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2.3 Hybrid computation of noise: Acoustic Analogies 29
2.3.2 Lighthill’s Analogy
When considering frozen vortical structures or a frozen flame, clearly all the forces implied in
each of these physical mechanisms are totally balanced and no sound is produced. But once
unsteadiness occurs, these forces are not balanced anymore. In the case of turbulence, for example,
the reciprocal motion of the surrounding gas tries to compensate for this ‘unexpected’
dynamics. Nevertheless, this reciprocal motion fails in completely absorbing the vortical energy
and therefore local compressions and dilatations due to compressibility are transmitted
to the adjacent particles: sound has been generated. In the source region (vortex/flame dynamics)
the flow can be considered incompressible, forasmuch as the compressible part of the
pressure fluctuation p ′ a is much smaller than the pressure fluctuations generated by both unsteady
vorticity p ′ h (if sound is generated by turbulence) or unsteady temperature p′ f
(if sound
is generated by combustion). In this region, pressure fluctuations can be then computed by
solving a Poisson equation ∇ 2 p ′ = ψ where ψ is a function of the implied sources. In the
far-field, on the contrary, the fluctuations due to acoustics p ′ a are much more important than
the significatively much smaller p ′ h ,p′ f
. In the case of turbulence, for instance, it is known that
hydrodynamic pressure fluctuations are negligible in the far field since they typically decay at
least as the inverse third power of the distance to the sources [80]. Pressure fluctuations p ′ a may
be assessed then by solving an acoustic wave equation.
Lighthill’s analogy is considered as the starting points of aeroacoustics as a research field. The
Navier-Stokes equations are rearranged so that the resulting formulation has the ‘shape’ of a
inhomogeneous wave equation: in the left hand side (LHS) the acoustic operator and in the
right hand side (RHS) the sources responsible for noise generation. Neglecting volume forces,
the mass and momentum equations (Eqs. 2.4 and 2.5) can be expressed in their conservative
forms:
∂ρ
∂t + ∂ρu j
∂x j
= 0 (2.29)
∂ρu i
∂t
+ ∂ρu iu j
∂x j
= − ∂p
∂x i
+ ∂τ ij
∂x j
(2.30)
to Eq. (2.30) and sub-
Applying the time derivative ∂ ∂t
stracting them results in
to Eq. (2.29), the spatial derivative
∂
∂x i
∂ 2 ρ
∂t 2 − ∂2 p
∂xi
2 = ∂2 ( )
ρui u j − τ ij
∂x i ∂x j
(2.31)
In order to construct the D’Alembertian operator, the term 1
c 2 ∞
∂ 2 p
∂t 2 is added to both RHS and LHS
of Eq. (2.31). c ∞ represents the sound velocity in the quiescent propagation media (far field). It