Civil Engineering Reference
In-Depth Information
+
+
∂ω
'
∂ω
'
+
+
dU
∂
v
'
1
y
y
[5.69]
+
2
+
+
U
= −
+
∇
ω
'
y
+
+
+
+
∂
t
∂
x
dy
∂
z
Re
τ
We can see that the term representing production by
+
+
dU
∂
∂
v
'
lifting of the mean spanwise vorticity
plays the
−
+
+
dy
z
role of nonlinear forcing. Consider the wave-type solutions
proposed by [LAN 80]
ˆ
v
(
y
,
t
)
e
i
(α
x
+ β
z
)
v
'(
x
,
y
,
z
,
t
)
=
y
(
y
,
t
)
e
i
(α
x
+β
z
)
[5.70]
ˆ
ω
'
y
(
x
,
y
,
z
,
t
)
=
ω
It is easy to show that equation [5.69] then takes the form
⎡
⎤
⎡
2
⎤
∂
1
d
(
)
[5.71]
(
+
iU
α
++
)
−
−
α
+
2
+
β
+
2
ω
ˆ
+
=−
i U
β
+ ++
'
ν
ˆ
⎢
⎥
⎢
⎥
y
+
+
2
∂
t
Re
dy
⎢
⎥
⎣
⎦
⎣
⎦
τ
with
ˆ
at the wall. The solution to this equation in the
absence of viscous diffusion within the limit of
Re
τ
→∞
ω =
0
y
is
+
t
+
dU
+++
+++
+++
ˆ
+
ˆ
+
−
i
α
Ut
+
−
i
α
Ut
∫
ˆ
(,
+
+
+
−
i
α
Ut
'
+
[5.72]
ωω
=
e
−
i
β
e
v
y
t
' )
e
t
'
y
y
0
+
dy
0
where
. The first term
in expression [5.72] represents the advection of the initial
wall-normal vorticity by the mean flow. The second term
corresponds to the integrated effect of the wall-normal
velocity [LAN 80]. If we suppose that the normal velocity
remains constant over time, we obtain
ˆ
y
+
is the initial vorticity field at
ω
t
=
0
0
+
dU
+
ˆ
+
+
ˆ
+ +
[5.73]
ωω β
=−
i
v t
y
y
0
0
dy
+
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