Civil Engineering Reference
In-Depth Information
u
'
'
⎡
u
⎤
⎛⎞
⎛⎞
⎜⎟
⎢
⎜⎟
⎥
v
v
⎢
⎥
⎜⎟
⎜⎟
(
)
(
)
(
)
i
αβ γ
x z
+
t
[5.57]
x yzt
,,;
=
⎢
R
yz e
,
e
⎜⎟
⎜⎟
⎥
w
'
'
w
⎜⎟
⎢
⎜⎟
⎥
p
p
⎝⎠
⎢
⎝⎠
⎥
⎣
⎦
(
′
indicates the disturbances,
R
is the real part, the
wavelengths
where
is complex
and
q
is a periodic function in
z
whose periodicity is
α
and
are real, the eigenvalue
β
γ
.
The system [5.57], coupled with the basic flow [5.55], is
solved by DNS. The basic flow representative of the streaks
is fixed, and therefore does not evolve to begin with.
Linearization of the Navier-Stokes and continuity equations,
coupled with equation [5.57], gives us
β
s
ˆ
(
)
∂
UU
∂
1
ˆ
2
γ
+
iUu v
+
+
w
α
= −
i p
+
∇
u
α
∂
y
∂
z
Re
∂
p
1
(
)
ˆ
2
[5.58]
γα
+
iUv
= −
+
∇
v
∂
∂
y
e
p
1
(
)
ˆ
2
γ
+
iUw
= −
α
−
i p
+
∇
w
β
∂
z
Re
where
∂ ∂ ∂
αβ β
∂∂ ∂
f
2
f
2
f
2
2
2
[5.59]
∇≡− − +
f
f
f
2
i
+
+
2
2
z
y
z
are fixed for a given mode.
The results discussed in the sections below relate strictly to
sinuous modes, with
The wavenumbers
α
and
β
(
)
(
)
(
)
(
)
(
)
uyz
,
−=− −
uyz
,
;
vyz
,
−=− −
vyz
,
;
wyz
wy z
,
−
[5.60]
(
)
=
,
−
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