Publishers version - DTU Orbit

Bottom of rotor
Φ
rotation
r
w
u
Φ
Top of rotor
rotation
r
Figure 88: Speed triangles at the top and bottom of the rotor showing the effect of wind
speed shear. The speed triangle at hub height is shown with dashed arrows
dFTkNm 0.18
0.16
0.14
0.12
0.10
0.08
0.06
150 100 50 50 100 150
Figure 89: Local tangential force seen from a point r = 30 m on a rotating blade as function
of θ. Black curve: no shear; Gray curve: power law profile with shear exponent of 0.5
where ρ is the air density, R the rotor radius and c is the chord (of the circle defined by the
rotor swept area) as function of z which varies between the bottom and the top of the rotor:
w
Θ
c = 2 2Rz −z 2 . (162)
In order to compare to the power output variations, Figure 90 also shows the difference
between the kinetic energy flux for a power law profile and a constant profile, normalised with
the power obtained with a constant profile KEprofile−KEhub/KEhub (see gray dashed line)
5 . Figure 90 shows two other interesting results:
1. The kinetic energy flux varies with shear exponent.
2. The power output of the turbine does not follow the same trend as the kinetic energy
flux.
Despite the high uncertainty in the modeled power output for a sheared inflow, the results
highlight that the influence of the shear on the power performance of a turbine can be seen
as the combination of two effects:
• The variation in kinetic energy flux (power input).
• The ability of the turbine to extract the energy from the wind, which depends on the
details of the design and the control strategy of the turbine.
5 The kinetic energy flux is not an output of HAWC2Aero, it has been here estimated and for power law
profiles, the normalized difference does not depend on the wind speed
DTU Wind Energy-E-Report-0029(EN) 147
u