Civil Engineering Reference
In-Depth Information
()
(
)
2
3/4
7/4
Ek
=
ρ
K
'
εν
Θ
k
η
[2.62]
ppx
p
px
o
at high wavenumbers. In isotropic homogeneous turbulence,
the
function
p
behaves
like
a
power
law
Θ
(
)
(
)
γ
−
7/3
Θ
k
η
∝
k
η
, with
γ
p
approaching
when the
p
px
o
x
o
Reynolds number is large [TSU 03].
The measurements performed in a turbulent boundary
layer by Tsuji
et al.
[TSU 07] with a large Reynolds number
based on the momentum thickness of
clearly
demonstrate the existence of a spectral zone corresponding
to overlap zone I in Figure 2.17, where
Re
θ
=
12, 000
at
−
1
−
1
y
+
k
=
k
>
30
x
1
(Figure 2.18). However, we do not see Kolmogorov
overlapping in
k
x
−7/3
, as the slope of -1.5 observed with high
values of the
k
x
y
is significantly lesser. Tsuji
et al.
[TSU 07]
attribute this behavior to the anisotropy caused by the mean
shearing. Another plausible explanation is the notable
interaction between the components with large and small
pressure scales. These interactions necessitate the inclusion
of a large-scale component in the inertial zone, which results
in a slope
k
x
−3/2
instead of
k
x
−7/3
[ALB 98].
[PAN 74] estimated that a zone with
k
x
−1
should emerge in
the spectral density distribution of the wall pressure under
the influence of the logarithmic sublayer. This zone only
appears clearly when the Reynolds number is very large. The
measurements recorded by Tsuji
et al.
[TSU 07] indicate a
slope
k
x
−0.7
at
Re = 12,000
. Figure 2.19 shows the pre-
multiplied spectral density
f
θ
as a function of
+
E
p
0
p
0
+
10
5
. We
can see a clear behavior in
k
x
−1
over more than a decade.
f
+
δ
+
f
+
Re
, obtained by [KLE 08]
14
at
Re
10
6
=
τ
=
±
2
×
τ
14 These very high Reynolds numbers were attained when measuring the
atmospheric turbulent boundary layer over the Utah desert.
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