Civil Engineering Reference
In-Depth Information
⎡
∂
d
2
d U
2
+
(
+
iU
α
++
)(
−
α
+
2
−
β
+
2
)
−
i
α
⎢
+
+
2
+
2
∂
t
dy
dy
⎣
[5.76]
2
⎤
1
d
−
(
−
αβ
+
2
−
+
2
)
2
v
ˆ
+
=
0
⎥
⎦
+
2
Re
dy
τ
which is subject to the boundary conditions
ˆ
at
the wall. Expressions [5.71] and [5.76] are sufficient to
represent the system [5.74]. They can be rewritten in matrix
form as follows:
ˆ
v vdy
=
=
0
ˆ
ˆ
⎡⎤ ⎡⎤
⎢⎥ ⎢⎥
=
ν
ν
d
[]
[5.77]
A
⎢⎥ ⎢⎥
⎢⎥ ⎢⎥
⎣⎦ ⎣⎦
dt
ˆ
ˆ
ω
ω
y
y
where
⎡
⎤
L
0
os
⎢
⎥
[]
A
=
⎢
[5.78]
⎥
⎢
⎥
L
L
⎣
⎦
c
sq
()
and
q
indicates the Fourier transform of the quantity in
question. The operators figuring in the matrix
[ ]
A
are,
respectively, the Orr-Sommerfeld operator
, the Squire
L
os
operator
and the coupling operator
L
, respectively,
L
sq
defined by
2
+
dU
−
1
+
+
+
2
L
=Δ −
(
i
α
U
Δ+
i
α
+
1 /
Re
)
Δ
)
os
τ
+
2
dy
Li
=−
α
++
U
+
(1 /
e
)
Δ
[5.79]
sq
τ
+
dU
+
Li
=−
β
c
dy
+
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