Civil Engineering Reference
In-Depth Information
In the zone
A-B
in Figure 2.17, equation [2.54] is coupled
with equation [2.58], and the overlap gives us
(
)
()
E
pp
k
x
Λ
0
E
pp
k
x
(
)
=
0
=
f
1
k
x
Λ
0
4
4
ρ
u
ρ
u
Λ
τ
τ
()
()
E
pp
k
x
y
E
pp
k
x
()
=
=
f
2
k
x
y
4
4
y
ρ
u
ρ
u
τ
τ
simultaneously.
C
1
,
where
C
1
can only be a constant. From this, we deduce the
spectral behavior in the first overlap zone,
Consequently,
(
)
=
(
Λ
0
f
1
k
x
Λ
yf
2
k
x
y
=
0
(
)
E
pp
k
x
Λ
0
C
1
k
x
Λ
0
=
2
u
4
ρ
τ
()
E
pp
k
x
y
C
1
k
x
y
=
2
u
4
ρ
[2.59]
τ
Note the emergence of a power law
k
x
−
1
in the short-
wavelength spectral zone.
The approach taken in the inertial overlap zone II
(Figure 2.17) is similar. The equality between equations
[2.55] and [2.58] gives us
()
= ρ
2
4
(
)
= ρ
2
u
4
yf
2
()
E
pp
k
x
ν
Ko
η
Ko
f
3
k
x
η
Ko
k
x
y
[2.60]
τ
However, in light of equation [2.57]
7/4
u
5/2
ν
4
τ
ν
η
=
Ko
Ko
3/4
()
κ
y
and the last term on the right-hand side of equation [2.60] is
independent of the viscosity. In addition,
3/4
according
to equation [2.57]. The only possibility for the term
ρ
η
∝ ν
Ko
2
4
(
)
to be independent of the viscosity is if
ν
Ko
η
Ko
f
3
k
x
η
Ko
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