Civil Engineering Reference
In-Depth Information
Launder
et al.
[LAU 75] directly drew inspiration from
Rotta's original proposal to model the rapid terms
π
ij
,
R
. They
supposed that
π
ij
,
R
is approximately expressed by
=
∂
U
l
x
m
α
l
mi
π
ij
,
R
∂
and that the fourth-order tensor
mi
is a linear function of
the Reynolds stresses
u
i
u
j
. These hypotheses lead to the
closure
α
lj
⎧
⎫
(
)
(
)
C
+
8
⎡
⎣
⎤
30
C
−
2
⎪
x
j
+
∂
U
j
∂
⎪
2
3
P
K
δ
ij
∂
U
i
∂
π
ij
,
R
=−
P
ij
−
−
K
⎨
⎬
⎢
⎥
11
⎦
55
⎪
x
i
⎪
⎩
⎭
[2.73]
⎧
⎫
⎛
⎞
(
)
8
C
−
2
⎪
⎪
u
i
u
k
∂
U
k
∂
u
j
u
k
∂
U
k
∂
2
3
P
K
δ
ij
⎜
⎟
−
⎨
−
x
j
+
−
⎬
⎜
11
x
i
⎪
⎪
⎝
⎠
⎩
⎭
where it should be recalled that
P
K
is a term for production
of kinetic energy
K
(see equation [2.20]). The coefficient
C
0.4
is chosen so that the model accurately reflects the
behavior of the isotropic homogeneous turbulence, subjected
to rapid distortion.
=
Equation [2.67] needs to be modified to take account of
the reflection of the source terms from the wall. [LAU 75]
suggest replacing equation [2.67] with
⎧
⎫
′
′
′
⎛
⎞
G
⎛
2
⎞ ⎛ ⎞
⎛
⎞
4
π
⎪
∂
uu
∂
U
∂
u
⎪
⎜
1
1
(
)
[2.74]
⎟
∫
p xt
;
=
lm
+
2
l
m
−
dV
⎨
⎜
⎟ ⎜ ⎟ ⎜ ⎟
⎬
⎜
GG GG
ρ
∂
xx
∂
∂
x
∂
x
x y
−
*
⎟
xy
−
⎪
⎝
⎠ ⎝ ⎠ ⎝ ⎠
⎪
⎝
V
l
m
m
l
⎠
⎩
⎭
G
is the image of the point
y
G
and the
integral is calculated in the flow at
(
)
*
where
y
=−
x
,
y z
,
. After certain
considerations and hypotheses, [LAU 75] propose wall
correction functions, which take account of equation [2.74].
The results found by Mansour
et al.
[MAN 88] with a small
y
>
0
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