Civil Engineering Reference
In-Depth Information
vorticity is indicated by
()
= ω
z
. The mean of the
two terms appearing after the viscous term is obviously
cancelled out because
Ω
z
y
;
t
0
by definition. Hence
∂ ∂
z
=
∂
Ω
∂ω
∂
v
∂
⎛
⎞
∂
w
2
[4.12]
z
+
v
z
= −
2
ω
−
u
+
ν
∇ Ω
⎜
⎝ ⎠
z
z
∂
t
∂
y
∂
y
∂
z
∂
y
Equation [4.12] is exact for a local flow with
0
. It is
∂ ∂
x
=
reduced to
∂
Ω
[4.13]
z
0
2
=∇Ω
ν
z
0
∂
t
at the wall, where the spanwise vorticity simply diffuses.
Orlandi and Jimenez [ORL 94] overlook viscosity and
consider an inviscid flow with
u
0
at the wall.
We then obtain a fundamental relation, which is
0
, but
w
=
v
=
≠
∂
Ω
∂
v
z
0
=−
2
ω
[4.14]
z
∂
t
∂
y
0
because the shear
y
is then taken to be null. This
equation can also be written as
∂
w
∂
∂
Τ
∂
v
0
=−
2
τ
'
[4.15]
p
∂
t
∂
y
0
()
where
is the spanwise mean of the wall friction
'
Τ
t
=
τ
0
p
() (
)
. Relation [4.15] clearly shows that the
'
τ
z t
;
=
μ
∂
u
∂
y
p
0
drag
Τ
0
increases over time in zones where
(
)
, in
τ∂∂
'
vy
<
0
p
0
other words, when the zones of friction
0
are correlated
τ
'
<
(
)
with
, and
vice versa
. Consequently, the ejections
caused by the QSVs increase the zones where
∂∂
vy
>
0
0
τ
'
<
0
p
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