Civil Engineering Reference
In-Depth Information
where
v
is the velocity and
λ
is the volume viscosity.
Mass balance in
f
dξ
d
t
+
ρ
0
∇
·
υ
=
0
(5.63)
where
ξ
is the acoustic density.
Adherence condition on
s
υ
/
s
=
0
(5.64)
Heat conduction equation
κτ
=
jω(ρ
0
c
p
τ
−
p)
(5.65)
Air state equation
P
0
ξ
,
τ
T
0
p
=
ρ
0
+
(5.66)
Thermal boundary condition
τ/
s
=
0
(5.67)
In order to express these equations in a dimensionless form, the reference length
L
is
chosen, and the dimensionless space variable will be
x
X
/L
. Adequate characteristic
values
υ
c
,p
c
,
...
of the quantities
υ,p
,
...
are used to derive the dimensionless quantities
υ
∗
,p
*,
...
obtained from
υ
=
ν
c
υ
∗
...
and similar relations for the constant parameters.
The above set of equations introduces several dimensionless numbers whose orders of
magnitude are related to the characteristic values by
=
|
∇
p
|
Lp
c
η
c
v
c
Q
L
=
υ
|
=
(5.68)
|
η
ρ
0
∂
∂t
η
ρ
0
c
ω
c
L
2
η
c
Rt
L
=
=
(5.69)
υ
d
t
d
ξ
ω
c
ξ
c
L
ρ
0
c
v
c
S
L
=
ρ
0
∇
υ
|
=
(5.70)
|
The flow is forced by the macroscopic pressure gradient, and
|
∇
p
|=
O(p
c
/L)
.The
flow occurs in the pores, and the viscous forces satisfy
|
η
υ
|=
O
η
c
υ
c
l
2
.
At radian frequency
ω
,
=
ρ
0
∂υ
∂t
O(ρ
0
c
ω
c
υ
c
).
The wavelength
λ
=
2
πL
, leading to
O
ρ
0
c
υ
c
L
.
|
ρ
0
(
∇
·
υ
)
|=
