Civil Engineering Reference
In-Depth Information
Figure 5.38.
Maximum growth rate corresponding to the sinusoidal
instability of the basic flow
(
)
s
Uyz
from equation
[5.94]
, according to
[WAL 97]. The Reynolds number based on the half-height of the channel
and the speed of movement of the walls of the Couette flow studied by
Waleffe [WAL 97] is 405. The solid lines correspond to the non-slip
boundary conditions at the wall
,
, and the dotted lines
uvw
== =
0
correspond to the free-slip boundary conditions
∂∂
uyv wy
==
∂ ∂
=
0
Consider the overall mean of the streamwise vorticity in
the direction of the flow
x
. This mean is written as
L
1
x
∫
q
=
lim
qdx
[5.97]
L
→∞
L
x
x
0
By definition,
. Let us take equation [5.43]
∂
qx
∂ =
0
completely:
∂ω
∂ω
∂ω
∂ω
x
+
u
x
+
v
x
+
w
x
=
∂
t
∂
x
∂
y
∂
∂
z
[5.98]
(
)
Uu
+
∂
u
∂
w
∂
v
∂
u
∂
2
ω
−
+
+
ν
ω
x
x
∂
x
∂
x
∂
y
∂
x
∂
z
∂
x
∂
x
l
l
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