Civil Engineering Reference
In-Depth Information
The wall is a source of vorticity where the fluxes are
determined by the local pressure gradients
via
⎛
⎞
∂ω
∂
p
⎛
⎞
∂ω
∂
p
⎛
⎞
⎛
⎞
x
=
;
z
= −
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
∂
y
∂
z
∂
y
∂
x
⎝ ⎠
⎝ ⎠
⎝ ⎠
⎝ ⎠
y
=
0
y
=
0
y
=
0
y
=
0
and
⎛
⎝
⎞
⎠
∂ω
y
∂
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
y
= 0
=−
∂ω
∂ω
z
∂
x
y
= 0
−
y
= 0
This last relation arises from
⎜
⎟
⎜
⎟
⎜
⎟
y
∂
x
z
G
, and the wall-
normal vorticity flux is linked to the local gradients of the
streamwise and spanwise components in the directions
x
and
z
. The vorticity diffuses from the wall toward the flow; it
is advected and, in turn, modifies the structures, which then
give rise to a new pressure distribution. This causes a new
distribution of the pressure flux at the wall, and the process
continues
ad vitam aeternam
[SHE 90]. There is,
a priori
, no
specific reason why the fluctuations in the pressure gradient
should coincide with
∇•
ω
=
0
i
.
18
Furthermore, it is important not to
confuse the fluxes with the local fluctuations in vorticity at
the wall. Consequently, although the equations given above
are accurate, there are no direct relations between
ω
ω
x
0
and
)
y
= 0
, or between
z
0
and
)
y
= 0
.
(
(
∂
p
∂
z
ω
− ∂
p
∂
x
Nevertheless, the results recorded by Kim [KIM 89] show
a close correspondence between the instantaneous contours
of the streamwise vorticity and the local spanwise pressure
gradient (Figure 2.24). However, the contours of
are totally dissimilar to , as shown in Figure 2.25. The
contours of the streamwise pressure gradient show local
structures of medium size, while the spanwise vorticity
18 The relations between the fluctuations of pressure, vorticity and the
quasi-streamwise vortices are complex and will be discussed in detail in
Chapter 3.
(
)
y
= 0
∂
p
∂
x
ω
z
0
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