Civil Engineering Reference
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where the quantity
q
on the right-hand side encapsulates the
nonlinear terms
2
y
∂∂
⎡
∂
⎤
() ()
[5.27]
q
≡
uv
+
vw
+
....
⎢
⎥
2
∂
∂
x
∂
z
⎣
⎦
Relation [5.26] is a non-homogeneous Orr-Sommerfeld
equation, which is extremely complex to solve because of the
nonlinear forcing
q
. The idea is to exploit the highly
intermittent nature of the fluctuations in wall-normal
velocity and rewrite equation [5.26] as
2
2
Dv
∇
∂∂
v
U
Qxyz
(
) (
)
[5.28]
−
=
,,
δ
tt
−
n
n
2
Dt
∂
x
∂
y
where the nonlinearity is modeled in the form of a temporal
Dirac comb, distributed in space. Note also that we consider
a non-viscous disturbance to begin with. The integration of
the above relation at
(which is merely the time of the
passage of the structure
8
) gives us
tt
=
n
2
y
∂∂
Ul
∂
l
(
)
[5.29]
2
∇=
vQ
ξ
,,
yz
+
,
ξ
=−
xU
nn
2
n
∂
∂
x
∂
x
where we can see the emergence of the Prandtl mixing
length with
t
(
)
[5.30]
∫
l
=
vxyz t
(,,,) ,
t
xxUt
= −
−
t
1
1
1
1
1
t
n
These equations describe an initial value problem for
, with
tt
>
n
Dv
∇
2
y
∂∂
2
U
v
[5.31]
−
=
0
2
Dt
∂
∂
x
8 These times can also be interpreted as the temporal occurrences of the
ejections.
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