Civil Engineering Reference
In-Depth Information
Fig. 3.8
Spatial illustration of the integral length scales
The spatial properties of turbulence are strongly dependent of the fetch, i.e. the up-
wind terrain. In general, the determination of spatial properties of the turbulence compo-
nents should be based on full scale recordings on the site in question. However, for a
first approximation and under homogeneous conditions not unduly close to the ground,
the following may be adopted
nuvw
sxyz
,,
,,
⎧
⎪
⎨
=
(
)
(
)
0 p /
s
s
,
s
L
ρΔτ
=≈
−
Δ
(3.35)
nn
n
=
⎪
⎩
f
f
f
y
f
⎡
⎤
L
u
⎢
⎥
⎡ ⎤
1/3
1/4
1/4
1/4
1/12
1/12
1/16
1/16
z
f
⎢
⎥
⎢ ⎥
L
u
x
f
()
(
0.3
⎧
⎢
⎥
⎢ ⎥
Lz
z
z
⎛ ⎞
uf
x
f
f
⎢
⎥
⎢ ⎥
⎪
L
L
≈
⎜
⎜ ⎟
v
)
⎪
x
f
⎢
⎥
⎢ ⎥
Lz
⎝ ⎠
f
0
y
f
uf
0
⎢
⎥
⎢ ⎥
⎪
⎪
⎨
⎪
x
v
f
L
⎢
⎥
≈
⋅
where:
(3.36)
⎢ ⎥
u
z
f
⎢
⎥
⎢ ⎥
L
v
zz
≥=
10
m
⎢
⎥
⎢ ⎥
⎪
⎪
f
f
0
x
f
⎢
⎥
⎢ ⎥
L
L
x
f
(
)
w
Lz
100
m
=
⎢
⎥
⎢ ⎥
⎪
⎩
uf
0
y
f
⎢
⎥
⎢
⎣ ⎦
w
⎢
⎥
z
f
⎢
L
⎥
⎣
⎦
w
While cross covariance functions (or coefficients) represent the time and space domain
properties of the turbulence components, it is the auto and cross spectral densities that
describe the frequency-space domain properties. In text books on mathematics, the dou-
ble sided cross spectra are usually defined with
ω
as the frequency variable, in which
case (see chapter 2.6 - 2.8)
