Environmental Engineering Reference
In-Depth Information
Longitudinal Correlation
:
g
x
(
x,y,z
)
w ith g
x
(
x +
z
,y,z
)
g
y
(
x,y,z
)
w ith g
y
(
x,y +
z
,z
)
Transverse Correlation:
g
x
(
x,y,z
)
with g
x
(
x,y +
z
,z
)
g
y
(
x,y,z
)
with g
y
(
x +
z
,y,z
)
A
correlation coefficient of unity indicates that wind fluctuations are identical at the two
points in question, while a negative correlation coefficient suggests structured reverse flow.
The calculation of correlation coefficients can be illustrated by considering a HAWT
with a rotor diameter
D
,
hub elevation
h
,
and time-histories of the longitudinal wind speed
at three locations in space: the center of the rotor, (
x = y =
0,
z
=
h
);
upwind from the rotor
center a distance 2
D
;
and laterally outward from the rotor center a distance 0.5
D.
Assum-
ing isotropic turbulence (
i.e.
,
equal in all directions), the correlation coefficients are
k
L
(-2
D
) =
g
x
(0, 0,
h
)
g
x
(- 2
D
, 0,
h
)/s
0,
x
(8-26a)
Longitudinal
:
k
T
(0.5
D
) =
g
x
(0, 0,
h
)
g
x
(0, 0.5
D
,
h
) /s
0,
x
Transverse
:
(8-26b)
where
k
L
= longitudinal correlation coefficient
k
T
= transverse correlation coefficient
overbars = time averages
Three mathematical forms of the correlation coefficients are the
von Karman
[Hinze
1975, p. 247], the
Dryden
[
loc. cit.
,
p. 58], and the
Batchelor and Townsend
[
loc. cit.
,
p. 202]. All can be written in terms of a parameter called the
integral length scale, Z,
as
follows:
0.333
z
Z
von Karman
:
k
L
= 0.593
K
1/3
(z/
Z
)
(8-28a)
0.333
z
Z
z
Z
K
-
2/3
(z/
Z
)
k
T
= 0.593
K
1/3
(z/
Z
) -
(8-28b)
Dryden
:
k
L
= exp(-1.50 z/
Z
)
(8-29a)
k
T
= (1 - 1.50 z/2
Z
) exp(-1.50 z/
Z
)
(8-29b)
k
L
= exp[- (z /
Z
)
2
]
Batchelor
-
Townsend
:
(8-30a)
k
T
= [1 - (z/
Z
)
2
]exp[- (z/
Z
)
2
]
(8-30b)
where
z =
separation distance between two points in space (m)
Z
= integral length scale of isotropic turbulence (m)
K
v
( )
=
modified Bessel function of the second kind of ( ), of fractional order
v
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