Aeroacoustics is a branch of acoustics that studies noise generation via either turbulent fluid motion or by aerodynamic forces interacting with surfaces. Noise generation can also be associated with periodically varying flows. Although no complete rational scientific theory of the generation of noise by aerodynamic flows has yet prevailed, most practical aeroacoustic analyses rely upon the so-called aeroacoustic analogy whereby the governing equations of fluid dynamics are combined with the wave equation of classical acoustics. Although various such analogies currently exist, by far the most common and widely used is Lighthill's aeroacoustic analogy, first proposed by James Lighthill in the 1950s when noise generation associated with the jet engine was beginning to be placed under scientific scrutiny. Computational Aeroacoustics (CAA) is the application of computers to solve numerically aeroacoustics problems.

Lighthill's equation

Lighthill essentially rearranged the Navier Stokes equations of viscous fluid flow into an inhomogeneous wave equation therefore making an acoustic analogy with fluid mechanics. For non-relativistic velocities, the exact governing equation for the conservation of mass is given by:

$\backslash frac\{\backslash partial\; \backslash rho\}\{\backslash partial\; t\}\; +\; \backslash nabla\backslash cdot\backslash left(\backslash rho\backslash mathbf\{v\}\backslash right)=\backslash frac\{D\backslash rho\}\{D\; t\}\; +\; \backslash rho\backslash nabla\backslash cdot\backslash mathbf\{v\}=\; 0$

where

$\backslash rho$ is the density of the fluid.

The exact equation for the conservation of momentum is given by:

$\{\backslash rho\}\backslash frac\{\backslash partial\; \backslash mathbf\{v\}\}\{\backslash partial\; t\}+\{\backslash rho\backslash mathbf\{v\}\backslash nabla\backslash mathbf\{v\}\}+\backslash nabla\; p=0$

Multiplying the conservation of mass equation by $\backslash mathbf\{v\}$ and adding it to the conservation of momentum equation gives

$\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\backslash left(\backslash rho\backslash mathbf\{v\}\backslash right)\; +\; \backslash nabla(\backslash rho\backslash mathbf\{v\}\backslash mathbf\{v\})+\{\backslash nabla\; p\}=0$

Note that $\backslash mathbf\{v\}\backslash mathbf\{v\}$ is a tensor. Differentiating the conservation of mass with respect to time and differentiating the conservation of momentum with respect to space and subtracting gives

$\backslash nabla^2\; p+\backslash frac\{\backslash partial^2\backslash rho\}\{\backslash partial\; t^2\}=\backslash nabla\backslash cdot\backslash left(\backslash nabla\; p+\backslash frac\{\backslash partial\}\{\backslash partial\; t\}(\backslash rho\backslash mathbf\{v\})\backslash right)$

Combining both these expressions gives Lighthill's equation

$\backslash frac\{\backslash partial^2\backslash rho\}\{\backslash partial\; t^2\}-c^2\_0\backslash nabla^2\backslash rho=\backslash nabla\backslash cdot\backslash left(\backslash frac\{\backslash partial\}\{\backslash partial\; t\}(\backslash rho\backslash mathbf\{v\})+\backslash nabla(\backslash rho\backslash mathbf\{v\}\backslash mathbf\{v\})-\backslash frac\{\backslash partial\}\{\backslash partial\; t\}(\backslash rho\backslash mathbf\{v\})\backslash right)$

which simplifies to become

$\backslash frac\{\backslash partial^2\backslash rho\}\{\backslash partial\; t^2\}-c^2\_0\backslash nabla^2\backslash rho=\backslash nabla^2(\backslash rho\backslash mathbf\{v\}\backslash mathbf\{v\})$

The above expression is the famous Lighthill's equation of aeroacoustics. This equation is the same as the well-known wave equation except for the source term on the right-hand side. The product $\backslash mathbf\{v\}\backslash mathbf\{v\}$ represents the so-called Lighthill’s stress tensor, commonly expressed as $T\_\{ij\}$. Lighthill’s equation can be written using indicial tensor notation as

$\backslash frac\{\backslash partial^2\backslash rho\}\{\backslash partial\; t^2\}-c^2\_0\backslash nabla^2\backslash rho=\backslash frac\{\backslash partial^2T\_\{ij\}\}\{\backslash partial\; x\_i\; \backslash partial\; x\_j\}$

where $T\_\{ij\}$ is given by

$T\_\{ij\}=\backslash rho\; u\_i\; u\_j+(p-\backslash rho\; c^2\_0)\backslash delta\_\{ij\}+\backslash sigma\_\{ij\}$

and $\backslash delta\_\{ij\}$ is the Kronecker delta.

Each of these acoustic source terms may play a significant role in the generation of noise depending upon flow conditions considered. It is generally accepted however, that the term $\backslash sigma\_\{ij\}$ representing the effects of viscosity on noise generation is orders of magnitude less than the other terms and can be consequently neglected in most situations. In most aeroacoustic studies, both theoretical and computational efforts are made to solve for the acoustic source terms in Lighthill's equation to make statments regarding the relevant aerodynamic noise generation mechanisms present.

External references

• Aeroacoustics at the University of Mississippi
• International Journal of Aeroacoustics

Acoustics | Aerodynamics

Aeroakustik | Aéroacoustique | Aeroakustyka | Aeroacústica