THESE de DOCTORAT - cerfacs

4.1 Fundamental validation cases 61
150
210
120
240
90
270
0.008
60
0.006
0.004
0.002
180 0
300
30
330
r = 0.05
r = 0.15
Pressure Fluctuation (Pa)
0.03
0.02
0.01
0
−0.01
−0.02
Analytics
AVSP−f
−0.03
−0.2 −0.1 0 0.1 0.2
X axis (m)
a) | ˆp| Directivity of a monopole b) Pressure fluctuation
for two different radii.
along the horizontal e 1 axis
[ —— Analytics, (◦△) AVSP-f ] [ —— Analytics, (◦) AVSP-f ]
Figure 4.4: Analytical solution vs Numerical solution.
4.1.2 A Dipole in free space
Once pure radiation is tested in section (4.1.1), the influence on the acoustic field of another
monopole is tested. If this monopole radiates sound out of phase with respect to the first
monopole, a substraction of acoustic fluctuations will occur leading to a diminution of the
acoustic amplitude in some regions. If this phase shift is equal to π a perfect cancelation should
occur in the plane of antisymmetry between the two monopoles. Two monopoles with such a
shift phase conform a dipole. The source is defined as
Ŝ(y, ω) = Ŝ 0 δ(y − y 1 )e −iφ 1
+ Ŝ 0 δ(y − y 2 )e −iφ 2
(4.5)
where φ 1 = φ 2 + π. From Eq. (4.3)
ˆp(x, ω) =
i ∫
4c 2 0
V 0 (y)
(Ŝ0 δ(y − y 1 )e −iφ 1
+ Ŝ 0 δ(y − y 2 )e 2)
−iφ H0(kr)dV 2 (4.6)
Finally
ˆp(x, ω) =
i (Ŝ0 δ(y − y 1 )e 1)
−iφ H0(kr 2 1 )dV 1 +
4c 2 0
i
4c 2 0
(Ŝ0 δ(y − y 2 )e 2)
−iφ H0(kr 2 2 )dV 2 (4.7)