Civil Engineering Reference
In-Depth Information
4.5
High- and low-frequency approximation
Tubes having a circular cross-section
For small and large values of
s
, asymptotic expressions of
J
1
/J
o
can be used in Equation
(4.19). With the Prandtl number close to 1, in the same ranges of frequencies, asymptotic
expressions can be used simultaneously in Equation (4.48) to evaluate the bulk modulus
K
. The quantity
s
is large if
η
ωρ
o
1
/
2
R
(4.57)
The range of frequencies where Equation (4.57) is valid is called the high-frequency
range. The length
δ
2
η
ωρ
o
1
/
2
δ
=
(4.58)
is called the viscous skin depth. This length is approximately equal to the thickness of the
layer of air close to the surface of the tube where the velocity distribution is considerably
perturbed by the viscous forces generated by the motionless frame. If Equation (4.57)
is valid, the effect of the viscous forces is negligible in a large part of the tube centred
around the axis of symmetry and a flat central core appears in the velocity distribution
over the cross-section of the tube. Similarly, a thermal skin depth
δ
canbedefinedby
the relation
1
/
2
1
/
2
2
η
ωB
2
ρ
0
κ
ωρ
0
C
p
2
δ
=
=
(4.59)
At high frequencies, choosing
=
(
−
1
+
j)/
√
2
(
−
j)
1
/
2
(4.60)
J
1
/J
o
is equal to
j
and Equation (4.18) approximates to
√
2
(
1
+
ρ
0
=
1
1
−
J
1
(s
√
−
ρ
2
s
√
−
j)
j)
J
0
(s
√
−
=
1
+
(4.61)
j
j)
js
The Newton equation (4.19) becomes
j)υ
3
2
η
R
2
ρ
o
ω
1
/
2
∂p
∂x
=
−
jωρ
o
υ
3
+
(
1
+
(4.62)
and the bulk modulus (Equation 4.48) can be written
γP
o
[1
+
√
2
(
K
=
−
1
+
j)(γ
−
1
)/(Bs)
]
(4.63)
