THESE de DOCTORAT - cerfacs

2.3 Hybrid computation of noise: Acoustic Analogies 27
h = h s + ∆h 0 f (2.14)
Both h s and ∆h 0 f
can be expressed as the sum of the contributions of each species k. Hence:
h = ∑
k
h s,k Y k + ∑ ∆h 0 f ,k Y k (2.15)
k
Introducing now the specific heat of species k at constant pressure c p,k , the material derivative
of sensible enthalpy h s can be obtained by
Dh s
Dt
= D Dt ∑ k
= ∑ c p,k T DY k
k
h s,k Y k = D Dt ∑ k
Dt + ∑
k
DY
= ∑ h k
s,k
Dt + c p
k
DT
Dt
c p,k TY k (2.16)
c p,k Y k
DT
Dt
(2.17)
(2.18)
And as a consequence
Dh
Dt = ∑
k
DY
h k
s,k
Dt + c DT
p
Dt + ∑
k
∆h 0 DY k
f ,k
Dt
(2.19)
Expression (2.19) can be re-stated considering changes in pressure , density and mass fraction
rather than changes in temperature. Considering the ideal gas equation,
P = rρT =
(
R ∑
k
)
Y k
ρT (2.20)
W k
where W k is the molecular weight of species k and R = 8.314 J mol −1 K −1 , the differential of
temperature yields
dT = ∂T
∂p ⏐ dp + ∂T
⏐ ∂T
ρ,Yk
∂ρ ⏐ dρ + ∑
dY
p,Yk
∂Y k (2.21)
k
⏐⏐⏐ρ,p,Yl̸=k
dT = 1 ρr dp − T ρ dρ − T ∑
k
k
W
W k
dY k (2.22)
The term T ∑ k
W
W k
dY k can be expressed as Td(ln r). Replacing Eq. (2.22) into Eq. (2.19) leads to