THESE de DOCTORAT - cerfacs

4.1 Fundamental validation cases 63
150
210
120
240
90
270
0.03
0.02
0.01
180 0
60
300
30
330
r = 0.05
r = 0.15
Pressure Fluctuation (Pa)
0.04
0.02
0
−0.02
Analytics
AVSP−f
−0.04
−0.2 −0.1 0 0.1 0.2
X axis (m)
a) | ˆp| Directivity of a dipole b) Pressure fluctuation
for two different radii.
along the horizontal e 1 axis
[ —— Analytics, (◦△) AVSP-f ] [ —— Analytics, (◦) AVSP-f ]
Figure 4.7: Analytical solution vs Numerical solution.
turbulent flame is acoustically equivalent to a set of different monopole sources, each of them
radiating at different strength, phase and frequencies. Therefore, for a last test of this first
validation, it is interesting to evaluate the noise radiation and directivity that results from three
monopoles with arbitrary positions and phase shifts. The contribution of a third pole is just
added as a third term in Eq. (4.6). After developing, the analytical solution reads
ˆp(x, ω) = i (Ŝ0
4c 2 δ(y − y 1 )e 1)
−iφ H0(kr 2 1 )dV 1 +
i (Ŝ0
0
4c 2 δ(y − y 2 )e 2)
−iφ H0(kr 2 2 )dV 2
0
+ i
(4.8)
(Ŝ0
4c 2 δ(y − y 3 )e 3) −iφ H0(kr 2 3 )dV 3
0
where Ŝ 0 = 1E10 W/m 3 s; y 1 = [−0.015, 0.015] m, y 2 = [0.015, 0.015] m and y 3 = [0.0, −0.015]
m; φ 1 = 0 rad, φ 2 = 1 rad and φ 3 = 2 rad; ω = 2π · 5000 rad/s. Fig. (4.8) shows the solution
given by the code AVSP-f. The reconstruction of the acoustic field at t 0 = 0 is evaluated by
p ′ = | ˆp| cos(arg( ˆp)). This acoustic field is shown in Fig. (4.9). Fig. (4.10) shows that a good
agreement is obtained between both analytical and numerical results. The directivity pattern
produced by these three monopoles is satisfactorily well recovered.