Improved energy forecast for Offshore Wind farms Siri

Cascade Analysis of a Floating Wind Turbine
Lene Eliassen & Jasna B. Jakobsen, University of Stavanger
Finn Gunnar Nielsen & Andreas Knauer, Statoil
Mounting a wind turbine on a floating foundation introduces new challenges to the aerodynamic loading. The floater motion may contain a wide range of frequencies. To study some of
the basic dynamic load effect on the blade due to these motions, a two-dimensional cascade approach combined with a potential vortex method is used. This is an alternative method
to study the aeroelastic behaviour of wind turbines that is different to the traditional blade element momentum method. The analysis method demands little computational power, and
can handle unsteady flows.
ROTATION
WIND
MOTION OF
WIND TURBIN
Figure 1: An illustration of the wing, with the surface divided into
panel elements
Dynamics of a Floating Wind Turbine
Figure 3: The two dimensional panel vortex method with
three airfoils moving in the rotational direction
Figure 2: One of the blades and the wind modelled in
the program
Discussion and Conclusion
The floating wind turbines have in general six additional degrees of
freedom compared to the bottom fixed wind turbines. These rigid body
motions, if significant, will alter the angle of attack and the wind velocity.
As a result the shape and strength of the wake behind the rotor will also
vary. The aerodynamic loads are dependent on both the incoming wind
and the wake shed behind, and therefore both wind and wake should be
included in the simulation.
The most common simulation tools for rotor loads, such as HAWC2 [1]
and AeroDyn [2], are based on the blade element momentum theory,
which does not directly include the wake strength and shape in the load
estimations. In this work, the aerodynamic loading on a wind turbine in
surge and pitch motion is studied, see Figure 4. The rotor loads on the
oscillating wind turbine are investigated using a two-dimensional panel
vortex method. This method includes the development of the wake, and
calculates the aerodynamic loads based on the geometry of the airfoil.
Method
The nature of the investigated motion-dependent loads is such that there
is a linear relationship between the velocity of the wind turbine and the
associated force. The linear relationship has a negative slope, and the
negative correlation between the axial rotor velocity and the associated
loading corresponds to a positive aerodynamic damping, as expected
according to the quasi-steady theory.
We would further like to investigate analytically the aerodynamic
1
damping, based on the definition of the lift force; 𝐿 = 𝜌𝑈 2 𝑐𝑙𝐶𝐿 . In the
2
equation, both the lift force coefficient, 𝐶𝐿 , and the relative wind speed,
U, will vary with the surge velocity. 𝜌 is the air density, 1.225 kg/m3, c is
the chord length and l is the unit length 1/m. A linearized approximation
of the 𝐶𝐿 curve about the mean angle of attack is used: 𝐶𝐿 𝛼 ≈ 𝐶𝐿0 +
Figure 4: Forced surge harmonic
response of a wind turbine
𝜕𝐶𝐿 𝑥
,
𝜕𝛼 𝑈𝑟𝑜𝑡
relative wind velocity, U, is (𝑈𝑤𝑖𝑛𝑑 −𝑥)2 + (𝑈𝑟𝑜𝑡 )2 , where 𝑈𝑤𝑖𝑛𝑑 is the
wind velocity and 𝑈𝑟𝑜𝑡 is the velocity of the blade rotation. The lift force
can be rewritten as 𝐿 ≈ 𝐿1 + 𝑙2 𝑥 , where:
The two dimensional panel vortex method applied is
based on the potential flow theory, and does therefore
not consider boundary layer development and coupled
viscous effects including stall. For higher angles of
attack the method is invalid. The position and the
strength of the vortex elements in the wake is included
in the calculation of the aerodynamic forces.
The method is coded in Matlab, and not optimized with
regards to computational speed. Three blades are
modelled in a cascade configuration, and it is assumed
that there is no flow in the spanwise direction of the
blade. As the blades are rotating along a repeating path
of length 2*pi*R, the blades will be influenced by the
wake of the upstream blade, see Figure 3. The aerofoils
are modelled by using linear panel elements, see Figure
1. The panels have a constant distribution of source and
doublet elements. The wake is modelled similarly, but
using only doublet elements [3]. The position in the
flow of the wake elements is part of the solution.
To simulate the oscillation of a floating wind turbine,
the incoming wind velocity is reduced to a harmonic
component, representing the horizontal velocity of the
rotor plane, see Figure 2. The harmonic rotor plane
motion in axial direction is simulated with the
amplitude of 1.5 m and the period of 12 sec, see Figure
4. The rotor used in the calculation is similar to the
5MW rotor used in the code comparison project OC3
[4]. The wind condition is set to 6 m/s, this is below
rated speed and there is no pitching of the blades. The
rotor has a constant speed of 7.96 rpm.
where 𝑥 is the axial velocity of the wind turbine rotor. The
𝐿1 = 1 2 𝜌c𝑙(𝑈𝑤𝑖𝑛𝑑 2 + 𝑈𝑟𝑜𝑡2) 𝐶𝐿0
𝑙2 =
−1
2 𝜌𝑐𝑙[
2
𝑈𝑤𝑖𝑛𝑑 +
and
2 𝜕𝐶𝐿 1
𝑈𝑟𝑜𝑡
𝜕𝛼 𝑈𝑟𝑜𝑡
+ 2𝑈𝑤𝑖𝑛𝑑 𝐶𝐿0 ].
The coefficient dependent on the rotor axial motion, 𝑙2 , is
the aerodynamic damping coefficient. Using the information
given in [4], as listed in Table 1, the first term of the lift force,
𝐿1 , is 2.87 kN, and the aerodynamic damping, 𝑙2 , is -0.39
kN/(m/s). The two values based on the lift coefficient
obtained by the vortex panel method, on a single blade, are
𝐿1 = 2.88 kN and 𝑙2 =-0.46 kN/(m/s).
Figure 5: Normal force on one airfoil, with respect to time. The
red markers indicate each rotation of the rotor.
The results from the vortex panel method, presented in
Figure 5 and 6, show reasonable comparison with the
theoretically based values. The mean normal load is 2.73 kN,
and the aerodynamic damping coefficient is -0.44 kN/(m/s).
In the further work we will extend these two-dimensional
results to obtain damping force of a full rotor.
[4]
Table 1: The airfoil lift coefficient
Figure 6: Normal force from one airfoil, with respect to the velocity of the wind
turbine motion.
Results
The result for a node located at a radius of 44.55 m (0.7R) is presented in the
following. A comparison between the lift coefficients as calculated with the
vortex panel code, and the lift coefficient given in [4] are shown in Table 3.
The force normal to the rotor plane is shown as a function of time in Figure 5.
The trend is similar for the loading of all sections along the blade. The force is
also plotted against the surge velocity of the wind turbine, see Figure 6.
References
[1] Larsen, Torben J. Welcome to HAWC2, DTU Risø Campus, Denmark. Accessed 13.09.2012. Available
from http://www.hawc2.dk/HAWC2%20Info/Aerodynamic%20model.aspx
[2] Laino, David J. NWTC Design Codes (AeroDyn), NREL, Boulder, Colorado, USA. Accessed 13.09.2012.
Available from http://wind.nrel.gov/designcodes/simulators/aerodyn/
[3] Katz, J. Plotkin, A. (2008). Low-Speed Aerodynamics (2nd Edition). New York, USA: Cambridge University
Press
[4] Jonkman, J. Butterfield, S. Musial, W. Scott, G. Definition of a 5-MW Reference Wind Turbine for
Offshore System Developement. (2009). NREL, Boulder, Colorado, USA. NREL/TP 500-38060.
Corresponding author:
Lene Eliassen, e-mail: lene.eliassen@uis.no