Civil Engineering Reference
In-Depth Information
q
0
=
φ
2
/
8 of a porous medium with circular cylindrical pores having a radius
R
=
leading to
1
+
1
/
2
4
2
jω
ν
8
ν
jω
2
α
(ω)
=
+
1
(5.38)
It will be shown in Section 5.6 that this arbitrary choice for
q
0
can lead to a large
error in the localization of the transition frequency where the imaginary part of the
bulk modulus reaches its maximum. This does not necessarily lead to a large error in the
evaluation of a surface impedance because the damping is mainly created by the viscosity
via the effective density.
5.5.3 The Wilson model
In the model due to Wilson (1993), the effective density and the bulk modulus are
given by
jωτ
vor
)
1
/
2
(
1
+
jωτ
vor
)
1
/
2
(
1
+
ρ(ω)
=
φρ
∞
(5.39)
−
1
jωτ
ent
)
1
/
2
(
1
+
K(ω)
=
φK
∞
(5.40)
(
1
+
jωτ
ent
)
1
/
2
+
γ
−
1
The parameters
τ
vor
,and
τ
ent
, are the vorticity-mode relaxation time, and the
entropy-mode relaxation time, respectively. The model is intended to match the
middle-frequency behaviour, and not to fit the asymptotic behaviour at high and low
frequencies. Therefore
φρ
∞
can be different from the effective density
ρ
when
ω
→∞
,
and
φK
∞
can be different from the bulk modulus
K
when
ω
→∞
, due to the fact that
the adjustment does not concern the high- and low-frequency asymptotic expressions.
5.5.4 Prediction of the effective density with the Pride
et al.
model
and the model by Johnson
et al.
In Figure 5.6, the effective density
ρ
is successively predicted with Equation (5.32) for
b
=
1. The other parameters used for the prediction are
q
0
=
1
.
23
×
10
−
10
m
2
,
=
0
.
6and
b
q
0
=
5
×
10
−
10
=
31
µ
m,
=
90
µ
m, and
α
∞
=
1
.
37. These param-
eters have been measured for a washed quarry sand (Tizianel
et al
. 1999). A noticeable
difference exists for both evaluations of Re
ρ
.
m
2
,
φ
=
0
.
37
,
5.5.5 Prediction of the bulk modulus with the simplified Lafarge
model and the Champoux-Allard model
Measurements of the static thermal permeability are not easy and with the
Champoux - Allard model, the thermal permeability is set equal to that of the porous
material having identical circular cross-sectional shaped pores with a radius
R
.
=
0
.
95, and
=
For a porosity
φ
=
610
µ
m, the static thermal permeability of this
porous medium
q
0
=
10
−
8
m
2
. The bulk modulus of the air saturating the medium
is represented in Figure 5.7 for
q
0
=
4
.
4
×
10
−
8
m
2
1
.
3
×
with the simplified Lafarge model
