Civil Engineering Reference
In-Depth Information
The problem is greatly simplified if the cross coupling between
q
and
q
θ
may be
disregarded, in which case
⎡
⎤
00
0
⎢
⎥
(
)
x
,
0
S
0
S
Δ≈
ω
⎢
⎥
(6.72)
qq
q
zz
⎢
⎥
00
S
⎢
⎥
⎣
qq
⎦
θθ
where the cross spectra
S
and
S
are given by
q
z z
qq
θθ
ˆ
()
( )
⎫
S
=
S
ω
⋅
Co
Δ
⎪
⎬
x
qq
q
q
zz
z
z
(6.73)
ˆ
()
( )
S
=
S
ω
⋅
Co
Δ
x
⎪
⎭
qq
q
q
θθ
θ
θ
The single point spectra
S
and
S
are defined in Eq. 5.33, while the reduced co-
q
z
θ
spectra
ˆ
and
ˆ
Co
Co
θ
are defined in Eq. 5.34. Thus, the elements of
S
(see Eqs. 4.75
q
z
q
- 4.78) are reduced to
T
i
()
(
) ()
∫∫
x
x
,
x
dx dx
φ
⋅
S
Δω
⋅
φ
1
qq
j
2
1
2
L
exp
()
S
ω
=
(
) (
)
ˆˆ
Q
ij
2
2
MM
ω
⋅
ω
i
i
j
j
{
}
() ()
() ()
∫∫
φ
x
φ
x
S
+
φ
x
φ
x
S
dx dx
i
1
j
2
q q
i
1
j
2
q q
1
2
z
z
z z
θ
θ
θ
θ
L
exp
=
(
) (
)
2
2
MM
ω
⋅
ω
i
i
j
j
ˆ
ˆ
() ()
() ()
S
∫∫
x
x
Co
dx dx
S
∫∫
x
x
Co
dx dx
φ
φ
+
φ
φ
q
i
1
j
2
q
12
q
i
1
j
2
q
12
z
z
z
z
θ
θ
θ
θ
L
L
exp
exp
=
(
) (
)
2
2
MM
ω
⋅
ω
i
i
j
j
(6.74)
Furthermore, it is a reasonable assumption that the integral length-scale of the vortices
D
L
of the structure, and since
q
and
q
θ
are caused by the same vortices their coherence properties are likely to be
is small as compared to the flow exposed length
λ
exp
∞
ˆ
(
)
(
)
identical, in which case [recalling that
∫
0
Co
x d
x
D
(see Eq. 5.34) and
ΔΔλ
≈
q
m
adopting the integration procedure presented in example 6.1] the following is obtained:
