Civil Engineering Reference
In-Depth Information
{
}
{
}
T
⎡
⎤
1
1
−
−
T
(
)
(
)
E
⎡ ⎤
E
Cov
=
r r
⋅
=
K
−
K
R
⋅
K
−
K
R
⎢
⎥
rr
⎣ ⎦
ae
ae
⎣
⎦
(9.93)
(
)
T
−
1
−
1
(
)
T
(
)
⎡
⎤
E
=−
KK
⋅
RR
⋅
⋅
KK
−
ae
ae
⎣
⎦
N
()
∑
T
()
()
()
ˆ
Since
R
t
=
A R
t
and
RBψ v
t
t
(see Eqs. 9.49 and 9.76) then
n
n
Q
n
n
n
n
1
=
T
⎡
⎤
N
N
N
N
⎧
⎫ ⎧
⎫
T
⎪
⎪ ⎪
⎪
⎡
(
) (
)
⎤
T
∑
T
∑
T
∑∑
T
T
⎡
⎤ ⎢
⎥
E
RR
⋅
=
E
AR
⋅
AR
=
E
AR
⋅
AR
⎨
⎬ ⎨
⎬
⎣
⎦
nn
nn
⎢
nn
mm
⎥
⎢
⎥
⎣
⎦
⎪
⎪ ⎪
⎪
⎩
⎭ ⎩
⎭
n
1
n
1
n
1
m
1
=
=
=
=
⎣
⎦
NN
∑∑
T
T
⎡
⎤
=
ARRA
⋅
E
⋅
⋅
n
⎣
n
m
⎦
m
nm
NN
11
==
T
⎡
(
)
(
)
⎤
∑∑
T
n
E
ˆ
ˆ
=
A
⋅
B ψvBψvA
⋅
⋅
(9.94)
Q
Q
m
⎢
⎥
⎣
n
m
⎦
nm
NN
==
11
(
)
∑∑
T
⎡
ˆ
ˆ
T
⎤
T
T
E
=
ABψ vvψ BA
⋅
nQn
⎣
nmmQ
⎦
m
n
m
nm
==
11
NN
(
)
∑∑
ABψ Cov ψ BA
T
T
T
=
⋅
⋅
⋅
⋅
⋅
⋅
ˆˆ
n
Qn
vv
mQ
m
n
m
n
==
11
where
…
are dummy summation variables (
N
is the
number of elements in the system) and where
n
1,2,
,
N
m
1,2,
,
N
=
…
and
=
ˆ
T
E
⎡
⎤
Cov
=
v
⋅
v
is the covariance
vv
ˆˆ
⎣
n
m
⎦
matrix of the reduced flow components contained in
v
, i.e.:
()
()
()
()
()
()
ut
vt
wt
ˆ
ˆ
ˆ
⎡
⎤
1
⎢
⎥
⎢
⎥
1
⎢
⎥
[
1
()
()
()
()
()
()
]
E
utvtwtutvtwt
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
Cov
=
⋅
⎡
⎤
(9.95)
⎢
⎥
⎣
⎦
vv
ˆˆ
1
1
1
2
2
2
ut
vt
wt
ˆ
ˆ
ˆ
m
⎢
⎥
2
⎢
⎥
2
⎢
⎥
⎢
⎥
⎣
⎦
2
n
where indices 1 and 2 refer to element ends 1 and 2. By performing the multiplications
and the relevant statistical calculations, and assuming that all cross covariances between
different flow components are negligible, i.e.
Cov
Cov
Cov
0
=
=
≈
(9.96)
uv
uw
vw
