The QBlade implementation of the lifting-line free vortex wake (LLFVW) method was tested in conditions analogous to floating platform motion. Comparisons against two independent test cases using a variety of simulation methods show good agreement in thrust forces, rotor power, blade forces and rotor plane induction. Along with the many verifications already undertaken in the literature, it seems that the code performs solidly even in these challenging cases. Further to this, the key steps are presented from a new formulation of the instantaneous aerodynamic thrust damping of a wind turbine rotor. A test case with harmonic platform motion and collective blade pitch is used to demonstrate how combining such tools can lead to a better understanding of aeroelastic stability. A second case demonstrates a non-harmonic blade pitch manoeuvre showing the versatility of the instantaneous damping method.

The proliferation of large wind turbine rotors has been accompanied by the
need for accurate and computationally inexpensive aeroelastic simulation
tools. For aeroelastic simulations, the aerodynamics of the wind turbine are
most typically calculated using methods based on blade element momentum (BEM).
In the scenario of offshore wind, particularly when designing for floating
platforms, the significant motion of the rotor leads to complicated
aerodynamics.

The lifting-line free vortex wake (LLFVW) method uses non-linear polar
data

Including viscous effects such as separation.

to calculate the blade forces coupled with a free vortex wake formulation and serves as a good method for simulating cases in which large rotor displacements and yaw misalignments occur (see Fig.LLFVW
formulations inherently account for attached flow unsteadiness; the unsteady
aerodynamic model mentioned here only includes terms for detached flow and
leading edge vorticity. The details are given by

After the validation of the LLFVW code for simulations involving a moving
rotor plane, the aerodynamic damping of the rotor is investigated. For this
analysis, a new formulation is presented for the instantaneous aerodynamic
damping of the fore–aft motion degree of freedom. The formulation is a
modification of an existing formulation that was first presented by

For the scope of this paper, two varieties of prescribed motion are
considered. The first variety, pitching, is the more realistic representation
in which the rotor plane undergoes both pitch and linear translation (see
Fig.

The two different assumed motions for the wind turbine:

The QBlade LLFVW implementation has been previously tested for a range of
standard HAWT and VAWT cases as can be found in the existing literature
(

Snapshot of LLFVW simulation during pitching platform motion; vorticity isosurface of the wake coloured with velocity magnitude.

Two different papers were used as a basis for comparison, both investigating
the NREL 5MW reference turbine (^{™} with
Star-CCM^{™} for meshing. It appears from the
presented information that the simulations should be high quality and within the
limitations of URANS.

Thrust and power over phase angle for pitching platform motion
(4

Thrust and power over phase angle for pitching platform motion
(1

Integrated blade forces at blade root for pitching platform motion
(4

Induction plot compared to moving actuator disc CFD hybrid from

The QBlade simulations were run with the same conditions as described above.
The unsteady aerodynamic model from Wendler was enabled without vortex lift
as the wind turbine is operating at near rated speed without yaw
(

The standard 5MW project file is available for download with the standard QBlade package.

.The simulation settings can be found in Table

QBlade FVLLW wake simulation settings.

The LLFVW (Fig.

In the publication chosen for a second comparison, ^{™}). In practice this
means that the actuator disc acts as a volume force onto the surrounding
cells. It is argued by

The QBlade simulations were conducted again using the settings stated above,
with prescribed linear rotor plane movement. Like

From the two verifications performed here it seems that the QBlade LLFVW simulation model produces results that are comparable to other higher-order or hybrid methods. These results and the results already published give a high degree of confidence in the simulation tool's ability to model wind turbines undergoing platform motion.

The following section briefly outlines a reformulation of the instantaneous
damping calculation method outlined by

Before setting out the derivation of the instantaneous damping coefficient it
is first essential to set out the cycle total aerodynamic damping. The
approach taken follows the derivation provided by

To begin the derivation, let us start with the homogeneous equation of the
wind turbine oscillating in linear fore–aft motion (denoted as

Now by introducing the aerodynamic terms into the equation we arrive at the particular equation of

Let us now take a different decomposition of the thrust force into its
constituent steady and unsteady parts

Two options exist for the normalization of the thrust force: the
freestream velocity or with the inflow velocity thus accounting for the
rotor movement. The former assumption simply implies that the unsteady
coefficient will contain the freestream effects for the expected velocity
ratios expected for wind turbine pitching movement and freestream velocities.
This may cause some peculiarities in the appearance of the data similar to
the lift coefficient overshoots seen by

This gives the coefficient form

Let us now inspect the unsteady thrust force terms further. If we assume the
thrust to be a sinusoidal time-dependent function, the unsteady thrust force
can be given in the Cartesian form

In most formulations, certainly as shown above, only a cycle-averaged value
of aerodynamic damping is found.

The inertia is not used in this derivation and the apparent mass terms are actually cancelled out later. However, it is important to note that the apparent mass analogy can be made for a rotor. If that were not true, then the first term would also be complex and this derivation would be invalidated.

Collective pitch damping cycles.

The time-averaged damping then gives us the cycle damping:

Negative blade pitch manoeuvre.

On the practical side, Hilbert transforms are intended to analyse narrowband
signals. It was previously established in

Having presented the analysis methods, it is possible to use these
methods to investigate an example case of floating platform wind turbine
aeroelasticity. A case was selected that should demonstrate more complicated
thrust damping behaviour. The case chosen is harmonic collective pitch in the
presence of platform translation. Further potential test cases for future
work would include harmonic platform movement in combination with the
following:

yawed inflow;

inflow turbulence;

gusts or sudden changes in direction;

changes in airfoil performance through simulated active flow control;

and/or non-synchronous pitch and platform movements.

The collective pitch cycle chosen is not a realistic control regime, but it was chosen to give a clear demonstration of the method. The LLFVW simulation was run for 60 s with a single cycle chosen for analysis after the initial wake effects had died out. The instantaneous damping was calculated from the thrust data using the method already discussed. As a verification the two cycle-averaged values were compared and had good agreement; the values are presented in Table 1.

Demonstration case 1 simulation settings.

The cycle-averaged aerodynamic damping values do in fact show that collective
pitch has an effect. While thrust forces tend to be positively damped
(with this sign convention, that means good damping), we can see that the
magnitude of the damping is altered. In Fig.

Comparison of cycle damping values.

A closer inspection reveals an interesting feature: a

i.e. structural or that provided by the floating platform.

to reduce the amplitude of oscillation.Positive blade pitch manoeuvre.

In the opposite case with a pitch phase shift of

The second case attempts to demonstrate one of the most useful aspects of the instantaneous damping approach. Again in this case, we will look at a collective blade pitch manoeuvre, but this time the pitching will not be periodic. This highlights one of the advantages of the assumptions made in the derivation. The prescribed motion of the platform must be periodic, but the thrust force response has no restriction. This in turn means that we are free to try out any control manoeuvres as long as the platform motion remains periodic.

The second demonstration case will consist of a simple 5.2

The results are presented in two sets with the negative blade pitch manoeuvre
in Fig.

Demonstration for case 2 simulation settings.

The QBlade implementation of the lifting-line free vortex wake (LLFVW) method proved to be a useful tool for analysing floating platform wind turbines. Comparisons against two independent test cases using a variety of methods showed relatively good agreement in thrust forces, rotor power, blade forces and rotor plane induction. Along with the many verifications already undertaken in the literature, it seems that the code will perform solidly even in these challenging cases. Further work is required to extend the same analysis with flexible blades, tower and eventually platform rather than prescribed motion; research on some of these topics is already under way.

A new formulation of the instantaneous aerodynamic thrust damping of a wind turbine rotor was described. The first demonstration case was used to verify that the cycle-averaged damping values line up with well-established methods. The case also showed how the system alternated between being stable and unstable within a single cycle. The second demonstration case showed a more complicated pitch manoeuvre; the instantaneous damping method was useful in understanding the system but provided helpful information for designing control strategies. It would be useful in future work to generalize the method so that any mode shape could be analysed without having to undertake the extensive derivation described in this paper.

The full implementation of QBlade is available for download
at

No additional data were generated that have not been included in this article.

The authors declare that they have no conflict of interest.

This article is part of the special issue “The Science of Making Torque from Wind (TORQUE) 2016”. It is a result of the The Science of Making Torque from Wind (TORQUE 2016), Munich, Germany, 5–7 October 2016.

We acknowledge support by the German Research Foundation and the open-access publication funds of the Technische Universität Berlin. Edited by: Carlo L. Bottasso Reviewed by: Vasilis A. Riziotis and two anonymous referees