Aeroacoustics

Aeroacoustics is a branch of acoustics that studies noise generation via either turbulent fluid motion or aerodynamic forces interacting with surfaces. Noise generation can also be associated with periodically varying flows. A notable example of this phenomenon is the Aeolian tones produced by wind blowing over fixed objects.

Although no complete scientific theory of the generation of noise by aerodynamic flows has been established, most practical aeroacoustic analysis relies upon the so-called aeroacoustic analogy, proposed by Sir James Lighthill in the 1950s while at the University of Manchester. whereby the governing equations of motion of the fluid are coerced into a form reminiscent of the wave equation of "classical" (i.e. linear) acoustics in the left-hand side with the remaining terms as sources in the right-hand side.

The modern discipline of aeroacoustics can be said to have originated with the first publication of Lighthill in the early 1950s, when noise generation associated with the jet engine was beginning to be placed under scientific scrutiny.

Lighthill rearranged the Navier–Stokes equations, which govern the flow of a compressible viscous fluid, into an inhomogeneous wave equation, thereby making a connection between fluid mechanics and acoustics. This is often called "Lighthill's analogy" because it presents a model for the acoustic field that is not, strictly speaking, based on the physics of flow-induced/generated noise, but rather on the analogy of how they might be represented through the governing equations of a compressible fluid.

The continuity and the momentum equations are given by

where ${\displaystyle \rho }$ is the fluid density, ${\displaystyle \mathbf {v} }$ is the velocity field, ${\displaystyle p}$ is the fluid pressure and ${\displaystyle {\boldsymbol {\tau }}}$ is the viscous stress tensor. Note that ${\displaystyle \mathbf {v} \mathbf {v} }$ is a tensor (see also tensor product). Differentiating the conservation of mass equation with respect to time, taking the divergence of the last equation and subtracting the latter from the former, we arrive at

Subtracting ${\displaystyle c_{0}^{2}\nabla ^{2}\rho }$, where ${\displaystyle c_{0}}$ is the speed of sound in the medium in its equilibrium (or quiescent) state, from both sides of the last equation results in celebrated Lighthill equation of aeroacoustics,

where ${\displaystyle \nabla \nabla }$ is the Hessian and ${\displaystyle \mathbf {T} }$ is the so-called Lighthill turbulence stress tensor for the acoustic field. The Lighthill equation is an inhomogenous wave equation. Using Einstein notation, Lighthill’s equation can be written as