Civil Engineering Reference
In-Depth Information
1.16
Hélium
1.12
1.08
Air
1.04
0
1e-3
2e-3
3e-3
4e-3
f
−
1/2
(Hz
−
−
1/2
)
Measured
n
2
for a foam of porosity
φ
=
0
.
98 and viscous static permeability
q
0
=
3
.
08
×
10
−
9
m
2
(Leclaire
et al
., 1996). Reprinted with permission from Leclaire, Ph.,
Kelders, L., Lauriks, W., Melon, M., Brown, N. & Castagnede, B. Determination of the
viscous and thermal characteristic lengths of plastic foams by ultrasonic measurements
in helium and air. J. Appl. Phys. 80, 2009-2012. Copyright 1996, American Institute of
Physics.
Figure 5.5
mand
=
Figure 5.5 gives
m. The intercepts of the straight lines that
extrapolate the measurements toward increasing frequencies give a tortuosity
α
∞
=
=
202
µ
367
µ
1
.
05.
These measurements can easily be performed on foams of low flow resistivity. For
granular media, diffusion in the high-frequency regime can modify wave propagation.
More generally, it has been shown by Ayrault
et al
. (1999) that measurements can be
improved by increasing the static pressure. The coupling of air with the transducers is
improved. The viscous skin depth decreases when the static pressure increases, and the
high-frequency regime appears at lower frequencies.
If the frame does not conduct electricity and can be saturated by a conducting fluid
with no modification of the microscopic geometry, tortuosity can be evaluated from
conductivity measurements, as indicated in Section 4.9, with no problem concerning
diffusion.
5.3.6 Static tortuosity
When the viscous skin depth is much larger than the characteristic dimensions, Equation
(5.7) can be used to predict the effective density. Lafarge (1993) has shown that the
real constant on the right-hand side is equal to the product
ρ
0
α
0
where
α
0
is the static
tortuosity. The static tortuosity is given by Equation (5.21), similar to
α
∞
. The difference
is that the velocity field in Equation. (5.21) is now the static field at
ω
=
0modifiedby
viscosity. It was shown by Lafarge (2006) that
α
0
≥
α
∞
.
