Combustion Noise

The question

How can a perfectly steady supply of fuel and air produce a pressure oscillation violent enough to crack metal?

It needs no external shaker at all. The flame and the chamber drive each other, and the physics is the same as pushing a child on a swing.

Roar and song

Combustors make two very different kinds of sound, and they have very different consequences.

Broadband combustion noise is the roar. Turbulent flames release heat in a chaotic, patchy way, and every patch that suddenly expands radiates a little pressure pulse. The result is sound across a wide range of frequencies. It adds to community noise around airports, but it rarely threatens the hardware.

Combustion instability is the song. Here the flame locks onto one of the chamber's natural acoustic frequencies and pumps energy into it every cycle. The amplitude grows until something nonlinear limits it, sometimes at pressure oscillations of several percent of the mean pressure, enough to fatigue liners, loosen fuel injectors and damage turbine blades within hours.

Rayleigh's criterion: pushing the swing

In 1878 Lord Rayleigh stated the condition — now called the — for a heat source to drive a sound wave:

If heat be given to the air at the moment of greatest condensation, or be taken from it at the moment of greatest rarefaction, the vibration is encouraged.

Think of a swing. Push at the right moment in each cycle and every push adds energy; push at the wrong moment and you slow it down. Adding heat to gas makes it expand, which is a push. So the flame's heat-release fluctuation $\cflame{q'}$ adds energy to the pressure oscillation $\cair{p'}$ when the two are in step. Formally, the oscillation grows when

$\oint_\text{cycle} \int_V \cair{p'}(\mathbf{x},t)\, \cflame{q'}(\mathbf{x},t)\; dV\, dt \;>\; \text{acoustic losses}$

The cleanest way to see this is to draw each oscillation as the shadow of a rotating arrow, a phasor. The pressure and the heat release rotate at the same frequency, separated by a phase angle. Their product, averaged over a cycle, is simply proportional to the cosine of that angle.

VisualiseRayleigh's criterion: pushing the swing

Phase lag of heat release30°

0° = heat released exactly at peak pressure

Acoustic damping0.25

Losses at the walls, exit and in liners

Rayleigh index0.39∝ ∮ p'q' dt ∝ cos(phase)

Net growth rate0.26Rayleigh − damping

Unstable. The flame gives the pressure wave a push at just the right moment every cycle, faster than damping can remove energy. The amplitude grows until something nonlinear stops it — a limit cycle.

Challenges0/2

Find a phase lag where the flame neither feeds nor starves the oscillation.
Make the oscillation grow even with damping of 0.6 or more.

Each wave is the shadow of a rotating arrow. The blue arrow is the pressure oscillation, the red arrow is the flame's heat-release oscillation, lagging by a phase angle. Their product, averaged over a cycle, is proportional to cos(phase): positive feeds the oscillation, negative starves it.

Phase lag near 0°: heat is released when pressure is high. The flame pushes the swing at the top of every swing, and the oscillation grows.
Phase lag near 180°: heat arrives when pressure is low. The flame pushes against the motion, and the oscillation dies.
Around 90°: the pushes cancel over a cycle, with no net driving.

Going deeper
Why the driving goes as cos(phase)

Let $p' = \hat p\cos\omega t$ and $q' = \hat q \cos(\omega t - \varphi)$ . Averaging their product over one period $T = 2\pi/\omega$ :

$\frac{1}{T}\oint p'q'\,dt = \frac{\hat p\,\hat q}{T}\oint \cos\omega t\,\cos(\omega t - \varphi)\,dt = \frac{\hat p\,\hat q}{2}\cos\varphi$

Using $\cos a\cos b = \tfrac12[\cos(a-b) + \cos(a+b)]$ , the product splits into a constant $\tfrac12\cos\varphi$ and a term oscillating at $2\omega$ that averages to zero. Only the in-phase component of $q'$ survives. In the phasor picture it is the projection of the heat-release arrow onto the pressure arrow. Add the time lag, $\varphi = \omega\tau$ , and the driving becomes $\propto \cos(2\pi f \tau)$ : periodic in $\tau$ , which is why stable and unstable bands alternate.

And even when the flame is driving, the oscillation only grows if the driving beats the damping (losses at walls, through the exit, in acoustic liners). Instability is always this contest: driving versus damping.

What sets the phase? Time lags

Why would a flame's heat release follow the pressure at all, and with what delay? Several mechanisms link them:

Fuel-feed coupling. The pressure oscillation in the chamber modulates the pressure drop across the injector, so the fuel flow oscillates. That fuel takes time to travel to the flame, evaporate and burn.
Equivalence-ratio waves. In a premixer, a velocity oscillation at the fuel injection point creates alternately rich and lean slugs of mixture, which convect to the flame and burn at different rates. This is the dominant mechanism in lean premixed combustors.
Vortex shedding. Swirlers and bluff bodies shed vortices periodically; when a vortex reaches the flame it wraps up fresh mixture and burns it in a burst.

Each introduces a time lag $\tau$ . If the chamber rings at frequency $f$ , the lag becomes a phase angle $2\pi f\tau$ . So whether the flame drives or damps an oscillation depends on how the lag compares with the period. That is why small changes in geometry or operating point can switch a combustor between quiet and screaming.

The combustor as an organ pipe

The chamber itself decides which frequencies are available. Closed at the dome and effectively closed at the choked turbine nozzle, a combustor of length $L$ supports longitudinal modes at

$f_n = \frac{n\,c}{2L}, \qquad c = \sqrt{\gamma R \cflame{T}}$

Estimate first

What is the fundamental frequency of a 0.5 m long combustor full of 1800 K gas?

101001,00010,000100,000

Drag along the line to place your guess — each tick is ten times the last.

With hot gas ( $c \approx 800$ m/s) and a length of half a metre, the fundamental sits in the high hundreds of hertz. An annular combustor also rings around its circumference, and those azimuthal modes are often the troublemakers in modern engines.

Put the pieces together and experiment. The tube's pressure pattern is the mode shape; the flame pulses with the heat release; the growth-rate plot shows each mode as the time lag varies.

SimulateA combustor that sings

Time lag τ (injector → flame)1.50 ms

Combustor length L0.60 m

Flame position15% along

At a pressure node the flame can't drive the mode

Gas temperature1700 K

sound speed 796 m/s

Natural damping0.35

Most dangerous mode664 Hzbuzz / howl

Growth rate0.65

Instability at 664 Hz: the flame's delayed response lands in phase with the pressure. Try shifting τ by a fraction of a period (1/f ≈ 1.51 ms), moving the flame toward a pressure node, or adding a resonator tuned to this frequency.

Challenges0/3

Silence the screaming combustor by changing only the time lag τ.
Silence it with a Helmholtz resonator instead.
Find a screech: an unstable mode above 1000 Hz.

The liner is an organ pipe. Closed at the dome and effectively closed at the choked turbine nozzle, its pressure modes ring at f = n·c/2L. The flame responds to pressure with a time lag τ (fuel-feed and convection delays); whether that response lands in phase depends on ωτ — so tiny changes in τ or length flip the combustor between quiet and screaming.

Some things to try:

Sweep τ and watch the modes slip in and out of instability as $2\pi f\tau$ passes through multiples of $2\pi$ . Stable and unstable bands alternate.
Move the flame position toward the middle of the tube. At a pressure node $p'$ is zero, so the flame cannot drive that mode however it responds.
Change the length: every mode shifts frequency, and a stable combustor can become unstable.
Add a Helmholtz resonator tuned to the offending frequency and watch its growth rate collapse.

Three frequency families

Lefebvre groups instabilities by frequency, which hints at their mechanism:

Regime	Frequency	Typical mechanism
Rumble / growl	below ~200 Hz	Whole-system coupling: fuel lines, plenum and combustor acting as a lumped system
Buzz / howl	~200–1000 Hz	Longitudinal modes driven by convective lags and vortex shedding
Screech	above ~1000 Hz	Transverse and azimuthal modes of the chamber; the most destructive

Fighting back

Design out the coupling. Change the time lag by moving the injector or altering the swirler, move the flame away from pressure antinodes, or change the chamber length so no mode lines up. Prevention in the design phase is by far the most effective remedy.

Add damping. , small cavities connected by a neck, absorb energy strongly near their tuned frequency. Perforated, cavity-backed liners absorb across a broader band. Both raise the damping side of the contest.

Active control. Measure the pressure and modulate the fuel flow out of phase to cancel the driving. It has been demonstrated convincingly in laboratories and some ground turbines, but actuator bandwidth and reliability make it difficult for flight.

The lean premixed trap

Here the course's threads begin to knot together. The lean premixed flames that make low NOₓ possible (Lessons 2 and 9) are exactly the ones most prone to instability:

They burn near the lean limit, where the flame speed and heat release are extremely sensitive to small changes in φ, which is the equivalence-ratio-wave mechanism at full strength.
Their heat release is compact and intense, a strong, concentrated acoustic source.
They lack the broad, diffusive reaction zone of a diffusion flame, which naturally smears out fluctuations.

Check your intuition

Why are modern low-NOₓ combustors more prone to combustion instability than older diffusion-flame combustors?

Recap

What to carry forward

Broadband roar is a noise nuisance; self-excited instability is a structural threat.
Rayleigh's criterion: a flame drives a mode when heat release is in phase with pressure (the cosine of the phase lag is positive) and the driving beats the damping.
The phase comes from time lags (fuel feed, equivalence-ratio waves, vortices): phase $= 2\pi f \tau$ .
The chamber is an organ pipe with modes at $f_n = nc/2L$ , plus azimuthal modes; a flame at a pressure node cannot drive a mode.
Remedies: shift the lag or geometry, add Helmholtz resonators and damping liners, or use active control.
Lean premixed flames are the most instability-prone, so low-NOₓ and quiet must be co-designed.

Next, the survival problem in its most literal form: how does a thin metal liner live for years just millimetres from 2300 K gas?