THESE de DOCTORAT - cerfacs

32 Chapter 2: Computation of noise generated by combustion
ings [106]. As stated in the introductory part of this section, the acoustic operator of Lighthill’s
analogy does not account for acoustic-flow interactions. All these possible interactions are implicitly
included in the source terms and therefore density fluctuations on the sources cannot
be neglected. However, in order to compute this compressible sources it would be necessary to
resolve the pressure field in which these sources are embedded or, in other words, to solve the
problem itself. Some simplifications then must be considered.
Simplifying Lighthill’s Analogy
Equation (2.41) is an exact result derived from the conservation equations for mass, momentum,
energy and species and therefore until now no assumptions or simplifications have been
introduced. Nevertheless, this equation in this way does not present a significant utility. Assumptions
must be considered so that Lighthill’s equation becomes tractable and therefore
solvable. The idea is then only to consider the predominant mechanisms generators of noise.
In doing so, diffusive fluxes of species J k and the dissipation function τ ∂u i
∂x j
˙Q vanishes if no external energy sources are applied to the system and ∂q i
∂x i
are first neglected.
is zero if heat fluxes
are considered small. Moreover, at high Reynolds number the viscous tensor τ ij is negligible if
compared to the Reynolds stresses ρu i u j and the propagation medium is considered homogenous
(p, ρ, c → p ∞ , ρ ∞ , c ∞ ) so that all the third line disappears. Further on, combustion can be
considered isomolar if air is used as oxidizer so that D Dt
(ln r) vanishes and for low Mach numbers
M the acceleration of density inhomogeneities
∂2
∂x j ∂t (ρ eu j ) can be neglected . The simplified
Lighthill Equation then reads
1
c 2 ∞
∂ 2 p
∂t 2 − ∂2 p
∂xi
2 = ∂2 ( ) (γ − 1) ∂ ˙ω
ρ∞ u i u j + T
∂x i ∂x j c 2 ∞ ∂t
(2.42)
Hassan demonstrates in his work [32] that noise produced by combustion is much more important
than the one generated by turbulence. Note that the combustion source of noise is a
monopole ( ∂ ∂2
∂t
) whereas the aerodynamic source of noise is a quadrupole ( ). A quadrupolar
∂xi
2
source of noise is known to be a bad radiator of sound if compared to a monopolar source and
therefore the aerodynamic source can be neglected. Finally, heat release ˙ω T and pressure p are
decomposed by its mean and fluctuating parts (p = ¯p + p ′ , ˙ω T = ¯˙ω T + ˙ω
T ′ ). Combustion
noise in a quiescent medium is given then by
1
c 2 ∞
∂ 2 p ′
∂t 2 − ∂2 p ′
∂x 2 i
=
(γ − 1)
c 2 ∞
∂ ˙ω ′ T
∂t
(2.43)
where ∂ ¯p
∂x i
≈ 0 due to the M = 0 assumption. It is useful also to solve this wave equation in
the frequency domain. Considering harmonic perturbations of pressure p ′ and the heat release