Civil Engineering Reference
In-Depth Information
5.9
Homogenization
Separation of scales
The semi-phenomenological models in the present chapter can be valid only under the
long-wavelength condition. Let
L
be the macroscopic size defined by
L
=
O
|
λ
|
2
π
,
(5.60)
where
λ
is the complex wavelength in the porous medium. Let
l
be the microscopic
size which characterizes the representative elementary volume (see Figure 5.9). The
long-wavelength condition corresponds to
L
l
. The use of homogenization methods
is based on the same condition. Among these methods, the homogenization method
for periodic structures (HPS) has been used to describe sound propagation in ordinary
porous media (Sanchez Palencia 1974, Sanchez Palencia 1980, Keller 1977; Bensoussan
et al
.1978, Auriault 1991, Auriault 2005) and in double porosity media (Auriault and
Boutin 1994; Boutin
et al
, 1998; Olny and Boutin, 2003) where two networks of pores
of very different characteristic sizes are interconnected. The period is defined by the
representative elementary volume and the period characteristic size is
l
.
Porous media are generally not periodic, but for random microscopic geometries the
method gives supplementary information about the parameters presented in the context
of the semi-phenomenological models. In what follows, some results obtained for simple
porosity media are presented. The fundamental parameter is
ε
given by
ε
=
l/L
(5.61)
This parameter characterizes the separation of scales and the method can be used
under the condition
ε
1.
Two dimensionless space variables are used. Let
X
be the physical space vari-
able. The dimensionless macroscopic space variable is
x
=
X
/L
, and the dimensionless
microscopic space variable is
y
=
X
/l
. The dependence on
x
corresponds to the slow
macroscopic variations and the dependence on
y
corresponds to the fast variations at the
microscopic scale.
Governing equations describing the fluid displacements in a motionless frame
The Navier - Stokes equation in
f
η
υ
+
(λ
+
η)
∇
(
∇
·
υ
)
−∇
p
=
ρ
0
∂
v
∂t
(5.62)
Macroscopic
scale
Microscopic
scale
Ω
f
Γ
s
Ω
s
L
l
Figure 5.9
A porous medium on the macroscopic scale and on the microscopic scale.
