Combustion instability — low-frequency (chugging), high-frequency (screaming): Question bank

3.3.32 · D5 · Physics › Rocket Propulsion › Combustion instability — low-frequency (chugging), high-freq

Neeche har "answer" side sirf reasoning deta hai, bare yes/no nahi — yahi concept trap ka poora point hai.

Is page par use hue Symbols — pehle padho

Traps se pehle, neeche har symbol ko plain words aur units ke saath pin kiya gaya hai, taaki kisi bhi reveal mein koi undefined cheez na aaye.

Definition Symbol glossary

$p^{'}$ — chamber pressure ka fluctuating part, yaani mean ke around ka wobble: $p_{c} (t) = \overset{p}{ˉ}_{c} + p^{'} (t)$ . Units: pascals (Pa). Swing analogy mein yahi "swing" karta hai.
$q^{'}$ — fluctuating heat-release rate per unit volume, mean burn rate ke around ka wobble. Units: W/m³. Yeh "push" hai.
$ϕ$ — phase lag (degrees ya radians mein) jitna heat wobble $q^{'}$ , pressure wobble $p^{'}$ se peeche rehta hai. $ϕ = 0$ matlab dono ek saath peak karte hain; $ϕ = 18 0^{\circ}$ matlab bilkul ulta.
$ω$ — oscillation ki angular frequency, rad/s mein. Yeh ordinary frequency $f$ (Hz, cycles/s mein) se $ω = 2 π f$ ke zariye related hai: ek full cycle $2 π$ radians of phase hoti hai.
$τ$ — combustion/transit lag: propellant ke andar aane aur uski heat actually release hone ke beech ka time (seconds, typically ms). Yeh chugging ki slow clock hai.
$Δ p_{inj}$ — injector pressure drop, $p_{f ee d} - p_{c}$ : feed kitna zyada push karta hai chamber ke push-back se. Units: Pa. Bada drop = "stiff" injector.
$\overset{m}{˙}_{in}$ — injector se chamber mein propellant mass flow (kg/s). Dot ka matlab "per second" hai. Yeh injector drop ke saath badhta hai: $\overset{m}{˙}_{in} \propto Δ p_{inj} = p_{f ee d} - p_{c}$ .
$\overset{m}{˙}_{o u t}$ — nozzle se mass flow out (kg/s). Choked throat ke liye $\overset{m}{˙}_{o u t} = p_{c} A_{t} / c^{*}$ , isliye yeh chamber pressure ke saath badhta hai.
$A_{t}$ — nozzle throat area (m²), sabse narrow cross-section jahan flow choke hota hai. Typical: kuch cm² se tens of cm² tak.
$c^{*}$ — [[Characteristic Velocity c|characteristic velocity]]* (m/s), ek propellant-aur-chamber property jo measure karta hai ki combustion kitni efficiently pressure raise karta hai; typical $\sim 1500$ – $1800$ m/s. Yeh $\overset{m}{˙}_{o u t} = p_{c} A_{t} / c^{*}$ mein aata hai.
$k$ — feed-coupling gain: ek dimensionless number jo batata hai ki chamber-pressure wobble $p^{'}$ kitni strongly injector ke zariye heat-release wobble mein feedback karta hai. Bada $k$ = tight coupling = easy chugging; $k \to 0$ karo aur loop band nahi ho sakta. Yeh $Δ p_{inj}$ badhne se shrink hota hai kyunki stiff injector $p_{c}$ ke wobble hone par $\overset{m}{˙}_{in}$ mein zara sa bhi change nahi karta.

$f_{c h ug} \approx 1/ (4 τ)$ Derive Karna — Quarter-Cycle Rule

Figure — Combustion instability — low-frequency (chugging), high-frequency (screaming)

Formula Phase se frequency tak

Pressure dip extra propellant flow banata hai, lekin woh heat sirf lag $τ$ ke baad aati hai. Us late heat ka next pressure rise ke saath in phase land karna — swing-pushing condition — ke liye, delay roughly quarter of a cycle ke barabar hona chahiye. Ek full cycle $2 π$ radians of phase hai; quarter cycle $π /2$ radians hai. Time $τ$ mein accumulated phase $ω τ$ hai (angular frequency $\times$ time). Toh marginal condition hai: $ω τ \approx \frac{π}{2} .$ $ω = 2 π f$ substitute karo: $2 π f τ = \frac{π}{2} .$ Dono sides ko $2 π τ$ se kyon divide karte hain? Hume $f$ akela left side par chahiye; $f$ abhi $2 π τ$ se multiply ho raha hai, toh exactly us factor se divide karne par left side par cancel ho jata hai aur same operation right side par bhi laagu hota hai (equation sahi rehti hai): $f = \frac{π /2}{2 π τ} = \frac{1}{4 τ} .$ Yahi chug frequency hai: $f_{c h ug} \approx \frac{1}{4 τ}$ . $τ = 2.5$ ms ke liye yeh $f = 1/ (4 \times 0.0025) = 1/0.01 = 100$ Hz deta hai — bilkul chugging band mein.

True ya False — justify karo

Ek rocket engine unstable ho sakta hai chahe cycle mein release ki gayi total heat exactly average design value ho.

True — instability ko timing se matlab hai, amount se nahi; wahi average heat, agar jab pressure high ho tab bunched ho, Rayleigh Criterion satisfy karta hai aur oscillation badhata hai.

Chugging aur screaming alag physical laws se govern hote hain.

False — dono identical Rayleigh condition

\oint p^{'} q^{'} d t > 0

obey karte hain; sirf time-delay reservoir alag hota hai (feed lag ms mein vs acoustic transit μs mein).

Heat ko exactly pressure ke saath phase mein add karna (

ϕ = 0

) sabse zyada destabilising case hai.

True — kyunki driving overlap

cos ϕ

se scale hota hai, zero phase lag se maximum positive

\oint p^{'} q^{'} d t

milta hai, toh swing apne forward peak par sabse zyada push hoti hai.

Agar heat release pressure se

18 0^{\circ}

lag kare, toh oscillation phir bhi badhti hai, bas slower.

False —

ϕ = 18 0^{\circ}

par,

cos ϕ = - 1

, toh

\oint p^{'} q^{'} d t < 0

; heat tab aati hai jab pressure low hota hai, mode ko feed karne ki bajaye actively damp karti hai.

Ek perfectly stiff injector (infinite

Δ p_{inj}

) chugging feedback loop hata deta hai.

True — infinite drop

\overset{m}{˙}_{in} \propto p_{f ee d} - p_{c}

p_{c}

se practically independent bana deta hai, toh feed-coupling gain

k

collapse hokar zero ho jata hai aur delay loop band nahi ho sakta. Dekho Injector Design & Pressure Drop.

First tangential (1T) mode mainly isliye dangerous hai kyunki yeh sabse high-frequency mode hai.

False — yeh isliye dangerous hai kyunki iska pressure side to side slosh karta hai, hot gas ko wall ke khilaf scrub karta hai; yeh actually lower transverse modes mein se ek hai, sabse high nahi.

Injector face par baffles chugging ka achha ilaj hain.

False — baffles transverse acoustic modes (screaming) ko disrupt karte hain; chugging feed system mein rehta hai aur unse untouched hai. Chug ka ilaj injector

Δ p_{inj}

hai.

Chamber pressure

p_{c}

badhana hamesha stability improve karta hai.

False — jo matter karta hai woh ratio

Δ p_{inj} / p_{c}

hai; drop badhaye bina

p_{c}

badhane se injector "softer" ho jata hai aur chugging worse ho sakti hai.

Nozzle radiation aur viscous losses hi instability ki final steady amplitude set karte hain.

True — Rayleigh growth linearly unbounded hai; nonlinear damping (nozzle radiation, viscosity) hi use finite limit cycle mein cap karta hai.

Ek feed system sirf hamesha ek frequency

1/ (4 τ)

par chug kar sakta hai.

False —

(2 n + 1) π /2

radians ka koi bhi delay bhi heat ko in phase land karta hai, isliye

f \approx (2 n + 1) / (4 τ)

par higher tongues (

3/ (4 τ)

5/ (4 τ)

, …) possible harmonics hain.

Error Pakdo

"Chugging kHz par hoti hai kyunki feed lines lambi hain, toh sound ko unhe travel karne mein time lagta hai."

Do errors hain: chugging low frequency hoti hai (10–400 Hz), aur iska clock combustion/transit lag

τ

hai, acoustic travel nahi — kHz screaming aur chamber acoustics se belong karta hai.

"Screaming khatam karne ke liye, chugging ki tarah injector pressure drop badha do."

Wrong reservoir — screaming chamber gas acoustics ke zariye feed hota hai, feed system ke zariye nahi; ilaj acoustic damping hai (Injector Baffles, Helmholtz Resonator cavities), jo us mode mein loss add karta hai.

"Rayleigh integral

\oint p^{'} q^{'} d t

total heat

q

use karta hai, toh bada engine hamesha zyada unstable hota hai."

q^{'}

fluctuating heat-release rate hai, mean

q

nahi; badi steady heat jisme koi fluctuation na ho woh integral mein zero contribute karti hai.

f_{c h ug} \sim 1/ (4 τ)

, toh feed line short karna

τ

short karta hai aur hamesha stabilise karta hai."

τ

combustion + transit + ignition lag hai, sirf line length se dominate nahi hota; aur

τ

shift karna sirf resonant frequency ko move karta hai, guarantee nahi karta ki phase growth window se bahar nikal jaye.

"Chamber mein sound speed ambient air use karta hai,

a \approx 340

m/s."

Gas

\sim 3000

K par combustion products hai, toh

a = γ R_{g a s} T_{c} \sim 1000

–

1200

m/s — roughly teen guna zyada, jo exactly isliye screech kHz band mein land karta hai. Dekho Acoustic Modes of a Cylindrical Cavity.

"Choked nozzle matlab

\overset{m}{˙}_{o u t}

fixed hai, toh woh chug loop mein participate nahi kar sakta."

Choked nozzle ke liye

\overset{m}{˙}_{o u t} = p_{c} A_{t} / c^{*}

(throat area

A_{t}

fixed,

c^{*}

ek propellant constant) phir bhi $p_{c}$ ke saath badhta hai — mass flow chamber pressure track karta hai, mass balance mein restoring/emptying term provide karta hai.

Why Questions

Rayleigh ke criterion mein sum ki bajaye product

p^{'} q^{'}

kyon aata hai?

Kyunki product phase mein overlap measure karta hai: yeh large-positive tab hi hota hai jab dono signals ek saath high hain, "heat added while pressure is already high" ko capture karta hai — swing-pushing condition.

τ

akela slow clock kyon hai jo chug frequency set karta hai?

Timescales compare karo:

\sim 1100

m/s par

\sim 0.1

m chamber mein acoustic transit

\sim 0.1

ms hai, aur chamber fill/empty bhi fast gas dynamics follow karta hai, jabki combustion lag

τ \sim 2

–3 ms

\sim 20 \times

slower hai — toh

τ

woh bottleneck delay hai jiske saath pressure cycle sync karni padti hai,

f \sim 1/ (4 τ)

deta hai.

Quarter-cycle (

ω τ \approx π /2

) marginal chug instability kyon mark karta hai?

Quarter-cycle phase shift delayed extra burning ko theek tab arrive karta hai jab pressure phir se chadh raha hota hai, heat ko pressure ke saath in phase rakh kar positive feedback close karta hai.

Screech chugging se ek hazaar guna faster kyon hai?

Iska clock chamber mein acoustic transit time hai (

\sim R / a

, microseconds), jabki chugging ka clock combustion lag hai (

\sim

milliseconds) —

\sim 1000

ka factor difference.

Ek Helmholtz cavity ek target frequency absorb karne ke liye kaise tune ki ja sakti hai?

Ek Helmholtz Resonator ka ek single natural frequency hota hai jo uske neck aur volume se set hota hai; us frequency par uske neck mein gas chamber

p^{'}

se quarter-cycle out of step oscillate karta hai, toh joh flow woh draw karta hai woh mode par negative work karta hai — energy exactly wahan nikalta hai jahan

p^{'}

sabse bada hai, local

\oint p^{'} q^{'} d t

ko negative kar deta hai.

Flame ko pressure antinode par rakhna screaming risk kyon maximise karta hai?

Antinode wahan hai jahan

p^{'}

sabse zyada swing karta hai, toh wahan release ki gayi heat ke paas multiply karne ke liye sabse bada possible

p^{'}

hota hai — Rayleigh overlap aur drive maximise hota hai.

Edge Cases

Agar combustion lag

τ \to 0

ho (instant burning), toh chugging ka kya hoga?

Delayed heat term present pressure par collapse ho jata hai; phase lag ke bina growth window tak pahunchne ke liye, classic

1/ (4 τ)

chug band infinite frequency tak push hota hai aur low-frequency loop effectively vanish ho jata hai.

Agar flame ek acoustic mode ke pressure node par exactly ho, kya woh mode scream kar sakta hai?

Nahi — node par

p^{'} \approx 0

, toh

p^{'} q^{'} \approx 0

aur Rayleigh integral ko flame se koi drive nahi milta; woh particular mode us location se pump nahi ho sakta.

Marginal-stability boundary par jahan

\oint p^{'} q^{'} d t = 0

exactly ho, engine kya kar raha hai?

Neutral oscillation — har cycle mein feed ki gayi energy exactly lost energy ke barabar hai, toh disturbance na badhti hai na ghatti hai; constant amplitude par hover karti hai aur koi bhi chhoti si change use kisi bhi taraf tip kar sakti hai.

Agar chugging khatam karne ke liye

Δ p_{inj}

bahut bada kar dein — koi hidden cost hai?

Bahut stiff injector bahut zyada feed/pump pressure demand karta hai (heavier turbopumps, zyada mass aur power), toh stability ek real system-mass aur performance ki keemat par milti hai. Dekho Injector Design & Pressure Drop.

Agar damping (nozzle radiation, viscosity) exactly zero ho, toh limit cycle kaisa dikhega?

Koi limit cycle nahi hoga — growth cap karne ke liye kuch nahi hoga, positive Rayleigh integral amplitude ko tab tak drive karega jab tak nonlinear/thermal failure (metal pighle), isliye real caps thermoacoustic losses se aate hain.

Extremely high mean chamber pressure par proportionally scaled

Δ p_{inj}

(constant ratio) ke saath, kya stability change hoti hai?

First order par nahi — kyunki feedback gain

k

ratio

Δ p_{inj} / p_{c}

par depend karta hai, use fixed rakhne se chug margin roughly constant rehta hai absolute

p_{c}

chahe kuch bhi ho.

Recall Fast self-test

Chugging aur screaming dono ke peeche ek rule ::: Rayleigh's criterion, $\oint p^{'} q^{'} d t > 0$ — grow tab hota hai jab heat high pressure ke saath in phase mein add ho. Dono mein jo ek cheez alag hoti hai ::: Time-delay reservoir: feed/combustion lag $τ$ (ms) chugging ke liye vs acoustic transit (μs) screaming ke liye. Chug cure vs screech cure ::: Chug → injector $Δ p_{inj}$ badhao; screech → acoustic damping (baffles, Helmholtz cavities). Quarter-cycle chug rule aur yeh kahan se aata hai ::: $f_{c h ug} \approx 1/ (4 τ)$ , $ω τ \approx π /2$ se jahan $ω = 2 π f$ . Higher chug harmonics ::: $f \approx (2 n + 1) / (4 τ)$ , $ω τ \approx (2 n + 1) π /2$ se, kyunki whole cycles add karne par heat in phase rehti hai.