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mechanism in a turbulent Couette flow.
11
They began by
showing t
ha
t the modes independent of
x
by way of -
(
make the greatest contribution to the
formation of the streaks. The term
v
in this relation is the
Fourier transform of the velocity component defined by
)
vykUy
0,
,
∂∂
z
∞∞
(
)
(
)
(
)
∫∫
⎡
⎤
uk ykt
,, ;
=
uxyzt
,,; exp
−
ikx kz dxdz
+
⎣
⎦
i
x
z
i
x
z
−∞
−∞
The flow induced by and around streamwise structures is
assumed to be independent of
x
.
12
Next, the authors conduct
a numerical experiment, with a view to artificially
eliminating the streaks by cancelling out all the modes
(0,
uy
. As the velocity
field is assumed to be independent of
x
, with
except for the mean flow
uyk
,
)
(0,
,0)
z
(
)
uyzt
,
,;
(
)
wyzt
, this elimination does not affect the
QSVs.
13
Indeed, the spanwise flow is then decoupled from
u
,
and is described by the system
vyzt
and
(,;)
,;
⎛
2
2
⎞
∂
v
∂
v
∂
v
1
∂
p
∂
v
∂
v
+
v
+
w
= −
+
ν
+
⎜
⎟
2
2
∂
t
∂
y
∂
z
ρ ∂
y
∂
y
∂
z
⎝
⎠
∂
ww w
∂
∂
1
∂
p
⎛
∂
2
ww
∂
2
⎞
[5.46]
+
v
+
w
= −
+
ν
+
⎜
⎟
∂
t
∂
y
∂
z
ρ ∂
z
∂
y
2
∂
z
2
⎝
⎠
∂∂
∂∂
v
w
+=
0
y
z
11 The authors studied a turbulent Couette flow in the minimum channel
configuration of [JIM 91]. Although the Couette flow is very different,
structurally - particularly in the outer sublayer - we can state that the
inner layer is dynamically similar to that of a classic fully developed
turbulent channel flow.
12 Note that a dependency on
x
is crucial for the regeneration of the
structures. Here, we are dealing with the intermittent stage which follows
the birth of the structures.
13 The situation is somewhat similar to the analysis of the response of the
local flow to an Oseen vortex discussed in section
5.3.1
.
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