## Abstract

A free-vortex wake (FVW) model is developed in this paper to analyse the unsteady aerodynamic performance of offshore floating wind turbines. A time-marching algorithm of third-order accuracy is applied in the FVW model. Owing to the complex floating platform motions, the blade inflow conditions and the positions of initial points of vortex filaments, which are different from the fixed wind turbine, are modified in the implemented model. A three-dimensional rotational effect model and a dynamic stall model are coupled into the FVW model to improve the aerodynamic performance prediction in the unsteady conditions. The effects of floating platform motions in the simulation model are validated by comparison between calculation and experiment for a small-scale rigid test wind turbine coupled with a floating tension leg platform (TLP). The dynamic inflow effect carried by the FVW method itself is confirmed and the results agree well with the experimental data of a pitching transient on another test turbine. Also, the flapping moment at the blade root in yaw on the same test turbine is calculated and compares well with the experimental data. Then, the aerodynamic performance is simulated in a yawed condition of steady wind and in an unyawed condition of turbulent wind, respectively, for a large-scale wind turbine coupled with the floating TLP motions, demonstrating obvious differences in rotor performance and blade loading from the fixed wind turbine. The non-dimensional magnitudes of loading changes due to the floating platform motions decrease from the blade root to the blade tip.

## 1. Introduction

Rapid progress has been made in offshore floating wind turbine (OFWT) technology. The world's first floating full-scale wind turbine, Hywind [1], operated successfully in the North Sea off Norway in 2009. The NREL offshore 5 MW baseline wind turbine [2] was developed by US NREL to provide realistic input values for offshore wind energy analysis. Many studies of OFWTs based on the NREL offshore 5 MW baseline wind turbine were done, including aerodynamics [3–5], dynamic response analysis [6,7], control strategy [8], etc. The highly complex ocean environments, e.g. wave, ocean current, typhoon, bring challenges to the study of OFWT technologies (table 1).

The aerodynamic issues of large-scale OFWTs are more complicated than those of onshore wind turbines, because offshore wind turbines on floating platforms experience larger motions than comparable fixed-bottom wind turbines—for which the majority of the industry standard design codes have been developed and validated. Wind turbine aerodynamic modelling relies mainly on three approaches: blade element momentum (BEM) theory, vortex wake method and computational fluid dynamics (CFD). BEM theory is a simple engineering model based on the strip theory. Analysis codes used to verify the stability and the ability of a floating wind turbine structure to withstand experienced loads (e.g. Bladed [9], FAST [10] or HAWC2 [11]) all base aerodynamic calculations on the BEM theory. CFD methods, in which Euler or Navier–Stokes equations are solved, provide more physically realistic simulations. However, the expensive computational cost and numerical problems, such as turbulence modelling, transition modelling, dynamic mesh, limit its application. Unlike BEM theory where annular average induction is found, the vortex wake method can determine vortical induction directly at blade elements from the effect of the modelled wake and is more efficient than the CFD method.

Free-vortex wake (FVW) methods have successfully been applied in helicopters [12–14], of which the experiences can be introduced into wind turbine aerodynamic study due to the similarity between helicopter rotors and wind turbine rotors. Therefore, they have been widely used for wind turbines in recent years. Garrel [15] introduced a nonlinear vortex-line strength model into the FVW method to represent the rotor wake rollup and the nonlinear aerodynamic characteristics of blades; Sant *et al*. [16,17] used a FVW code to estimate the angles of attack from blade pressure measurements, respectively, in the axial conditions and the yawed conditions, and gave us new insights into how circulation formed at the blades; Gupta [18] used the predictor–corrector second backward (PC2B) scheme for wind turbine application and studied its stability in [19]; moreover, they validate the FVW model [20,21] in steady and unsteady conditions. In the field of OFWTs: Sebastian & Lackner [4] developed a Wake Induced Dynamic Simulator (WInDS) based on the FVW method to simulate the wake development of a rotor oscillating under platform motions and validated WInDS with the MEXICO and TUDelft experiments for fixed wind turbine rotors. Farrugia *et al*. [22] modified the WInDS code to enable parallel processing and used it to simulate a small-scale rigid wind turbine coupled with a TLP using a dedicated wind-wave generating facility.

In this paper, the FVW model is proposed for investigating the aerodynamic response of a large-scale OFWT when the hydrodynamic response is assumed to be known. Other studies in [5] rely on the same assumption. A three-step and third-order predictor–corrector (D3PC) algorithm is developed for the difference approximation of the wake governing equation. The blade inflow conditions and the positions of initial points of vortex filaments are modified for the floating platform motions in the model. Firstly, the simulation model is validated by comparison with experimental data of the University of Malta (UoM) OFWT [22] and Tjæreborg test turbine [23]. Then, the aerodynamic performance is simulated in a yawed condition of steady wind and in an unyawed condition of turbulent wind, respectively, for a large-scale wind turbine coupled with the floating TLP motions. For completeness, the results are compared to those for the fixed wind turbine.

## 2. Methodology

The FVW model used in the wind turbine aerodynamic calculation is introduced in this section. A time-marching difference approximation of the wake governing equation used in the FVW method is developed and its accuracy is analysed. For the OFWT, the treatments in the unsteady inflow of blade and the unsteady wake due to the floating platform motions are given in detail, followed by descriptions of the three-dimensional rotational effect model and the dynamic stall model for the unsteady calculation.

### (a) Difference approximation of wake governing equation

The vortex method assumes that the flow field is incompressible and potential. The vortex filaments, extending downstream from the blade, are allowed to freely distort under the influence of local velocity field. The convection of these vortex filaments can be described by the Helmholtz equation. In the blade fixed coordinates, the governing equation of the vortex filaments can be written as the partial differential form
2.1
where *V*_{ind} equals the mean value of the values at the surrounding four grid points.

To solve the convection equation of the vortex filaments numerically, finite difference approximations are used to approximate the derivatives. In this study, a time-marching FVW method is used for the unsteady aerodynamic analysis of the OFWT. For the spatial derivative (*ζ*), a five-point central difference approximation [12] has been used in this paper. The accuracy of the temporal derivative (*ψ*) approximation is a significant part in the time-marching free-vortex method. Equation (2.1) can be written in another form
2.2
Equation (2.2) can be written in a general form of an ordinary difference equation
2.3
Introduce the linear multistep method to solve equation (2.3). Assuming the step lengths are equal, the general form of the linear multistep method for equation (2.3) is written as
2.4
where *y*_{n+j} and *f*_{n+j} substitute, respectively, for the terms of *y*(*x*_{n+j}) and *f*(*x*_{n+j}, *y*_{n+j}). When *k*=3, it is the three-step linear multistep method. The values of *α*_{j} and *β*_{j} can be calculated by the method of undetermined coefficients. When *α*_{0}=1/2, *α*_{1}=−3, *α*_{2}=3/2, *α*_{3}=1, *β*_{0}=0, *β*_{1}=0, *β*_{2}=3 and *β*_{3}=0, an explicit form of the linear multistep method is written as
2.5
When *α*_{0}=−2/11, *α*_{1}=9/11, *α*_{2}=−18/11, *α*_{3}=1, *β*_{0}=0, *β*_{1}=0, *β*_{2}=0 and *β*_{3}=6/11, an implicit form of the linear multistep method is written as
2.6

Use the explicit linear multistep to predict an intermediate solution, and then use the implicit linear multistep method to correct it. This is a predictor–corrector algorithm used for the temporal derivative approximation in this study. Combine the temporal derivative approximation with the spatial derivative approximation and, the difference approximation of wake governing equation is obtained
2.7
2.8
where the subscripts *i* and *j* indicate the temporal step and the spatial step, respectively (figure 1); and the superscript *n*−1 means the induced velocities are calculated via the old wake geometry of last iteration step. Note from figure 1 that the stencil for the time-marching method uses points from three steps to evaluate the solution at the new time step.

### (b) Accuracy and stability of difference approximation

A Taylor-series expansion of the difference approximation identifies the errors in such approximations. Expand both sides of equation (2.5) around *x*_{n}, and the local truncation error can be identified as the difference between left- and right-hand sides
2.9
The leading term of the local truncation error is , of which the order number of *y* is four, so the difference approximation equation (2.5) is third-order accurate. In the same way, the local truncation error of equation (2.6) is
2.10
which shows equation (2.6) is also third-order accurate. So this predictor–corrector algorithm above is referred to as the three-step and third-order predictor–corrector (D3PC) algorithm.

Table 2 gives the step number and order number of five different schemes. The predictor–corrector central (PCC) [12] scheme is a one-step method, but its stability is not good. Bhagwat & Leishman [13] showed that a second-order scheme is a minimum accuracy and stability requirement for a numerical integration approach, but is not sufficient. In the wind turbine field, second-order Runge–Kutta (RK2) [4], fourth-order Adams–Moulton (AM4) [24] and PC2B [18] schemes have been studied in the previous work. The Runge–Kutta scheme will become complex when its accuracy is increased. The AM4 scheme is formally fourth-order accurate, and for small step lengths it is only mildly unstable. However, for practical rotor applications with typical time-steps of 5°–15°, this scheme may not be suitable because it is unstable for higher values of the step length. The PC2B scheme is a three-step predictor–corrector method, but it is only two-order accurate.

To better understand the concept of the nonlinear stability of the D3PC algorithm in the FVW method, the wake convergence calculation test was performed. A representative three-bladed Grumman wind turbine [18] with a nominal power output of 15 kW was used for the calculation. Table 3 gives some key parameters of the turbine. The present calculation was performed for a steady wind speed of 13 m s^{−1}.

Figure 2 shows the time histories of the *L*_{2}-norm of the change in the wake geometry of unyawed and yawed conditions using three difference approximations. Note that the PCC scheme shows a converging trend, but there is still an accumulation of numerical errors in the two conditions. This is because the discrete approximation of the temporal and spatial derivatives results in a larger error compared with the D3PC scheme. Because the implicit dissipation in both the D3PC scheme and the PC2B scheme damps out any numerical errors, the wake geometries stabilize quickly. However, the D3PC scheme is found to produce a stabler and more quickly convergent wake system than the PC2B scheme for both unyawed and yawed conditions.

### (c) Vortex core model

The vortex filaments comprise a series of straight-line vortex elements. The induced velocities at the control nodes of vortex elements are calculated using the Biot–Savart law. However, when a collocation point is very close to the vortex-line segment, a very high induced velocity will be obtained. In addition, the self-induced velocity has a logarithmic singularity. These two phenomena will cause convergence problems. To avoid the numerical problems, a viscous vortex core model is used in this paper. The core radius growth similar to the Lamb–Oseen [25] vortex model is used in the Biot–Savart law and modified by an empirical viscous growth model [26]. The core radius model is given by
2.11
where *r*_{0} equals to 5% of the rotor radius.

The vortical wake behind a wind turbine expands, which stretches the vortex filaments, so it is important to consider the effects of stretching of the vortex filaments for wind turbine application. In this study, stretching effects are taken into account by an application of a model developed by Ananthan & Leishman [27]. Assuming a change in the length because of filament straining to be *ϵ*=Δ*l*/*l*, which occurs over a time step, the effective core radius at the wake age angle *ζ* can be written as
2.12

### (d) Blade model

The simplest representation of the blade model is the classical lifting line model. However, this model does not capture the three-dimensional effects on a wind turbine blade. Using a lifting surface model, where the blade is divided into a matrix of spanwise and chordwise panels, has been shown to better represent the three-dimensional effects. However, the computational cost of using a lifting surface model is much higher than a lifting line model. A good compromise between the lifting line and the lifting surface models is the Weissinger-L model [28], in which bound vortices are located at the 1/4-chord and the control points are located at the 3/4-chord at the centre of each panel. The wake vortices extend downstream from the 1/4-chord forming a series of horseshoe filaments. In unsteady cases, shed vorticity is introduced into the wake to account for the temporal changes in bound circulation. A Weissinger-L blade model has been shown to give much better representation of the aerodynamics of a blade compared with a lifting line model and at the same time with a much lower computational cost than a lifting surface model.

The trailing filaments will roll up and form a single tip vortex filament in the far-wake. The trailing filaments cut off at a wake age angle of 60° in the near-wake. The strength of the tip vortex equals the global maximum bound vorticity over the span of the blade. The release point of the tip vortex is the tip of the blade.

### (e) Unsteady inflow under platform motions

Assuming that the wind turbine is rigid, the platform motions are defined as six types, i.e. surge, sway, heave, pitch, roll and yaw, as shown in figure 3. Aeroelastic effects may influence the accuracy of aerodynamic load and can be taken into account in the FVW method via coupling with the structural dynamics. However, aeroelasticity has a small effect on the tendency of results as the changes of blades are small compared with a floating platform's unsteady motions. In the rotor coordinate system (*x*, *y*, *z*), the three components of the inflow velocity at the control point of the blade element are
2.13

If it is assumed that there are no aerodynamic interactions between the blade elements, the axial and tangential components of the resultant velocity at the control point of the blade element (figure 4) are 2.14 and 2.15

### (f) Unsteady wake under platform motions

In the FVW scheme, the velocity fields and the vorticity strengths of the vortex filaments are unsteady both for floating wind turbines and for fixed-bottom wind turbines. However, the positions of the initial points of vortex filaments on the blade change with floating platform motion. This is different from the fixed-bottom wind turbine. The positions of the nodes of vortex filaments are described in the fixed coordinate system (*x*_{F}, *y*_{F}, *z*_{F}). The origin of the fixed coordinate system is the same as that of the platform coordinate system. The directions of *x*_{F}, *y*_{F}, and *z*_{F} are the same as *x*, *y* and *z* only when the wind turbine is completely stationary. The three components of the changed value of the initial point position compared with the value at the static condition can be written as
2.16

### (g) Three-dimensional rotational effect model and dynamic stall model

The resultant velocity and the attack angle *α* at the blade element can be obtained via the velocity triangle (figure 4). Then the blade loading and rotor performance can be calculated based on the blade element theory. Generally, the aerofoil data used in the FVW model are two-dimensional experimental or calculated data. When unsteady conditions are considered, the two-dimensional aerofoil data should be corrected for the three-dimensional rotational effect and the dynamic stall phenomenon.

The three-dimensional rotational effect is one of the typical differences between rotating rotor and fixed wing, resulting in stall delay, which is characterized by significantly increased lift coefficient compared with the corresponding two-dimensional case, and by a delay in the occurrence of flow separation to higher angles of attack. In [29], a comparison of six models is made to illustrate the significance of stall delay models and some defects in these models. The Du-Selig stall-delay model [30], which is coupled into the FVW model, is used to modify the aerofoil aerodynamic data by consideration of the three-dimensional rotational effect in this work.

Dynamic stall can exist for a high angle of attack, whereas stall hysteresis occurs even for small angles of attack under dynamic conditions. The delay in the flow field and the dynamic stall vortex as the shedding of a strong vortical disturbance from its leading edge are two important characteristics of dynamic aerofoils. Many mathematical models for its engineering applications have been provided, among which the well-known Beddoes–Leishman model [31] has widely been applied to predict the blade unsteady loads and is also used in this paper. In modelling the performance of an aerofoil in unsteady flow, the indicial technique used in the Beddoes–Leishman model implicitly includes the induced effect of the shed wake structure downstream of the aerofoil. Meanwhile, the wake structure generated in the free wake method contains filaments of shed vorticity and their induced effect on the blade flow field is calculated directly by application of the Biot–Savart law. Consequently, coupling of the free wake scheme and the dynamic stall model is hindered by this duplicative effect. This can be overcome by selectively neglecting the shed wake terms from the free wake model and calculating the induced effect via the dynamic stall model [32,33].

## 3. Unsteady calculation procedure

The initial wake geometry (a regular helix) and the parameters of the initial field are calculated according to rotor properties and the inflow when *t*=0. The predictor–corrector method is used to speed up the wake iteration until convergence, at which the wake geometry of *t*=0 is obtained and the flow field is updated. Then the time marches to the next step, at which the unsteady inflow and the unsteady wake due to the floating platform motions obtained from the computation of the hydrodynamic response are considered using equations (2.14)–(2.16). The predictor–corrector method is used again to update the wake geometry and the flow field at the new time step. The induction at the blade elements can be found from the new vortex wake. The information required by the dynamic stall model, such as mean angle of attack, change amplitude in angle of attack and the reduced frequency, can be obtained. Then the dynamic stall model is coupled in this procedure to calculate the blade unsteady aerodynamic loads. After this, the gross aerodynamic performance of the rotor can be calculated through integrals. Then the time step marches continually until the time course is completed.

## 4. Validation of the implemented model

The implemented FVW model is validated in this section by comparison between the calculated results and experimental data on a small-scale rigid wind turbine coupled with a TLP [22] (case I) and the Tjæreborg test turbine [23] (case II). Table 3 lists some key parameters of both turbines. Assuming that aerodynamic loads do not influence the hydrodynamic response, the aerodynamic performance of case I is investigated under given floating motions from the experimental measured data, so no hydrodynamic equations are solved. However, such an assumption will not be appropriate when doing the hydrodynamic analysis of the floating platform or investigating some extreme conditions such as extreme operating gust conditions. Although the Tjæreborg test turbine is a land-based turbine, a comparison with such tests is useful to validate the proposed aerodynamic model for the pitching and yawed cases.

### (a) Case I: effect of floating platform

The experiment using an OFWT model was conducted at the Fluids Laboratory at the University of Malta. In the first phase of the experiment, consisting of power measurements, the wind speed was kept constant at 8 m s^{−1}, whereas the rotor speed was varied over the range 360–1800 r.p.m. to alter the rotor tip speed ratio λ. The aerodynamic characteristics were compared to the fixed platform condition. Details of the main components of the experimental set-up are listed in [22]. Here, the results for only the one-dimensional regular wave frequency of 0.36 Hz translating into peak surge velocity of 0.06 m s^{−1} from the experiment are used to compare with the calculations.

Figure 5 shows the mean power coefficients, including fixed and floating platform conditions, at different tip speed ratios. The calculation results are obtained from the FVW model presented in §2. The unsteady inflow and the unsteady wake are considered in the simulation under the floating platform motions. The other calculation results are from WInDS, which is also based on the FVW method [22]. Generally speaking, the calculated results from the present model compare well with the experimental data both for floating and for fixed configurations and are better than those from WInDS at high λ. It is also shown that, at high λ, the FVW model predicts a difference in the mean power coefficient between the fixed and floating conditions. This is consistent with the experimental measurements, indicating that the present FVW model for OFWTs can availably predict the effect of the floating platform.

### (b) Case II

#### (i) Pitching case

When the rotor experiences a transient flow, the wake distorts firstly and achieves a new equilibrium condition after a while. This phenomenon can be called dynamic inflow effect. Unlike BEM theory in which a dynamic inflow model is required to handle the transient flow, the FVW method can represent the dynamic inflow effect by itself because the vortex filaments are allowed to freely distort under the influence of local velocity field. In this section, a pitching transient on the Tjæreborg test turbine [23] is simulated to illustrate the dynamic inflow effect. After an initial period with *θ*_{1}=0.1°, the blade pitch is rapidly increased to *θ*_{2}=3.7°, maintained at this level for 30 s and then rapidly decreased to its original value. The pitch change takes place while the wind turbine is operating in a steady wind speed of 8.7 m s^{−1} with a tip speed ratio of 8.1.

The rotor shaft torque calculated from different methods and the experimental data are plotted as non-dimensional values in figure 6. The quantities were non-dimensionalized according to
4.1
where *M*_{s2} is the stationary value after step in *θ*_{2}. The results all indicate an initial overshoot of the rotor shaft torque directly after a fast pitch change that gradually decays to steady-state values. The calculation result from the present FVW model agrees well with the experimental result. The results from TUDk code [23] and GH code [23] which were all based on the BEM method show higher overshoots than the experimental result.

#### (ii) Yawed case

The case of the Tjæreborg at a yaw angle of 32° is used for simulation to validate the present FVW model. The rotor speed is 22 r.p.m., the wind speed 8.5 m s^{−1} and the pitch angle 0.5°. Structure flexibility, tower shadow and tilt angle are not taken into account in the FVW model. The wind shear according to power law with exponent 0.31 is considered.

Figure 7 shows the variation of the flapping moment at *r*=2.75 m (blade root) along the azimuthal angles from 0° to 360°. The Unist model [23] is also based on the FVW method. The qualitative trend in the measured flapping moment is visible in three calculations. The results from the present FVW model and the Unist model [23] are closer to the measured results than the GH results [23]. The GH model that has the structure flexibility included predicts the measured shape better than the other models. The GH mean flapping moment is consistently lower than the measured level, which is explained by the higher induction from this model.

## 5. Unsteady aerodynamic analysis for large-scale offshore floating wind turbine

In this section, the FVW model is used to simulate the unsteady aerodynamic performance of a large-scale OFWT in yaw and turbulent wind. These data are compared and discussed to demonstrate differences in rotor performance and blade loading between floating and fixed conditions. The platform motion in this example is simulated by FAST rather than by the solution of hydrodynamic equations.

### (a) NREL 5 MW wind turbine and platform

The NREL 5 MW wind turbine, of which the rotor radius is 63 m, is studied as a calculation example in this paper. Details of the aerodynamic properties at the blade sections are in [2]. The aerofoils used in the blade are DU40, DU35, DU30, DU25, DU21 and NACA64. The lift and drag coefficients of these aerofoils used in the calculation model are also obtained from [2]. The mean wind speed inputted in the present simulations is kept at the rated speed of 11.4 m s^{−1}, while the rotor speed is 12.1 r.p.m.

The NREL 5 MW wind turbine is located on a floating TLP. The resulting FAST-simulated platform kinematics (figure 8 [3]) of the TLP for the rated operating condition, lasting 600 s for simulation, is adopted in this study.

### (b) Steady performance

The rotor power and rotor thrust of the NREL 5 MW wind turbine are calculated by the present FVW model at a number of given, steady, uniform wind speeds. The power production control strategy of this wind turbine is pitch-regulated and speed-variable. The control curves of the pitch angle and the rotor speed are from the NREL report [2] and are optimized by the FAST simulator in which the BEM theory is used. The results from FAST are used to compare with those from the present FVW model.

Figure 9 shows the rotor powers from 5 to 20 m s^{−1}. The rotor power increases cubically with wind speed before the rated wind speed. Above the rated wind speed, the rotor power remains at about 5300 kW. The agreement between the predicted powers from FVW and BEM is noted to be very good. The rotor thrusts from the FVW model (figure 10) also agree well with those from the FAST simulator.

### (c) Yawed condition and platform motions

The accuracy of the FVW model in unsteady conditions has been validated by case I and case II and the calculation results of the steady performance also agree well with the results obtained from the FAST simulator. This section and next section focus on the difference of the unsteady aerodynamic performance due to the floating platform motions, so no comparison with the other models will be made.

The yawed direction of the rotor is the same as the yawed direction of the platform. The yaw angle in this case is *γ*=10°. The simulations are run for the steady wind speed of 11.4 m s^{−1}. Figure 11 gives the variation of the rotor power and the rotor thrust along the time from 200 to 400 s calculated using the FVW model. Comparing the floating and fixed conditions, it is noted that both the power and the thrust have a great difference because of the platform motions shown in figure 8. Therefore, the control strategy of the OFWT should be correctly designed according to the floating platform motion effects on power output and aerodynamic loading.

Figure 12 shows the normal force coefficient *C*_{n} and the tangential force coefficient *C*_{t} at 94% radial position of the blade calculated from the FVW model. In the yawed conditions, the aerodynamic force coefficients of one blade vary in a cycle of 360°. The floating platform motions result in increase in both *C*_{n} and *C*_{t} and the amplitudes of these two coefficients are no longer kept constants in yawed conditions.

The figures showing the trends of load changes at other blade stations resulting from the floating platform motions are available in the electronic supplementary material. They show that the trends at other blade stations are similar to the change at 94% radial position of the blade, so the non-dimensional magnitudes of changes, which are the differences between fixed and floating coefficients divided by the one for the fixed condition, is given in figure 13. Based on the rigid assumption of the rotor, because the tangential velocity increases from the blade root to the blade tip, the change of the inflow angle resulting from the floating platform motion at the inner blade section is larger than that resulting from the same floating platform motion at the outer blade section. On the other hand, the nonlinearity of the aerodynamic coefficient of the inner section may be more complex than that of the outer section because of the larger angle of attack of the inner section. Therefore, non-dimensional magnitudes of changes decrease from the blade root to the blade tip.

### (d) Turbulence condition and platform motions

One-dimensional velocity change (axial component) is considered at the hub centre height of 90 m in the turbulence condition. The wavelet inverse transformation method [34] is used to create the turbulence wind field according to the advanced Von Karman power density spectrum. The axial velocity varies around the rated wind speed of 11.4 m s^{−1} along the time from 0 to 600 s (see figure 14 for this simulation). The roughness of the ocean surface is 0.001, the turbulence intensity is 0.0933 and the wind shear power law exponent is 0.2.

Figure 15 shows the variations of the rotor power and the rotor thrust along the time from 200 to 400 s in a turbulent wind with a mean value of 11.4 m s^{−1} calculated from the FVW model. The rotor power of the floating wind turbine oscillates around the mean value 5390 kW and the rotor thrust around 807 kN because of the axial turbulent inflow. The difference between the floating wind turbine and the fixed wind turbine is sometimes obvious owing to the large motions of the floating platform. This difference is also observed in *C*_{n} and *C*_{t} at 94% radial position (figure 16) and other blade stations (see the electronic supplementary material) calculated from the FVW model. Figure 17 shows that the non-dimensional magnitudes of loading changes due to floating platform motions also decrease from the blade root to the blade tip.

## 6. Conclusion

In the FVW model, the D3PC algorithm is developed for the difference approximation of the wake governing equation. The blade inflow conditions and the positions of initial points of vortex filaments are modified to consider the complex floating platform motions. The accuracy of the FVW model is validated by the experimental data for two test wind turbines. One of the turbines is a small-scale rigid wind turbine coupled with a TLP for the effect of the floating platform. The other is the Tjæreborg test turbine for the pitching transient and the yawed load validations. The calculation results for the two wind turbines agree well with the experimental data.

The unsteady aerodynamic performance of a large-scale OFWT, NREL 5 MW wind turbine with floating TLP is predicted by the FVW model. Yawed and turbulence conditions are considered in the calculations. The comparisons between the floating wind turbine and the fixed wind turbine indicate that there are obvious differences between them, both in rotor performance and in blade loading. The magnitudes of loading changes due to the floating platform motions decrease from the blade root to the blade tip.

## Funding statement

This work was funded jointly by the National Basic Research Program of China (973 Program) (2014CB046200), China Postdoctoral Science Foundation (2014M551495) and the Programme of Introducing Talents of Discipline to Universities (111 Project) (B14022).

## Acknowledgements

We thank prof. Tonio Sant for providing the original data of the wind turbine experiment conducted at the Fluids Laboratory at the University of Malta.

## Footnotes

One contribution of 17 to a theme issue ‘New perspectives in offshore wind energy’.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.