Combustion Instability

Every chamber is an organ pipe

If you have followed the gas turbine course, this lesson will feel familiar: combustion instability in a rocket is the same physics as in a gas turbine combustor, only more violent. Sutton describes three families:

• Low-frequency instability, or chuggingChuggingLow-frequency instability (10–400 Hz) coupling the chamber with the propellant feed system through the injector. (roughly 10–400 Hz): the chamber pressure couples with the propellant feed system.
• Intermediate frequency, or "buzzing": coupling with parts of the feed system such as manifolds and pumps.
• High-frequency instability, or screamingHigh-frequency instabilityCombustion coupled to the chamber’s acoustic modes (kHz). Transverse modes can destroy an injector in a fraction of a second. (1–20 kHz): the combustion couples with the chamber's own acoustic modes. This is the dangerous one.

A combustion chamber is a closed can of hot gas, open only at the throat, which for acoustic purposes behaves almost like a wall. Like any enclosed volume, it rings at particular frequencies.

Longitudinal modes slosh gas from injector to nozzle, with frequency $f = n\,c/2L$. Transverse modes live in the cross-section: tangential modes sweep around the circumference, radial modes breathe in and out. Their frequencies follow from the zeros of Bessel-function derivatives,

$f_{mn} = \frac{\alpha_{mn}\,c}{\pi D},\qquad \alpha_{1T} = 1.841,\;\alpha_{2T} = 3.054,\;\alpha_{1R} = 3.832.$

With the speed of sound in chamber gas above 1000 m/s, a 0.5 m chamber's first tangential mode sits near 1.5 kHz. Tangential modes are the most dangerous: they drive gas back and forth across the injector face, stripping away the cool boundary layer, and the heat flux there can rise tenfold within milliseconds.

The Rayleigh criterion, rocket edition

A sound wave in a chamber will grow if the combustion adds heat in phase with its pressure oscillation — Rayleigh's criterion, which explains every thermoacoustic instability from singing flames to screaming rockets:

$\oint p'\,\dot q'\,dt > \text{losses}$

In a liquid rocket, the link between pressure and heat release runs through the injector and the spray. Luigi Crocco modelled it with just two numbers: when the pressure changes, the combustion rate responds with a gain $n$, the interaction index, after a delay $\tau$, the time lag it takes to atomize, evaporate, mix and burn. This is the n–τ modeln–τ modelCrocco’s time-lag model: combustion responds to a pressure disturbance with gain n after a delay τ..

The time lag is the key. If $\tau$ is close to a whole number of periods of some mode, the heat arrives in phase with that mode's pressure peaks, and the mode is driven. Change the injector — finer atomization, different element spacing — and $\tau$ changes, and so does which mode is dangerous.

Chugging: when the chamber talks to the feed lines

At low frequency the loop runs through the injector. If the chamber pressure rises, the pressure drop across the injector falls, less propellant flows in, the chamber pressure falls, more propellant flows in… With the right time lag this becomes a self-sustaining oscillation, the chug.

The standard cure is a stiff injector: a pressure drop across the injector of at least 15–20% of chamber pressure, so that chamber pressure fluctuations barely change the flow. Stiffness costs pump pressure — one more reason pump discharge pressures are so much higher than chamber pressures.

How the F-1 was fixed

When Rocketdyne's F-1 began destroying itself on the test stand in 1962, the analytical tools could not predict which designs would be stable. So NASA and Rocketdyne did it empirically. Small explosive charges — bombs — were fixed inside the chamber and set off during firings, deliberately kicking the chamber at every frequency at once. A stable design had to damp the disturbance quickly.

Over some two thousand full-scale firings, engineers tried injector variations, eventually settling on a design with a pattern of baffles — copper walls sticking out of the injector face, dividing it into compartments — and modified orifice patterns. The baffles break up transverse modes and move their frequencies. The final F-1 damped bomb disturbances within a small fraction of a second, and never suffered an instability in flight.

Modern engines add acoustic cavities: resonators around the injector's edge, tuned like the gas turbine's Helmholtz dampers to absorb energy at a specific frequency.

Next: the machine that makes it all possible — the turbopump — and the engine cycles that decide how to power it.