Archivo General

The field of aeroacoustics began with the work by Lighthill (1952) on aerody- namically generated sound. There were other researchers at the time, such as Moyal (1952) who were also exploring this field, by way of the spectra of turbu- lence in a compressible fluid media, but Lighthill’s particular attention to sound generation led him to rearrange the fluid equations of motion (conservation of mass and conservation of momentum) to form an inhomogeneous wave equation as shown in Equation 2.8, ∂2_ρ ∂t2 −c 2 ∞∇2ρ= ∂2_T ij ∂xi∂xj (2.8) where ρ is the density, c∞ is the ambient speed of sound and Tij is the Lighthill

stress tensor defined by Equation 2.9,

Tij =ρνiνj+ (p−ρc2∞)δij −τij (2.9)

where νi is the velocity, p is the pressure, τij is the viscous stress and δij is the

Kronecker delta.

bance is dependent on a non-linear term. The left hand side represents the acoustic wave propagation and the right hand side represents noise generating sources. The source terms are second spatial derivatives and are referred to as quadrupoles.

Lighthill identified this to be applicable to noise produced by jets, in which a freely propagating sound field is induced by a heavily rotational region domi- nated by non-linearity. With this, the Acoustic Analogy Theory was established.

The noise scaling law is an important result obtained from the Acoustic Analogy Theory by solving Equation 2.8 using the Green’s function of the wave equation. Lighthill found, by applying dimensional analysis, that the acoustic power radi- ated from a jet, scales with the eighth power of jet velocity, namely P ∼V8

j .

The quadrupole sources are acoustically compact, meaning that cancellations in sound wave emissions (due to time differences) from the same eddy are ignored. They are moving sources and are convected downstream by the mean flow for which Lagrangian dynamics predict the sound field. Ffowcs-Williams (1963) in- vestigated the effect source convection has on the directivity of jet noise and found by extending Lighthill’s dimensional analysis work, that for high speed jets the power of the radiated noise P follows the third power of the jet velocity. That is P ∼V3

j .

As the sound field is convected downstream it is refracted away from the jet axis. This is because the wave front is dependent on the local speed of sound and the local flow velocity. As the centre flow of the jet is faster than other regions, a wave front emanating from the mixing layer for example, will tilt and bend out-

wards. This leads to the next development in this field regarding flow-acoustic interaction, where sound is refracted away from the jet axis.

With the refraction of quadrupole sources, Lighthill determined a cone of rel- ative silence which exits the nozzle at small angles to the jet axis. This is a relatively quiet region in which the sound pressure levels drop by more than 20 dB as shown by Atvars et al. (1965). However, when reviewing this analytically by collecting together all the non-linear terms (from the flow equations) on the right hand side of Lighthill’s equation, the flow-acoustic interaction terms are not obviously placed. Ribner (1962) started a discussion on this topic and upon inspection, Lilley (1974) separated the flow-acoustic interaction from the pro- duction mechanisms to produce a modified wave equation. He proposed the use of linearized Euler equations as the wave operator. This improved the under- standing for many researchers of sound mechanisms in free jets from compact, convected noise sources.

However, a source term representing temperature fluctuations was also believed to exist by Fisher et al. (1973), for which Tester & Morfey (1976) found ana- lytical solutions and did further investigations on the scattering of turbulence pressure fields by density inhomogeneities, which they claimed were temperature dependent. Their far field solutions for sound, gave way to outlining the shape of radiated sound at the sidelines, based on a master spectrum and dipole term which scaled to the sixth power of velocity.

These findings were widely accepted until recently when Viswanathan (2004a) questioned the experimental data that was used to show a dipole master spec- trum and a sixth power velocity term. Viswanathan showed that the ‘hump’

in far field spectra is not due to the dipole master spectrum but is based on the Reynolds number. By keeping the Mach number constant, he compared the spectra of a cold jet with a high Reynolds number, to that of a hot jet with a lower Reynolds number, which showed that temperature alone does not alter the profile of the sideline noise radiation or side lobes. Because of these discrepancies, further work is needed for clarity on the effects of hot jets.

The heating of the jet does however, affect the sound field radiated at small angles to the jet axis. It also increases/decreases the sound power levels of the jet flow, for the respective heating of jets with low/high velocities, as also described by Tanna et al. (1975) and Tanna (1977), with the effects of heating being reversed at a critical point of Mach 0.7.

There have also been a number of significant acoustic theories proposed, such as Powell (1964), Howe (1975) and Mohring (1978) who expressed the source term, in terms of the vorticity of the flow, which is simpler to deal with mathematically than the double divergence of the stress tensor. This analogy is supported by Ewert & Schroeder (2003), Cabana et al. (2006) and Schramet al. (2005).

These analogies differ to Lighthill and Lilley, in that they use an implicit lin- earization about a given base flow. In Lighthill based analogies, the base flow is homogeneous and uniform, and with Lilley the base flow is parallel and sheared. In each case, the source is defined such that it holds the position of driving or exciting the base flow. But as the source definitions are different, there is some lack of agreement regarding the noise mechanisms. There is also a degree of redundancy in each in the dynamics of the noise mechanisms. In Lighthill’s anal- ogy, the flow-acoustic effects are largely ignored. With Lilley, these effects are

accounted for by the mean flow-acoustic effects of the base flow, however some information is still lost. This is where controversy lies, in which approach to use to describe the physics of sound production. Although in both cases, the solution to the base flow is found using the Green’s function.

Goldstein (2003) believed the Lighthill and Lilley analogies are particular cases of a more general solution to the production of noise. He showed the linearization of any base flow can be done on a generalized basis. In a later paper Gold- stein (2005a), proposed an acoustic analogy showing no redundancy in the sound source term, however this would not be practically possible in a periodic, homo- geneous, unsteady flow system where it would contradict locality constraints as described by Jordan & Gervais (2008).