Publishers version - DTU Orbit

angle of attack
4.0
3.5
3.0
2.5
150 100 50 50 100 150
Θ
Vrelms
45.0
44.5
44.0
43.5
43.0
150 100 50 0 50 100 150
Figure 86: phi (left) and w (including induction) (right) as a function of θ, seen from a point
at radius r = 30 m on a rotating blade. Black curve: no shear; grey curve: power law profile
with shear exponent of 0.5
Φ
rotation
w
r
Figure 87: Speed triangle for a blade element at radius r. rΩ is the blade speed and w
corresponds to the sum of the pitch angle, the twist angle and φ. As the twist angle is
constant for a given position on the blade and the pitch angle is 0 ◦ for wind speeds below
rated speed, φ represents the variation of the angle of attack
et al., 2002):
dFT = dFLsinφ−dFDcosφ. (160)
As the wind speed increases with height (e.g. in the case of the power law profile), the
amplitude of the variations of the free wind speed seen by a rotating point increases with the
radius (not shown here). The local tangential force consequently varies with the radius too.
As the torque results from the integral of the tangential force over the whole rotor, it thus
depends on the wind speed profile.
7.2.2 Consequences on the power production
A series of cases were simulated with theoretical wind speed shear defined from the power law
in Eq. (159), with −0.1 < α < 0.5 and 5 m s−1 < uhub < 10 m s−1 . Therelative variationsin
power (defined as the percentage difference between the power outputs obtained with a shear
inflowand an uniform inflow)are shownin Figure 90.Accordingto the simulationsresults, the
power output obtained with shear inflow is generally smaller than the power output obtained
with an uniform inflow. Moreover, it decreases as the shear exponent increases except at 5 m
s−1 where the power output reaches a minimum for α = 0.2 and increases for larger shear
exponents, even exceeding the power output from uniform inflow with α = 0.5.
The first difference between a sheared and an uniform inflow is the kinetic energy flux. In
case of horizontally homogeneous inflow, the kinetic energy flux can be expressed by:
H+R
KEprofile =
H−R
Θ
u
0.5ρu 3 c(z −(zhub −R))dz, (161)
146 DTU Wind Energy-E-Report-0029(EN)