THESE de DOCTORAT - cerfacs
THESE de DOCTORAT - cerfacs THESE de DOCTORAT - cerfacs
62 Chapter 4: Validation of the acoustic code AVSP-f where Ŝ 0 = 1E10 W/m 3 s, y 1 = [−0.02, 0.0] m, y 2 = [0.02, 0.0] m, φ 1 = 0 rad, φ 1 = π rad, and ω = 2π · 5000 rad/s. Fig. (4.5) 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. 4.6. It can be observed from Fig. 4.6 and Fig. 4.7 that AVSP-f succeeds in recovering the acoustic field produced by a dipole. The zone of silence is well seen in the line of antisymmetry between the two monopoles, i.e., the 90 ◦ and 270 ◦ directions. a) Modulus of Pressure | ˆp| b) Argument of pressure arg( ˆp) Figure 4.5: Outputs of AVSP-f. Dipole case Figure 4.6: Acoustic field produced by a dipole. p ′ = | ˆp| cos(arg( ˆp)) 4.1.3 Three poles out of phase in free space From sections (4.1.1) and (4.1.2) it has been shown that AVSP-f is able to compute correctly the radiation and cancellation between two monopoles. Smith and Kilham [95] stated that a
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.
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62 Chapter 4: Validation of the acoustic co<strong>de</strong> AVSP-f<br />
where Ŝ 0 = 1E10 W/m 3 s, y 1 = [−0.02, 0.0] m, y 2 = [0.02, 0.0] m, φ 1 = 0 rad, φ 1 = π rad, and<br />
ω = 2π · 5000 rad/s. Fig. (4.5) shows the solution given by the co<strong>de</strong> AVSP-f. The reconstruction<br />
of the acoustic field at t 0 = 0 is evaluated by p ′ = | ˆp| cos(arg( ˆp)). This acoustic field is shown<br />
in Fig. 4.6. It can be observed from Fig. 4.6 and Fig. 4.7 that AVSP-f succeeds in recovering the<br />
acoustic field produced by a dipole. The zone of silence is well seen in the line of antisymmetry<br />
between the two monopoles, i.e., the 90 ◦ and 270 ◦ directions.<br />
a) Modulus of Pressure | ˆp| b) Argument of pressure arg( ˆp)<br />
Figure 4.5: Outputs of AVSP-f. Dipole case<br />
Figure 4.6: Acoustic field produced by a dipole. p ′ = | ˆp| cos(arg( ˆp))<br />
4.1.3 Three poles out of phase in free space<br />
From sections (4.1.1) and (4.1.2) it has been shown that AVSP-f is able to compute correctly<br />
the radiation and cancellation between two monopoles. Smith and Kilham [95] stated that a
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