Civil Engineering Reference
In-Depth Information
2
D uu
dU
∂
2
∂
u
∂
uu
11
=−
2
uu
1
−
u u u
+
p
1
+
ν
11
12
1 1 2
D t
dx
∂
x
ρ
∂
x
∂
x
∂
x
2
2
1
2
2
P
Τ
π
D
11
11
11
11
[2.6]
∂∂
u
u
xx
−
2
ν
1
1
∂∂
ε
l
l
11
The terms which come into play in the transport of
u
1
u
1
are
the production
P
11
, the turbulent transport
T
11
, the
pressure/velocity gradient interactions
π
11
, the molecular
diffusion
D
1
1
and the dissipation
ε
11
. The production term
(
)
is positive because the mean turbulent
stress
u
1
u
2
is negative. This does not rule out the possibility of
positive values of
u
1
u
2
locally and instantaneously but, on
average, the negative
P
11
=−
2
u
1
u
2
dU
1
dx
2
uu
contributions are dominant, leading
12
to a balance of
u
1
u
2
0
. There are no
pure
ly mathematical
arguments capable of demonstrating that
u
1
u
2
<
0
. The term
P
11
is the only term containing the mean shear
dU
1
dx
2
, which is
supposed to be positive. Without mean shear
3
, the turbulence
cannot be sustained. Hence, the production must be a source
term in the process of regeneration - all the
mo
re so
whe
n it
only
appears in the transport equation for
u
1
u
1
(not
u
2
u
2
or
u
3
u
3
), as we will see later on. The contribution of the
dissipation
<
ε
11
is clearly negative.
The ter
m f
o
r the
press
ure
/v
eloc
ity gradient correlations is
π
11
<
u
3
u
3
. The transport equations
for
u
2
u
2
and
u
3
u
3
are, respectively
0
i
f
u
1
u
1
≥
u
2
u
2
and
u
1
u
1
≥
D uu
∂
u
∂
2
uu
∂
u
∂
u
∂
2
22
=−
0
uuu
+
p
2
+
ν
22
−
2
ν
2
2
222
D t
∂
x
ρ
∂
x
∂
x
∂
x
∂
x
∂
x
[2.7]
2
2
2
2
l
l
P
Τ
π
D
ε
22
22
22
22
22
3 More specifically, without mean vorticity.
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