Physics of Wind Parks - Wind Energy Meteorology: Atmospheric Physics for Wind Power Generation - page 141

Fig. 6.4 Wind speed u hn (from u hno at the rear end of the wind park to the original undisturbed
wind speed further down to the left u h0 ) in the wake of an indefinitely broad wind park

Du hn
Dt

¼ ju z
Dz 2

ð u h0 u hn Þ¼ ju zu h0
Dz 2

ju zu hn
Dz 2

ð 6 : 29 Þ

This is a first order difference equation of the form:

Du hn
Dt þ au hn ¼ au h0

ð 6 : 30 Þ

with a = ju * z/Dz 2 and the time-dependent solution:

u hn ð t Þ¼ u h0 þ C exp a t t 0 ð ð Þ Þ ð 6 : 31 Þ

The constant of integration C can be determined from the initial condition:

u hn ð t ¼ t 0 Þ¼ u hn0 ¼ u h0 þ C

ð 6 : 32 Þ

Please note the difference between the undisturbed wind speed u h0 at hub height
and the wind speed at hub height directly behind the wind park u hn0 . Inserting
( 6.32 )in( 6.31 ) yields:

u hn ð t Þ¼ u h0 þ u hn0 u h0 ð Þ exp at ð Þ ð 6 : 33 Þ

Dividing by u h0 gives the ratio R n between the wind speed at hub height in the
wake to the undisturbed wind speed in the same height u h0 :
R n ¼ u hn ð t Þ
u h0

exp at

u hn0
u h0

¼ 1 þ 1 ð Þ ð 6 : 34 Þ