- Aeroacoustics
**Aeroacoustics**is a branch ofacoustics that studies noise generation via eitherturbulent fluid motion oraerodynamic forces interacting with surfaces. Noise generation can also be associated with periodically varying flows. 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 "Acoustic Analogy", whereby the governing equations of motion of the fluid are coerced into a form reminiscent of thewave equation of "classical" (i.e. linear) acoustics. The most common and a widely-used of the latter is "Lighthill's aeroacoustic analogy"Fact|date=June 2008. It was proposed byJames Lighthill in the 1950s 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.] when noise generation associated with thejet engine was beginning to be placed under scientific scrutiny.Computational Aeroacoustics (CAA) is the application of numerical methods and computers to find approximate solutions of the governing equations for specific (and likely complicated) aeroacoustic problems.**Lighthill's equation**The first equation of interest is the

conservation of mass equation, which reads:$frac\{partial\; ho\}\{partial\; t\}\; +\; ablacdotleft(\; homathbf\{v\}\; ight)=frac\{D\; ho\}\{D\; t\}\; +\; ho\; ablacdotmathbf\{v\}=\; 0,$

where $ho$ and $mathbf\{v\}$ represent the density and velocity of the fluid, which depend on space and time, and $D/Dt$ is the

substantial derivative .Next is the

conservation of momentum equation, which is given by:$\{\; ho\}frac\{partial\; mathbf\{v\{partial\; t\}+\{\; ho(mathbf\{v\}cdot\; abla)mathbf\{v\; =\; -\; abla\; p+\; ablacdotsigma,$

where $p$ is the thermodynamic

pressure , and $sigma$ is the viscous (or traceless) part of the stress tensor.Now, multiplying the conservation of mass equation by $mathbf\{v\}$ and adding it to the conservation of momentum equation gives

:$frac\{partial\}\{partial\; t\}left(\; homathbf\{v\}\; ight)\; +\; ablacdot(\; homathbf\{v\}otimesmathbf\{v\})\; =\; -\; abla\; p\; +\; ablacdotsigma.$

Note that $mathbf\{v\}otimesmathbf\{v\}$ is a

tensor (see alsotensor product ). Differentiating the conservation of mass equation with respect to time, taking thedivergence of the conservation of momentum equation and subtracting the latter from the former, we arrive at:$frac\{partial^2\; ho\}\{partial\; t^2\}\; -\; abla^2\; p\; +\; ablacdot\; ablacdotsigma\; =\; ablacdot\; ablacdot(\; homathbf\{v\}otimesmathbf\{v\}).$

Subtracting $c\_0^2\; abla^2\; ho$, where $c\_0$ is the

speed of sound in the medium in its equilibrium (or quiescent) state, from both sides of the last equation and rearranging it results in:$frac\{partial^2\; ho\}\{partial\; t^2\}-c^2\_0\; abla^2\; ho\; =\; ablacdotleft\; [\; ablacdot(\; homathbf\{v\}otimesmathbf\{v\})-\; ablacdotsigma\; +\; abla\; p-c^2\_0\; abla\; ho\; ight]\; ,$

which is equivalent to

:$frac\{partial^2\; ho\}\{partial\; t^2\}-c^2\_0\; abla^2\; ho=(\; ablaotimes\; abla)\; :left\; [\; homathbf\{v\}otimesmathbf\{v\}\; -\; sigma\; +\; (p-c^2\_0\; ho)mathbb\{I\}\; ight]\; ,$where $mathbb\{I\}$ is the identity tensor, and $:$ denotes the (double)

tensor contraction operator.The above equation is the celebrated

Lighthill equation of aeroacoustics. It is awave equation with a source term on the right-hand side, i.e. an inhomogeneous wave equation. The argument of the "double-divergence operator" on the right-hand side of last equation, i.e. $homathbf\{v\}otimesmathbf\{v\}-sigma+(p-c^2\_0\; ho)mathbb\{I\}$, is the so-called "Lighthill turbulence stress tensor for the acoustic field", and it is commonly denoted by $T$.Using

Einstein notation , Lighthill’s equation can be written as:$frac\{partial^2\; ho\}\{partial\; t^2\}-c^2\_0\; abla^2\; ho=frac\{partial^2T\_\{ij\{partial\; x\_i\; partial\; x\_j\},quad\; (*)$

where

:$T\_\{ij\}=\; ho\; v\_i\; v\_j\; -\; sigma\_\{ij\}\; +\; (p-\; c^2\_0\; ho)delta\_\{ij\},$

and $delta\_\{ij\}$ is the

Kronecker delta . 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.In practice, it is customary to neglect the effects

viscosity of the fluid, i.e. one takes $sigma=0$, because it is generally accepted that the effects of the latter on noise generation, in most situations, are orders of magnitude smaller than those due to the other terms. 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.**Related model equations**In their classical text on

fluid mechanics , Landau and LifshitzL. D. Landau and E. M. Lifshitz, "Fluid Mechanics" 2ed., Course of Theoretical Physics vol. 6, Butterworth-Heinemann (1987) §75.] derive an aeroacoustic equation analogous to Lighthill's (i.e., an equation for sound generated by "turbulent " fluid motion) but for theincompressible flow of aninviscid fluid. The inhomogeneous wave equation that they obtain is for the "pressure" $p$ rather than for the density $ho$ of the fluid. Furthermore, unlike Lighthill's equation, Landau and Lifshitz's equation is**not**exact; it is an approximation.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 $p-p\_0=c\_0^2(\; ho-\; ho\_0)$, where $ho\_0$ 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 $(*)\; ,$ we obtain the equation:$frac\{1\}\{c\_0^2\}frac\{partial^2\; p\}\{partial\; t^2\}-\; abla^2p=frac\{partial^2\; ilde\{T\}\_\{ij\{partial\; x\_i\; partial\; x\_j\},quad\; ext\{where\}quad\; ilde\{T\}\_\{ij\}\; =\; ho\; v\_i\; v\_j.$

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

:$frac\{1\}\{c\_0^2\}frac\{partial^2\; p\}\{partial\; t^2\}-\; abla^2p=\; ho\_0frac\{partial^2hat\{T\}\_\{ij\{partial\; x\_i\; partial\; x\_j\},quad\; ext\{where\}quadhat\{T\}\_\{ij\}\; =\; v\_i\; v\_j.$

A similar approximation [in the context of equation $(*),$] , namely $Tapprox\; ho\_0hat\; T$, is suggested by Lighthill [see Eq. (7) in the latter paper] .

Of course, one might wonder whether we are justified in assuming that $p-p\_0=c\_0^2(\; ho-\; ho\_0)$. The answer is in affirmative, if the flow satisfies certain basic assumptions. In particular, if $ho\; ll\; ho\_0$ and $p\; ll\; p\_0$, 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 $ho$ that we assumed is just alinear approximation to the genericbarotropic 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 Ostrovsky [*K. 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.*] .**References****External links*** M. J. Lighthill, "On Sound Generated Aerodynamically. I. General Theory," "Proc. R. Soc. Lond. A"

**211**(1952) pp. 564-587. [*http://links.jstor.org/sici?sici=0080-4630(19520320)211%3A1107%3C564%3AOSGAIG%3E2.0.CO%3B2-7 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. [*http://links.jstor.org/sici?sici=0080-4630(19540223)222%3A1148%3C1%3AOSGAIT%3E2.0.CO%3B2-2 This article on JSTOR*] .

* L. D. Landau and E. M. Lifshitz, "Fluid Mechanics" 2ed., Course of Theoretical Physics vol. 6, Butterworth-Heinemann (1987) §75. ISBN 0750627670, [*http://www.amazon.com/gp/reader/0750627670/ 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. ISBN 052139984X, [*http://books.google.com/books?vid=ISBN052139984X 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. ISBN 0123218608, [*http://books.google.com/books?vid=ISBN0123218608 Preview from Google*] .

* [*http://www.olemiss.edu/depts/ncpa/aeroacoustics/ Aeroacoustics at the University of Mississippi*]

* [*http://www.mech.kuleuven.be/mod/aeroacoustics/ Aeroacoustics at the University of Leuven*]

* [*http://www.multi-science.co.uk/aeroacou.htm International Journal of Aeroacoustics*]

* [*http://www.grc.nasa.gov/WWW/microbus/cese/aeroex.html Examples in Aeroacoustics from NASA*]**See also***

Acoustic theory

*Computational Aeroacoustics

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