*2.6. Computational Methods*

The flow field of the wind turbine is simulated in ANSYS Fluent software, and the unsteady Reynolds-Averaged Navier–Stokes equations (RANS) is solved by using the SST *k-w* turbulence model considering the transition effects on the blade surface. The sliding grid method is used to simulate the rotation of the wind turbine, which is set as a transient calculation. The pressure-based solver is used, the SIMPLEC algorithm is selected for pressure-velocity coupling, in which the second-order upwind scheme is used for pressure term, convection term, turbulent kinetic energy equation and turbulent dissipation rate. Based on the steady calculation results, the unsteady calculation for 20 s is carried out, the additional calculation for 40 s is carried out in the case of platform motion, and the last platform motion period after convergence is taken as the analysis result. The residual of the continuity equation is reduced by at least four orders of magnitude in the process of calculation.

Figure 1 has defined the form of platform motions, including surge, pitch and yaw, which are the main factors affecting the aerodynamic performance of FOWTs. According to the situation of the literature [36], the suitable amplitude and frequency parameters of the floating foundation are selected, and the main motion parameters are shown in Table 1. In order to realize the above platform motions, the additional speed change is compiled by

the DEFINE\_PFORILE macro of UDF. It is assumed that the motion of the FOWT floating foundation is harmonic; its expression can be written as:

$$
\beta\_i(t) = A\_i \sin(\varepsilon\_i t),
\tag{7}
$$

where *βi*(*t*), *Ai* and *ε<sup>i</sup>* (*i* = surge, pitch, yaw) denote the angle/displacement, amplitude and frequency of the platform motion.

**Figure 3.** Computational mesh; (**a**) External domain; (**b**) Internal domain; (**c**) Hub surface mesh; (**d**) Blade surface mesh.


**Table 1.** Floating foundation motion conditions.

The motion of the floating foundation is described by the matrix between coordinate systems [37]. The incoming wind speed is set to *V* = [*Vx*, *Vy*, *Vz*] T, where *Vx*, *Vy*, *Vz* represents the partial velocity in the direction of X, Y and Z, respectively. the relative inflow velocity of the platform during surge motion is converted into *VS*, which can be written as

$$V\_S = V(t) + \begin{bmatrix} 0\\ \mathcal{S}'\_{\text{surgc}}(t) \\ 0 \end{bmatrix} = \begin{bmatrix} 0\\ V\_y + \varepsilon\_{\text{surgc}} A\_{\text{surgc}} \cos(\varepsilon\_{\text{surgc}} t) \\ 0 \end{bmatrix},\tag{8}$$

where *β surge*(*t*) denotes the velocity of surge motion at each moment.

The pitch motion corresponds to the rotation *βpitch* of the floating platform around the *x*-axis, and the incoming flow velocity is converted to the relative velocity *VP*, which can be presented as

$$V\_P = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \beta\_{pitch} & \sin \beta\_{pitch} \\ 0 & -\sin \beta\_{pitch} & \cos \beta\_{pitch} \end{bmatrix} V = \begin{bmatrix} 0 \\ V\_y \cos(A\_{pitch} \sin(\varepsilon\_{pitch} t)) \\ -V\_y \sin(A\_{pitch} \sin(\varepsilon\_{pitch} t)) \end{bmatrix} \tag{9}$$

The yaw motion corresponds to the rotation *βyaw* of the floating platform around the *z*-axis, and the incoming flow velocity is converted to the relative velocity *VY*, which can be presented as

$$V\_Y = \begin{bmatrix} \cos \beta\_{yww} & \sin \beta\_{yaw} & 0 \\ -\sin \beta\_{yaw} & \cos \beta\_{yaw} & 0 \\ 0 & 0 & 1 \end{bmatrix} V = \begin{bmatrix} V\_y \sin(A\_{yaw} \sin(\varepsilon\_{yaw} t)) \\ V\_y \cos(A\_{yaw} \sin(\varepsilon\_{yaw} t)) \\ 0 \end{bmatrix} \tag{10}$$

#### **3. Numerical Model Verification**

*3.1. Model Grid Independence Verification*

Table 2 exhibits the numerical value of the wind turbine torque under different meshing precision, which are refined along the circumference of wind turbine blades. The number of simulated grids is as follows: 2.86 million, 4.77 million, 6.71 million, 8.50 million and 10.08 million.


**Table 2.** Torque of different gird sizes.

It can be found that with the refinement of the mesh, the error value of torque decreases constantly, and the decrease becomes smaller and smaller. The mesh of 8.50 million is selected for in-depth research due to the consideration of calculation accuracy and efficiency.

#### *3.2. Model Power Verification*

Due to the lack of feasible experimental data, the experimental results of Jonkman [38] were selected for verifying the accuracy of the simulation. The cut-in wind speed of the NREL 5 MW wind turbine is 3 m/s. When the wind speed reaches more than 3 m/s, the wind turbine starts to operate. The rated wind speed is 11.4 m/s, and the rated speed is 12.1 rpm. When the wind speed exceeds the rated wind speed, the constant power control is realized by changing the pitch angle and reducing the lift-drag ratio.

Figure 4 illustrates the comparison of stable power between numerical simulation and experimental data under different incoming wind speeds. All of the results exhibited a high consistency before the incoming wind speed reached 11.4 m/s, indicating that the current simulation method has good accuracy and reasonable results.

#### *3.3. Validation Considering the Tower under the Unsteady Condition*

It should be noted that the change of additional wind speed caused by the motion of the platform is synchronously added to the inlet velocity. The numerical model is verified with and without tower under unsteady calculation, respectively. A sufficient degree of steady condition calculation is carried out for the numerical model under rated conditions first, and the results of the steady calculation are taken as the initial flow field under unsteady wind speed conditions.

**Figure 4.** Power comparison of different wind speeds.

The uniform wind speed is 11.4 m/s, which is set at the inlet of the flow field. The speed of the wind turbine is 12.1 rpm. The time-step corresponding to 10◦ azimuth increment with 40 pseudo-time sub-iterations is 0.137741 s. The rotation time of the wind turbine for 4 cycles is calculated under unsteady conditions. Figure 5 shows the power fluctuation with and without the tower and it indicated that the power of the wind turbine tends to be stable after the wind turbine rotates for three cycles. Furthermore, a periodic rotation variation of the wind turbine in stable operation was observed locally. Compared with the non-tower, the existence of the tower makes the power decrease dramatically with an azimuth of 120◦ apart. This phenomenon is consistent with Wen [39]'s research, which verifies the influence of the tower shadow effect on the wind turbine.

**Figure 5.** Power variation under unsteady rated condition.

#### **4. Results and Discussion**

Due to the obstruction of the tower to the airflow, the interference effect of the tower on the blades is different when the blades rotate to different azimuth angles. This paper takes the blades rotating to the azimuth directly above the tower as the starting point. The schematic diagram of the azimuth angle of the wind turbine blades is shown in Figure 6. *θ* is the angle between the two blades, *γ* is the influence area of the tower shadow effect and the wind turbine rotates counterclockwise.

**Figure 6.** Schematic diagram of wind turbine blade-rotation phase angle.

#### *4.1. Unsteady Aerodynamic Analysis under Surge Motion*

#### 4.1.1. Total Performance Analysis

Figure 7 illustrates the total power and thrust comparison with and without the tower under surge motion. It can be observed that the fluctuation period of power and thrust is consistent with the harmonic surge. The equilibrium position divides the power and thrust curve into rising and falling parts, mainly because the power and thrust are proportional to the relative wind speed. For f = 0.05 Hz, As = 5 m, as shown in Figure 7b, the maximum power of the wind turbine without the tower is 6.98 MW, while the wind turbine with the tower is 6.85 MW. Both of these were larger than the rated power, while the existence of the tower reduced the maximum wind turbine power by 1.86%. The minimum power was 3.37 MW and 3.24 MW, respectively, and the power of the wind turbine with the tower was reduced by 3.86%. The average power generation was 5.12 MW and 5.01 MW, respectively; the latter was reduced by 2.15%. Further, the power fluctuated at a frequency thrice that of the rotation frequency under the combination of surge and tower shadow, while the surge motion played a leading role in the influence of power fluctuation. As shown in Figure 7c,d, the trend of the axial thrust fluctuation was similar to the power. The peak value of thrust without the tower was 888.3 KN; the wind turbine with the tower was 881.4 KN. Additionally, the valley value of thrust was 772.7 KN and 764.7 KN, respectively. It can be seen that the peak and valley values of thrust were basically the same with or without the tower. In addition, the average thrust was 772.7 KN and 764.7 KN, separately; the latter was reduced by 1.04%. It can be concluded that the influence of the tower shadow effect on thrust was less than that on power under the same condition of surge motion.

To further explore the variation mechanism of the aerodynamic load of the wind turbine under platform motion, the airfoil-induced velocity distribution under surge motion was analyzed, as shown in Figure 8. The surge motion of the platform produces an additional induced speed *Vind* to the wind turbine, which is obtained by superposing the surge speed with the free flow wind speed. When the platform moves forward, the *Vind* is opposite to the incoming wind speed, and the relative wind speed increases. When the platform moves backward, the *Vind* is in the same direction as the incoming wind speed, and the relative wind speed decreases. *Vind* can be decomposed into chord velocity *Vc* and radial velocity *Vr* on the rotating plane, the magnitude and direction of the relative velocity *Vrel* acting on the rotating plane of the airfoil changes and the angle of attack changes accordingly. This theoretically expounds the essence of the fluctuation of the aerodynamic performance of the wind turbine under surge motion.

**Figure 7.** Total aerodynamic comparison of surge motion; (**a**) power versus azimuth angle; (**b**) extreme and average values of power; (**c**) thrust versus azimuth angle; (**d**) extreme and average values of thrust.

**Figure 8.** Schematic diagram of airfoil-induced velocity under surge motion.

#### 4.1.2. Distribution of Pressure on the Blade Surface

Figure 9 shows the surge amplitude of the platform motion with reference to time. The second period of stable surge motion of the wind turbine is selected as the research object. Two typical positions were selected to analyze the pressure distribution on the blade surface.

**Figure 9.** The two typical positions during surge motion.

The most fundamental influence of the tower shadow effect on the aerodynamic performance of the wind turbine is the interference of the tower on the blade; the blade surface pressure is the basic parameter to characterize the aerodynamic performance of the blade. This section selects two typical positions in which the blade rotates to the front of the tower under surge motion, and analyzes the pressure distribution under the root (r/R = 0.32), middle (r/R = 0.63) and tip (r/R = 0.94) sections of the blade. The abscissa is dimensionless as x/c (the abscissas of different points on the section/chord length of the section).

Figure 10 shows the distribution of pressure in each section of the blade with and without the tower at two typical positions in a surge cycle. It can be found that the pressure difference distribution at position 1 at the corresponding section is greater than that at position 2. It can be explained that the *Vrel* at each blade section reaches larger values due to the forward surge velocity of the platform, while the *Vrel* decreases when the platform surges backward. From the numerical point of view, the maximum pressure difference of the wind turbine without the tower at 0.32 R, 0.63 R, 0.94 R section is 1952 Pa, 4240 Pa and 7910 Pa, respectively. It can be seen that the closer to the tip of the blade, the greater the pressure difference on the blade surface. In addition, the tower shadow effect mainly affects the negative pressure value of the suction leading edge and the absolute value of the maximum negative pressure difference in each section is reduced by 10.56%, 7.61% and 5.36%, respectively. It can be inferred that the closer to the tip of the blade, the less obvious the interference of the tower shadow effect on the negative pressure of the suction surface.
