Civil Engineering Reference
In-Depth Information
For simplicity it is in the following assumed that cross spectra between the
u
,
v
and
w
components are negligible, i.e. that
x
½
( )
S
ωΔ ≈
,
s
0
=
uvw
,,
(A.10)
¾
¿
xy
y
where
y z
−
plane. We will then only need
information about the cross spectra of the turbulence components themselves,
Δ
is the spatial separation in the
f
f
( )
ωΔ
.
S
,
s
xx
()
()
Let
x
m
S
ω
the corresponding cross spectral
density between two arbitrary points
m
and
n
. As shown in chapter 2.6 these quantities
constitute a Fourier transform pair. An
M
by
M
cross spectral density matrix
Cov
τ
be the covariance and
x
mn
S
S
S
ª
" "
# %# $#
"
º
xx
xx
xx
11
1
n
1
M
«
»
«
»
«
»
()
S
S
S
"
S
ω
=
«
(A.11)
xx
xx
xx
xx
»
m
1
m n
m M
«
»
# $# %#
"
«
»
S
S
S
"
«
»
¬
¼
xx
xx
xx
M
1
M n
M M
will then contain all the space and frequency domain information that is necessary for a
time domain simulation of
M
time series with the correct statistical properties for a
special representation of the process. It follows from the assumptions of stationarity and
homogeneity that
Cov
=
Cov
(A.12)
x x
x x
mn
nm
*
S
S
and thus,
=
(A.13)
x x
x x
mn
nm
()
This implies that
S
xx
ω
is Hermitian and non-negative definite. A Cholesky
decomposition of
x
S
will then render a lower triangular matrix
G
GG
0
0
0
0
0
"
"
ª
º
xx
11
«
»
0
0
0
0
"
"
«
»
xx
xx
21
22
«
»
#
#
#
#
#
«
»
()
G
ω
=
«
(A.14)
xx
»
GG G G
0
0
"
"
xx xx
xx
xx
m
1
m
2
m n
m m
«
»
«
»
#
#
#
#
#
«
»
GG G G G
"
"
"
«
»
¬
xx xx
xx
xx
xx
¼
M
1
M
2
M n
M m
M M
