Civil Engineering Reference
In-Depth Information
*
T
⎧
⎫
N
N
1
⎡
⎤ ⎡
⎤
⎪
(
)
(
)
⎪
()
∑
T
∑
T
S
ω
=
lim
A
⋅
B ψ a
⋅
A
⋅
B ψ a
⎨
⎬
⎢
⎥ ⎢
⎥
R
n
Qn
v
ˆ
n
Qn
v
ˆ
n
n
n
n
π
T
T
→∞
⎣
⎦ ⎣
⎦
⎪
⎪
n
1
n
1
=
=
⎩
⎭
NN
1
⎡
(
)
⎤
∑∑
T
*
ˆ
T
T
T
ABψ
lim
aa ψ BA
(9.114)
=
⋅
⋅
⋅
⋅
⎢
⎥
ˆ
n
Qn
v
v
mQ m
n
n
m
m
T
π
⎣
T
→∞
⎦
nm
NN
11
==
∑∑
T
()
T
T
⎡
⎤
=
ABψ S
⋅
⋅
ω
⋅
ψ BA
n
⎣
Qn
vv
ˆˆ
mQ m
⎦
n
m
nm
11
==
where
1
(
)
()
*
T
lim
S
ω
=
a
⋅
a
vv
ˆˆ
v
ˆ
nm
ˆ
T
π
T
→∞
⎛
*
ˆ
1
*
ˆ
1
*
ˆ
⎞
⎡ ⎤
a
u
⎜
⎟
⎢ ⎥
⎜
⎟
⎢ ⎥
a
v
⎜
⎟
⎢ ⎥
(9.115)
⎜
⎟
⎢ ⎥
a
SS
⎡
⎤
1
⎜
w
⎟
vv
ˆ ˆ
vv
ˆ ˆ
⎢ ⎥
⎡
1
11
12
⎤
lim
aaa aaa
=
⋅
=
⎢
⎥
⎜
⎣
u
ˆ
v
ˆ
wu
ˆ
ˆ
v
ˆ
w
m
ˆ
⎦
⎟
⎢ ⎥
T
*
1
1
1
2
2
2
π
SS
T
a
⎢
⎥
→∞
⎣
vv
ˆˆ
vv
ˆˆ
⎦
⎜
⎟
⎢ ⎥
u
ˆ
21
22
2
*
ˆ
2
*
ˆ
2
⎜
⎟
⎢ ⎥
a
⎜
⎟
⎢ ⎥
v
⎜
⎟
⎢ ⎥
⎜
a
⎟
⎢
⎣ ⎦
w
⎝
⎠
is the cross spectral density matrix of the reduced turbulence components. Assuming that
all cross spectral densities between two different flow components are negligible, and
then its content is given by
(
)
ˆ
⎡
⎤
⎫
Ss
,
0
0
Δ
ω
uu
11
⎪
⎢
nm
⎥
⎪
(
)
⎢
⎥
ˆ
0
Ss
,
0
S
=
Δ
ω
⎪
vv
ˆˆ
⎢
vv
11
⎥
⎪
11
nm
⎢
⎥
(
)
ˆ
0
0
Ss
Δ
,
ω
⎪
⎢
⎥
⎪
ww
11
⎣
⎦
⎬
nm
(9.116)
(
)
ˆ
⎡
⎤
⎪
Ss
,
0
0
Δ
ω
uu
22
⎢
nm
⎥
⎪
(
)
⎢
⎥
⎪
ˆ
0
Ss
,
0
S
=
Δ
ω
vv
ˆˆ
⎢
vv
22
⎥
⎪
22
nm
⎢
⎥
⎪
(
)
ˆ
0
0
Ss
,
Δ
ω
⎢
⎥
⎪
ww
22
⎣
⎦
⎭
nm
(
)
ˆ
⎡
⎤
Ss
,
0
0
Δ
ω
uu
12
⎢
nm
⎥
(
)
⎢
ˆ
⎥
0
Ss
,
0
S
=
Δ
ω
=
S
(9.117)
vv
ˆˆ
⎢
vv
12
⎥
vv
ˆˆ
12
nm
21
⎢
⎥
(
)
ˆ
0
0
Ss
Δ
,
ω
⎢
⎥
ww
12
⎣
⎦
nm
