THESE de DOCTORAT - cerfacs

2.3 Hybrid computation of noise: Acoustic Analogies 31
and resolving for ∂ρ e
∂t
, one finds:
∂ρ e
∂t
= ρ ∞
ρ
Dρ
Dt + 1
c 2 ∞ρ (p − p ∞) Dρ
Dt − 1 Dp
c 2 ∞ Dt − ∂ (ρ e u j ) (2.39)
∂x j
The energy equation in terms of the material derivative of the density Dρ
Dt
section 2.3.1. Introducing Eq. (2.28) into Eq. (2.38) yields:
was obtained in
∂ρ e
∂t
= ρ ∞
ρ
(
(γ − 1)
c 2 − ˙ω T − ∑
+ 1
c 2 ∞ρ (p − p ∞) Dρ
Dt − 1
c 2 ∞
k
)
∂J
h k
k + ∂q i ∂u
− i
τ ij − ˙Q
∂x i ∂x i ∂x j
(
1 − ρ ∞c 2 )
∞ Dp
ρc 2 Dt − ∂ (ρ e u j )
∂x j
D
− ρ ∞ (ln r)
Dt
(2.40)
The complete equation is then obtained in the frame of Lighthill’s analogy. This exact reformulation
of the Navier-Stokes equations reads
1
c 2 ∞
∂ 2 p
∂t 2 − ∂2 p
∂xi
2 = ∂2
+ ∂ ∂t
( )
ρui u j − τ ij
∂x i ∂x j
[
(
ρ ∞ (γ − 1)
ρ c 2
+ 1
c 2 ∞
[(
∂
1 − ρ ∞c 2 ∞
∂t ρc 2
+ ∂2
∂x j ∂t (ρ eu j )
˙ω T + ∑
k
∂J
h k
k − ∂q i ∂u
+ i
τ ij + ˙Q
∂x i ∂x i ∂x j
]
) Dp
Dt − p − p ∞
ρ
Dρ
Dt
)
+ ρ ∞
D
Dt (ln r) ]
(2.41)
As it can be observed, the sources contributing to noise in reacting flows are multiple. It is
observed that the heat release ˙ω T , the diffusion flux of species J k as well as the non-isomolar
combustion due to changes in r generate noise. The heat flux q i , the dissipation function τ ij
∂u i
∂x j
and the energy sources applied to the system ˙Q are also responsible for noise generation. The
terms present on the third line of Eq. (2.41) are related to the inhomogeneities of the flow with
respect to the propagation media. Finally, the last term of Eq. (2.41) stands for the noise produced
by the acceleration of density inhomogeneities and should be related to the indirect
combustion noise.
Lighthill’s analogy is a suitable formulation when sources are compact and the radiation of
sound takes place in a quiescent field. Extensions of Lighthill’s analogy in order to account for
turbulence-body interactions have been derived by Curle [20] and Ffowcs-Williams and Hawk-