Aeroacoustics

 ( Acoustics ) 

The continuity and the momentum equations are given by

where $\backslash rho$ is the fluid density, $\backslash mathbf\{v\}$ is the velocity field, $p$ is the fluid pressure and $\backslash boldsymbol\backslash tau$ is the viscous stress tensor. Note that $\backslash mathbf\{v\}\backslash mathbf\{v\}$ 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

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

Subtracting $c\_0^2\backslash nabla^2\backslash rho$, where $c\_0$ 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,

$\backslash frac\{\backslash partial^2\backslash rho\}\{\backslash partial\; t^2\}-c^2\_0\backslash nabla^2\backslash rho\; =\; \backslash nabla\backslash nabla\; :\backslash mathbf\; T,\; \backslash quad\; \backslash mathbf\; T\; =\; \backslash rho\backslash mathbf\{v\}\backslash mathbf\{v\}\; +\; (p-c^2\_0\backslash rho)\backslash mathbf\; I\; -\backslash boldsymbol\backslash tau,$

where $\backslash nabla\backslash nabla$ is the Hessian matrix and $\backslash mathbf\; T$ 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

$\backslash frac\{\backslash partial^2\backslash rho\}\{\backslash partial\; t^2\}-c^2\_0\backslash nabla^2\backslash rho=\backslash frac\{\backslash partial^2T\_\{ij\}\}\{\backslash partial\; x\_i\; \backslash partial\; x\_j\},\backslash quad\; T\_\{ij\}=\backslash rho\; v\_i\; v\_j\; +\; (p-\; c^2\_0\backslash rho)\backslash delta\_\{ij\}-\backslash tau\_\{ij\}.$

Each of the acoustic source terms, i.e. terms in $T\_\{ij\}$, may play a significant role in the generation of noise depending upon flow conditions considered. The first term $\backslash rho\; v\_i\; v\_j$ describes inertial effect of the flow (or Reynolds' Stress, developed by Osborne Reynolds) whereas the second term $(p-\; c^2\_0\backslash rho)\backslash delta\_\{ij\}$ describes non-linear acoustic generation processes and finally the last term $\backslash tau\_\{ij\}$ corresponds to sound generation/attenuation due to viscous forces.

In practice, it is customary to neglect the effects of viscosity on the fluid as it effects are small in turbulent noise generation problems such as the jet noise. Lighthill provides an in-depth discussion of this matter.

In aeroacoustic studies, both theoretical and computational efforts are made to solve for the acoustic source terms in Lighthill's equation in order to make statements regarding the relevant aerodynamic noise generation mechanisms present. Finally, it is important to realize that Lighthill's equation is exact in the sense that no approximations of any kind have been made in its derivation.

If one is to allow for approximations to be made, a simpler way (without necessarily assuming the fluid is incompressible) to obtain an approximation to Lighthill's equation is to assume that $p-p\_0=c\_0^2(\backslash rho-\backslash rho\_0)$, where $\backslash rho\_0$ and $p\_0$ are the (characteristic) density and pressure of the fluid in its equilibrium state. Then, upon substitution the assumed relation between pressure and density into $(*)\; \backslash ,$ we obtain the equation (for an inviscid fluid, σ = 0)

$\backslash frac\{1\}\{c\_0^2\}\backslash frac\{\backslash partial^2\; p\}\{\backslash partial\; t^2\}-\backslash nabla^2p=\backslash frac\{\backslash partial^2\backslash tilde\{T\}\_\{ij\}\}\{\backslash partial\; x\_i\; \backslash partial\; x\_j\},\backslash quad\backslash text\{where\}\backslash quad\backslash tilde\{T\}\_\{ij\}\; =\; \backslash rho\; v\_i\; v\_j.$

And for the case when the fluid is indeed incompressible, i.e. $\backslash rho=\backslash rho\_0$ (for some positive constant $\backslash rho\_0$) everywhere, then we obtain exactly the equation given in Landau and Lifshitz, namely

$\backslash frac\{1\}\{c\_0^2\}\backslash frac\{\backslash partial^2\; p\}\{\backslash partial\; t^2\}-\backslash nabla^2p=\backslash rho\_0\backslash frac\{\backslash partial^2\backslash hat\{T\}\_\{ij\}\}\{\backslash partial\; x\_i\; \backslash partial\; x\_j\},\backslash quad\backslash text\{where\}\backslash quad\backslash hat\{T\}\_\{ij\}\; =\; v\_i\; v\_j.$

A similar approximation in, namely $T\backslash approx\backslash rho\_0\backslash hat\; T$, is suggested by Lighthill see.

Of course, one might wonder whether we are justified in assuming that $p-p\_0=c\_0^2(\backslash rho-\backslash rho\_0)$. The answer is affirmative, if the flow satisfies certain basic assumptions. In particular, if $\backslash rho\; \backslash ll\; \backslash rho\_0$ and $p\; \backslash ll\; p\_0$, then the assumed relation follows directly from the linear theory of sound waves (see, e.g., the linearized Euler equations and the acoustic wave equation). In fact, the approximate relation between $p$ and $\backslash rho$ that we assumed is just a linear approximation to the generic barotropic equation of state of the fluid.

However, even after the above deliberations, it is still not clear whether one is justified in using an inherently linear relation to simplify a nonlinear wave equation. Nevertheless, it is a very common practice in nonlinear acoustics as the textbooks on the subject show: e.g., Naugolnykh and OstrovskyK. Naugolnykh and L. Ostrovsky, Nonlinear Wave Processes in Acoustics, Cambridge Texts in Applied Mathematics vol. 9, Cambridge University Press (1998) chap. 1. and Hamilton and Morfey.M. F. Hamilton and C. L. Morfey, "Model Equations," Nonlinear Acoustics, eds. M. F. Hamilton and D. T. Blackstock, Academic Press (1998) chap. 3.

• Acoustic theory
• Aeolian harp
• Computational aeroacoustics

Williams, J. E. Ffowcs, "The Acoustic Analogy—Thirty Years On" IMA J. Appl. Math. 32 (1984) pp. 113-124.

M. J. Lighthill, "On Sound Generated Aerodynamically. I. General Theory," Proc. R. Soc. Lond. A 211 (1952) pp. 564-587.

M. J. Lighthill, "On Sound Generated Aerodynamically. II. Turbulence as a Source of Sound," Proc. R. Soc. Lond. A 222 (1954) pp. 1-32.

• M. J. Lighthill, "On Sound Generated Aerodynamically. I. General Theory," Proc. R. Soc. Lond. A 211 (1952) pp. 564–587. This article on JSTOR.
• M. J. Lighthill, "On Sound Generated Aerodynamically. II. Turbulence as a Source of Sound," Proc. R. Soc. Lond. A 222 (1954) pp. 1–32. This article on JSTOR.
• L. D. Landau and E. M. Lifshitz, Fluid Mechanics 2ed., Course of Theoretical Physics vol. 6, Butterworth-Heinemann (1987) §75. , Preview from Amazon.
• K. Naugolnykh and L. Ostrovsky, Nonlinear Wave Processes in Acoustics, Cambridge Texts in Applied Mathematics vol. 9, Cambridge University Press (1998) chap. 1. , Preview from Google.
• M. F. Hamilton and C. L. Morfey, "Model Equations," Nonlinear Acoustics, eds. M. F. Hamilton and D. T. Blackstock, Academic Press (1998) chap. 3. , Preview from Google.
• Aeroacoustics at the University of Mississippi
• Aeroacoustics at the University of Leuven
• International Journal of Aeroacoustics
• Examples in Aeroacoustics from NASA
• Aeroacoustics.info