Civil Engineering Reference
In-Depth Information
Fig. 3.6
Cross covariance of along-wind u component
The corresponding covariance coefficients are defined by
(
)
Cov
s
,
mn uvw
sxyz
,
=
, ,
,
Δ
τ
⎧
⎪
⎨
mn
(
)
s
,
ρ
Δ=
τ
(3.31)
mn
Δ=Δ
Δ Δ
,
σσ
⋅
⎪
⎩
mn
f
f
f
should be taken at position
s
, while
(If the process is not ergodic, then
σ
should
σ
n
m
be taken at
ss
+Δ
.) The situation is illustrated in Fig. 3.6 with
mu
=
and
sy
=
,
f
which is most relevant for a horizontal structure where time series of the turbulence
components have been recorded at various positions
a
and
b
along the span of the struc-
ture. As can be seen from Eq. 3.29 (and 3.30), there are 27 possible covariance func-
tions. However, it is a usual assumption that cross covariance between two different
turbulence components may be neglected, at least beyond a certain distance above the
ground. All off-diagonal terms in Eq. 3.29 will then become zero, and the number of
possible covariance estimates is reduced to nine:
(
)
()(
)
()(
)
Cov
s
,
ust us
,
st
,
ust us
,
st
,
⎡
Δ
τ
⎤ ⎡
⋅
+ Δ
+
τ
⎤
⎡
⋅
+ Δ
+
τ
⎤
uu
T
1
⎢
⎥ ⎢
⎥
⎢
⎥
(
)
()(
)
()(
)
Cov
s
,
E v st v s
,
st
,
∫
v st v s
,
st
,
dt
Δ
τ
=
⋅
+ Δ
+
τ
=
⋅
+ Δ
+
τ
⎢
⎥ ⎢
⎥
⎢
⎥
vv
T
(
)
() (
)
() (
)
⎢
⎥ ⎢
⎥
⎢
⎥
Cov
s
,
wst ws
,
st
,
0
wst ws
,
st
,
Δ
τ
⋅
+ Δ
+
τ
⋅
+ Δ
+
τ
⎣
⎦ ⎣
⎦
⎣
⎦
ww
(3.32)
