Civil Engineering Reference
In-Depth Information
A rough estimate of the transition angular frequency
ω
ν
between the viscous and the
inertial regimes is obtained when
q
0
=
q(
∞
)
and is given by
O
η
l
2
ρ
0
α
∞
ω
ν
=
(5.103)
The wavelength and the macroscopic size can be evaluated from the wave number
k
in both regimes. The complex wave number
k
ω
[
ηφ/(jωq(ω)K(ω))
]
1
/
2
can be evaluated with the adiabatic bulk modulus, the order of magnitude is not modified
by the factor
√
γ
ω(ρ/K)
1
/
2
=
=
=
1
.
18. The low frequency estimation of the wave length is given by
=
O
l
λ
2
π
λ
0
√
2
π
(5.104)
δ
ν
where
λ
0
is the wave number in the free air. Tortuosity is generally close to 1 and the
high-frequency estimation is given by
γP
0
ρ
0
=
O
1
=
O
λ
0
2
π
λ
2
π
(5.105)
ω
Realistic double porosity medium
Three conditions must be satisfied;
(i) The wavelength is much larger than the mesoscopic size
l
p
in the whole audible
frequency range
→
l
p
≤
10
−
2
m.
(ii) The microporous medium must be sufficiently pervious to acoustical waves
→
l
m
≥
10
−
5
m.
(iii) The separation of both smaller scales must be sufficient
→
l
p
/l
m
>
10.
Two different cases are selected, the first with a low contrast between static per-
meabilities
l
p
10
−
3
10
−
4
=
m, and
l
m
=
m, the second one with a high contrast,
l
p
=
10
−
2
10
−
5
m. The moduli of the wavelengths
λ
p
and
λ
m
predicted with
Equations (5.104) - (5.105) are shown in Figure 5.11(a) for the low-contrast medium and
in Figure 5.11(b) for the high-contrast medium.
In Figure 5.11(a) there is a domain for
ω>ω
νm
where the wavelengths are similar in
the porous medium and the microporous medium, and a strong coupling can be supposed
between pores and micropores. In Figure 5.11(b) at
ω
m, and
l
m
=
ω
d
the modulus of the complex
wavelength in the micropores is equal to the characteristic size of the pores. There
is a domain for
ω>ω
d
where the wavelength in the micropores is smaller than the
characteristic size of the pores. The regime in the micropores is diffusive, and fast spatial
variation of pressure can occur around the pores in the microporous medium at the scale
of the REVp.
=
5.10.3 Asymptotic development method for double porosity media
Wave propagation in double porosity media has been described with the homogenization
method for periodic structures (HPS) (Olny and Boutin 2003, Boutin
et al
. 1998, Auriault
and Boutin 1994). The porous medium presents a double periodicity at the mesoscopic
