Civil Engineering Reference
In-Depth Information
where
F(B
2
ω
)isgivenby
1
+
js
2
B
2
G
c
(Bs)
−
1
1
+
jωB
2
ρ
o
G
c
(Bs)
−
1
8
c
2
σφ
F(B
2
ω)
=
=
(4.121)
4.7.3 Effective density and bulk modulus of air in cylindrical pores
of arbitrary cross-sectional shape
Using for the slit the effective density given by
φσG
c
(s)
jω
ρ
=
ρ
0
+
s
=
c
8
ωρ
0
σφ
1
/
2
(4.122)
=
√
2
/
3 gives the right asymptotic expression for Re
(ρ)
and Im
(ρ)
when
ω
tends
to infinity,
ρ
with
c
ρ
0
[1
+
√
2
δ/(
√
j
¯
r)
], and the right imaginary part when
ω
tends to zero,
=
ρ
φσ/(jω)
. In the whole frequency range, Equation (4.117) gives the effective density
to a good approximation. The slit and the circular cross-section are very different, and
it can be guessed that the use of Equation (4.117) for other pore shapes is possible. The
parameter
c
must be chosen by adjusting the high-frequency limit. The right limit is
obtained by using in Equation (4.115)
c
satisfying
=
8
η
σφ
1
c
¯
r
=
(4.123)
When
ω
tends to zero,
ρ
tends to
ρ
0
1
c
2
3
j
σφ
ω
ρ
=
+
−
(4.124)
The limit of the imaginary part is right. As a test for the validity of the general
formulation, the limit of the real part,
ρ
0
(
1
c
2
/
3
)
, is compared in Table 4.1 with
evaluations performed by Craggs and Hildebrandt (1984, 1986) using the finite element
method.
It appears that Equations (4.123) - (4.124) can be used to predict the limit Re
(ρ/ρ
0
)
when
ω
+
→
0 with a good precision for the cross-sectional shapes considered.
Ta b l e 4 . 1
Flow resistivity as a function of the hydraulic radius,
c
obtained from
Equation (4.123), (1
+
c
2
/3), and Re
(ρ/ρ
0
)
obtained with the finite element method.
c
2
/3
Cross-sectional shape
σφ
c
1
+
Re(
ρ
/
ρ
0
)
8
η/r
2
Circle
1
1.33
1.33
7
η/r
2
Square
1.07
1.38
1.38
6
.
5
η/r
2
Equilateral triangle
1.11
1.41
1.44
12
η/r
2
Rectangular slit
0.81
1.22
1.2
