Civil Engineering Reference
In-Depth Information
The inner scales
()
are not perfectly adequate for the
ν
,
u
correlations
ij
, which thus depend on the Reynolds number.
For example, Figure 2.23 shows the distributions of
π
33
for
different Reynolds numbers varying by a decade, from
186
π
17
, obtained by DNSs and published
in [HOY 08]. This behavior is not, in itself, surprising, as the
pressure is an integral quantity controlled both by the inner
and outer scales. We note a significant increase of
to
Re
τ
=
Re
τ
=
2, 003
π
33
in the
buffer sublayer when
Re
increases from 180 to 600, beyond
that limit the variations become more moderate. The
pressure/velocity correlati
ons
expressed in relation to the
local intensity values
pp
collapse relatively well at
different Reynolds numbers, as shown in Figure 2.23.
τ
It is more instructive to express the system [2.65] in the
Fourier domain [KIM 89]. The Fourier transform of system
[2.65] in the homogeneous directions
x
and
z
enables us to
write
⎡
2
⎤
−+
d
(
)
(
)
(
)
2
2
[2.75]
kk pkyk Fkyk
,,
=
,,
⎢
⎥
x
z
x
z
x
z
dy
2
⎣
⎦
(
)
=
∫∫
is clearly the
Fourier transform of the pressure field at a given
y
value,
and
(
)
(
)
−
kx kz
+
where
pk yk
,,
pxyze
,,
dxdz
x
z
x
z
(
)
is
the
forcing
term,
which
is
Fx
,
y
,
z
∂
u
∂
∂∂
u
2
dU
∂
∂
v
(
)
j
i
either
(
)
or
for
p
R
and
Fxyz
,,
=−
Fxyz
,,
=−
x
x
dy
x
j
i
p
S
, respectively. In both cases, the boundary conditions are
homogeneous at the wall with
[2.76]
(
)
pk y
,
=±
1,
k
=
0
x
z
17 Remember that all the quantities analyzed in this section have already
been rendered dimensionless by the inner scales, and the notation
+
is
()
omitted for the sake of simplicity.
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