Civil Engineering Reference
In-Depth Information
⎛
⎝
⎞
⎛
⎞
Du
1
u
2
Dt
dU
1
∂
p
ρ
1
ρ
∂
u
2
x
1
+
∂
u
1
=−
u
2
u
2
dx
2
−
u
1
u
2
u
2
+
+
p
⎜
⎟
⎜
⎟
∂
x
2
∂
∂
x
2
⎝
⎠
⎠
P
12
Τ
12
π
12
2
u
1
u
2
∂
ν
∂
u
1
∂
u
2
[2.9]
+ ν
x
2
−
2
∂
x
2
∂
∂
x
l
∂
x
l
D
12
ε
12
We ca
n cl
early see that
P
12
0
. This suggests that, on
<
average,
u
1
u
2
0
if we consider the production term to be
dominant. Figure 2.1 illustrates the mechanism of inter-
component transfer for a flow which is homogeneous in
directions
x
1
an
d
x
3
, non homogeneous in
x
2
and subject to a
shear effect
dU dx
2
.
<
2.3. Models of return to isotropy
π
12
plays a crucial role in the mechanism of
transfe
r. It
s effect is to decrease the intensity of th
e Re
ynolds
stress
u
1
u
2
. Indeed, we can envisage that
The correlation
u
1
u
2
, if we
accept the postulate that the pressure/velocity gradient
correlations essentially render the turbulence isotropic.
π
∝−
12
One of the simplest ways of looking at models of return to
isotropy is to consider homogeneous anisotropic turbu
le
nce,
which develops in the absence of any mean gradient
x
j
[CHO 01].
5
The transport equation [2.2] thus takes the exact
form
∂
U
i
∂
⎛
⎞
∂
uu
∂
u
∂
u
1
∂
u
∂
u
[2.10]
i
j
j
j
=
p
+
i
−
2
ν
i
⎜
⎟
⎜
⎟
∂
t
ρ
∂
x
∂
x
∂
x
∂
x
⎝
⎠
i
j
l
l
5 A considerable number of studies have been published on this subject.
Readers can consult [CHO 01] for a variety of representative references.
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