### Using Maths to Reduce Aircraft Noise

If you have ever tried to sleep under a flight-path near an airport, you will know how serious the problem of aircraft noise can be. Aircraft noise is amongst the loudest sounds produced by human activities. The noise is over a broad range of frequencies, extending well beyond the range of hearing. The problem of aviation noise has become more severe as aircraft engines have become more powerful  [TM180 or search for “thatsmaths” at irishtimes.com].

Engine inlet of a CFM56-3 turbofan engine on a Boeing 737-400 [image Wikimedia Commons].

Sources of noise include exhaust jet noise, which is dominant during take-off, engine noise important during the in-flight phase and airframe noise important during landing. Airframe noise may be generated by the flaps, wing-tips or landing-gear. Researchers have to consider all these sources, and also allow for noise-cancellation mechanisms. For supersonic flight, the problem is complicated by shock-waves or sonic booms. However, with the withdrawal of Concorde from service, this is no longer an issue for civilian flight.

Lighthill’s Eighth Power Law

The British applied mathematician James Lighthill initiated the field of aero-acoustics. His starting point was the Navier-Stokes equation, the basic equation of fluid flow first formulated rigorously by Sligo-born mathematician and physicist George Gabriel Stokes. Lighthill isolated the terms of the equation that act as sources of sound waves. He was then able to calculate the sound field generated by the airflow.

By means of an elegant qualitative argument, Lighthill showed how the noise intensity changes with the speed of the aircraft. For subsonic flight, the power of the propagating sound varies as the eighth power of the jet speed. This mathematical result has major engineering consequences. It implies that a 10% increase in speed gives rise to more than double the sound energy radiated. Thus, it is vital to limit the velocity of the exhaust gases as much as is feasible.

Quieter Engines

In pure turbojets, all the air in the exhaust passes through the compressor and exits at high velocity. The result is severe noise due to the jet. The requirement to limit noise was one of the factors leading to the development of high-bypass turbofan engines to replace the older turbojets. In these more modern engines, only some of the ingested air passes through the compressors and combustion chambers; the remainder is propelled through surrounding channels by a large fan driven by the compressor. The large diameter of turbofan engines enables the mass-flow and thrust to be maintained at much lower exhaust speeds and therefore greatly reduced noise production.

Schematic diagram of a high-bypass turbofan engine [image Wikimedia Commons].

In much of computational fluid dynamics, the fluid is treated as incompressible and pressure fluctuations are balanced by changes in fluid velocity. This effectively filters out the sound waves, so it is inadequate for aero-acoustics. Following Lighthill’s work, the discipline of computational aero-acoustics emerged. This allows for the imbalances that produce sound waves. Although the energy of these waves is only a tiny fraction of total energy, they are responsible for all the problems arising from aircraft noise. Very substantial noise reductions have been achieved by engineering innovations based on computational aero-acoustics.

There are many unsolved problems, and research is world-wide, at universities, government laboratories and commercial aviation companies. Computational aero-acoustics is now a dominant tool in airframe and aero-engine development, and the design process is intensely mathematical in character.

Sources

Chapman, C. J., Aircraft Noise. Pp 783-786 in Princeton Companion to Applied Mathematics (2015), Princeton University Press, 994 pp.