Aeroacoustics is a branch of
acousticsthat studies noise generation via either turbulentfluid motion or aerodynamicforces 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 the wave equationof "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 by James Lighthillin 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 the jet enginewas 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 rearranged the
Navier–Stokesequations, which govern the flowof a compressible viscous fluid, into an inhomogeneous wave equation, thereby making an analogy between fluid mechanicsand acoustics.
The first equation of interest is the
conservation of massequation, which reads
where and represent the density and velocity of the fluid, which depend on space and time, and is the
Next is the
conservation of momentumequation, which is given by
where is the thermodynamic
pressure, and is the viscous (or traceless) part of the stress tensor.
Now, multiplying the conservation of mass equation by and adding it to the conservation of momentum equation gives
Note that is a
tensor(see also tensor product). Differentiating the conservation of mass equation with respect to time, taking the divergenceof the conservation of momentum equation and subtracting the latter from the former, we arrive at
Subtracting , where is the
speed of soundin the medium in its equilibrium (or quiescent) state, from both sides of the last equation and rearranging it results in
which is equivalent to
:where is the identity tensor, and denotes the (double)
The above equation is the celebrated
Lighthill equationof aeroacoustics. It is a wave equationwith 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. , is the so-called " Lighthill turbulence stress tensorfor the acoustic field", and it is commonly denoted by .
Einstein notation, Lighthill’s equation can be written as
and is the
Kronecker delta. Each of the acoustic source terms, i.e. terms in , may play a significant role in the generation of noise depending upon flow conditions considered.
In practice, it is customary to neglect the effects
viscosityof the fluid, i.e. one takes , 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 the incompressible flowof an inviscidfluid. The inhomogeneous wave equation that they obtain is for the "pressure" rather than for the density 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 , where and 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
Of course, one might wonder whether we are justified in assuming that . The answer is in affirmative, if the flow satisfies certain basic assumptions. In particular, if and , 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 and that we assumed is just a linear approximationto the generic barotropic equation of stateof 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 acousticsas 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.] .
* 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]
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