Civil Engineering Reference
In-Depth Information
reveals that t
he
dominant terms are those that contain the
mean shear
Uy
∂∂
. Put differently
w
U
PP
x
∂∂
∂∂
≈
−
ω
I
ω
y
x
x
In other words, streamwise vorticity production is
essentially due to the tilting of the shear layers
wx
. The
spanwise structure, therefore, plays a very significant role.
The process of regeneration of the layers
wx
∂∂
is complex
and will be analyzed in detail in Chapter 5. The mean
vorticity at the wall runs in the spanwise direction and is
nothing but
∂∂
⎛
⎞
Ω=−
2
∂
U
y
τ
τ
u
x
0
[1.65]
=− =− =−
τ
⎜
⎝ ⎠
z
0
∂
μμν
0
where the subscript 0, as usual, refers to the wall and
is
the mean wall shear stress (in the direction of the mean
flow). Two fluctuating vorticity components in the
streamwise and spanwise directions, respectively, remain at
the wall:
τ
0
(
)
(
)
′
′
τ
xzt
,,
τ
xzt
,,
⎛
⎞
∂
w
(
)
z
z
ω
xzt
,,
=
=
=
⎜
⎝ ⎠
x
0
∂
y
μ
μ
0
[1.66]
(
)
(
)
′
′
τ
xzt
,,
τ
xzt
,,
′
⎛ ⎞
= −
∂
u
τ
(
)
x
x
ω
xzt
,,
=−
=−
=−
0
⎜
⎝ ⎠
z
0
∂
y
μ
μ
μ
0
whereas
ω
=
0
. The turbulent intensity
τ′′
of the wall
00
y
0
shear stress is directly linked to
. We have precisely:
ωω
z
00
z
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