Aero and Hydroacoustics
PictureSketch of thin airfoil theory implementation
Brief description

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 \(T_{ij}=\rho v_i v_j + \sigma_{ij}+(p-c^2_0\rho)\delta_{ij}\). 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 Navier-Stokes equation. However, under certain circumstances, simplifications can be made that allows to find scalings for the far-field 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.

PictureAcoustic field generated by a thin airfoil
Nevertheless, in many practical applications, turbulent flows interact with solid boundary to generate noise. This interaction can occur through many different physical phenomena and it is therefore impossible to dress an exhaustive list. Through extensive studies based on Lighthill’s work, it is known that strong noise amplification occurs when turbulent pressure fluctuations are scattered by geometrical singularities such as the trailing edge of an aerofoil or of a propeller blade.

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 so-called wall pressure spectrum to the acoustic far-field 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