Civil Engineering Reference
In-Depth Information
Figure 1.10, adap
te
d from [MAR 13], shows three
experimental profiles
(
)
+
+
, where
, in the
κ =
0.39
U
−
1/
κ
ln
y
range
98,190
. We can see that the profiles do
indeed reach plateau regions in the zone
3
Re
≤≤
18, 010
.
Re
≤≤
y
+
0.15
Re
τ
τ
The velocity in the outer su
bl
ayer depends on the friction
velocity
u
τ
in the potential zone
of a bou
nd
ary layer (or the velocity at the center of the
channel
U
), the distance
y
from the wall and the external
length scale
, the velocity
U
∞
of the boundary layer or the
half-
he
ight
h
of channel). A simple similarity analysis shows
that
Λ
(the thickness
δ
0
()
Uy
is of the form:
()
UUy
−
⎛⎞
y
UUyg
()
∞
+
+
+
[1.28]
=−
=
⎜⎟
Λ
∞
u
τ
⎝⎠
0
in that zone. More generally, and taking account of
uU
τ
,
∞
we can write
()
UUy
−
⎛
⎞
y
UUyg
u
()
∞
=−
+
+
+
=
,
U
τ
[1.29]
⎜
⎟
∞
u
Λ
⎝
⎠
τ
0
∞
The introduction of the Rotta thickness [ROT 50]
()
∞
UUy
dy
u
τ
∞
−
∫
[1.30]
δ
=
R
0
enables us to express the velocity defect distribution as
()
UUy
−
⎛⎞
y
UUyg
()
∞
+
+
+
[1.31]
=−
=
⎜
⎝⎠
∞
u
τ
δ
R
with the function
g
from equation [1.29] repeated here for
ease of notation.
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