 # Green's function of pressure in both free space and a duct

• James Mathews (CCA/DAMTP)
• Wednesday 30 April 2014, 16:00-17:00
• MR14, Centre for Mathematical Sciences.

In this talk I will look to develop an understanding of the noise in an aircraft engine due to interactions with the fan blades and lining of the engine casing, which are the main contributions to the engine noise.

We use a model of an infinite duct as the engine casing and consider the fan blades as some rotating surface, although more sophisticated models could be considered as a later date. To understand the noise, we derive a differential equation for pressure in the duct and then we can use the formula L = 20log10(prms/pref) to calculate the sound level in decibels, where pref is usually 20dB and prms is the root mean square of unsteady pressure.

Although we can solve the differential equation numerically, we would like some analytic results, so we introduce the Green’s function. By assuming the frequency to be large we can make progress towards an analytic solution using standard asymptotic methods. The pressure is then calculated by integrating the Green’s function against the source terms, which will be done in the future and not in this talk.

The assumption of an infinite duct adds significantly more complications than free space, so we first calculate the Green’s function in free space, which is discussed in the first half of the talk. In the second half of the talk I will point out the difficulties with an infinite duct and how I have overcome most of them to analytically calculate the Green’s function in an infinite duct.

Key words: Partial differential equations, Green’s functions, asymptotics, fluid dynamics.

This talk is part of the Cambridge Analysts' Knowledge Exchange series.