THESE de DOCTORAT - cerfacs

88 Chapter 5: Assessment of combustion noise in a premixed swirled combustor
u ′ i,LES = u′ i,hyd + u′ i,ac (5.6)
Appling the operator ∂/∂t to Eq. (5.6) leads to
∂u ′ i,LES
∂t
= ∂u′ i,hyd
∂t
+ ∂u′ i,ac
∂t
(5.7)
From linear acoustics, the momentum equation is given by
¯ρ ∂u′ i,ac
∂t
= − ∂p′ ac
∂x i
(5.8)
where ¯[] and [] ′ represent respectively the mean and fluctuating flow. Combining Eq. (5.8) and
Eq. (5.7) leads to
− 1¯ρ
∂p ′ ac
∂x i
+ ∂u′ i,hyd
∂t
= ∂u′ i,LES
∂t
(5.9)
Finally the divergence operator to this equation is applied to yield
− ∂ (
∂p
∂x i
1¯ρ
′ ) ( )
ac
+ ∂ ∂u
′
i,hyd
= ∂2 u
i,LES
′
∂x i ∂x i ∂t ∂x i ∂t
(5.10)
5.4.1 Finding ∂u i,hyd
∂x i
Neglecting viscosity, species diffusion and heat conduction, the Navier-Stokes equations for
reacting flows read
∂ρ
∂t + ρ ∂u j
∂x j
+ u j
∂ρ
∂x j
= 0 (5.11)
ρ ∂u i
∂t + ρu ∂u i
j = − ∂p
(5.12)
∂x j ∂x i
ρc p
∂T
∂t + ρc pu j
∂T
∂x j
= ˙ω T (5.13)
In the low-Mach number approximation, the thermodynamic pressure ¯p only depends on tem-