THESE de DOCTORAT - cerfacs

162 Chapter B: Publications
The temporal derivative of the density in Eq. 10 is combined with Eq. 14. After some algebra, the
divergence of the velocity can therefore be expressed as:
∂u j
= 1 ˙q (15)
∂x j c p K 0
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. One can state that the divergence of the
fluctuating velocity is
B. Finding the Acoustic Pressure
Injecting Eq. 16 into Eq. 9 leads to
− ∂ ( 1 ∂p ′ )
ac
= ∂ ( ∂u
′ )
i,LES
− ˙q′
∂x i ρ 0 ∂x i ∂t ∂x i c p K 0
or in the frequency domain
Finally, multiplying everywhere by γP 0
∂u ′ j,hyd
∂x j
= 1
c p K 0
˙q ′ (16)
(17)
( )
∂ 1 ∂ ˆp ac
= iω ∂û i,LES
− iω ˆ˙q
(18)
∂x i ρ 0 ∂x i ∂x i c p K 0
(
∂
c 2 ∂ ˆp )
ac
∂x i ∂x i
= iωγP 0
∂û i,LES
∂x i
} {{ }
T 1
− iω γP 0ˆ˙q
c p K 0
} {{ }
T 2
It has been found by the author that the contribution of T 2 can be neglected. As a consequence, one can
state that for a reactive flow the divergence of the hydrodynamic velocity field ( ∂u hyd
∂x i
) can be considered
zero as typically done for non-reactive flows. Equation 19 simplifies
F(ˆp ac ) =
∂ (
c 2 ∂ ˆp )
ac
∂û i,LES
= iωγP 0 (20)
∂x i ∂x i ∂x i
(19)
VI.
LES Vs Hybrid Results
Figures 7 and 8 shows the Sound Pressure Level (SPL) given by the solution of Eq. 20 (LES ‘Filtered’)
and the hybrid computation for microphones 5, 6 and 7.
SPL (dB) − micro 5
180
160
140
120
100
80
Hybrid Computation
Direct Computation: LES (Filtered)
SPL (dB) − micro 6
180
160
140
120
100
80
Hybrid Computation
Direct Computation: LES (Filtered)
60
0 500 1000 1500 2000 2500
Frequency (Hz)
60
0 500 1000 1500 2000 2500
Frequency (Hz)
Figure 7.
Sound Pressure Levels from the direct and hybrid approaches
After extracting the acoustic field from the complete pressure fluctuation field, it is seen that results match
pretty well with the values predicted by the hybrid method, especially for microphones 6 and 7. There is a
high contain of hydrodynamic fluctuations at low frequencies (before 400 Hz) that has been removed. Also
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