Civil Engineering Reference
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distributions of the Reynolds stress
uv
+
−
and the total stress
τ
+
obtained by direct numerical simulations (
DNS
s) in a
fully developed turbulent channel flow at a low Reynolds
number. The results demonstrate perfect linearity of
tot
τ
+
,
and this type of analysis enables us to check the quality of
the results (such as the statistical convergence, for example),
both experimental and numerical.
DNS
s resolve the
integrality of the scales defined by discretization of the
domain of calculation, and of course, do not require closure.
We can go one step further and also determine the friction
coefficient. Integration of equation [1.17], from the lower
wall of the channel to
y
+
in the flow, gives us the velocity
distribution.
tot
+
y
y
+
2
()
()
+
∫
++
+
+
+
[1.18]
Uy y
=− +
v d
η
η
2
h
+
0
By integrating this relation from the wall to the center
(
yh
+
+
=
), we arrive at:
+
+
⎡
+
⎤
y
h
+
2
h
h
()
()
+
∫
++ +
∫ ∫
+
+ +
U y dy
=+
⎢
uv
ηη
d
⎥
dy
3
⎢
⎥
⎣
⎦
0
0
0
The latter integral can be calculated by integration by
parts:
⎡
+
⎤
+
+
h
y
h
()
(
) ()
+
+
∫∫
∫
uv
ηη
+
d
+
dy
+
=
h
+
−
y
+
uv
y
+
dy
+
⎢
⎥
⎢
⎥
⎣
⎦
00
0
Using
the
definition
of
the
bulk
velocity
+
h
1
()
+
∫
++ +
, we find:
U
=
U y dy
m
h
+
0
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