0}$ conserved hota hai — koi budget kharach nahi hota.
Shock ke across $p_{0}$ girata hai ( $p_{0, 2} < p_{0, 1}$ ): kuch ordered energy heat ban gayi aur kabhi poori recover nahi ho sakti. Woh drop hi loss hai.
Velocity coefficient $C_{v} = \frac{v _{e, actual}}{v _{e, ideal}} = \frac{p _{0, actual}}{p _{0, chamber}}$ flow ke surviving pressure budget ko chamber ke original budget se compare karta hai; kyunki $p_{0}$ sirf gir sakta hai, $C_{v} \leq 1$ hamesha.

Plume-structure figure dekho yeh dekhne ke liye ki yeh sab ek real over-expanded jet ke andar kahan rehta hai.

Us figure mein baayi taraf navy bell nozzle hai; magenta curves plume boundary hai jo andar ki taraf pinch ho rahi hai (navy arrows $p_{a}$ ko andar push karte dikhaate hain, kyunki $p_{e} < p_{a}$ ); har violet line ek oblique shock hai pehli figure ki $β$ -line ki tarah; axis par orange bar Mach disk hai — ek chota normal shock jahan opposite oblique shocks milte hain. Ek bright "diamond" cell ek poora shock-then-expand cycle hai, aur poori train axis ke neeche repeat hoti hai.

Aakhri figure dikhata hai loss kaise accumulate hota hai — ek oblique shock ke across stagnation-pressure ratio $M_{1 n}$ ka function hai, aur teen aaise cuts kaise multiply karte hain. Jab bhi koi trap "compounding losses" mention kare toh ise refer karo.

Formula Ek oblique shock ke across stagnation-pressure loss

Normal-shock loss law ko normal component $M_{1 n} = M_{1} sin β$ par apply karo, jahan $p_{0, 1}$ shock ke bilkul upstream stagnation pressure hai aur $p_{0, 2}$ bilkul downstream: $\frac{p _{0, 2}}{p _{0, 1}} = (\frac{( γ + 1 ) M _{1 n}^{2}}{2 + ( γ - 1 ) M _{1 n}^{2}})^{\frac{γ}{γ - 1}} (\frac{γ + 1}{2 γ M _{1 n}^{2} - ( γ - 1 )})^{\frac{1}{γ - 1}}$ Yahan $γ$ gas ke specific heats ka ratio hai. $M_{1 n} = 1$ par dono factors alag alag 1 ke barabar hote hain: pehle factor ka numerator aur denominator dono $(γ + 1)$ ban jaate hain, aur doosre factor ka denominator $2 γ - (γ - 1) = γ + 1$ uske numerator ko cancel karta hai — toh unka product $1$ hai (ek Mach wave, zero loss). Jaise $M_{1 n}$ 1 se aage badhta hai product 1 se neeche girta hai. $N$ ko un oblique shocks ki number maano jo flow diamond train ke ek reflection pattern mein cross karta hai; bachne wale $(p_{0, 2} / p_{0, 1})^{N}$ ki tarah multiply karte hain — 1 se neeche ke fractions saath mein multiply karke sirf shrink hote hain, toh losses stack hoti hain, kabhi cancel nahi hoti.

Har symbol ab words, ek picture, aur ek formula se anchored hai, isliye neeche ke traps self-contained hain.

True or false — justify karo

Over-expansion ka matlab hai gas ke paas nozzle se nikalne ki energy bahut kam hai.

False — gas ke paas kaafi energy hai; over-expansion matlab hai woh pressure ambient se neeche expand hua (

p_{e} < p_{a}

), yeh nahi ki usmein speed ki kami hai. Yeh ek pressure-matching problem hai, energy shortage nahi.

p_{e} = p_{a}

par pressure-thrust term exactly zero hota hai.

True — thrust

F = \overset{m}{˙} v_{e} + (p_{e} - p_{a}) A_{e}

hai, aur

p_{e} - p_{a} = 0

doosre term ko khatam karta hai, sirf momentum thrust

\overset{m}{˙} v_{e}

bachta hai. Yeh perfectly-matched, maximum-efficiency condition hai.

Ek over-expanded nozzle hamesha same altitude par perfectly matched nozzle se kum total thrust produce karta hai.

True us altitude par — pressure term

(p_{e} - p_{a}) A_{e}

negative hai kyunki

p_{e} < p_{a}

, isliye yeh momentum thrust se subtract karta hai. Matching

p_{e} = p_{a}

restore karega aur woh khoya hua term recover karega.

Over-expanded plume mein shocks isentropic hote hain.

False — shocks inherently irreversible hote hain aur entropy raise karte hain, isliye

p_{0}

drop karte hain. Woh entropy rise exactly hai kyun over-expansion usable energy destroy karta hai, unlike under-expansion ke reversible expansion fans.

Over-expansion aur under-expansion equal efficiency penalties ke saath symmetric off-design conditions hain.

False — under-expansion pressure ko expansion fans ke through relieve karta hai, jo infinitely weak Mach waves ka ek spread-out family hai: har ek entropy ko negligible amount se raise karta hai aur total isentropic (reversible) hai, isliye

p_{0}

preserved hota hai. Over-expansion ki jagah compression finite shocks ke through force karta hai, jo

p_{0}

irreversibly kaate hain. Supersonic flow mein compression hamesha lossy hota hai; smooth expansion essentially free hai.

Same upstream Mach number par ek normal shock aur ek oblique shock same stagnation-pressure loss cause karte hain.

False — loss law

M_{1 n} = M_{1} sin β

par depend karta hai, aur ek oblique shock (

β < 90°

) ka

M_{1 n}

normal shock (

β = 90°

, toh

M_{1 n} = M_{1}

) se chota hota hai. Chota normal component → chota

p_{0}

drop, isliye nature oblique shocks ko compress karne ke liye prefer karta hai.

Nozzle area ratio bada karna hamesha thrust increase karta hai.

False — yeh sirf vacuum mein sach hai. Nozzle Area Ratio fix karta hai gas kitna expand hoga, isliye

p_{e}

: bada ratio

p_{e}

ko lower karta hai. Kam altitude par woh

p_{e}

p_{a}

se aur neeche le jaata hai, over-expansion aur uske shock losses ko gehri karta hai; har altitude ke liye ek optimum ratio hota hai (dekho Altitude Compensation).

Shock diamonds decorative aur harmless hain.

False — har bright diamond ek shock/expansion cycle mark karta hai, aur har shock (khaaskar central Mach disk, axis par ek chota normal shock)

p_{0}

bleeds karta hai aur ordered kinetic energy ko heat mein convert karta hai.

Flow separation guaranteed hai jis instant

p_{e}

thodi bhi

p_{a}

se neeche girti hai.

False — mild over-expansion tolerate hoti hai. Wall par patli, slow boundary layer sirf ek limited adverse pressure jump climb kar sakti hai detach hone se pehle; empirical criteria (jaise Summerfield) detachment ko

p_{e} / p_{a} \approx 0.3 - 0.4

ke paas rakhte hain kyunki wahan required pressure rise woh exceed kar leti hai jo near-wall low-momentum gas push kar sakti hai. Us ratio se upar layer attached rehti hai.

Severely over-expanded nozzle ke liye velocity coefficient

C_{v}

1 se zyada ho sakta hai.

False —

C_{v} = p_{0, actual} / p_{0, chamber}

aur stagnation pressure real flow mein sirf gir sakta hai ya equal reh sakta hai, kabhi rise nahi kar sakta, isliye

C_{v} \leq 1

hamesha. Over-expansion use 0.8 ya aur bura ki taraf push karta hai.

Error dhundo

p_{e} < p_{a}

ka matlab hai nozzle throat bahut hard choked hai, isliye hume throat chauda karna chahiye."

Throat ka over-expansion se koi lena dena nahi hai; over-expansion diverging section ke area ratio se set hoti hai, jisne

p_{e}

bahut neeche fix kar di. Throat chauda karna mass flow

\overset{m}{˙}

change karta hai, exit-to-ambient pressure match nahi.

"Oblique shocks flow ko bahar bend karte hain, plume ko spread karte hain."

Woh ise andar bend karte hain. Ambient hawa sides se squeeze karti hai, isliye compression streamlines ko centerline ki taraf turn karta hai — woh inward turning exactly figure mein deflection angle

θ

hai.

"Kyunki pressure-thrust term negative hai, rocket net backward thrust experience karta hai."

Nahi — momentum term

\overset{m}{˙} v_{e}

kisi bhi real engine ke liye dominate karta rahta hai, isliye net thrust

F

forward rehta hai. Negative pressure term sirf positive total ko reduce karta hai, reverse nahi.

"Kyunki shocks pressure raise karte hain, woh stagnation pressure bhi raise karte hain, isliye

p_{0}

increase hota hai."

Static pressure

p

raise hota hai lekin stagnation pressure

p_{0}

har shock ke across girta hai — kuch ordered energy heat ban gayi.

p_{0}

drop hi irreversible loss ka sahi measure hai;

p

p_{0}

se confuse karna classic trap hai.

"Sea level par test kiya hua vacuum engine sirf thoda kam thrust deta hai — koi badi baat nahi."

Pressure-thrust penalty

(p_{e} - p_{a}) A_{e}

tens of kilonewtons ho sakti hai, aur stacked shock losses

C_{v}

ko 0.78 tak gut kar sakti hain. Severe over-expansion flow ko bhi separate kar sakti hai aur nozzle ko physically shake kar ke tod sakti hai.

"Hum oblique-shock relations use karte hain kyunki flow subsonic hai aur turn kar rahi hai."

Shocks sirf supersonic flow mein exist karte hain (

M_{e} > 1

); ek subsonic exit bina shock ke smoothly adjust karta hai. Saara shock discussion

M_{e} > 1

presuppose karta hai Isentropic Flow se diverging cone ke through.

"Shock angle

β

deflection angle

θ

se bada hota hai sirf coincidence se."

Yeh structural hai, coincidental nahi: shock ko flow compress karne ke liye itna steep hona chahiye, lekin streamlines shock line se kam turn hoti hain, isliye attached oblique shock ke liye

β > θ

hamesha. Geometry Gas Dynamics se fix hai, luck se nahi.

Why questions

Atmosphere ek over-expanded plume ko kyun compress karta hai instead of plume simply hawa ko side mein push karne ke?

Kyunki

p_{a} > p_{e}

: zyada bahar ka pressure plume boundary par force balance jeetta hai aur andar push karta hai, aur ek supersonic flow us inward push ka jawab sirf shock waves ke through de sakta hai.

Compression smooth pressure rise ke through kyun nahi ho sakta, shock ke zariye kyun hona chahiye?

Supersonic flow mein gas pressure signals (jo sound speed par travel karte hain) se tez move karti hai, isliye information upstream propagate nahi ho sakti flow ko "warn" karne ke liye — pressure raise karne ka ek hi tarika hai discontinuous shock.

Shocks oblique (angled) kyun appear hote hain normal (perpendicular) ki jagah?

Compression sides se aati hai (nozzle lip par lateral squeeze), aur ek angled shock ko sirf flow ka normal component

M_{1 n} = M_{1} sin β

process karna hota hai, jo nature ka lower-entropy, sasta compression route hai.

Plume ke centerline par Mach disk kyun banta hai?

Opposite lips se oblique shocks andar travel karte hain aur axis par milte hain; jahan woh cross karte hain, ek single continued oblique turn flow ko over-constrain kar dega, isliye ek chota normal shock (Mach disk) ise resolve karta hai, flow ko locally subsonic reset karta hai.

Over-expansion rocket ke early climb ke dauran buri kyun hoti hai phir upar jaane par better?

Sea level par

p_{a}

sabse zyada hota hai, isliye ek fixed low

p_{e}

ambient se sabse door hota hai — deepest over-expansion. Jaise altitude badhti hai,

p_{a}

p_{e}

ki taraf girta hai, mismatch shrink hoti hai, aur eventually

p_{e} = p_{a}

(matched) phir

p_{e} > p_{a}

(under-expanded).

Multiple shocks loss ko average out karne ki jagah kyun compound karte hain (

p_{0}

ke terms mein)?

Kyunki

p_{0}

ratios successive shocks ke across multiplicative hain, additive nahi; har shock jo bachta hai uska ek fixed fraction remove karta hai, isliye

N

shocks

(p_{0, 2} / p_{0, 1})^{N}

dete hain, ek ever-shrinking product jo kabhi recover nahi kar sakta.

Under-expansion engine hardware ke liye "safer" off-design failure kyun hai?

Under-expansion expansion fans produce karta hai (reversible, no separation) aur plume simply bahar ki taraf bulge karta hai, jabki over-expansion bell ke andar flow separate kar sakti hai, asymmetric side-loads cause karte hue jo nozzle crack kar sakti hain. Isliye altitude-compensating designs (dekho Altitude Compensation) prized hain.

Vacuum-optimized engine ko sea level par fire karna tumhe do baar kyun punish karta hai?

Ek baar negative pressure-thrust term

(p_{e} - p_{a}) A_{e}

ke through (pressure mismatch ko raw force kho deta hai), aur dobara stacked shock losses ke through jo

C_{v}

ko drag karte hain — momentum thrust khud pressure penalty ke upar degrade hota hai.

Edge cases

Jab engine perfect vacuum mein fire karta hai toh

p_{e} - p_{a}

kya hota hai?

Tab

p_{a} = 0

, isliye pressure term

+ p_{e} A_{e}

ban jaata hai, hamesha positive — vacuum-designed nozzle space mein kabhi over-expanded nahi hota, exactly yahi uske liye banaya gaya tha.

Kya ek nozzle ek hi ascent ke dauran ek altitude par over-expanded aur doosri par under-expanded ho sakta hai?

Haan — ek fixed-geometry nozzle ka fixed

p_{e}

hota hai, isliye jaise

p_{a}

altitude ke saath girta hai woh

p_{e} < p_{a}

(over) se

p_{e} = p_{a}

(matched) se

p_{e} > p_{a}

(under) tak pass karta hai. Ek design point har jagah optimal nahi ho sakta, jo Rocket Staging aur altitude compensation motivate karta hai.

Exactly

p_{e} = p_{a}

par, kya lip par koi oblique shocks form hote hain?

Nahi — perfect matching ke saath koi pressure jump accommodate karne ki zarurat nahi, isliye plume ek clean parallel column ki tarah nikalti hai bina shock ya expansion fans ke. Yeh ideal, loss-free boundary hai do off-design regimes ke beech.

Extreme over-expansion ki limit mein,

p_{e} / p_{a} \to 0

par kya hota hai?

Adverse pressure ratio itna violent ho jaata hai ki boundary layer attached nahi reh sakti; flow nozzle ke andar separate ho jaati hai, effective exit upstream shift ho jaata hai, aur internal shock structure aur side-loads hardware destroy kar sakte hain.

Agar exit flow subsonic hoti (

M_{e} < 1

), toh over-expansion kaisi dikhti?

Koi bhi shocks nahi hote — subsonic flow nikalne se pehle apna pressure smoothly ambient ke saath adjust kar leta hai. Shock-based over-expansion losses sirf supersonic exits ke liye unique hain, isliye poori plume-shock story tab hi apply hoti hai jab

M_{e} > 1

ho.

Ek single oblique shock ke liye, jaise

β

Mach angle ki taraf shrink hota hai, limiting deflection

θ

kya hai?

Jaise

β \to arcsin (1/ M_{1})

normal component

M_{1} sin β \to 1

hota hai, shock ek Mach wave tak weak hota hai, aur deflection

θ \to 0

— ek infinitely weak shock flow ko kuch nahi turn karta aur essentially koi loss nahi cause karta.

Recall Ek-line self-test is page band karne se pehle

Woh single inequality naam batao jo over-expansion define karti hai, aur woh single physical process jo ise lossy banata hai. Answer ::: $p_{e} < p_{a}$ ise define karta hai; irreversible shock compression ( $p_{0}$ drop, entropy rise) ise lossy banata hai.