Civil Engineering Reference
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at
y
0
. Onl
y
the d
iss
ipation components in the transport
equation for
uu
and
ww
are significant near to the wall. Note
also that the turbulent intensities of the field of vorticity at
the wall are directly linked to the turbulent intensities of
vorticity by
=
+
+
2
ε
11,0
=−
2
σ
ω
z
0
[2.50]
+
+
2
ε
33,0
=−
2
σ
ω
x
0
where it must be recalled that
x
+
are, respectively, the turbulent intensities of the vorticity in
the spanwise and streamwise directions. We showed in
Chapter 1 that these intensities increase in line with the
Reynolds number [TAR 11b], but did not specify whether
that increase was continuous or eventually reaches a
universal peak value, independent of
Re
τ
and
+
2
z
+
z
+
+
2
x
+
σ
ω
z
0
= ω
ω
σ
ω
= ω
ω
x
0
. The immediate
consequence of these results is the equivalent behavior of
−ε
1
+
15
[HOY 08].
Figure 2.11, which is adapted from [HOY 08], shows the
distributions of
4
and
−ε
3
+
in the sublayer
y
+
≤
=−ε
11
ν
/
u
τ
−ε
33,
+
as a function of the Reynolds
number. We can indeed see that the dissipation at the wall of
the transport equations for
u
and
w
increase with
Re
τ
+
and
−ε
11,0
. The
()
()
increase is logarithmic with
at
the beginning, but the slope changes and the quantities
appear to reach a plateau. Hoyas and Jiménez [HOY 08]
attribute this behavior to the effect of large-scale passive
structures coming from the outer zone.
12
The arguments put
forward by these authors can be summarized as follows. The
large-scale structures reach the wall. However, they are
irrotational, and for this reason we cannot see their
trajectory at
10
+
and
+
−ε
11,0
∝
ln
Re
−ε
33,0
∝
ln
Re
y
+
14
on 2D spectral densities of enstrophy
<
<
GG
, in the upper right-hand part of the line
L
+
=
L
+
,
ωω ωω
•=
i
i
12 From the middle of the logarithmic layer, according to Marusic's group
(University of Melbourne). These aspects will be discussed in greater
detail in Chapter 6.
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