THESE de DOCTORAT - cerfacs

2.3 Hybrid computation of noise: Acoustic Analogies 35
reacting flow, the energy equation Eq. (2.28) is combined with Eq. (2.47) leading to
[
1 Ds (γ − 1)
=
c p Dt ρc 2
Phillip’s equation for reacting flows reads
D 2 π
Dt 2 − ∂ (
c 2 ∂π )
= ∂u j ∂u i
− ∂
∂x i ∂x i ∂x i ∂x j
+ D Dt
]
∂J
˙ω T + ∑ h k
k − ∂q i
+ τ ij : ∂u i
+ ˙Q + D (ln r) (2.58)
∂x
k i ∂x i ∂x j Dt
∂x i
( 1
ρ
[ (
(γ − 1)
ρc 2
)
∂τ ij
∂x j
∂J
˙ω T + ∑ h k
k − ∂q i
+ τ ij : ∂u i
+ ˙Q (2.59)
∂x
k i ∂x i ∂x j
Dt 2 (ln r)
+ D2
Equation (2.59) was introduced by Chiu & Summerfield [14] and Kotake [39]. This expression
accounts for some acoustic-flow interactions, since gradients of the sound velocity c are
included in the acoustic operator. It has several advantages compared to Lighthill’s analogy,
since mean temperature and density of the propagation medium are not assumed homogeneous
in space, which is clearly the case for reacting flows. The term responsible for noise
generation due to hydrodynamic fluctuations is now defined as ∂u j
∂x i
∂u i
∂x j
. The density contribution
in this term has been removed, contrary to its form in Lighthill’s analogy
∂2
∂x i ∂x j
(ρu i u j ). It is
clear then that for low Mach numbers there is not anymore ambiguity in this source definition
since acoustic-flow interactions are not anymore included in this term [4]. Nevertheless, Lilley
[52] demonstrated that when the Mach number is not negligible, acoustic-flow interactions
are still contained in the compressible part of u i . Another interesting remark done by Bailly
[4] is that in this equation the source responsible for indirect noise (the acceleration of density
inhomogeneities) no longer appears. It is recommended therefore that if indirect noise is not
negligible and no strong changes in the mean flow are present, Lighthill’s analogy should be
the formulation under consideration.
)]
Simplifying Phillips’ Analogy
Phillips’ equation (Eq. 2.59), as Lighthill’s formulation, is an exact rearrangement of the fluid
dynamics equations. Similar assumptions as those made in section 2.3.2 are considered here.
It is assumed then that the diffusion of species J k , the dissipation function τ ∂u i
∂x j
, external energy
sources ˙Q, heat fluxes ∂q i
∂x i
, the viscosity stresses τ ij , changes in the molecular weight of
the mixture D Dt (ln r) and the Reynolds stress tensor ρu iu j are neglected in comparison to the
monopolar source of noise ∂ ˙ω T
∂t
. The simplified Phillips’ equation then reads: