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

3.3.32 · Physics › Rocket Propulsion

WHAT is combustion instability?

WHY do naam kyun hain? Kyunki do alag physical reservoirs hain jo disturbance ko time-store karke phase mein wapas de sakte hain. Slow reservoir = propellant feed lines. Fast reservoir = chamber ke andar ke standing sound waves.

The universal condition: Rayleigh's Criterion

First principles se derivation (yeh master rule kyun hai): Ek gas parcel ke liye linearized energy equation lo. Acoustic energy $E$ isliye change hoti hai kyunki unsteady heat release acoustic field pe kaam karti hai. Ek perfect gas ke liye acoustic energy addition ki rate per unit volume hai: $\frac{dE}{dt}=\frac{\gamma -1}{\gamma \bar{p}}{p}^{\prime }{q}^{\prime },$ jahan $q^{\prime }$ per unit volume fluctuating heat-release rate hai, $\bar{p}$ mean pressure hai, $\gamma$ specific-heat ratio hai.

• WHY $p^{\prime }{q}^{\prime }$ product? Gas mein heat add karne se pressure badhta hai ($p\propto$ fixed volume pe heat). Agar aap heat add karo jab pressure already high ho, toh pressure aur bhi zyada push hoga — constructive. Donon signals ka overlap unke product se measure hota hai.
• HOW hum ek cycle criterion lete hain: Ek cycle mein net energy gain $\oint \frac{dE}{dt}dt\propto \oint p^{\prime }{q}^{\prime }dt$ hai. Agar positive hai, toh har cycle pichhle se zyada energy chhodti hai → growth, jab tak nonlinear losses (nozzle radiation, viscous damping) amplitude ko ek limit cycle mein nahi rok deti.

Toh dono chugging aur screaming same Rayleigh condition hain — ye sirf time delay kis cheez se set hota hai is baat mein differ karte hain, yaani pressure blip aur resulting heat release ke beech ka delay. Yeh time delay feed lag se aata hai ya acoustic transit se.

Low-frequency: CHUGGING

HOW frequency set hoti hai — chug frequency derive karo.

Chamber mass balance: stored gas $\propto p_c$ hai, toh: $\frac{V}{a^2}\frac{d{p}_c}{dt}={\dot{m}}_{in}-{\dot{m}}_{out}.$

• ${\dot{m}}_{out}$ choked nozzle se: ${\dot{m}}_{out}=\frac{p_c{A}_t}{c^{\ast }}$ ($p_c$ ke saath badhta hai).
• ${\dot{m}}_{in}$ injector se: ${\dot{m}}_{in}\propto \sqrt{\Delta p_{inj}}=\sqrt{p_{feed}-p_c}$, lekin combustion lag $\tau$ se delayed: yeh $p_c(t-\tau )$ pe respond karta hai.

Linearize karo ($p_c={\bar{p}}_c+p^{\prime }$) aur aapko ek delay-differential equation milta hai: ${\theta }_c\frac{d{p}^{\prime }}{dt}+p^{\prime }(t)=-k{p}^{\prime }(t-\tau ),$ jahan ${\theta }_c=V/(a^2)\cdot (\text{gain terms})$ chamber time constant hai aur $k$ feed-coupling gain hai. $p^{\prime }\sim e^{i\omega t}$ maan kar, marginal instability tab hoti hai jab delayed feedback phase mein hoti hai, roughly jab: $\boxed{\omega \tau \approx \frac{\pi }{2}\text{–}\pi \Rightarrow f_{chug}\sim \frac{1}{4\tau }}$

• WHY $1/\tau$ scaling? Loop mein sirf ek hi slow clock hai — combustion/transit delay $\tau$ (milliseconds). $\tau \sim 2$ ms → $f\sim 125$ Hz. Yahi chugging band hai.
• Fix (WHY it works): Injector $\Delta p_{inj}$ badhao. Ek stiff injector ${\dot{m}}_{in}$ ko $p_c$ se nearly independent bana deta hai — aap feedback gain $k$ tod dete ho. Rule of thumb: rakho $\Delta p_{inj}\gtrsim 0.2p_c$.

High-frequency: SCREAMING (screech)

HOW frequency set hoti hai — chamber ke acoustic modes. Chamber ek resonant cavity hai jisme sound speed $a=\sqrt{\gamma R{T}_c}$ hai. Iske natural frequencies cavity geometry se set hote hain, jaise cylinder ke liye transverse (tangential/radial) aur longitudinal modes: $\boxed{f_{mn\ell }=\frac{a}{2}\sqrt{{\left(\frac{{\alpha }_{mn}}{\pi R}\right)}^2+{\left(\frac{\ell }{L}\right)}^2}}$ jahan $R$ chamber radius hai, $L$ length hai, $\ell$ longitudinal index hai, ${\alpha }_{mn}$ transverse pattern ke liye Bessel roots hain.

• WHY Hz ki jagah kHz? Hot gas mein $a$ $\sim 1000$ m/s hai aur $R\sim 0.1$ m hai, toh $f\sim a/(2\pi R)\sim$ kuch kHz. Yahan clock chamber ke across acoustic transit time hai, microseconds — feed lag se hazaar guna tez → screaming.
• Sabse dangerous: first tangential (1T) mode, kyunki iska pressure side to side slosh karta hai, wall ke against hot gas scrub karta hai.
• Fix (WHY it works): Acoustic damping — injector face pe baffles lagao (transverse modes tod do) aur chamber liner mein Helmholtz-resonator acoustic cavities lagao (target frequency pe energy absorb karo). Ye loss add karke $\oint p^{\prime }{q}^{\prime }dt$ ko net-negative banate hain.

Worked Examples

Common Mistakes

Flashcards

Connections

• Rayleigh Criterion — sabhi thermoacoustic growth ke liye master energy condition.
• Injector Design & Pressure Drop — chugging ka cure yahan hai.
• Acoustic Modes of a Cylindrical Cavity — screaming frequencies set karta hai.
• Helmholtz Resonator aur Injector Baffles — high-frequency damping.
• Choked Nozzle Flow — chug mass balance mein use hone wala ${\dot{m}}_{out}\propto p_c$ set karta hai.
• Characteristic Velocity c* — chamber balance mein combustion efficiency term.
• Thermoacoustics — is topic ko encompass karne wala general field.

Concept Map