Physics of Wind Parks - Wind Energy Meteorology: Atmospheric Physics for Wind Power Generation - page 138

R t ¼ R u ð c teff Þ
R u ð c s ; h Þ

ð 6 : 17 Þ

using R u (c s,h ) = u h0 /u 0 . Inserting for the exchange coefficient K m ( 6.9 ) and the
effective drag coefficient ( 6.15 )in( 6.7 ) yields:

c teff u h ¼ ju zu 0 u h

ð Þ

ð 6 : 18 Þ

Dz/ m

The height z in ( 6.18 ) is essentially h ? Dz, so that the ratio z/Dz can be
approximated by a constant value:

z
Dz ¼ f h ; Dz

ð 6 : 19 Þ

The horizontal turbulence intensity I u at hub height h is defined by:
I u ¼ r u
u h

ð 6 : 20 Þ

The standard deviation of the horizontal wind speed can be parameterized using
the friction velocity u * :

r u ¼ 1

j u

ð 6 : 21 Þ

which yields the following relation between friction velocity, u * and turbulence
intensity, I u :

u ¼ jr u ¼ ju h I u ð 6 : 22 Þ

Inserting of ( 6.19 ) and ( 6.22 )in( 6.18 ) yields finally:

c teff u h ¼ j 2 u h u 0 u h f h ; Dz I u ¼ j 2 u h u 0

u 0 / m

f h ; Dz I u j 2 u h
/ m

ð Þ

f h ; Dz I u ð 6 : 23 Þ

/ m

Rearrangement leads to:

¼ u 0

c teff u h þ j 2 u h
/ m

c teff þ j 2
/ m

R u j 2
/ m

f h ; Dz I u ¼ u h f h ; Dz I u f h ; Dz I u ð 6 : 24 Þ

and finally to an expression for the ratio ( 6.16 ):

j 2

/ m f h ; Dz I u
c teff þ j 2

R u ¼ u h
u 0

u h
R u u 0

f h ; Dz I u
f h ; Dz I u þ / j 2 c teff

¼

¼ ¼ ð 6 : 25 Þ

/ m f h ; Dz I u

Thus, the ratio ( 6.17 ) between the wind speed at hub height inside the wind park
to the undisturbed wind speed upstream is: