Civil Engineering Reference
In-Depth Information
surfaces
A
in the representative elementary volume. The velocity
υ
i
(
r
)
is the velocity
inside the pores, the integral in the denominator is performed over the volume
V
of
the pore. The parameter
given by Equation. (5.24) only depends on the geometry of
the frame. With the factor 2,
is equal to the hydraulic radius for identical cylindrical
pores. It has been noted by Johnson
et al
. (1986) that
and the flow resistivity
σ
are
related by
8
ηα
∞
σφ
1
/
2
1
c
=
(5.25)
with
c
close to 1. It has been shown by Johnson
et al
. (1987) that the effective density
can be written at high frequencies at first-order approximation in 1
/
√
ω
α
∞
ρ
0
1
+
j)
δ
ρ
=
(
1
−
(5.26)
A previous expression of the dynamic viscous permeability obtained by Auriault
et al
. (1985) for anisotropic media leads to a similar result where
ρ
is the sum of a real
inertial term and a correction proportional to 1
/
√
ω
due to viscosity.
5.3.3 Thermal characteristic length
It has been shown by Champoux and Allard (1991) that the high-frequency behaviour
of the bulk modulus
K
can be characterized by a second length denoted as
and
given by
A
d
A
2
=
A
S
V
d
V
=
(5.27)
The integral in the numerator is performed over the pore surfaces
A
in the elementary
representative volume and the integral in the denominator is performed over the volume
V
of the pore, as for Equation (5.24), but there is no weighting by the squared velocity. As
shown in Appendix 5.B, the bulk modulus
K
is given at high frequencies to first - order
approximation in 1
/
√
ω
by
γP
o
γ
−
(γ
−
1
)
1
−
(
1
−
j)
δ
B
K
=
(5.28)
For identical cylindrical pores,
=
¯
r
. This is a direct consequence of the def-
inition of these quantities and this can be verified by comparing Equation (4.156) with
Equation (5.26), and Equation (4.108) with Equation (5.28).
=
5.3.4 Characteristic lengths for fibrous materials
As indicated in Chapter 2, fibres in layers of fibreglass generally lie in planes parallel to
the surface of the layers. At normal incidence, the macroscopic air velocity is perpendic-
