Civil Engineering Reference
In-Depth Information
transport contains only the triple correlations, and the
pressure terms are separated into two. We can define
∂
Τ
i
*
=−
(
u
i
u
u
k
)
j
∂
x
k
⎛
⎝
⎞
∂
u
1
ρ
x
i
+
∂
u
i
j
= π
i
s
[2.4]
π
ij
=
p
⎜
⎟
⎟
∂
∂
x
j
⎠
⎛
⎞
1
ρ
∂
∂
π
i
d
=
⎜
pu
j
+
pu
i
⎟
⎜
⎟
∂
x
∂
x
⎝
⎠
i
j
where
Τ
i
*
is the turbulent diffusive flux,
π
i
s
is the
pressure/strain correlation term and
i
d
is the turbulent
π
pressure diffusion.
Consider an internal wall flow
1
, heterogeneous in the
wall-normal direction
x
2
and homogeneous in
the streamwise and spanwise directions
x
1
and
x
3
, as is the
case in a 2D channel. We are going to determine
th
e
tran
spo
rt e
quations applying to the R
eyn
olds stresses
u
1
u
1
,
u
2
u
2
,
u
1
u
2
which plays a
fundame
nt
ally important role in
w
all turbulence. The
velocity
U
1
and the mean shear
dU
1
dx
2
>
u
3
u
3
, and the correlation
−
0
depend only
on
x
2
. The mean flow is two-dimensional (2D), with
0
. The derivatives of the quantities averaged in
relation to
x
1
and
x
3
are cancelled out because of the
homogeneity, while
U
2
=
U
3
=
0
. Additionally, the flow is
symmetrical in relation to the planes perpendicular to the
direction
x
3
. Consequently, all the correlations containing
u
3
and the odd-numbered derivatives in relation to
x
3
(except
∂
()
∂
x
2
≠
1 By “internal flow”, we mean turbulent flows in channels (or pipes). In
line with this terminology, turbulent boundary layers are outer flows. The
internal (or outer) flows must not be confused with the inner (or outer)
sublayers.
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