3.1.1. Grid Independence

Table 2 shows the di fferent torques calculated for five di fferent numbers of cells. Every mesh was refined along the airfoil circumferential direction. The grid numbers were computed in the T-SST turbulent mode and simulations were performed with the following grid numbers: 1.6 million, 3.09 million, 3.67 million, 5.31 million, and 6.05 million. In addition, two grid numbers (1.6 million and 3.67 million) were investigated under di fferent yaw angles (10◦ and 20◦), as shown in Table 2. For non-yaw condition, the design torque at 11.4 m/s is about 4.08 × 10<sup>6</sup> N·m, which is taken as a reference value. The relative errors for the results using other four mesh are 1.9~0.49%. The relative error for mesh of 3.67 million cells is smaller than 1% and the computational cost using this mesh is moderate. Thus this mesh was selected for further investigation.


**Table 2.** Computed torque of di fferent grid size averaged in one revolution.

#### 3.1.2. Turbulent Model Studies

In this section, we present results of the unsteady independence study based on four turbulent models: SST, *k*-*kl*-ω, Reynold stress, and T-SST. The rotor torque under axial flow obtained for each turbulent model is presented in Table 3. The computed torque was similar for each of the four turbulent models. The simulation cases using the SST model and Reynold stress model were used to interpret the fully developed flow mechanism without transient processes, resulting less torque compared to using Transient SST turbulent model. When the wind turbine operates under low wind speeds, transient phenomena occur on the suction side of the blade; therefore, the T-SST turbulent model was selected for static and yawing process studies.

**Table 3.** Time averaged rotor torque calculated using di fferent unsteady turbulence models.


#### 3.1.3. Time Step Studies

In the unsteady numerical simulation, the time step could influence the overall performance of the model, therefore, it is necessary to investigate time-step independence. Table 4 lists the averaged rotor torques during one revolution for non-yawed case obtained using the T-SST turbulent model with three different time steps (72, 144, and 360). The comparisons show slight differences among the results (less than 3%). The result using 72 time steps in one revolution gave a better agreemen<sup>t</sup> comparing to the designed value. Considering the computational cost, a time step of 72 during one revolution was selected for subsequent simulations.

**Table 4.** Rotor torque calculated using different time step sizes averaged over one revolution.


#### *3.2. The Validation of Numerical Simulation Results for Yawed Wind Turbine*

After the verification of the simulation, the subsections below will show some aerodynamic analysis of wind turbine under yawed and yawing simulation.

#### 3.2.1. Overall Performance Analysis for Different Yaw Angles

Figure 7 shows the overall performance of the rotor under yaw. As shown in Figure 7a, only a small amount of variation in rotor power can be observed under small yaw angles (≤5◦). When the yaw angle exceeds 5◦, the rotor torque significantly decreases as the yaw angle increases. The torque value computed using the BEM method, with or without the Beddoes stall model, is higher than the value computed by CFD since the BEM computation does not consider flow separation and flow transient phenomena. Zhu [39] extensively investigated the combined effects of rotational augmentation and dynamic stall, and found that the hysteresis loop of aerodynamic load is much larger compared to 2D simulations. In fact, for the NREL 5-MW wind turbine and a wind speed of 11.4 m/s, flow separation and 3D radial flow mainly occur on the inner board region, resulting in a lower torque and lower thrust. The deviation in the power and thrust between CFD and BEM was 4.1% and 5.91%, respectively. Compared with the axial free inflow, the three functions on cos(γ), cos2(γ), cos3(γ) of the power and thrust are shown with dashed, dotted and dash-dotted line, respectively. The variation of averaged power in yaw conditions will decrease by cos2(γ); the averaged thrust agrees well with cos(γ).

**Figure 7.** Variation of the power and thrust of the wind turbine under yaw.

#### 3.2.2. Aerodynamic Load Analysis along the Span of the Blade

Theoretical analysis is necessary to determine the mechanism behind the yawed effect. Figure 8 shows the velocity diagram under the yawed condition. A velocity component exists in the blade revolution plane and can be projected into radial and chordwise components denoted *Vr* and *Vc*, as shown in Figure 8a. The retreating and advancing effects occur because of *Vc*, which causes the sectional load and AOA to fluctuate periodically. The maximum AOA and maximum load occur when the blade is in the 12 o'clock direction in the current simulation setup. Additionally, loads on the retreating side are much larger than on the advancing side. Figure 9b,c quantify the influence of *Vc*, which is calculated by:

(2)

**Figure 8.** Section airfoil induced velocity *Vc* under stable yaw: (**a**) magnitude of radial and chordwise velocity; (**b**) direction of radial and chordwise velocity; (**c**) velocity diagram of blade element.

(1) Time-averaged load analysis on spanwise section under yawed condition

Figures 9–11 show the analysis of the variation of averaged AOA, *Cn*, and *Ct* under one rotor revolution. The value of *Cn* and *Ct* is calculated by:

$$\mathcal{C}\_n = \int\_{\frac{\pi}{c} = 0}^{\frac{\pi}{c} = 1} \mathcal{C}\_{p - up} d(\frac{\chi}{c}) - \int\_{\frac{\pi}{c} = 0}^{\frac{\pi}{c} = 1} \mathcal{C}\_{p - down} d(\frac{\chi}{c}) \tag{3}$$

$$\mathcal{C}\_{1} = \int\_{y0}^{y1} \mathcal{C}\_{p-up} d(\frac{y}{c}) - \int\_{y0}^{y1} \mathcal{C}\_{p-down} d(\frac{y}{c}) \tag{4}$$

where *Cp-up* and *Cp-down* denote the pressure coefficients in the upper and down side of the airfoil, respectively; *y*0 and *y*1 mean the leading edge position and trailing edge position of the airfoil.

The AOA distribution presented in Figure 9 takes into account the advancing and retreating effects, as previously shown by Castellani [20] in the yawed simulation of a HAWT using the BEM model. Comparing the simulation results for CFD and FAST, the same trends can be observed along the blade span, whereas the AOA value is much larger than the FAST result. Figures 10 and 11 show the variation of aerodynamic load spanwise along the blade. For the axial flow, some large abnormal fluctuations in the AOA and aerodynamic loads can be observe d. Differences in the results of the CFD and FAST methods mainly occur along the inner board, owing to flow separation, which lead to smaller aerodynamic loads compared to those computed by the BEM model that does not consider flow separation.

**Figure 9.** The profile of AOA along the blade span under yawed cases.

**Figure 10.** The profile of *Cn* along the blade span under yawed cases.

**Figure 11.** The profile of *Ct* along the blade span under yawed cases.

(2) The spanwise section aerodynamic load analysis under yawed conditions

Figures 12 and 13 show the distribution of the AOA with respect to the azimuth angle under a yaw angle of 15◦ and 30◦. Fluctuations become larger as the yaw angle increases due to the

advancing and retreating effect, which is partly caused by the inflow velocity component under yaw. After post-processing, the AOA using the combination of the CFD sectional airfoil aerodynamic loads and the BEM method, the maximum and minimum AOA during one revolution occur at an azimuth angle of 0◦ and 180◦, respectively. Wen [36] found that under the impact of non-uniform effects (due to variations of the induced factor caused by the radial position and azimuth angle in the rotational plane), the maximum AOA tends to occur at 90◦ for inboard airfoils. The maximum and minimum aerodynamic loads occur at an azimuth angle of 90◦ and 270◦, respectively. Thus, future computations of the AOA should consider non-uniform effects.

**Figure 12.** Azimuthal aerodynamic loads at different spanwise sections during one revolution under a yaw angle of 15◦.

**Figure 13.** Azimuthal aerodynamic loads at different spanwise sections during one revolution under a yaw angle of 30◦.

Figure 14 shows the variation of the averaged AOA with aerodynamic load at five typical spanwise sections with respect to yaw angle. Fluctuations of the aerodynamic performance are clearly observed and become larger as the yaw angle increases.

**Figure 14.** Variation of time-averaged AOA and aerodynamic load under yaw during one revolution.

The mean aerodynamic performance remains relatively constant with less decrease due to the yaw effect. Variation of the AOA and aerodynamic loads exhibit much higher fluctuations in the inner board

compared to the middle and outer board. With the increasing of the yaw angle, the circumferential loads and AOA also cause higher maximum loads.

#### *3.3. Numerical Simulation of Dynamic Yawing Wind Turbine*

This section presents results of the numerical analysis of the NREL 5-MW RWT under the dynamic yawing process.

#### 3.3.1. Torque Characteristics of Wind Rotor

Figure 15 shows the distribution of the wind rotor torque under different start-stop yaw velocities. During the yawing process, fluctuation of the wind rotor torque is small. At the beginning of dynamic yaw, larger torques occur due to changes in the rotor position. As the yaw angle increases, the torque gradually decreases and is 5 × 10<sup>5</sup> N·<sup>m</sup> larger in the case of the 2-s duration compared to the 4-s duration. The 2-s duration may induce a larger rotor torque since the yaw angle changes more quickly than with 4-s duration. The reasons are given below. At the start and stop stage, the yaw velocity is changed with the sinusoidal variation law related to the frequency, the higher frequency of yaw velocity caused much larger of the power and thrust, which is similar to the variation of power under platform pitching [38,40]. When the wind turbine is yawing with the constant yaw velocity, the additional velocity inducing by yawing is the same, and the power is only decreased with the square cosine of yaw angle. When wind turbine yawed to the stage of stop, the power under the 2-s case decrease much faster than under 4-s cases, which is similar to the yaw start period.

**Figure 15.** Torque characteristics of rotor under two different yawing rotational velocities.

The results of 58 rotor revolutions were analyzed. During each revolution, 12 torque measurements were collected. Then, the fast Fourier transform was used to obtain the rotor aerodynamic frequency. Figure 16 shows the torque power spectra of the rotor under 2-s and 4-s duration yawing process. The blade passing frequency of the NREL 5-MW turbine is about 0.2017 Hz (1P fluctuation), and is clearly the main frequency of the two yawing processes. Both are 0.6 Hz (which is approximately a 3P rotor fluctuation), which is similar to results presented by Castellani [20]. Figure 16a shows a secondary frequency of 0.2 Hz (1P fluctuation). Due to the less sampling data in the sinusoidal stage, the yawing start-stop frequency (2-s and 4-s duration with a corresponding main frequency of 1/8 Hz and 1/16 Hz, respectively) cannot be captured in the computation of the torque power structure.

**Figure 16.** Frequency of rotor torque for a yawing start-stop process with a duration of 2-s and 4-s.

#### 3.3.2. Torque Characteristics of Blade

Figure 17 shows the variation of torque under different yaw rates in two periodic processes of start-stop yawing. Similarly, the torque of the blade under a 2-s duration is larger than the torque of the 4-s duration. Interestingly, the torque of the blade in the forward yaw stage (shown in Figure 18 within 0~66.8 min, yaw angle from 0◦ to 20◦) is larger than that of the backward yaw stage (shown in Figure 18 within 66.8~133.6 min, yaw angle from 20◦ back to 0◦) since the dynamic yawing effect, which generates the dynamic velocity, reduces the relative velocity.

**Figure 18.** Six cases during the start-stop stage under dynamic yaw.

#### 3.3.3. Wake Flow Characteristics

Wake effects are important in analyzing wind turbine aerodynamics. For convenience, the dynamic yaw was classified into six cases, as illustrated in Figure 18. Cases 1,3,4,6 include the forward-start yaw, forward-stop yaw, backward-start yaw, and backward-stop yaw, respectively. For Cases 2 and 5, the yaw angle is 10◦ and they represent forward yaw and backward yaw, respectively. The grey line in the figure illustration the variation of yaw angular velocity during the simulation, and shows that except the Cases 2 and 5, other cases are all in the simulation about the variation of yaw angular velocity.

(1) Forward yaw-start stage (Case 1). The velocity contours at t = 1/8T (T = 8 s or 16 s) at the beginning of dynamic yaw are shown in Figure 19. The near wake velocity flow structure shows the same status at the beginning of the yawing process for both the 2-s and 4-s start-stop duration. The different yaw velocities have very little influence on the velocity distribution in the 2D range downstream.

**Figure 19.** Velocity contours for Case 1.

(2) Yaw angle of 10◦ under forward yawing stage (Case 2). Sketches of the velocity streamlines of the two dynamic yawing processes (yaw angle of 10◦) are shown in Figure 20a,b are similar to those obtained for the yawed case, as shown in Figure 20c. The dynamic yaw rotates with a fixed yaw velocity of 0.3◦/s, and similar results are observed for the dynamic process. More energy intermediate effects can be observed between the wake zone and the main flow zone than in the yawed case. This may be due to the effects of dynamic stall.

**Figure 20.** Velocity contours for Case 2.

(3) Forward-yaw-stop stage (Case 3). Figure 21 shows the instantaneous velocity contours for the dynamic yawing and yawed cases. Both dynamic yawing processes result in a much larger wake zones than under the yawed case. Meanwhile, some deflection occurs in the velocity wake, which is similar to the wake deflection effect reported in the work of Qian [18], which took into account the velocity deficit and turbulent intensity using a Gaussian-based wake model.

**Figure 21.** Velocity contours for Case 3.

(4) Backward-yaw start stage (Case 4). Figure 22 illustrates the velocity wake in the backward-yaw start stage. The wake zone velocity field retains almost the same flow structure. To refine the simulation results, large eddy simulations can be used to improve the interpretation of the flow process.

**Figure 22.** Velocity contours for Case 4.

(5) Yaw angle of 10◦ under backward yawing stage (Case 5). When the wind turbine rotates about the Y-axis with a yaw angle of 10◦, the velocity wake gradually become symmetrical and the 4-s duration simulation recovers to the symmetry state faster than 2-s duration simulation, as shown in Figure 23.

**Figure 23.** Velocity contours for Case 5.

(6) Backward-yaw stop stage (Case 6). As shown in Figure 24, the wind turbine returns to its initial state, the wind direction is normal to the rotor rotational plane, and the three-dimensional flow structure is symmetrical in streamwise.

**Figure 24.** Velocity contours for Case 6.

In summary, dynamic yawing based on two start-stop durations shows the approximate flow structure, and the upwind and downwind effect induced by yawing process expands the wake zone earlier and much larger than the yawed condition.

#### 3.3.4. Aerodynamic Characteristics along Blade Spanwise Section

To investigate the yawing effect, additional velocity induced by the yawing wind rotor should also be examined. Figure 25 illustrates the velocity triangle along the span of the blade under dynamic yawing. The dashed line of the yawing zone indicates positive yaw (negative yaw was not investigated in the present simulation). Four process variations can be extracted according to the dynamic yawing stage. Note that *R* = 0 means that initial position of the rotational axis of the wind turbine is the rotor hub position, which is different from the platform yawing process used for the offshore wind turbines. The direction of *<sup>V</sup>*dyn is different on both sides of the yaw axis. In the process of yawing counterwise, *<sup>V</sup>*dyn and the inflow wind speed create an acute angle in the right side of yaw axis, while *<sup>V</sup>*dyn and the inflow wind speed create an obtuse angle in the left one of yaw axis, as shown in Figure 26a. The process of yawing clockwise is just the contrary to the case of yawing counterwise. The absolute formulation of *<sup>V</sup>*dyn can be written as:

$$V\_{\rm dyn} = \left| \omega\_{\rm ywn} r \sin(\varphi) \right| \tag{5}$$

where, *<sup>V</sup>*dyn become zero at the azimuth angle of 0◦ and 180◦ in the current setup of wind turbine.

**Figure 25.** Velocity diagram for a blade section under dynamic yaw at radius *r*: (**a**) sketch of the yawing process for the velocity dynamic analysis; (**b**) velocity diagram along the span of the blade.

> The relative velocity of the section under dynamic yawing can be defined as:

$$V\_{rel} = \sqrt{\left(V\_o(\cos\gamma - a) + \eta V\_{\rm dyn}\right)^2 + \left(\omega r (1 + b) - \beta V\_0 \sin\gamma \cos\varphi\right)^2} \tag{6}$$

where η = 1 if the direction of *<sup>V</sup>*dyn and inflow wind velocity creates an acute angle and η = −1 when the direction of *<sup>V</sup>*dyn and inflow wind velocity creates an blunt angle. In the present yawing simulation, if the wind turbine is rotating in the positive yaw direction, β = 1.

Figure 26 shows the variation of the AOA in three typical section (*r*/*R* = 0.21, 0.4, or 0.67), with respect to yaw angles 0–20–0–20–0◦. The upper abscissa is the wind rotor yaw angle, while the bottom abscissa is the wind turbine rotational cycle. The vertical coordinate indicates the AOA distribution along the radial direction of the blade. The three sections are indicated by solid lines of different thickness of airfoil. Inner board airfoils are the thickest and outer board one are thinnest.

The AOA gradually decreases along the span of the blade. Due to the effect of dynamic yawing, the AOA oscillates during the rotational period of the rotor, with the combined effects of both retreating & advancing and upwind & downwind. Figure 26a shows the calculated results of 2-s start-stop yaw rate. In this case, the start-stop velocity is faster, and the overall fluctuation are larger than those in the 4-s scenario. The start-stop effect mainly affects the AOA near the outer board but has less influence on other areas. In addition, the start-stop process leads to changes in the AOA.

**Figure 26.** Comparison of AOA under different start-stop durations for three different spanwise sections under dynamic yawing.

Figure 27 show the distribution of the normal force coefficient at three typical sections (*r*/*R* = 0.21, 0.4, 0.67) within the dynamic process, including two forward yawing and backward yawing states with a yaw angle of 20◦. Variation of *Cn* under dynamic yawing with a 2-s duration causes larger differences at the radial position (*r*/*R* = 0.4) than the 4-s duration. Under the yawed case, the average normal force coefficient of the *r*/*R* = 0.21 section with a yaw angle of 20◦ is 1.1, and the maximum and minimum load coefficients are 1.7 and 0.5, respectively (see Figure 14). In the current yawing case, the maximum and minimum load factors are 1.9 and 1.0, which are due to the downwind and upwind effects of the yaw dynamics, similar to the horizontal wind shear effect, but more pronounced than typical horizontal wind shear effects. In the start and stop duration of yawing, the yawing velocity of 2-s case has much higher frequency than 4-s case, resulting to much higher additional velocity and aerodynamic loads. Thus the overall performance of 2-s yawing presents much larger fluctuation in the process of yawing.

**Figure 27.** Comparison of *Cn* under different start-stop durations for three different spanwise sections under dynamic yawing.

Figure 28 shows how the tangential force coefficient varies with simulation time at three typical sections (*r*/*R* = 0.21, 0.4, and 0.67). The results sugges<sup>t</sup> the aerodynamic load inside the inner board is more influenced under dynamic yawing with the 2-s during than the 4-s duration. Fast shifts of the yaw angle under the yawing start-stop stage influence the aerodynamic load along the inner and middle blade span.

**Figure 28.** Comparison of *Ct* under different start-stop durations for three different spanwise sections under dynamic yawing.
