THESE de DOCTORAT - cerfacs

28 Chapter 2: Computation of noise generated by combustion
ρ Dh [ 1
Dt = ρc Dp
p
ρr Dt − T Dρ
ρ Dt − T D ]
Dt (ln r)
( )
+ ρ ∑ h s,k + ∆h 0 DYk
f ,k
Dt
k
(2.23)
The last term of Eq. (2.23) can be developed so that heat release due to combustion ˙ω T comes
up. This term is defined as ˙ω T = − ∑ k (h s,k + ∆h 0 f ,k ) ˙ω k [74]. Using the balance of species
(Eq. 2.6) results in
( ) (
∑ h s,k + ∆h 0 f ,k
˙ω k − ∂J )
k
∂x
k
i
= ∑
k
h s,k ˙ω k + ∑
k
∆h 0 f ,k ˙ω ∂J
k − ∑ h k
s,k
k
∂x i
− ∑
k
∆h 0 ∂J k
f ,k (2.24)
∂x i
∂J
= − ˙ω T − ∑ h k
k (2.25)
∂x
k i
Equation (2.8) is combined with Eqs. (2.23),(2.25) so that
c p
r
Dp
Dt − c pT Dρ
Dt − ρc pT D Dt (ln r) − ˙ω T − ∑
k
h k
∂J k
∂x i
= Dp
Dt − ∂q i
∂x i
+ τ ij
∂u i
∂x j
+ ˙Q + ρ f i u i
(2.26)
Dividing everywhere by (c p T) and neglecting volume forces ( f i = 0) yields
(
Dp 1
Dt
[
+ 1
c p T
rT − 1
c p T
)
− Dρ
Dt − ρ D (ln r)
Dt
∂J
− ˙ω T − ∑ h k
k + ∂q i ∂u
− i
τ ij − ˙Q
∂x
k i ∂x i ∂x j
]
= 0
(2.27)
A similar derivation for the material derivative of the density ρ can be found in [4]. The resulting
expression is really useful to study acoustics, as both Dρ
Dt
and 1 Dp
c 2 Dt
appear explicitly.
Reorganizing Eq. 2.27 and knowing that c p = rγ/(γ − 1)
[
Dρ
Dt = 1 Dp (γ − 1)
c 2 +
Dt c 2
]
∂J
− ˙ω T − ∑ h k
k + ∂q i ∂u
− i
τ ij − ˙Q
∂x
k i ∂x i ∂x j
− ρ D (ln r) (2.28)
Dt
where the velocity of sound c = √ γrT has been introduced.