Civil Engineering Reference
In-Depth Information
(Equation 13.13) can be written
∂p
a
∂x
i
·
∂(δp
a
)
∂x
i
1
ρ
0
ω
2
1
ρ
0
c
0
p
a
δp
a
)
d
−
a
1
jω
a
A(
x
,
y
)p(
y
)δp
a
(
x
)
d
S
y
d
S
x
−
(13.75)
a
1
jω
a
A(
x
,
y
)p
b
(
y
)δp
a
(
x
)
d
S
y
d
S
x
=
0
δp
a
,
+
∀
a
where
a
denotes the front interface between the fluid cavity part and the waveguide.
For a given number of kept normal modes
ϕ
mn
in the wave-guide, Equations (13.74)
and (13.75) are discretized using the finite element method and solved for the solid phase
displacement, interstitial pressure variables in the porous medium and acoustic pressure
in the holes. From nodal solutions, the normal incidence absorption coefficient can be
computed using a power balance (Equation 13.72) or the sum of the powers dissipated
inside the heterogeneous porous medium. The expressions of the powers dissipated in
the porous materials by structural damping, viscous and thermal effects are explained in
Section 13.7.
To illustrate the validity of the described model, a case taken from Atalla
et al
. (2001b)
is considered. For the normal incidence sound absorption prediction (Figure 13.15), it is
sufficient to consider one representative cell instead of the whole perforated material, pro-
vided that the perforation network is periodic. A rockwool sample made up of a generic
square cell of side
L
c
=
0
.
085
m
and a central perforation is considered. The acoustic and
mechanical properties of the material are given in Table 13.2. Sound absorption measure-
ments have been performed using a 1.2-m-long standing wave tube of 0
.
085
×
0
.
085 m
square cross section (Olny and Boutin 2003). The samples were slightly constrained
around their edges in order to avoid leaks, but no sealant was used. In the experiments,
a hole of radius 0.016 m has been cut out from the material. On the other hand, the
numerical model assumes an equivalent area square hole with side
a
0
.
0283 m. Linear
Hexa 8 (brick) elements were used for both the porous and mesoporous (hole) domains.
Figure 13.16 shows the comparison between simulation (analytical and numerical
model) and measurements for a thickness of 5.75 cm. Analytical simulations using the
model of Olny and Boutin (2003) are also presented. Based on the chosen perforation size
and sample side length, a double porosity of
φ
p
=
0
.
11 is used in the analytical model.
Recall from Chapter 5 that in the analytical model, the porous material is assumed rigid.
Moreover, the diffusion function was chosen to fit the experiments. Contrary to the
analytical model, calculations in the numerical model are based on Biot's theory with no
assumption or parameter adjustment. That is, no assumption is made on the motion of the
skeleton and in consequence the influence of the solid phase on the acoustic absorption
is assessed. Excellent agreement is found for the numerical model. It is observed in
=
Table 13.2
The parameters used to predict the absorption coefficients in Figure 13.16.
Material
φ
σ
α
∞
m
(
0
)
ρ
1
E
ν
s
(N.s/m
4
)
(m
2
)
(kg/m
3
)(P )
(
µ
m)
(
µ
m)
10
−
9
3
.
3
×
Rockwool
0.94
1 35 000
2.1
49
166
130
4400
0
0.1
