Nozzle Cooling, Noise and Infrared Signature

Lighthill's eighth-power law, why bypass makes engines quiet, chevrons, cooling hot nozzle walls, and infrared emission from Planck's law

1952

The first jet airliners are about to enter service, and nobody can say how loud they will be or why. In 1952 the 28-year-old British mathematician James Lighthill publishes "On sound generated aerodynamically" in the Proceedings of the Royal Society. Treating the turbulent jet as a source of sound in still air, he finds that the acoustic power radiated by a jet grows with the eighth power of its velocity (Lighthill's eighth power law). Lighthill is credited with founding aeroacoustics, the science behind every quiet engine since (James Lighthill). His law says something brutal: double a jet's speed and it radiates 256 times the sound power. It also says how to make engines quiet — and it is the same thing that makes them efficient.

P_{acoustic} \sim \rho\,\frac{V^8 D^2}{a_0^5}

Lighthill's scaling for the sound power of a jet of velocity V and diameter D in air of density ρ and sound speed a0.

Derivation

The eighth power, by dimensions

Lighthill rewrote the equations of fluid motion as a wave equation for the density, with the turbulence acting as a distribution of quadrupole sound sources. Their strength scales with the momentum flux,

\rho V^2

; their frequency with

V/D

. A quadrupole radiates very inefficiently at low Mach number: the fraction of the jet's mechanical power

\sim \rho V^3 D^2

that escapes as sound scales as

(V/a_0)^5

. Together:

P \sim \rho V^3 D^2 \left(\frac{V}{a_0}\right)^5 = \rho\,\frac{V^8 D^2}{a_0^5}.

At a fixed thrust,

T \approx \rho V^2 D^2

(Lesson 1-1), so

D^2 \propto 1/V^2

and

P \propto V^6

📐Same thrust, slower jet

Deliver 200 kN of take-off thrust with jets of different speeds (Lab 1):

600 m/s through 0.76 m: the reference.
300 m/s through 1.52 m: 18.1 dB quieter.
250 m/s through 1.82 m: 22.8 dB quieter.

The high-bypass turbofan, designed for propulsive efficiency, is quiet for the same reason: it makes its thrust with a slow, wide jet. That is why the JT3D of Lesson 1-2 was quieter than the turbojet it replaced, and why today's engines are quieter still.

Chevrons. Much jet noise comes from the shear layer where the jet meets still air. Saw-tooth chevrons on the trailing edges of nacelles and nozzles — visible on the GEnx engines of the 787 and the 747-8 — stir small streamwise vortices into that layer and make the mixing gentler, trading a little thrust for less low-frequency noise (chevrons).

Cooling. Afterburner liners and fighter nozzles see gas at nearly 2000 K. They survive behind films of cooler air bled from the fan, and increasingly by being made of ceramic matrix composites, which tolerate far higher temperatures than nickel alloys.

📐Cooling a nozzle wall

A wall exposed to 1900 K afterburner gas (Lab 3) settles where the heat arriving from the gas equals the heat removed by coolant air and by radiation:

Uncooled, it would run at 1632 K.
A nickel alloy, limited to about 1150 K, needs a coolant heat-transfer coefficient of about 1490 W/(m²·K) — a lot of cooling air.
A ceramic matrix composite, good to about 1600 K, needs almost none.

Every kilogram of cooling air is air that does not make thrust, which is why the hottest parts of new engines are turning to ceramics.

Infrared. Everything radiates according to its temperature (Planck's law), and the peak of the radiation moves to shorter wavelengths as it gets hotter (Wien's law,

\lambda_{max} = 2898\ \mu\mathrm{m\,K}/T

). Two bands pass through the atmosphere well: 3–5 µm, where heat-seeking missiles look for hot metal and hot exhaust gas (carbon dioxide emits strongly near 4.3 µm), and 8–12 µm, where thermal cameras see bodies near room temperature.

E_\lambda(T) = \frac{2\pi h c^2}{\lambda^5}\,\frac{1}{e^{hc/(\lambda k T)} - 1}, \qquad \int_0^\infty E_\lambda\,d\lambda = \sigma T^4

Planck's law for the spectral emissive power of a black body, and its integral, the Stefan–Boltzmann law.

📐Why hot parts are hard to hide

Integrating Planck's law over each band with scipy.integrate.quad (Lab 2):

A 300 K airframe peaks at 9.7 µm; only 1.3 % of its radiation falls in 3–5 µm.
A 900 K nozzle peaks at 3.2 µm, with 36 % of its radiation in 3–5 µm — and emits about 2300 times as much 3–5 µm power per square metre as the airframe.

Stealth designs therefore hide and cool the hot parts. The F-117's non-circular tail pipes mixed its hot exhaust with cool ambient air, and it had no afterburners at all (the F-117).

A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.

– Max Planck, Scientific Autobiography and Other Papers (1949) (Wikiquote)

💡Noise is not only the jet

On a modern high-bypass engine the jet is no longer the only loud source: the fan's blades, their supersonic tips at take-off ("buzz-saw" tones) and the airframe's flaps and landing gear all matter on approach. Acoustic liners in the nacelle absorb much of the fan noise.

💡Where you will use this

Lesson 6-4 develops turbine cooling, with the same heat balance as Lab 3. Lesson 10-2 asks whether open rotors, which have no nacelle to line, can be made quiet enough to certify.

📚Further reading

M. J. Lighthill, 'On sound generated aerodynamically. I. General theory', Proceedings of the Royal Society A 211 (1952), 564–587.
Lighthill's eighth power law and Planck's law on Wikipedia.
Michael J. T. Smith, Aircraft Noise (Cambridge University Press, 1989).

↓But there is a problem

For four units the compressor and the turbine have been boxes with an efficiency written on them: 0.90, 0.89, 0.91. Where do those numbers come from? In 1926, at the Royal Aircraft Establishment, a young engineer named A. A. Griffith opened the box and treated a compressor blade as a wing.