THESE de DOCTORAT - cerfacs

5.4 Filtering a LES pressure field to find the corresponding acoustic field 89
perature. The equation of state is simply
Replacing Eq. 5.14 in the left hand side of the Eq. (5.13) leads to
¯p
r = K 0 = ρT (5.14)
ρc p
∂K 0 /ρ
∂t
+ ρc p u j
∂K 0 /ρ
∂x j
= ˙ω T (5.15)
Developing Eq. (5.15), and combining it with Eq. (5.11) results in a simplified equation in which
density ρ is not anymore present. It reads
∂u j
∂x j
= 1
c p K 0
˙ω T (5.16)
This velocity field is supposed to be composed only by hydrodynamics, due to the fact that in
the low Mach number model the acoustic wave length is infinitely long and then ∂u ac /∂x j ≈ 0.
One can state that the divergence of the fluctuating velocity is
∂u ′ j,hyd
∂x j
= 1
c p K 0
˙ω ′ T (5.17)
5.4.2 Finding the Acoustic Pressure
Injecting Eq. (5.17) into Eq. (5.10) leads to
− ∂ (
∂p
∂x i
1¯ρ
′ ) ( )
ac
= ∂ ∂u
′
i,LES
− ˙ω′ T
∂x i ∂t ∂x i c p K 0
(5.18)
or in the frequency domain
Finally, multiplying everywhere by γ ¯p
( )
∂ ∂ ˆp ac
= iω
∂x i
1¯ρ
∂û i,LES
− iω ˆ˙ω T
(5.19)
∂x i ∂x i c p K 0