Civil Engineering Reference
In-Depth Information
the correlations which are linked to
x
3
) are null. The
best way to demonstrate this fact is to take the approach
adopted in [POP 00]. The invariance in relation to
x
3
means
that the joint probability density of the velocity components
(
∂
u
3
∂
GG
respects
)
0
and is therefore written as:
p
uxt
;,
∂
p
d
∂
x
3
=
(
)
=
(
)
[2.5]
p
d
u
1
,
u
2
,
u
3
;
x
1
,
x
2
,
x
3
,
t
p
d
u
1
,
u
2
,
−
u
3
;
x
1
,
x
2
,
−
x
3
,
t
Let us take the example of the correlation
u
1
u
3
at
x
3
for
fixed values of
x
1
,
x
2
and
t
. By definition
∞
∫
u
3
p
d
∞
∫
()
=
(
)
du
1
du
3
u
1
u
3
x
3
u
1
u
1
,
u
3
;
x
3
−∞
−∞
and at
x
3
−
∞
∫
u
3
p
d
∞
∫
(
(
)
()
u
1
u
3
−
x
3
=
u
1
u
1
,
−
u
3
;
x
3
du
1
du
3
=−
u
1
u
3
x
3
−∞
−∞
which gives us
u
1
u
3
0
.
By the same arguments, it is possible to state that
u
2
u
3
and
any correlation containing the odd-numbered derivatives in
relation to
x
3
are equal to zero.
2
Consequently, all the terms
in the transport equation [2.2] for
i
0
at an arbitrary positi
on
x
3
=−
u
1
u
3
=
=
3
are
null. The only Reynolds
cor
rel
atio
ns
whi
ch still
ne
ed to be
analyzed therefore are
u
1
u
1
,
u
1
u
2
,
u
2
u
2
and
u
3
u
3
. These
arguments are also valid in a 2D turbulent boundary layer,
which is not homogeneous in direction
x
1
.
1
,
j
3
and
i
2
,
j
=
=
=
=
The transport equation for
u
1
u
1
is obtained by substituting
1
in equation [2.2]
i
=
=
j
2 Except, as stated earlier, the correlations which contain
∂∂
ux
because
3
3
the antisymmetries of
u
and
x
neutralize each other.
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