Civil Engineering Reference
In-Depth Information
where
z
and
z
are defined in Eq. 3.1. Under isotropic conditions (e.g. high above
min
I
I
I
the ground)
. In homogeneous terrain up to a height of about 200 m and
not unduly close to the ground
≈≈
uv
w
I
3/4
1/2
⎡⎤⎡ ⎤
≈
v
I
⋅
(3.16)
⎢⎥
⎢
⎣ ⎦
u
I
⎣⎦
w
The auto covariance functions and corresponding auto covariance coefficients (see chap-
ter 2.2) are defined by
() (
)
⎡
Eut ut
⎤
()
()
()
⎡
⋅
+
τ
⎤
() (
)
Cov
u t
u t
⎡
τ
⎤
⎡
⋅
+
τ
⎤
⎣
⎦
u
T
⎢
⎥
1
⎢
⎥
⎢
⎥
() (
)
() (
)
Cov
E vt vt
∫
vt vt
dt
τ
=
⎡
⋅
+
τ
⎤
=
⋅
+
τ
(3.17)
⎢
⎥
⎢
⎥
⎣
⎦
⎢
⎥
v
T
⎢
⎥
⎢
⎥
⎢
()
(
)
⎥
Cov
0
w t
w t
τ
()
(
)
⋅
+
τ
Ewt wt
⎡
⋅
+
τ
⎤
⎣
⎦
⎣
⎦
w
⎢
⎥
⎣
⎣
⎦
⎦
()
Cov
τ
n
()
nuvw
,,
ρτ
=
where
=
(3.18)
n
2
σ
n
T
where
τ
is an arbitrary time lag that theoretically can take any value within
±
. At
τ
=
0
Eq. 3.17 becomes identical to 3.13, and thus
(
)
ρτ
==
01
where
nuvw
=
,,
(3.19)
n
At increasing values of
τ
the auto covariance of the turbulence components diminish,
and at large values of
τ
they asymptotically approach zero, i.e.
()
lim
0
nuvw
,,
ρτ
=
where
=
(3.20)
n
τ
→∞
As shown in Eq. 2.19,
()
( )
Cov
Cov
nuvw
,,
τ
=
−
τ
where
=
(3.21)
n
n
()
implying that also
is symmetric. A principal variation of the covariance coeffi-
cient for the along wind turbulence component is shown in Fig. 3.4. The time scale
ρ
τ
n
∞
=
∫
()
T
d
nuvw
,,
ρ
ττ
where
=
(3.22)
n
n
0
