Civil Engineering Reference
In-Depth Information
∂
u
1
u
1
∂
()
+ π
11
=−
2
u
1
u
2
β
t
t
∂
u
2
u
2
∂
= π
22
t
[2.93]
∂
u
3
u
3
∂
= π
33
t
∂
u
1
u
2
∂
()
+ π
12
=−
u
2
u
2
β
t
t
The comparison of this equation with relations [2.6]-[2.9]
shows that the linearization inherent to the RDT filters out
all the transport terms except for those of production and
pressure/strain correlation
ij
. [MAX 82] shows that
π
11
and
π
π
33
is positive, which indicates
tha
t
both
u
1
u
1
and
u
2
u
2
feed into the spanwise component
u
3
u
3
.
These behaviors are in qualitative agreement with the
“triangle” of inter-component transfer shown in Figure 2.1.
The RDT is a linear approach, and the correlations
π
22
a
re n
egati
ve, w
hile
ij
need
π
to be interpreted as the rapid component
π
ij
,
R
from equation
[2.68]. The exact expression of
π
ij
in terms of rapid distortion
is, in fact [MAX 82]
kk
⎡
∫
G
G
G
(
)
(
)
()
()
()
()
1
j
3
π
t
=
d mt
2
β
Amt
,
α
Amt
,
α
⎢
⎣
ij
2
p
iq
k
kk
2
G
G
G
⎤
⎥
⎦
(
)
(
)
()
()
()
+
1
i
Am t
,
α
Am t
,
α
Φ
m
2
p
jq
pq
2
k
is directly linked to the shear.
However, there is no accordance between the results found
with the RDT and certain existing models (see equation
[2.73] and [LAU 75]).
Consequently,
()
π
ij
∝ β
t
Rapid distortion describes the development and structure
of (homogeneous) turbulence subjected to shear, and thus the
results stemming from it cannot be directly applied to wall
turbulence. However, the nonlinear effects tend to limit the
development of the turbulence without radically changing its
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