Numerical analysis of aerodynamic performance of a floating offshore wind turbine under pitch motion

Journal Pre-proof Numerical analysis of aerodynamic performance of a floating offshore wind turbine under pitch motion Yuan Fang, Lei Duan, Zhaolong Han, Yongsheng Zhao, He Yang PII:

S0360-5442(19)32316-3

DOI:

https://doi.org/10.1016/j.energy.2019.116621

Reference:

EGY 116621

To appear in:

Energy

Received Date: 18 July 2019 Revised Date:

31 October 2019

Accepted Date: 23 November 2019

Please cite this article as: Fang Y, Duan L, Han Z, Zhao Y, Yang H, Numerical analysis of aerodynamic performance of a floating offshore wind turbine under pitch motion, Energy (2019), doi: https:// doi.org/10.1016/j.energy.2019.116621. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Numerical Analysis of Aerodynamic Performance of a Floating Offshore Wind Turbine under Pitch Motion Yuan Fang3, Lei Duan*1, 2, 3, Zhaolong Han1, 2, 3, Yongsheng Zhao1, 2, 3, He Yang4 1 State Key Laboratory of Ocean Engineering, Shanghai, 200240, China 2 Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai, 200240, China 3 School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China 4 Ocean Engineering Department, Texas A&M University, College Station, TX 77843, USA

ABSTRACT The aerodynamic performance of wind turbines is essential to evaluate their electricity-generating capability. Compared with the wind turbines with traditional fixed structures, the aerodynamic performance of floating offshore wind turbines (FOWTs) is affected by the additional platform motion, especially for the pitch motion. In view of this, the computational fluid dynamics (CFD) method with the turbulence model of improved delayed detached eddy simulation (IDDES) is applied. Before the numerical simulation, the CFD model of the redesigned 1:50 scale rotor from NREL 5MW wind turbine is verified with the available experimental data and numerical comparison. Then, the aerodynamics of the FOWT under the harmonic pitch motion with different periods and amplitudes is investigated. It is shown that the aerodynamic performance of the FOWT is sensitive to these parameters of the pitch motion. First, amplitudes of the rotor thrust and torque decrease with the increment of the pitch period. In particular, the wake interference phenomenon is most evident when the pitch period is short and the pitch amplitude is high, which may compensate some energy to the rotor. In addition, amplitudes of the rotor thrust and torque increase with the increment of the pitch amplitude. Also, the stall phenomenon happens when the pitch amplitude is high, which may impact FOWTs’ aerodynamic performance adversely. Finally, the rotor power increases under the periodic pitch motion, especially by decreasing the period or increasing the amplitude. It is concluded that the pitch motion of the platform will change the aerodynamic performance of a wind turbine and should be taken into consideration during design procedure.

KEYWORDS: floating offshore wind turbines; aerodynamic performance; pitch motion; CFD; IDDES 1. Introduction Floating offshore wind turbines (FOWTs) have been attracting attention from both academia and industry in the last decade. As complements to the offshore wind turbines with traditional fixed foundations which is only installed in the near and shallow sea area [1], FOWTs have many advantages by being installed in the far and deep sea area with stronger and more consistent wind, less interference to residents, and more available installation sites [2]. However, the shortages always come with advantages, such as its more complicated dynamic characteristics brought by the interaction between the wind turbine and its floating platform [3]. Therefore, researchers have been continuously dedicating to FOWTs, especially in the field of dynamics with their own properties [4-7]. Among the dynamics of FOWTs, their aerodynamics is quite important but different from that of wind turbines with fixed foundations, and therefore has interested researchers. For importance, the aerodynamics of FOWTs is inherently tied to their generating efficiency [8]. For difference, the aerodynamics is influenced by the oscillating motion brought by the sophisticated offshore environment [9], which may be decomposed into six degrees of freedom (DOFs) as surge, sway, heave, roll, pitch and yaw [2, 10, 11]. To shed light on the particular aerodynamics of FOWTs, many scholars conducted researches on the aerodynamic performance when a FOWT underwent different motion such as surge [12-14], heave [15], pitch [16-20], yaw [21] and the combined motion like pitch & yaw [22], pitch & surge [23] or even surge, heave & pitch [24]. The pitch motion is one of the most important of these 6 DOFs motion which may lead to significant influence on FOWTs’ aerodynamics. Three possible reasons are presented. First, the relative velocity between the flow and the rotor varies when a FOWT is pitching upwind or downwind. Bayati et al. performed tests on a 1:75 scale model of the DTU 10MW wind turbine and observed that the aerodynamics changed in accordance with the change of relative velocity caused by the pitch motion [25]. In addition, the flow field around the rotor may be switched when a FOWT undergoes the pitch motion. Sebasian and Lackner divided the complex flow field around the rotor of a FOWT into four states including the windmill braking state, the turbulent wake state, the vortex ring state and the normal working state

[26, 27]. These four wake states were further validated by Rockel et al. resorting to Particle Image Velocity [28]. Finally, the turbulence may be generated in the wake field when a FOWT moves into and out of it. Khosravi et al. conducted an experiment by a 1:300 scale model FOWT and found that the turbulent mixing process in the wake field was affected by the pitch motion [15, 16, 29]. However, it is still unclear how pitch motion would influence the aerodynamic performance of FOWTs. In order to study the aerodynamic performance of FOWTs, several numerical tools have been developed and utilized, based on the blade element momentum (BEM) theory, the generalized dynamic wake (GDW) model, the vortex lattice method (VLM), the cascade model and the free vortex method (FVM). Jonkman et al. developed a fully coupled aero-hydro-servo-elastic code based on the BEM theory and GDW model [10, 30, 31]. Vaal et al. utilized the BEM theory with a quasi-steady wake model to study FOWTs’ aerodynamics under the surge motion [32]. Jeon et al. employed the VLM to predict the aerodynamic loads of a FOWT in a turbulent wake state [33]. Shen et al. and Wen et al. used the FVM to study the unsteady aerodynamics of a FOWT under the pitch motion [19, 20, 34]. However, these approaches may only provide macro information such as thrust, torque and power of FOWTs but fail to explain their changes from the perspective of detailed flow field information [35]. Moreover, these techniques require empirical formula or correction models, and all of these assumptions and approximations restrict their use on analyzing and explaining the sudden change of the FOWT’s aerodynamic performance [33]. The computational fluid dynamics (CFD) method is presented as a reliable technique with provision of detailed flow information, in which the Reynolds averaged Navier-Stokes (RANS) model is regarded as an ideal model to conduct numerical simulations due to the relatively lower requirement of mesh and the proper accuracy [36]. Tran and Kim conducted a CFD study with the RANS model to investigate the unsteady aerodynamic interference on a FOWT by the pitch motion, which was the initiative research and made the foremost contribution in this specific field [17, 37]. Liu et al. also adopt the CFD method with the RANS model to study the aerodynamics of the same FOWT under superimposed motion containing surge, heave and pitch [24]. However, the results of the CFD method with the RANS model may not be accurate enough because important vortex structures (especially the ones around the rotor) would be ignored or averaged when the fluid appears intense unsteady characteristics [38]. Therefore, a turbulence model with more detailed information of the flow field and acceptable computational costs is desirable to accurately simulate or predict the FOWTs’ aerodynamic performance under the pitch motion. The improved delayed detached eddy simulation (IDDES) [39] may provide such a reliable and economical turbulence model, which was developed based on the detached eddy simulation (DES) [40] and delayed detached eddy simulation (DDES) [41]. In this model, the standard RANS model is adopted in the near wall and subgrid fields while the large eddy simulation (LES) model is utilized in the far field. Hence, the IDDES model owes the properties of higher accuracy than the RANS model and less computation costs than the LES model [39]. Because of these advantages, the IDDES model is widely and successfully applied in many fields. Krappel et al. employed the IDDES model to calculate the Francis pump turbine flow, and validated its results by the experiment [42]. Wang et al. also utilized the IDDES model to observe different scales of vortices and the turbulent transport process of the hydrogen-fueled supersonic combustion [43]. Moreover, it is important to note that Lei et al. adopted the IDDES model in investigating the aerodynamic loads on a vertical axis FOWT in the surge and pitch motion [44], which shows great potential of the IDDES model in studying the FOWTs’ aerodynamic performance under the pitch motion. However, to the authors’ best of knowledge, there seems little investigation on the aerodynamic performance of a horizontal-axis FOWT under the pitch motion by using the IDDES model, which attracts the authors’ interest and turns out to be the purpose of the present work. Hence, this paper will employ the IDDES turbulence model to study the aerodynamic performance of a FOWT under the pitch motion. In addition to the overall influence on the rotor thrust, torque and power by the pitch motion which is well studied by Tran et al. in previous studies [17, 37], we particularly focus on explaining the sudden change of the FOWT’s aerodynamic performance with the physical flow phenomenon around the rotor and in the wake, which has been not investigated clearly by the numerical tools with high fidelity as the CFD method with the IDDES model. The main content of this paper is organized as follows. In section 2, the IDDES turbulence model and the numerical settings including mesh, boundary conditions and solver, as well as dependency tests are introduced. In section 3, the validation of the numerical simulation is conducted. In section 4, the aerodynamics containing the thrust, torque and power of the FOWT under the pitch motion with different periods and amplitudes are obtained and discussed. Also, the related flow field information is also presented and compared with that of the wind turbine with fixed structures. Finally, the summary of the main conclusions is given in section 5.

2. Numerical simulation The numerical simulations are conducted on a 1:50 scale rotor, which is redesigned from the NREL 5MW wind turbine by Zhao [45]. In order to obtain the details of the flow field and to reduce the computational consumption, the CFD method with the IDDES turbulence model incorporated into the STAR-CCM+ software [SIEMENS, Germany, [46]], is used for all the following simulations. The overset mesh technique is used to realize the rotor’s rotation and platform’s pitch motion. The mesh topologies including the Cartesian grids, prism boundary layer cells, surface remesher and automatic surface repair are also utilized in the simulation [46]. 2.1 IDDES model The IDDES model is a hybrid model proposed by Travin in 2006 [39] and developed by Shur in 2008 [47] which combines the advantages of both the RANS and LES turbulence models. The turbulent kinetic energy equation in the IDDES model can be presented as

[48]

( ) + =



1
 
[( + ) ] 
   +   − =0
 

(1)

where , , , ,  ,  ,  and  are represented as density, turbulent kinetic energy, time, dynamic viscosity, tensor of stress, mean strain rate, the three components of the fluid particle’s velocity and position, respectively. Moreover,  is the IDDES length scale which can be written as  = "̅ (1 +

# )$%&

+ (1 − "̅ )'(&

(2)

where ̅ = )* ((1 −

"

$%&

"+ ),  )

(3)

-

 = .×0

'(& = 1(& ∆(&

(4)

(5)

In addition, the blending function  and the elevating function # in Equation 2 and 3 are introduced to the IDDES length scale to consider Wall-Modeled LES (WMLES) capability which can calculate turbulence in the boundary layer when the incoming stream is unstable and contains turbulence. Also, "+ is written as "+

= 1 − *3ℎ [(86"+ ) ]

(6)

in which 6"+ is the turbulent analogue of 6" that is the marker of the wall region. Moreover, the . in Equation 4 is a constant, whose value is 0.09 in the shear-stress transport (SST) k-ω model, and 0 is the specific dissipation rate. Finally, the 1(& in Equation 5 is the model coefficient, which blends the values obtained from the k-ɛ and k-ω model, and ∆(& is the mesh length scale of IDDES as ∆(& = )73 ()* (0.15:, 0.15∆, ∆;< ), ∆)

(7)

where ∆;< is the smallest distance between the cell centers of the one under consideration and its neighbors; ∆ is the grid scale; : is the distance to the wall. The detailed information can be obtained in Ref. [47]. 2.2 Geometrical model and grid strategy 2.2.1 Geometric model A 1:50 scale rotor of the NREL 5MW wind turbine meeting thrust similarity criterion [45] is employed. In detail, the rotor is 2.52m in diameter, which composes of three blades with 0° blade pitch angle, that is designed based on optimization of the NACA 4412 airfoil. The tower and the nacelle are not taken into consideration for the slight effects although they may result in aerodynamic fluctation, especially in the case with high wind velocity [37]. The three dimensional model built in SOILDWORKS [Dassault aircraft, France, [49]] is shown in Figure 1. The inertial coordinate system is defined as Figure 1 whose origin lies at the center of the rotor’s hub without any extra motion; the X axis is upwind; the Z axis is downward along with gravity; the Y axis is the cross product of the Z axis and the X axis.

Wind velocity

Y

X

Z

Diameter=2.52m

Figure 1. The computational model of the 1:50 scale rotor as built in SOILDWORKS 2.2.2 Mesh topology The whole computational domain, two refined parts and one rotation part are all cylinder shape due to consideration of rotor’s rotation, whose sizes are described as follows. First, the radius, inlet length and outlet length of the whole computational domain are 3D, 3D and 6D (D is the rotor diameter), respectively, shown in Figure 2(a). The outlet length is prolonged two times for better capturing the wake interaction. In addition, two refined cylinders are used for smooth transition. The first refined part keeps up with the whole computational domain in length for reducing the numerical diffusion of the inlet wind, meanwhile reduces its radius to 1.5D. The second refined part is designed based on the pitch motion so that the rotor always stays within this refined region. Specifically, it is applied with the radius of 1D and extended 1D and 2D in upstream and downstream direction, respectively. Finally, the rotation part is 0.8D in radius and 1D in length, at whose center the rotor is located. Furthermore, the Cartesian mesh type is applied in all of the computational fields whose details of mesh are presented as follows. To mention first, the basic mesh size of the whole computational domain is twice the length of the first refined part, which is well aligned as 1:2 ratio on each boundary of the refined part, shown in Figure 2(b). So does the ratio of basic mesh size between the first and second refined parts. Besides, the overset mesh is employed to simulate the rotor’s rotation and pitch motion, shown in Figure 2(c). In order to improve the interpolation precision, the mesh on both sides of the overset boundary remains the same Cartesian mesh with equal size, and the linear interpolation method is chosen on the overset boundary. What’s more, three surface controls are generated due to the complex shape of the rotor, for controlling the hub, main blades and trailing edges. The target surface size of the hub is 2.5×10-2m and the minimum one is 3.125×10-3m while the corresponding values of the main blades except trailing edges are 1.25×10-2m and 7.8125×10-4m, shown in Figure 2(d). The trailing edges are refined for avoiding blades’ deformation whose target and the minimum surface sizes are 1.5625×10-3m and 7.8125×10-4m. Last but not least, to meet the requirement of “Y+” less than 5 by STAR-CCM+ [46], the total thickness of the prism boundary layer is set at 0.005m with the first layer’s thickness of 3.0305×10-4 and the growth rate of 1.2, shown in Figure2(e). According this setting, the monitored “Y+” is less than 5, indicating that the near-wall cells are placed in the viscous sublayer. Pressure ontlet

Velocity inlet

Symmetry plane

(a)

The whole computational field

The first refined part

The second refined part

(b)

Overmesh boundary

Blade

(d)

Boundary layer cells

(c) (e) Figure 2. Mesh topology for the computational model: (a) the whole computation field; (b) two orthogonal sections of the whole mesh; (c) the position of the overmesh boundary; (d) mesh topology around one blade; (e) detailed mesh on one section of the blade 2.3 Boundary and solver setting Appropriate boundary conditions are set up. At the inlet of the computation domain, velocity inlet condition is set. The wind speed is 1.51 m/s, which is scaled down from the real wind speed, and the wind direction is contrary to the positive X axis. At the outlet of the domain, pressure outlet condition is set, where the static pressure equals to the standard atmospheric pressure. Around the computational domain, the symmetry condition is executed. This setting enlarges the domain that is already large enough to ignore the boundary constraint influence, which may well simulate the case in open air. On the rotor’s face, the non-slip condition is defined because there is no relative motion on it for the viscous effect. IDDES equations with the SST k-ω model is applied. For the three-dimensional solver, the segregate flow model and the unsteady implicit method are chosen in the solver setting. The second-order implicit time scheme and the second order upwind spatial discretization are applied to discrete equations that may guarantee the second-order accuracy. Furthermore, the maximum number of internal iteration is 10 in both time step and spatial discrete [50]. In order to save calculation time of the rotor under the pitch motion, a pre-calculated case without any extra motion is utilized, in which the rotor rotates for 17.25s (23 revolutions) with rotational speed at 80rpm according to our pretested results. Then, the cases under pitch motion start at 17.25s when the flow field in the pre-calculated case acts as their initial condition to quickly obtain the convergence results. The different cases are with different pitch amplitudes or periods, but with the same initial condition. This helps to quickly obtain stable results because the incoming flow field and wake flow field at 17.25s are more similar to the real situation than those at 0s. All of the computations are executed on a parallel computing sever with AMD Ryzen Threadripper 2950X 16-Core processor and 64 GB RAM. A case under a harmonic pitch motion can be completed in 24 hours. 2.4 Mesh and time dependency test To solve the nonlinear discrete differential equation, it is significant to select the size of mesh at different part in the computational domain and the time step for both calculation accuracy and efficiency [44]. Three different mesh cases, named coarse mesh, medium mesh and fine mesh are set to perform the mesh independency test before formal calculations. All of the three mesh cases ensure the satisfactory mesh transition and the rotor’s integrity without large deformation. The coarse mesh size increases 21/3 times of the medium mesh size while the fine mesh size decreases 21/3 times of it, which means increase and decrease 2 times of the medium mesh number, respectively. However, the boundary layer mesh is maintained the same because it needs to meet the requirement of “Y+” in the IDDES turbulence model. According to this strategy, the number of the coarse, medium and fine mesh cases are 4.89 million, 7.31 million and 11.53 million, respectively. Table 1 presents the calculated thrust of different mesh cases and their relative errors compared with the experimental data when the tip speed ratio (TSR) is 7. It is shown that the relative error of coarse mesh is larger than the ones of medium and fine mesh. Besides, the computation cost of fine mesh is huger as its total mesh number is 2.4 or 1.6 times more than the ones of coarse or medium mesh, shown in Table 1. Hence, the medium mesh is employed in all the following simulation cases due to the consideration of computation precision and consumption. Table 1. Mesh dependency test with TSR=7 Mesh Case name

Total number of mesh

CFD thrust

Experimental thrust (N)

Relative error

Coarse Medium Fine

4887115 7307967 11529186

（N） ） 8.116 8.097 8.029

[45] 7.892 7.892 7.892

2.838% 2.598% 1.736%

For the unsteady simulation, the time step can be regard as another key indicator for the consideration of both computation precision and costs. It is meaningful to select an appropriate time step because a small one may consume much computation time while a large one may pay the price of unreliable results [12]. Hence, three different time steps are considered to conduct the time independency test before formal numerical simulations. According to normal selection of the time step for rotating machinery [50], different rotational degrees as 1°, 3° and 4° per time step corresponding to ∆- , ∆ and ∆ are adopted. The calculational thrust of different rotational degrees per time step is compared by Figure 3 in which the three lines are quite similar. However, it is shown that the curve of thrust predicted by ∆ is relatively different from the ones by ∆- and ∆ , especially the part near the climax. Meanwhile, the computation consumption of ∆- is relatively huge because it costs 88500 iterations to simulate 23 revolutions (17.25s) while ∆ costs 43000 iterations and ∆ costs 22000 iterations. Consequently, ∆ is selected by the consideration of computational precision and consumption. What’s more, in order to obtain pictures in post-processing at every 1/8 times of the rotational period, ∆ is chosen considering that 45 may be divisible by 3.

Figure 3. Time dependency test with three different computational time steps (∆- , ∆ , ∆ ) when TSR=7 2.5 Pitch motion The harmonic pitch motion with different amplitudes and periods are assigned to the computational model by a user defined function. The pitch center is at (0, 0, 1.54) in the inertial coordinate system as mentioned above. In this study, the displacement of the rotor is assumed to comply with the principle of sine motion as 2E >?+@A = >B73 C G F

(8)

Thus, the form of the corresponding angular velocity is a cosine function as 2E> 2E 0?+@A = C G HIB C G F F

(9)

where > and F are the amplitude and period of the pitch motion;  is the time.

In this study, >=1.5°, 2.25° and 3° are chosen as the amplitude, which corresponds to the motion of the full-size FOWT pitching with 1.5°, 2.25° and 3° amplitude, based on the Froude similarity criteria. Similarly, F=0.375s, 0.75s, 1.50s and 3.00s are chosen as the period, corresponding to the motion of the full-size FOWT pitching with 2.652s, 5.303s, 10.607s and 21.213s periods, respectively. Figure 4 presents the harmonic angular displacement and angular velocity with 2.25° amplitude and 0.75s period, in which T1 ~ T9 divided the period into 8 equal parts. Also, the detailed flow field information will be recorded respectively at those time points later.

Figure 4. Variation of the angular displacement and velocity in a pitch period (>=2.25°, F=0.75s) The pitch motion may cause the variation of the distributed velocity around the rotor. To be exact, the induced velocity by the pitch motion increases linearly as the increasing of distance away from the pitch center. Therefore, the induced velocity of three blades is different at the same time, so does the different sections of the same blade.

3. Validation The CFD model is validated by both experimental and other numerical results. The first subsection provides validations of the rotor without any extra motion, in which the rotor thrust computed by the CFD model is compared with that of the available experimental data. For further validation, the rotor thrust under the pitch motion calculated by the CFD model, FVM and BEM theory are compared, presented in the second subsection. 3.1 Validation of the rotor without any extra motion In this subsection, the experimental results of the same scale rotor as the one in the CFD model without any extra motion [45] is adopted, as there is few experimental tests on FOWTs with the pitch motion only. In the experiment [45], the wind speed was constant 1.61m/s, and wind turbine was operated at different rotational speeds to obtain the TSR values, as summarized in Table 2. The thrust calculated by the present CFD model is compared with that obtained by the experiment in Figure 5.It shows that the variation trend of calculated thrust against TSRs fits well with the experiment data [45]. In addition, the relative error is also presented in Figure 5, in which the maximum relative error of thrust occurs at TSR=5 while the minimum one occurs at TSR=7. It also shows that the relative error is large at high and low TSRs, and is small at medium TSR. Among these TSRs, the following simulation cases in this paper only use TSR=7 with constant wind velocity and rotational speed of the rotor, at which the relative error is acceptable. Therefore, the present CFD model may be utilized in the subsequent simulations. Moreover, all these comparisons demonstrate the feasibility of the CFD method with the IDDES model to investigate the aerodynamic characteristics of the wind turbine. Table 2. Parameters of the experiment for validation [45] Case name Validation 1 Validation 2 Validation 3 Validation 4 Validation 5 Validation 6 Validation 7 Validation 8

Wind speed (m/s) 1.61 1.61 1.61 1.61 1.61 1.61 1.61 1.61

Rotational speed of the wind turbine (rpm) 36.61 48.81 61.01 73.21 85.41 97.61 109.8 122.0

TSR of the wind turbine 3 4 5 6 7 8 9 10

Figure 5. Validation of the present CFD model by comparing with experiment on thrust [45] 3.2 Validation of the rotor with pitch motion In this subsection, the numerical results by the FVM [51, 52] and BEM theory [52] are employed, as a compromise scheme instead of the experimental results of the same rotor under the pitch motion. In particular, the lift and drag coefficients of the airfoil NACA4412 used in the FVM and BEM theory were obtained from the website of Airfoil Tools [53] and then addressed by AirfoilPreppy master [National Renewable Energy Laboratory, United States of America, [54]]. In the numerical simulation, the wind speed is constant 1.51m/s. Meanwhile, the wind turbine is operated at 80rpm. Figure 6 presents the comparison of rotor thrust among the CFD model, FVM and BEM theory when the rotor undergoes the pitch motion with the same period of 1.5s and different amplitudes of 1.5°, 2.25° and 3°. As shown in Figure 6, these three lines have similar tendencies, which proves the reliability of the CFD model. However, large deviations among the three lines occur near the climax where the dynamic stall may happen which turns out to be the significance of numerical investigation resorting to the CFD method.

(a) (b) (c) Figure 6. Validation of the present CFD model by comparing with the FVM and BEM theory on thrust (a) >=1.5°, F=1.5s. (b) >= 2.25°, F=1.5s. (c) >=3°, F=1.5s

4. Results & discussion In this section, the FOWT’s aerodynamic performance affected by the pitch motion with different periods and amplitudes is presented, discussed and explained, not only the overall rotor thrust, torque and power, but also the sudden changes on them. Several hypotheses are given: (1) the fluid (air)’s density remains constant during the simulation process; (2) the inlet wind is a constant stream whose speed is 1.51m/s; (3) the rotor rotates at a constant rotational speed of 80rpm; (4) the rotor undergoes harmonic pitch motion whose rotating center is at (0, 0, 1.54m) in the inertial coordinate system. 4.1 Unsteady aerodynamics under pitch motion with different periods This subsection shows the results demonstrating how small pitch motion would affect the variation of FOWTs’ aerodynamics. To study the effect, harmonic motion with four different periods 0.375s, 0.75s, 1.5s and 3.0s are exerted on the FOWT. Unsteady aerodynamic performance with these pitch periods is compared and analyzed. The rotor vorticity during one pitch period with 2.25° amplitude and 0.75s period is shown from different views in Figure 7 ~ 9. The

value of Q criterion is 1.0/s2, to clearly present those instantaneous vortices at different pitch positions. 0.232D

(a)

(b)

(c)

(d)

0.161D

(e)

(f)

(g)

0.200 D

(h)

0.200 D

(i) (j) (k) Figure 7. The instantaneous vorticity of the rotor under pitch motion (>=2.25°, F=0.75s): (a) ~ (h) denote the instantaneous vorticity at T1 ~ T8 in (i), respectively; (j) and (k) denote the instantaneous vorticity without pitch motion at T1 and T3, as the comparison In Figure 7, strong blade tip vortices can be observed. Also, vortices detached from the hub and the transition part between the airfoil and cylinder are visible. Figure 7(a) ~ (h) show the vortices at different time points of the pitch case, and Figure 7(j) and (k) show the vortices of the reference case without any extra motion, in which the vorticity is conformed around the rotor and the gap between the first revolution to the next one is nearly constant. Unlike the reference case, however, inconformity velocity is distributed on the rotor’s surface due to the different distance from the pitch center. From point T1 to T3, the rotor moves forward (along with the X axis or opposite to the wind) from the equilibrium position and reaches the positive maximum distance. In the process, the distance of the gap between the first revolution and the next one becomes larger, shown by Figure 7(a) ~ (c), and the maximum value is 0.232D in Figure 7(c) (corresponding to the point T3). On the contrary, when it moves backward and reaches the negative maximum position, the corresponding distance narrows down and the minimum value is 0.161D in Figure 7(g) (corresponding to the point T7). Moreover, from point T7 to T8,

the gap increases again and two vortices merge together, shown in Figure 7(g) and (h). This wake interference may affect the rotor’s aerodynamic performance, which may only be observed by the advanced CFD method rather than the BEM theory, GDW model or FVM. To provide more insight into this phenomenon, the side-view of vorticity visualization is shown in Figure 8. In order to better contrast the vorticity, the side-view vorticity on every 1/8T is saved in the coordinate following the pitch motion, shown in Figure 8(a) ~ (h). Also, that of the reference case without any extra motion is shown in Figure 8(j) and (k). Similar to Figure 7, strong blade tip and near hub vortices are illustrated in Figure 8. As the rotor moves forward, the vorticity increases due to the increment of the relative velocity, since the velocity of the platform has the opposite direction to the wind velocity. Meanwhile, the gap distance between the first revolution and the second one changes from 0.200D to 0.232D, shown by Figure 8(a) ~ (c) (corresponding to the point T1 ~ T3). On the contrary, the vorticity becomes weaker and the distance decreases when the model moves backward, shown by Figure 8(d) ~ (g) (corresponding to the point T4 ~ T7).

0.200D

0.232D D

(a)

(b)

(c)

(d)

(g)

(h)

0.161D

(e)

(f)

0.200D

0.200D

(i) (j) (k) Figure 8. The side-view vorticity of the rotor under pitch motion (>=2.25°, F=0.75s): (a) ~ (h) denote the side-view vorticity at T1 ~ T8 in (i), respectively; (j) and (k) denote the side-view vorticity without pitch motion at T1 and T3, as the comparison The instantaneous vorticity at the distance of 0.1D downwind away from the rotor is given in Figure 9. Figure 9(a) ~ (h) show the vorticity at different time points of the simulation case with the pitch motion while Figure 9(a’) ~ (h’) show the vorticity at the same time points of the reference case without extra motion. It is noticed that the vorticity on the rotor plane of the reference case remains unchanged and it is distributed almost equally on each blade from Figure 9(a’) to (h’). However, it differs from that of the simulation case. First, the vortex becomes stronger when the rotor moves upwind, as shown in Figure 9(a) ~ (c) (corresponding to the point T1 ~ T3). Then, the vortex becomes weaker because of the decrement of the relative velocity when the rotor moves downwind, as shown in Figure 9(d) ~ (g) (corresponding to the point T4 ~ T7). Finally, the vorticity strengthens again with the change of the motion direction, as shown in Figure 9(g) ~ (h) (corresponding to the point T7 ~ T8). Besides, the vorticity varies among the three blades in each figure, due to the non-uniform relative velocity around the rotor plane because of the pitch motion.

with pitch motion, T1

(a)

without pitch motion, T1

(a’)

with pitch motion, T5

(e)

without pitch motion, T5

with pitch motion, T2

(b)

without pitch motion, T2

(b’)

with pitch motion, T6

(f)

without pitch motion, T6

with pitch motion, T3

(c)

without pitch motion, T3

(c’)

with pitch motion, T7

(g)

without pitch motion, T7

with pitch motion, T4

(d)

without pitch motion, T4

(d’)

with pitch motion, T8

(h)

without pitch motion, T8

(e’)

(f’)

(g’)

(h’)

(i) Figure 9. The vorticity at a distance of 0.1D downwind away from the rotor with pitch motion (>=2.25°, F=0.75s): (a) ~ (h) denote the vorticity with pitch motion at T1 ~ T8 in (i), respectively; (a’) ~ (h’) denote the vorticity without pitch motion at T1 ~ T8, as the comparison As shown in Figure 7 ~ 9, even small pitch motion may cause significant aerodynamic effects on FOWTs, not to mention the wake interference phenomenon which can be clearly seen in Figure 7(g). To figure out the unsteady aerodynamic performance, the rotor thrust and torque under different pitch periods are compared. The pitch amplitude is chosen as 2.25° and the periods are chosen as 0.375s, 0.75s, 1.5s and 3.0s. The angular velocity of the pitch motion varying with time is shown by Figure 10, in which the latter period is two times of the former one while the latter amplitude is half of the former one.

Figure 10. Angular velocity of pitch motion with variation of periods As shown in Figure 11, the general trend of the rotor’s aerodynamics is consistent with the platform’s pitch motion. The rotor thrust and torque responses are similar to the form of the pitch angular velocity. It may be explained by the change of relative velocity between the rotor and the flow. It is noticed that the variation of the relative velocity impacts the rotor thrust and torque directly. The periods of both the rotor thrust and torque increase with the increment of the pitch period from 0.375s to 3.0s, shown in Figure 11(a) ~ (d). At the meantime, the amplitudes of the rotor thrust and torque decrease with the increment of the pitch period, which may be explained by Figure 10 that the increment of the period results in the decrement of the relatively velocity’s amplitude.

(a)

(b)

(c)

(d) Figure 11. The unsteady aerodynamics of the rotor during pitch motion with different periods: (a) the thrust and torque when F=0.375s; (b) the thrust and torque when F=0.75s; (c) the thrust and torque when F=1.5s; (d) the thrust and torque when F=3.0s To obtain more information of the unsteady aerodynamics of FOWTs under the pitch motion, all the rotor thrust and torque with 0.375s, 0.75s, 1.5s and 3.0s pitch periods are placed in Figure 12. It is shown that the equilibrium positions of the rotor thrust and torque remain unchanged, indicating that pitch period variation causes no effect on the aerodynamics at the equilibrium position. When the curves of rotor thrust and torque come across their equilibrium positions, the pitch angular velocity is zero that the relative velocity is only composed of the uniform wind velocity. At these positions, the FOWT can be regard as the ones with fixed structures. Therefore, no matter what the pitch period is, the equilibrium positions of the rotor thrust and torque remain the same as the ones with fixed structures, respectively. However, there are differences at the maximum and minimum values, which are especially evident on the curve of the rotor torque. It implies that the rotor aerodynamics do not meet the standard harmonic function as the pitch motion. For the inconformity of the upper and lower part of the curves, it may provide a reasonable explanation that the rotor thrust and torque are proportional to the square of the relative velocity.

Figure 12. Comparison of the rotor’s unsteady aerodynamics under pitch motion with different periods: (a) the thrust; (b) the torque Moreover, the phenomenon of wake interference is discovered with the help of the advanced CFD method. The wake interference is the strongest when the pitch period is short and the amplitude is high. In this study, the shortest pitch period is 0.375s while the highest pitch amplitude is 3.0°. In this situation, the amplitude of the pitch angular velocity is 0.8773 rad/s and the longest distance away from the pitch center is 2.8m (the distance between the hub center and the pitch center 1.54m plus the rotor radius 1.26m). Therefore, the greatest speed induced by the pitch motion is 2.46m/s whose value is even bigger than the wind speed at 1.51 m/s.

(a) (b) Figure 13. Unsteady aerodynamics under pitch motion with >=3°, F=0.375s: (a) the thrust; (b) the torque The aerodynamic performance of the severe pitch case with 3.0° amplitude and 0.375s period is shown in Figure 13. Two phenomena are observed. For one thing, there are abrupt changes on curvature where the curves of rotor thrust and torque go downward and cross their equilibrium positions. The changes show that the rotor thrust and torque are a little bigger than those they should be. Corresponding to these positions, the pitch displacement reaches its positive maximum position and begins to move backward. Therefore, the phenomenon of wake interference leads to these changes by transfer of energy from the wake to the rotor. For another thing, severe fluctuations are found when the rotor torque is near its minimum position. Corresponding to these positions, the platform is pitching downwind with its greatest angular velocity so that the velocity induced by the pitch motion even overrides the wind velocity at some upper part of the rotor plane, which means the velocity of the rotor’s upper part is greater than the velocity of detaching vortices. Meanwhile, the severest phenomenon of wake interference happens at these places that the detached vortices provide energy to the rotor. Therefore, the rotor torque cannot reach the minimum values as it is supposed to be. 4.2 Unsteady aerodynamics under pitch motion with different amplitudes This subsection analyzes the aerodynamic performance when FOWTs under the pitch motion with different amplitudes, such as 1.5°, 2.25° and 3.0°. The numerical results of the rotor thrust and torque with different amplitudes are shown in Figure 14 and compared with those without any extra motion. Similar to the result in section 4.1, the variation trend of the rotor’s unsteady aerodynamics is similar to the prescribed harmonic motion. Specifically, their fluctuations conform to the change of the platform’s pitch angular velocity, which can be addressed with the relative velocity. It is noticed that the rotor thrust and torque increase with the increment of the pitch amplitude because bigger amplitude brings higher velocity. In addition, there is inconsistence between the positive and negative amplitudes from the equilibrium position, and inequality between the portions above and below the equilibrium position, which may also be explained by the relationship that the rotor thrust and torque are proportional to the square of the relative velocity. Furthermore, there are fluctuations near the maximum positions, among which the one in the case of 3.0° pitch amplitude is the most obvious. The relative velocity is the highest when the curve reaches its maximum position. Meanwhile, the attack angle also increases and reaches its maximum which may cause dynamic stall phenomenon and then damage the rotor’s aerodynamic performance.

(a)

(b)

(c)

Figure 14. The unsteady aerodynamics of the rotor under pitch motion with different amplitudes: (a) the thrust and torque when >=1.5°; (b) the thrust and torque when >=2.25°; (c) the thrust and torque when >=3° In order to verify the dynamic stall phenomenon, the velocity magnitude of one blade’s section at a distance of 0.9R away from the hub center is monitored and shown in Figure 15, where Figure 15(a) ~ (h) correspond to the point T1 ~ T8 in Figure 15(i), respectively. The case with 3.0° pitch amplitude and 0.75s pitch period is used because the dynamic stall phenomenon is the most evident. From point T1 to T5 (corresponding to the Figure 15(a) ~ (e)), the rotor moves forward and the velocity induced by the pitch motion is in the opposite direction of the wind velocity, therefore the relative velocity is higher than that without extra motion. It is seen that the velocity magnitude decreases in Figure 15(a) ~ (e) because the velocity induced by the pitch motion decreases during this time period. It reaches its maximum in Figure 15(a) and minimum in Figure 15(e), corresponding to the point T1 and T5, respectively. In Figure 15(a), it is noticed that the airflow on the upper surface of the blade section begins to detach rather than sticks to the surface where the attack angle reaches its maximum. As a result, the lift from the low-pressure part disappears which is known as the dynamic stall phenomenon. To be more precise, because the viscous zone’s vertical extent keeps the same order of the airfoil thickness, this scale of interaction is defined as the light stall, which may not only leads to the loss of lift and increase of drag as classic static stall, but also causes unsteady behavior by large phase lags and hysteresis [55]. By contrast, the velocity magnitude is increasing with the increment of the velocity induced by the pitch motion from point T5 to T8, corresponding the Figure 15(e) ~ (h). Therefore, the pitch motion causes the rotor’s oscillating in and out of dynamic stall, which is named stall flutter [55]. Compared with the surge motion, three blades in the pitch motion would experience more stall flutter brought by the more frequent change on the relative velocity during the rotor’s pitching and rotation. Hence, the instable vibratory would be introduced to the blades [56] and may result in fatigue load on them, which deserves due attention.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i) (j) (k) Figure 15. Velocity magnitude of 0.9R section of one blade (>=3°, F=0.75s): (a) ~ (h) denote the velocity magnitude with pitch motion at T1 ~ T8 in (i), respectively; (j) and (k) denote the velocity magnitude without pitch motion at T1 and T3, as the comparison. To shed more light of the light stall phenomenon, the pressure coefficients of sections at different distances away from the hub center are recorded in Figure 16. The distances of 0.5R, 0.6R, 0.7R, 0.8R, 0.9R and 0.95R are chosen. The definition of the pressure coefficient is written as 1? =

J − JK 0.5E3L 

(10)

where J is the absolute pressure; JK is the standard atmospheric pressure;  is the air density; 3 is the rotational speed of the rotor; L is the rotor diameter. As shown in Figure 16, the pressure difference between the upper and lower part is increasing with the increment of the distance away from the hub center. Besides, the absolute value of the pressure coefficient decreases with the decrement of the pitch angular velocity from point T1 to T5, and the minimum value is at point T5. Also, it increases with the increment of the pitch angular velocity from point T5 to T8, and the maximum value appears at point T1. If we take a further look, the pressure coefficients suddenly change at the trailing edge, such as the markers denoting the pressure coefficient at point T1 in Figure 16(f). Hence, the dynamic light stall phenomenon can be proved because its main characteristic is the fluctuation of pressure distribution for flow separation near trailing edges. Based on the analysis above, it is found that the dynamic stall phenomenon has adverse impact on FOWTs’ aerodynamic performance while the wake interference may compensate some energy to the rotor.

(a)

(b)

(c)

(d)

(e) (f) Figure 16. Pressure coefficient of different sections of a blade (>=3°, F=0.75s): (a) ~ (f) denote the pressure coefficient of r/R=0.5, 0.6, 0.7, 0.8, 0.9, 0.95, respectively The comparison of rotor’s unsteady aerodynamic performance under the pitch motion with different amplitudes is shown in Figure 17. Although the pitch amplitude varies from 1.5° to 3.0°, the pitch period remains 0.75s. The rotor thrust and torque of all the three cases have the same equilibrium position as the ones without any extra motion. What’s more, there still exists difference between the shape above and below the equilibrium positions which may be explained as the analysis in Figure 12.

(a) (b) Figure 17. Comparison of the rotor’s unsteady aerodynamics under pitch motion with different amplitudes: (a) comparison of the thrust; (b) comparison of the torque

4.3 Effects on power from different pitch periods and amplitudes In this subsection, the variation of the rotor power caused by platform’s pitch motion is shown. The average rotor power obtained with different pitch periods and without extra motion is compared in Figure 18, demonstrating that even small pitch motion may result in large variation of the unsteady aerodynamic responses. As mentioned above, although the equilibrium position of the rotor thrust and torque in all cases remain the same, the upper and lower portions change with the pitch period. The difference between the upper and lower portions increases with the decrease of the pitch period. Therefore, the rotor power decreases with the increment of the pitch period, shown in Figure 18. The increasing pitch period from 0.375s to 3.0s leads to the decrease of the amplitude of the pitch angular velocity, which cuts down the airflow’s momentum acting on the rotor. Thus, the average rotor power decreases. Also, the biggest difference of average rotor power under different pitch periods occurs when the pitch amplitude is 3.0°. The average rotor power obtained with different pitch amplitudes is shown in Figure 19. It is shown that the rotor power increases successively with the increment of the pitch amplitude from 1.5° to 3.0°, which may also be explained by the relative velocity increasing with the increment of the pitch amplitude. As mentioned above, there exists the inconsistency between the upper portion above the equilibrium position and the lower portion below it. Specifically, the area of the lower portion is less than that of the upper portion, and the area difference between them increases with the increment of pitch amplitude, shown in Figure 17. Therefore, the average rotor power increases with the increment of pitch amplitude, and is larger than that without extra motion. This is the most evident one in the cases with different pitch amplitudes when the pitch period is 0.375s.

Figure 18. Average rotor power under pitch motion with different periods, compared to that without pitch motion

Figure 19. Average rotor power under pitch motion with different amplitudes, compared to that without pitch motion

5. Conclusions In this study, the aerodynamic performance of a redesigned scale model from NREL 5MW wind turbine including rotor thrust, torque and power under the pitch motion is investigated by the CFD method with the IDDES turbulence model. To ensure simulation model’s accuracy, it is validated by the data of an experimental work and by numerical comparison using other methods of the same scale rotor. Results show that even small pitch motion may result in great fluctuations on the aerodynamic performance of FOWTs. Also, several

conclusions are obtained as follows. (1)

(2)

(3)

The amplitudes of the rotor thrust and torque decrease with the increment of the pitch period due to the decrement of the amplitude of the relative velocity. The wake interference phenomenon may happen when the FOWT is pitching downwind, which is captured by the CFD method when the pitch period is not long and the amplitude is high. In some cases, the induced velocity by the pitch motion may even exceed the wind velocity or the velocity of detaching vortices, then the severest phenomenon of wake interference occurs, which may provide momentum to the rotor, and thus enhance the minimum of the rotor thrust and torque. The amplitudes of the rotor thrust and torque increase with the increment of the pitch amplitudes because of the increment of the amplitude of the relative velocity. The dynamic light stall phenomenon may happen when the FOWT is pitching upwind, which is clearly observed by the CFD method when the pitch amplitude is relatively high. It may cause damage to the rotor thrust and torque. More seriously, the consequent stall flutter on blades may be more frequent under the pitch motion. Rotor power is regard as the mainly concerned character in the engineering field. The average rotor power under the pitch motion with different periods time and amplitudes are compared. It is found that the average rotor power is improved in the cases under the pitch motion, especially by the decreasing the pitch period and increasing the pitch amplitude.

It is concluded that the pitch motion of the platform will change the aerodynamic performance of a FOWT and should be taken into consideration during design procedure.

FUTURE WORK Future work will be conducted from two aspects. On the one hand, effects on aerodynamic performance by other single DOF motion or multiple DOFs motion of a FOWT will be studied in near future. One the other hand, the phenomenon of wake interference phenomenon due to the platform motion shown in this study is difficult captured by traditional numerical method such as BEM theory, GDW model and FVM. However, the advantage of these method is obvious that their computational efficiency is much higher than the CFD method. Hence, the BEM theory and FVM will be improved to well predict FOWTs’ aerodynamic performance.

ACKNOWLEDGEMENTS This work was partially supported by the Yangfan youth scientific a technological talent project of Shanghai (17YF1409500), by National Natural Science Foundation of China (Grant Nos. 51809171, 51879160, 51809170), Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning (No. ZXDF010037), and by the Open Project of State Key Laboratory of Ocean Engineering (1506). REFERENCES

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IDDES model is employed to analyze the aerodynamics of a FOWT under pitch motion.




Effects of pitch periods are studied, and wake interference phenomenon is observed.




Effects of pitch amplitudes are investigated, and stall phenomenon is revealed.




The rotor power under periodic pitch motion is investigated.

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: