Aero and Hydroacoustics

Aero and Hydroacoustics are the study of the noise generated by flows on their own (e.g. jet noise) or through their interaction with solid boundaries (e.g. trailing edge noise). The mathematical theory behind these phenomena was introduced by the pioneering work of Sir James Lighthill on free space jet noise in the early 50s. In this classical paper he introduced what is now commonly referred to as the Lighthill’s Acoustic Analogy
$$\begin{equation*} \frac{\partial^2\rho}{\partial t^2}c^2_0\nabla^2\rho=\frac{\partial^2}{\partial x_i\partial x_j}(T_{ij}), \end{equation*}$$ where \(T_{ij}=\rho v_i v_j + \sigma_{ij}+(pc^2_0\rho)\delta_{ij}\). Physically this analogy means that the turbulent motion of the fluid in free space is equivalent to a volume distribution of quadrupoles in a fluid otherwise at rest. Note that this reformulation is exact and therefore solving this equation is equivalent to solving the NavierStokes equation. However, under certain circumstances, simplifications can be made that allows to find scalings for the farfield noise (e.g. Lighthill's famous \(v^8\) power law for jet noise). This result encouraged the development of high bypass ratio jet engine for commercial aircraft which allowed to reduce the exhaust speed (and therefore the noise) for equivalent thrust.
The physical phenomenon behind the conversion of hydrodynamical pressure fluctuations into acoustic waves is still an open problem. However there exist models based on unsteady thin aerofoil theory that link the socalled wall pressure spectrum to the acoustic farfield by means of an acoustic transfer function. This formulation is equivalent to Lighthill’s however the input data are different. The calculation of this acoustic transfer function is not a simple task and only approximate results based on the Schwarzschild technique can be found in the literature.
 D. Yumashev, I.D. Abrahams, N. Peake, C.J. Chapman, P.F. Joseph, and P.A. Cotterill. Broadband
hydroacoustic research (Phase I). Technical report, TUSL, ref: 0026K7626,62461037,179,
2012.  D. Yumashev, I.D. Abrahams, N. Peake, C.J. Chapman, P.F. Joseph, and P.A. Cotterill. Broadband
hydroacoustic research (Phase II). Technical report, TUSL, ref: 0026K7626,62921576,575
2012.
Our collaborators
 Prof. Nigel Peake, University of Cambridge
 Prof. John Chapman, University of Keele
 Prof. Phil Joseph, University of Southampton