Over - under expanded nozzle flows — Hinglish notes

3.1.18 · Physics › Compressible Flow & Aerodynamics

YE problem hoti hi kyun hai?

Ek converging–diverging (de Laval) nozzle gas ko supersonic speeds tak accelerate karta hai. Lekin yahan ek catch hai:

Ek supersonic exit downstream conditions ke liye "behra" hota hai. Jab flow diverging section mein supersonic ho jaati hai, to pressure information (jo sound ki speed se travel karti hai) flow ke against upstream nahi ja sakti. Isliye nozzle exit pressure $p_{e}$ area ratio $A_{e} / A^{*}$ se fix hoti hai, back pressure $p_{b}$ se nahi.
Lekin bahar ka atmosphere $p_{b}$ par hota hai. Agar $p_{e} \neq = p_{b}$ , to nature ko inhe milana padta hai — aur woh yeh kaam oblique shocks se karta hai (agar exit pressure zyada kam ho) ya expansion fans se (agar exit pressure zyada ho), aur yeh sab nozzle ke bahar hota hai.

Har regime pressure mismatch ko kaise resolve karta hai?

Under-expanded ( $p_{e} > p_{b}$ )

Jet ambient se zyada pressure par hai, isliye exit hote hi woh aur expand hoti hai. Yeh Prandtl–Meyer expansion fans ke zariye hota hai jo nozzle lip par anchor hoti hain. Jet bahar ki taraf bulge karta hai (downstream mein diamond/shock-cell pattern banta hai jab fans free jet boundary se reflect hote hain).

Over-expanded ( $p_{e} < p_{b}$ )

Ambient pressure zyada hai, isliye woh jet ko andar ki taraf push karta hai. Compression lip se oblique shocks ke zariye hoti hai.

Mildly over-expanded: oblique shock pattern (lip par attached rehta hai).
Strongly over-expanded: required pressure rise itna bada hota hai ki ek normal shock ya Mach disk banta hai; agar back pressure kaafi zyada ho to shock diverging section ke andar chali jaati hai (flow separation walls se ho jaata hai).

DERIVATION — $p_{e}$ aata kahan se hai? (first principles)

Hum yeh dikhana chahte hain ki $p_{e}$ sirf geometry par depend karta hai. Isentropic, steady, 1-D, adiabatic flow of a perfect gas ke liye teen conservation ideas se shuru karte hain.

Step 1 — Stagnation relations. Kyun? Energy conservation ( $h_{0} = h + \frac{1}{2} V^{2}$ constant) aur isentropic link $p / ρ^{γ} =$ const milake, perfect gas ke liye deta hai: $\frac{p _{0}}{p} = (1 + \frac{γ - 1}{2} M^{2})^{\frac{γ}{γ - 1}}$ Yeh step kyun? Yeh local pressure ko local Mach number $M$ se jodata hai — yahi master relation hai.

Step 2 — Area–Mach relation. Kyun? Mass conservation $ρ A V =$ const ko usi isentropic relations ke saath combine karne par milta hai: $\frac{A}{A ^{*}} = \frac{1}{M} [\frac{2}{γ + 1} (1 + \frac{γ - 1}{2} M^{2})]^{\frac{γ + 1}{2 ( γ - 1 )}}$ Yeh step kyun? $A^{*}$ throat area hai (jahan $M = 1$ , choked). Exit-to-throat area ratio $A_{e} / A^{*}$ diya ho to is equation ka ek unique supersonic root $M_{e}$ hota hai.

Step 3 — Combine karo. $M_{e}$ (geometry se, Step 2) ko Step 1 mein dalo: $p_{e} = \frac{p _{0}}{( 1 + \frac{γ - 1}{2} M _{e}^{2} ) ^{γ / (γ - 1)}}$

Thrust: practical WHY yeh matter karta hai

Consequences:

Perfectly expanded ( $p_{e} = p_{b}$ ): pressure term zero ho jaata hai → diye gaye $V_{e}$ ke liye maximum thrust. Yahi optimal design point hai.
Under-expanded ( $p_{e} > p_{b}$ ): positive pressure term thrust add karta hai, lekin tumne "velocity table par chhod di" (aur expand kar sakte the) — net efficiency optimum se neeche.
Over-expanded ( $p_{e} < p_{b}$ ): pressure term negative hota hai → drag-jaisa loss; severe over-expansion separation aur instability cause kar sakta hai.

Common mistake Steel-manned misconceptions

"Back pressure ek supersonic nozzle ki exit pressure change kar deta hai." Yeh sahi kyun lagta hai: Everyday (subsonic) flow mein downstream pressure flow ko bilkul control karta hai — hose ka end dabaao aur pressure har jagah badhta hai. Isliye yeh obvious lagta hai. Fix: Subsonic info sound ki speed par upstream travel karti hai; supersonic flow use peeche chhod deta hai. Exit $p_{b}$ ko sun hi nahi sakta, isliye $p_{e}$ geometry se lock hai. Pressure mismatch external shocks/fans mein dump hoti hai.

"Over-expanded = pressure bahut zyada." Yeh sahi kyun lagta hai: "Over" bahut zyada pressure ki tarah sunai deta hai. Fix: "Over-expanded" = bahut zyada expand kiya ⇒ pressure bahut kam ( $p_{e} < p_{b}$ ). Upar [!intuition] dobara padho.

"Shocks is liye banti hain kyunki gas bahut fast hai." Yeh sahi kyun lagta hai: Shocks supersonic speed se associated hain. Fix: Yahan shocks is liye banti hain kyunki pressure ko $p_{b}$ se match karne ke liye badhna padta hai. Compression (over-expanded) → shocks; expansion chahiye (under-expanded) → fans. Driver pressure direction hai, speed akeli nahi.

Recall Feynman: ek 12-saal ke bacche ko samjhao

Ek water slide socho jo perfectly tuned hai taaki bacche pool mein gently splash karein. Slide sirf EK pool height ke liye bani hai. Agar pool neeche ho jitna slide expect karti hai, to bacche fast nikle aur hawa mein phail jaate hain (under-expanded → expansion fans). Agar pool upar ho, to paani peeche push karta hai aur bacche neeche splash mein ichad jaate hain (over-expanded → shocks). Slide apni shape beech ride mein nahi badal sakti, isliye bahar ki duniya ko adjust karna padta hai — sprays (fans) ya splashes (shocks) se.

Flashcards

Fully-supersonic nozzle ki exit pressure kya set karta hai?

Sirf area ratio

A_{e} / A^{*}

aur

p_{0}

(geometry), back pressure nahi, kyunki supersonic flow upstream info transmit nahi kar sakti.

Over-expanded flow define karo.

p_{e} < p_{b}

: nozzle ne gas ko bahut zyada expand kiya, exit pressure ambient se neeche; oblique shocks (ya Mach disk/separation) compress karne ke liye banti hain.

Under-expanded flow define karo.

p_{e} > p_{b}

: exit pressure ambient se upar; flow bahar Prandtl–Meyer expansion fans ke zariye expand hoti rehti hai, jet bulge karta hai.

Perfectly expanded flow ki condition kya hai?

p_{e} = p_{b}

, parallel exit flow, koi external waves nahi, diye gaye exit velocity ke liye maximum thrust.

Thrust equation likho aur pressure term explain karo.

F = \overset{m}{˙} V_{e} + (p_{e} - p_{b}) A_{e}

; pressure term exit pressure

p_{e}

minus ambient

p_{b}

A_{e}

par net force hai.

Rocket climb karte waqt over- se under-expanded kyun ho jaata hai?

p_{e}

geometry se fix hai, lekin

p_{b}

altitude ke saath girta hai; eventually

p_{b} < p_{e}

ho jaata hai, over-expanded se under-expanded flip ho jaata hai.

Strongly over-expanded jets mein kya dikhai deta hai?

Ek normal shock / Mach disk aur possibly diverging section ke andar flow separation.

Over-expanded ka memory hook kya hai?

"OLE": Over = Low exit pressure → shocks chahiye.

p_{0} / p_{e}

M_{e}

se jodne wala relation?

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

Connections

Isentropic Flow Relations
Area-Mach Number Relation
Choked Flow & the Throat (M=1)
Oblique Shocks
Prandtl-Meyer Expansion Fans
Normal Shock & Mach Disk
Rocket Nozzle Design & Thrust Optimization
Flow Separation in Nozzles

Over - under expanded nozzle flows