Over - under expanded nozzle flows: Question bank — Hinglish deep dive

3.1.18 · D5 · Physics › Compressible Flow & Aerodynamics › Over - under expanded nozzle flows

Shuru karne se pehle, ek shared vocabulary reminder taaki neeche kuch bhi unexplained na lage:

Over-expanded $= p_{e} < p_{b}$ (zyada expand ho gaya → bahut low → shocks). Under-expanded $= p_{e} > p_{b}$ (kam expand hua → bahut high → fans). Perfectly expanded $= p_{e} = p_{b}$ .

Exit-pressure formula kahan se aati hai (woh missing "why")

Neeche ke traps ek master formula par lean karte hain, toh aao ise sirf quote karne ki jagah samjhein.

YEH KYUN HOLD KARTA HAI? Do physical facts, chain kiye hue:

Energy conserved hoti hai adiabatic flow mein: "total energy per kilogram" $h_{0} = h + \frac{1}{2} V^{2}$ constant rehti hai. Jaise gas speed up hoti hai ( $V$ badhta hai), uska thermal part $h$ (aur hence temperature) drop hona chahiye. Mach number mein likha jaaye toh yeh banta hai $\frac{T _{0}}{T} = 1 + \frac{γ - 1}{2} M^{2}$ — jitna tez jaao, utna thanda hota hai gas.
Process isentropic hai (smooth, no shocks, no friction), toh pressure aur temperature $p / T^{γ / (γ - 1)} = const$ se lock hain. Fact 1 ko isme daalo aur pressure ko wahi bracket mila power $γ / (γ - 1)$ ke saath.

Yehi poora "why" hai: tez jaana gas ko thanda karta hai (energy), aur thanda hona pressure ko girata hai (isentropy) — dekho Isentropic Flow Relations. Toh bada $M_{e}$ hamesha chota $p_{e}$ matlab hai, aur $M_{e}$ khud purely area ratio $A_{e} / A^{*}$ se set hota hai Area-Mach Number Relation ke zariye. Curve dekho:

Figure — Over - under expanded nozzle flows

True ya false — justify karo

Ek supersonic nozzle ko downstream valve kholkar apne back pressure ke liye redesign kiya ja sakta hai.

False. Jab diverging section supersonic ho jaata hai, pressure signals (local speed of sound par travel karte hue) faster flow ke against upstream nahi ja sakte, toh downstream ki koi bhi cheez

p_{e}

nahi badal sakti; sirf geometry aur

p_{0}

ise set karte hain.

Over-expanded flow matlab exit pressure ambient se zyada hai.

False. Over-expanded matlab zyada expand ho gaya →

p_{e} < p_{b}

; exit pressure ambient se neeche hai, isliye ise wapas upar laane ke liye compression (shocks) chahiye.

Exact design point par thrust equation mein pressure term zero hoti hai.

True. Perfectly expanded matlab

p_{e} = p_{b}

, toh

(p_{e} - p_{b}) A_{e} = 0

aur saara thrust momentum term

\overset{m}{˙} V_{e}

se aata hai.

Under-expanded flow hamesha perfectly expanded flow se zyada thrust produce karta hai.

False us sense mein jo matter karta hai. Pressure term

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

positive hai aur thrust add karta hai, lekin tumne poora expand na karke exit velocity

V_{e}

sacrifice kiya, toh overall engine apni optimum efficiency se neeche hai.

Sea level par over-expanded rocket space mein bhi over-expanded rahega.

False.

p_{e}

geometry se fix hai, lekin

p_{b}

altitude badhne par girta hai; jab

p_{b}

p_{e}

se neeche gir jaata hai toh same nozzle under-expanded ho jaata hai, aur exactly yehi hota hai jab rocket climb karta hai.

Over-expanded jet mein shocks isliye aate hain kyunki gas bahut tez chal rahi hai.

False. Yeh isliye aate hain kyunki pressure ko

p_{e}

p_{b}

tak rise karna hota hai; compression shocks demand karta hai. Speed shocks ko exist karne deti hai (unhe supersonic flow chahiye) lekin driver pressure direction hai.

Expansion fans gas ko compress karte hain.

False. Prandtl–Meyer fans (dekho Prandtl-Meyer Expansion Fans) pressure lower karte hain aur flow ko accelerate karte hain; yeh under-expanded jets mein precisely isliye appear hote hain kyunki gas ke paas abhi bhi excess pressure hai jo baahir nikalna chahti hai.

Chamber pressure

p_{0}

badhana nozzle fix rakhते hue flow ko under-expansion ki taraf push karta hai.

True.

p_{e} = p_{0} / (1 + \frac{γ - 1}{2} M_{e}^{2})^{γ / (γ - 1)}

mein exit Mach

M_{e}

geometry se fix hai, toh

p_{e}

directly

p_{0}

ke saath scale karta hai; zyada

p_{0}

p_{e}

badhata hai, aur agar

p_{b}

unchanged hai toh

p_{e}

p_{b}

se upar chadh sakta hai → under-expanded.

Error dhundo

"Nozzle over-expanded hai, toh ise fix karne ke liye diverging section ko chhota karna chahiye."

Verdict sahi hai lekin sirf tab agar tum dikhaa sako kyun. Kam-flared exit matlab chota

A_{e} / A^{*}

; Area-Mach Number Relation se chota area ratio chota supersonic

M_{e}

deta hai; aur boxed formula mein chota

M_{e}

denominator ko shrink karta hai, toh

p_{e} = p_{0} / (1 + \frac{γ - 1}{2} M_{e}^{2})^{γ / (γ - 1)}

p_{b}

ki taraf rise karta hai. Chain

A_{e} / A^{*} ↓\Rightarrow M_{e} ↓\Rightarrow p_{e} ↑

asli reason hai.

"Kyunki

p_{e}

sirf geometry par depend karta hai, thrust altitude se independent hai."

Error:

p_{e}

fixed hai, lekin thrust mein

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

hai, aur

p_{b}

altitude ke saath change karta hai. Toh thrust rocket climb karne par badhta hai chahe

p_{e}

locked ho — woh pressure term hi poora reason hai ki altitude matter karta hai.

"Mach disk bas ek bada oblique shock hai."

Error: Mach disk strongly over-expanded jet ke core mein ek normal (perpendicular) shock hai, jo bada pressure jump deta hai; oblique shocks kamaore aur inclined hote hain. Inhe confuse karna pressure rise aur entropy loss ko under-predict karta hai.

"Area ratio se

M_{e}

find karne ke liye, bas area–Mach relation ka subsonic root lo."

Error: area–Mach relation diye gaye

A_{e} / A^{*}

ke liye do roots deta hai; supersonic nozzle supersonic root leta hai. Subsonic root ek venturi describe karta hai jo wapas decelerate karta hai, de Laval exit nahi.

"Kyunki flow throat par choked hai, exit bhi

M = 1

par hona chahiye."

Error: choking

M = 1

sirf throat par fix karta hai. Diverging section mein downstream supersonic branch

M = 1

se aage accelerate karta hai, toh

M_{e} > 1

aur area ratio ke saath badhta hai.

"Over-expanded jet ki jet boundary ek solid wall hai, toh shocks mirror ki tarah reflect karte hain."

Error: jet edge ek free (constant-pressure) boundary hai, wall nahi. Isse takraata hua shock ek expansion fan ke roop mein reflect karta hai (aur vice versa), jo alternating diamond/shock-cell pattern banata hai.

Why questions

Back pressure supersonic exit mein "pahunch" ke

p_{e}

kyun nahi badal sakta?

Pressure disturbances local sound speed par travel karte hain; supersonic flow mein gas usse tez move karta hai, toh upstream-directed signals downstream sweep ho jaate hain aur kabhi exit plane tak nahi pahunchte.

"Over-expanded" itna confusing naam kyun hai?

Yeh action ko naam deta hai (gas zyada expand ho gaya) resulting pressure ko nahi. Zyada expansion pressure ko ambient se neeche le jaata hai, toh "over" surprisingly low

p_{e}

ke saath pair karta hai.

Over-expanded nozzle mein flow separation ka risk kyun hai lekin under-expanded mein nahi?

Over-expansion ko flow direction ke against pressure rise chahiye; itna strong adverse gradient boundary layer ko nozzle ke andar wall se stall kar deta hai (dekho Flow Separation in Nozzles). Under-expansion ko pressure drop chahiye, jo flow khushi se accept karta hai aur nozzle ke baahir entirely karta hai.

Pressure-thrust term

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

thrust equation mein aata hi kyun hai?

Control-volume momentum balance mein exit plane par surface pressure forces shamil hain:

p_{e}

gas ko baahir push karta hai, atmospheric

p_{b}

same area

A_{e}

par andar push karta hai, aur unka net woh term hai.

Perfect expansion thrust optimum kyun hai na ki maximum-velocity expansion?

Aur expand karna

V_{e}

badhata hai lekin

p_{e}

p_{b}

se neeche le jaata hai, pressure term ko negative kar deta hai (ek drag-jaisi loss). Momentum thrust mein gain pressure penalty se zyada kha jaata hai, toh actual peak

p_{e} = p_{b}

par baithta hai (dekho Rocket Nozzle Design & Thrust Optimization).

Under-expanded jets visible "shock diamonds" kyun form karte hain agar pehli waves expansion fans hain?

Fans jet ko

p_{b}

se neeche over-expand karte hain; yeh free boundary se compression waves ke roop mein reflect hote hain jo oblique shocks mein coalesce ho jaate hain, jo phir fans ke roop mein reflect hote hain — yeh repeating over/under cycle hi diamond pattern hai.

Ek single fixed nozzle zyaattar time inevitably off-design kyun run karta hai?

Yeh exactly ek

p_{b}

ke liye perfectly expanded hai; jaise ambient pressure change hoti hai (altitude, throttling)

p_{b}

fixed

p_{e}

se drift ho jaata hai, toh real engines apni flight over- ya under-expanded guzaarte hain, sirf momentarily match karte hain.

Edge cases

Kya hota hai jab

p_{b} \to p_{e}

dono taraf se?

Baahri waves vanishing ki taraf kamzor hoti jaati hain;

p_{b} = p_{e}

par jet parallel aur wave-free nikalti hai (perfect expansion). Upar se approach karna kamazor shocks deta hai, neeche se kamazor fans.

Flow kaisi dikhti hai jab

p_{b}

bahut high ho — nearly

p_{0}

ke barabar?

Nozzle supersonic exit tak pahunch bhi nahi sakta; normal shock diverging section ke andar baithta hai (ya flow entirely subsonic hai), toh "supersonic-exit, geometry-locked

p_{e}

" wali story toot jaati hai aur back pressure wapas exit control karne lagta hai.

Extreme case mein kya hota hai

p_{b} > p_{0}

(ambient chamber pressure se bhi zyada)?

Ab gas ko baahir push karne ke liye koi pressure drop nahi raha — driving pressure difference reverse ho jaata hai. Nozzle forward supersonic flow establish hi nahi kar sakta; ya toh flow band ho jaata hai ya atmosphere gas ko backward chamber mein push kar deta hai. Har over/under-expanded concept

p_{0} > p_{b}

assume karta hai, toh yeh case regime map se bilkul baahir hai.

Limiting under-expanded case mein kya hota hai jab

p_{b} \to 0

(vacuum mein firing)?

p_{e} > p_{b}

hamesha, toh jet maximally under-expanded hai; exhaust plume strong fans ke saath bahut baahir expand karta hai, aur pressure-thrust term

(p_{e} - 0) A_{e} = p_{e} A_{e}

apni sabse badi positive value tak pahunchta hai.

Throat par khud, kya flow kabhi over- ya under-expanded hoti hai?

Nahi. Throat choked rahta hai

M = 1

par chahe koi bhi regime ho; over/under-expansion exit pressure vs ambient ka statement hai, jo diverging section mein aage decide hota hai.

Agar

A_{e} / A^{*} = 1

ho, toh kaunsa regime ho sakta hai?

Toh "diverging" section absent hai aur

M_{e} = 1

(sonic exit). Koi supersonic expansion nahi, toh classic over/under-expanded shock-and-fan behaviour develop nahi hota; exit bas choked aur sonic hai.

Agar

p_{b}

badhta rahe toh over-expanded oblique-shock pattern ka kya hota hai?

Required pressure jump badhta hai jab tak oblique shocks ise supply nahi kar sakte; ek stronger normal shock / Mach disk form hota hai, aur aur badhaane par shock nozzle ke andar upstream march karta hai, wall separation trigger karta hai (dekho Oblique Shocks aur Flow Separation in Nozzles).

Recall Ek-line summary lock karne ke liye

Geometry $p_{e}$ set karta hai; sky $p_{b}$ set karta hai; mismatch $p_{e}$ vs $p_{b}$ decide karta hai shocks (over, $p_{e} < p_{b}$ ) ya fans (under, $p_{e} > p_{b}$ ), aur $(p_{e} - p_{b}) A_{e}$ wahan hai jahan yeh tumhare thrust par lagta hai.