Civil Engineering Reference
In-Depth Information
Indeed, the transport equation for streamwise vorticity in
this precise configuration
31
, can be written as
(
)
ˆ
(
)
(
)
∂
Uu
+
'
⎡
∂
ww
'
+
⎤
∂
ww
'
+
D
ω
∂∂
v
u
'
'
CT
CT
x
=
−
−
⎢
⎥
Dt
∂
y
∂
z
∂
x
∂
x
∂
y
⎣
⎦
[5.89]
(
)
ˆ
∂
Uu
+
'
2
∂
v
x
'
∂
+
+
ν
ω
x
∂
∂
z
∂
x
∂
x
l
l
An analysis of the order of magnitude tells us that the
predominant production terms of
are the tilting of the
ω
x
ˆ
Uy
(
)
wall-normal shear layers
, and
the twisting of the spanwise vorticity component, which is
reduced to
by
∂
ww
'
+
∂
x
∂∂
TG
(
)
ˆ
. The difference in relation
to the normal modes lies in the regeneration of streamwise
vorticity by the tilting
∂∂∂
vx Uu z
'
×
+
'
∂
ˆ
induced by the
transient growth process. Therefore, it is not at all
surprising that the value of
−
∂
TG
wx
∂
×
∂
Uy
∂
is increased by the presence
of
T
w
, which is explicitly dependent upon
x
. However, the
real situation is more complex, and it is difficult, in a more
general framework, to imagine the regeneration of
ω
x
,
whose causality is called into question in zones near to
vortex structures that are greatly elongated in the
streamwise direction.
∂∂
x
Schoppa and Hussein [SCH 02] stated a different
interpretation of the regeneration of streamwise vorticity
based on a transposition of the transport equation for
into
ω
x
the coordinate system
(,,)
x ns
linked to the vorticity lines of
the basic flow (Figure 5.35). Thus, the vorticity is simply
written as
because, by definition, the vorticity
(
)
ˆ
ˆ
ωω
=
ns s
. Similarly, the velocity
lines are tangents to the vector
ˆ
ω
31 See Chapter 1.
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