Influence of Surging and Pitching Behaviors on the Power Output and Wake Characteristics of a 15 MW Floating Wind Turbine

Abstract

This study investigates the impacts of surging and pitching motions on the power generation performance and wake characteristics of an IEA 15 MW offshore wind turbine under specific inflow wind conditions. The three-dimensional, unsteady continuity equation, momentum equations, and SST k–ω turbulence model are solved numerically using the computational fluid dynamics software STAR-CCM+ (version 2206) to simulate the aerodynamic flow field around the turbine rotor and in its downstream wake region. Under the condition of an inflow wind speed of 9 m/s at hub height and a corresponding rotor rotational speed of 7.457 RPM, the surging and pitching motions of the turbine are prescribed by sinusoidal functions with a period of 45 s and amplitudes of 2.75 m and 5°, respectively. This study analyzes and quantifies the power output and wake characteristics of the turbine over a duration corresponding to 200 rotor revolutions, considering stationary, surging, and pitching conditions. The results indicate that the surging and pitching motions of the turbine cause reductions in the mean power output of 2.18% and 3.54%, respectively, compared to a stationary condition. The surging and pitching motions also lead to significant wake enhancement in the downstream region, and a minimum spacing of downstream wind turbines is suggested.

1. Introduction

In recent years, the research and development of wind energy has become a global trend driven by growing energy demands and shortages. In March 2023, the Global Wind Energy Council (GWEC) reported that more than 77.6 GW of onshore and offshore wind energy capacity was installed in 2022, and the total amount of global wind energy capacity exceeded 906 GW, representing a 9% increase over 2021. In 2022, onshore wind energy installations totaled 68.8 GW, reaching a cumulative capacity of 842 GW, making it the third-highest year for new installations. Offshore wind energy capacity added in 2022 was 8.8 GW, bringing the total to 64.3 GW, marking the second-highest year for new installations [1]. Although offshore wind capacity remains lower than onshore, its growth rate is significantly faster. Offshore wind power has become a strategic focus in many countries due to its ability to harness stronger winds with fewer geographical constraints. The Taiwanese government aims to install 20 GW of offshore wind energy capacity by 2035 and plans to develop deeper water areas with water depths exceeding 50 m, i.e., Hsinchu offshore area, after 2026 [2], to leverage the rich wind energy resources in the Taiwan Strait [3]. Given the advantages in offshore distance, water depth, and cost, a 15 MW semi-submersible floating offshore wind turbine system is considered a favorable option for the deeper water areas of the Taiwan Strait. However, the wake characteristics of a floating wind turbine system are strongly influenced by its motions [4], particularly surge and pitch responses [5].

Qiu et al. [6] established a numerical framework for predicting the unsteady aerodynamic loads of horizontal-axis wind turbines via the non-linear lifting line method and free-vortex wake method and computed the aerodynamic loads and rotor wake characteristics during blade pitching and rotor yawing for the NREL Phase VI, TU Delft, and Tjæreborg wind turbine in 2014. Ma et al. [7] employed the actuator line method and non-linear finite element beam theory to calculate the coupled aeroelastic wake behavior of an NREL 5 MW wind turbine in 2019. This study identified an underestimation of the velocity and vorticity recovery in the far-wake region more than five rotor diameters behind the wind turbine. Based on the free-vortex wake method, Rodriguez and Jaworski [8,9] simulated the interaction between the rotor and wake of an NREL 5 MW floating wind turbine without considering the movement of the turbine in 2019. The results showed good agreement with simulations using FAST under cut-in and rated wind speed conditions. However, under high-speed conditions, the rotor performance is underestimated, and the discrepancy increases with higher inflow speeds. Lee and Lee [10] established and successfully validated a framework based on the nonlinear vortex lattice method for predicting the aerodynamic performance and wake geometry using the MEXICO rotor model and analyzed complex unsteady wake structures using the vortex particle method in 2019. These studies laid the foundation for numerical methods for analyzing the aerodynamics and wake behaviors of floating wind turbines. However, the influence of the floating wind turbine platform movement caused by wind, wave, and current loads has not been considered in these studies.

Tran and Kim [11] analyzed the influence of the pitching motion on the wake characteristics around the rotor of an NREL 5 MW wind turbine using the SST k–ω turbulence model, overset mesh, and unsteady blade element momentum method in 2015. Using the same methods, Tran and Kim [12] then analyzed the influence of the surging behavior of an NREL 5 MW wind turbine in 2016. Cai et al. [13] employed the Reynolds-averaged Navier–Stokes equations to calculate the aerodynamics of a 1.5 MW customized turbine with consideration of wind shear, tower shadow, and yaw motion in 2016. The results showed that the blade–tower interaction has a significant impact on the power output. Wong and Chau [14] discussed the influence of the surging motion of an IEA 15 MW wind turbine on the wake characteristics and power output by solving the three-dimensional unsteady continuity equation and the momentum equations and applying the SST k–ω turbulence model in 2022. The results showed that the velocity in the near-wake region of a surging turbine is up to 3.13% higher than a stationary turbine but up to 8.44% lower in the far-wake region. Simulating the flow field around an operating wind turbine with movement through computational fluid dynamics considers the impact of the complex geometry of the wind turbine, as well as the tower shadow effect, which delivers a relatively accurate prediction.

To discuss the influence of surging and pitching motions on the power output and wake characteristics of an operating 15 MW floating wind turbine, this study employs the IEA 15 MW offshore wind turbine [15] as the research target, using STAR-CCM+ to predict the power output and wake characteristics over 200 revolutions in stationary, surging, and pitching conditions. The rotor speed is assumed to be 7.457 RPM, and the inflow wind speed at hub height is assumed to be 9 m/s.

2. Numerical Methods

2.1. Governing Equations

STAR-CCM+ is employed in this study to simulate the flow field around and behind the turbine rotor. Under the assumption of an incompressible fluid, the continuity equation is expressed as

$${\displaystyle \frac{\partial u_i}{\partial z_i}}=0,$$

(1)

where $\rho$ is the density of the fluid, $t$ is the time, and $u_i$ is the velocity component in the $z_i$-direction, and the momentum equations are expressed as

$${\displaystyle \frac{\partial \left(\rho u_i\right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho u_i{u}_j\right)}{\partial z_j}}=-{\displaystyle \frac{\partial p}{\partial z_i}}+{\displaystyle \frac{\partial }{\partial z_j}}\left[\mu \left({\displaystyle \frac{\partial u_i}{\partial z_j}}+{\displaystyle \frac{\partial u_j}{\partial z_i}}\right)\right]-{\displaystyle \frac{\partial \left(\rho \overline{u_i^{\prime }{u}_j^{\prime }}\right)}{\partial z_j}}+{\rho g}_i,$$

(2)

where $p$ is the pressure of the fluid, $\mu$ is the viscosity of the fluid, $u_i^{\prime }$ is the turbulent velocity component in the $z_i$-direction, and $g_i$ is the gravity component in the $z_i$-direction. The subscripts $i=\mathrm{1,2},3$ refer to the $x$-, $y$-, and $z$-direction. According to the Boussinesq eddy viscosity assumption 16, the Reynolds stress, i.e., $-\rho \overline{u_i^{\prime }{u}_j^{\prime }}$ in (2), is proportional to the mean strain rate tensor, where the proportional constant is the eddy viscosity ${\mu }_t$, which is calculated by the SST $k$–$\omega$ model [16,17] in this study and defined as

$${\mu }_t=\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\displaystyle \frac{\rho k}{\mathrm{m}\mathrm{a}\mathrm{x}(\frac{\omega }{{\alpha }^{*}},\frac{\left|\mathbf{S}\right|F_2}{a_1})}},{\displaystyle \frac{\rho k{C}_T}{\sqrt{3\left|\mathbf{S}\right|}}}\right\},$$

(3)

where $k$ is the turbulent kinetic energy, $\omega$ is the specific dissipation rate, $\overline{\mathbf{v}}$ is the mean velocity, and $\mathbf{S}$ is the mean strain rate tensor defined as

$$\mathbf{S}={\displaystyle \frac{1}{2}}\left(\nabla \overline{\mathbf{v}}+\nabla {\overline{\mathbf{v}}}^{\mathrm{T}}\right).$$

(4)

The other key model constants and functions refer to [17]. The turbulent kinetic energy equation and the specific dissipation rate equation are shown below, where ${\tau }_{ij}$ is the shear stress.

$${\displaystyle \frac{\partial \left(\rho k\right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho u_{j}k\right)}{\partial z_j}}={\tau }_{ij}{\displaystyle \frac{\partial u_i}{\partial z_j}}-{\beta }^{*}\rho \omega k+{\displaystyle \frac{\partial }{\partial z_j}}\left[\left(\mu +{\sigma }_k{\mu }_t\right){\displaystyle \frac{\partial k}{\partial z_j}}\right],$$

(5)

$${\displaystyle \frac{\partial \left(\rho \omega \right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho u_j\omega \right)}{\partial z_j}}={\displaystyle \frac{\rho \gamma }{{\mu }_t}}{\tau }_{ij}{\displaystyle \frac{\partial u_i}{\partial z_j}}-\beta \rho {\omega }^2+{\displaystyle \frac{\partial }{\partial z_j}}\left[\left(\mu +{\sigma }_{\omega }{\mu }_t\right){\displaystyle \frac{\partial \omega }{\partial z_j}}\right]+2\left(1-F_1\right){\displaystyle \frac{\rho {{\sigma }_{\omega }}_2}{\omega }}{\displaystyle \frac{\partial k}{\partial z_j}}{\displaystyle \frac{\partial \omega }{\partial z_j}},$$

(6)

2.2. Model Setup

Figure 1 shows the dimensions and boundary conditions of the computational domain in this study, where $D$ is the rotor diameter. The inflow boundary is located 2$D$ before the turbine, the outflow boundary is located 12$D$ behind the turbine, the top boundary is located 5$D$ above the mean sea level where the lower boundary is located, and the boundaries on the other two sides are located 2.5$D$ from the hub of the wind turbine. Note that real wake may extend farther than 12$D$ in a stable atmospheric boundary layer. However, regarding the unsteadiness of the motions in waves and that fact that the fluctuated aerodynamic loads on the blades require tremendous computational efforts, our simulation cropped the domain up to 12$D$. Another engineering practice of large offshore wind farm grid layouts is to use a spacing of about 7$D$.

The boundary condition of the object surface and the lower boundary of the computational domain are assumed to be no-slip walls:

$$U=0.$$

(7)

where $U$ is the inflow velocity. By considering the wind shear, the boundary condition of the top and inflow boundary of the computational domain are assumed to be velocity inlets:

$$U=V_w{\left({\displaystyle \frac{z}{H_h}}\right)}^{\alpha },$$

(8)

where $V_w$ is the wind velocity at hub height, $z$ is the height above the mean sea level, $H_h$ is the hub height, and $\alpha$ is the power law exponent, which is assumed to be 0.14 for offshore areas. The boundary condition of the outflow boundary of the computational domain is assumed to be an outlet:

$$\nabla U·\mathbf{n}=0,$$

(9)

where $\mathbf{n}$ is the normal vector. As shown in Figure 2, the size of the background mesh cells is refined from 72 m to 18 m. To better capture the boundary of the wake, the cell size at the boundary of the rotor is further refined to 9 m in the downstream direction, as the purple box represents. Due to the complex geometry of the turbine, the cell size around the turbine is refined to 4.5 m. The total number of computational mesh cells is around 13 million. A grid dependency study was conducted by systematically varying the spacing. Power output was selected as the target function, and five meshes from coarse at 7 million to fine at 15 million were tested. The grid-independent solution was estimated by Richardson’s Extrapolation method, as per Figure 3. At 13 million, the discretization error is 0.2%, which is considered acceptable. Due to the unsteadiness of motions, time-marching iteration errors were controlled by the residuals of field variables. The minimum of the continuity residual was 9 × 10⁻³, that of the turbulence dissipation rate residual was 1 × 10⁻⁵, and that of the turbulence kinetic energy residual was 2 × 10⁻⁴. The X-, Y-, and Z-momentum residuals ranged from 4.4 × 10⁻³ to 7.4 × 10⁻¹.

In this study, overset mesh is used to simulate the surging and pitching behaviors of the floating wind turbine, as the red box represents, and sliding mesh is used to simulate the rotation of the rotor and the blade pitch, as the green box represents. The surging and pitching trajectory of the floating wind turbine is defined as a sinusoidal function.

3. Simulating Conditions

3.1. Wind Turbine

The IEA 15 MW offshore wind turbine is adopted as the reference turbine in this study. The specifications and dimensions of the IEA 15 MW wind turbine are shown in

Table 1. Specifications of the IEA 15 MW offshore wind turbine [18].

Parameter	Units	Value
Rated Power	MW	15
Rotor Orientation	-	Upwind
Blade configuration	-	3
Blade Length	m	120
Hub Height	m	150
Hub/Rotor Diameter	m	7.94/240
Tower Base Diameter	m	10
Hub Overhang	m	11.35
Rated Wind Speed	m/s	10.59
Rated Rotor Speed	RPM	7.56
Shaft Tilt/Pre-Cone Angle	°	6/4

and Figure 4 [18]. The hub height and rotor diameter are 150 m and 240 m, respectively. The cut-in, rated, and cut-out wind speeds are 3 m/s, 10.59 m/s, and 25 m/s, respectively. The cut-in and rated rotor speeds are 5 RPM and 7.56 RPM, respectively.

3.2. Case Description

In this study, the hub-height wind speed is assumed to be 9 m/s, and the rotor speed of the wind turbine is set to be 7.457 RPM. Note that this is not the rated condition of the target wind turbine. Due to the motion-induced relative inflow velocity, we selected 9 m/s and the corresponding 7.457 RPM such that the condition meets the turbine’s operational control system [18] and the power output does not exceed 15 MW.

The simulation first considers 200 revolutions under a stationary condition. After the wake fully develops and reaches the outflow boundary of the computational domain, simulations are conducted for 200 rotor revolutions under stationary, surging, and pitching conditions, separately, to investigate the effects of these motions on the turbine’s wake characteristics and power output. The amplitudes of the sinusoidal functions describing the surging and pitching motions are 2.75 m and 5°, respectively, with a common period of 45 s. The amplitude of the surging trajectory is based on the maximum mean offset of a 15 MW floating offshore wind turbine system installed on the original semi-submersible TaidaFloat platform, developed by a research team at National Taiwan University. This value corresponds to the platform’s response under a typical wave condition in the Hsinchu offshore area, as shown in Figure 5, where the dominant motion frequency is 45 s [19]. The amplitude of the pitching trajectory is determined based on the allowable mean tilt angle of a floating wind turbine during normal operation [20].

Table 2. Case description.

Parameter	Value
Turbine Rotor Speed	7.457 RPM
Wind Speed at Hub Height	9 m/s
Turbine Condition	Stationary	Harmonic Surge	Harmonic Pitch
Solution Time	200 Rotor Rev.	200 Rotor Rev. after Stationary	200 Rotor Rev. after Stationary
Motion Period	-	45 s	45 s
Motion Amplitude	-	2.75 m	5°

shows the details of the three studied cases.

4. Result and Discussion

4.1. Power Output

This study first validates the numerical methods used to predict the power output of the wind turbine under stationary conditions. Figure 6 shows the comparison of the generator torque of the wind turbine under different conditions, where $Q$ is the generator torque and $t/t_p$ denotes the normalized time in terms of rotor revolutions. With the following equation, the power output $P$ is obtained with an energy conversion efficiency of 95% [15]:

$$P=2\pi nQ·\eta ,$$

(10)

where $n$ represents the rotor speed in rad/s and $\eta$ denotes the energy conversion efficiency. Figure 7 illustrates the comparison of the wind turbine power output under different conditions. According to [15], the designed power output of the IEA 15 MW offshore wind turbine at a hub-height wind speed of 9 m/s is 8.603 MW, as indicated by the yellow reference line in the figure. The power output of the wind turbine under stationary, surging, and pitching conditions, as predicted by STAR-CCM+ in this study, is represented by the red, blue, and green lines, respectively. The corresponding power output amplitudes are 0.165 MW for the stationary case, 1.390 MW for surging, and 5.712 MW for pitching. As shown in

Table 3. Comparison of average generator torque and average power output.

Turbine Condition	Ref. [15]	Stationary	Surging	Pitching
$\overline{Q}$ (MN∙m)	-	11.874	11.613	11.452
$\overline{P}$ (MW)	8.603	8.808	8.616	8.496

, the average power output of a stationary turbine is 8.808 MW, which is approximately 2% higher than the designed value of 8.603 MW. This result confirms the validity of the numerical methods used in this study. The average power outputs under surging and pitching motions are 8.616 MW and 8.496 MW, respectively, representing a reduction of 2.18% and 3.54% compared to the stationary condition.

4.2. Wake Field

The normalized axial velocity distributions on the $x$-$y$, $x$-$z$, and $y$-$z$ planes at $t/t_p=200$ are shown in Figure 8 and Figure 9. The near-wake region is defined as the area within 5$D$ downstream of the turbine, while the far-wake region i refers to the area beyond 5D. The normalized axial velocity $V_x^{*}$ is defined as:

$$V_x^{*}={\displaystyle \frac{V_x}{V_w}},$$

(11)

where $V_x$ is the axial velocity component of the flow field in the $x$-direction. In Figure 8, a low-speed region induced by rotor rotation is observed in the negative y-direction of the flow field under all three turbine operating conditions. Additionally, a low-speed region caused by the tower shadow effect appears below the hub in all cases. Figure 9 shows that after 200 rotor revolutions, the wake behind the stationary turbine begins to deform at 4$D$ and to form an irregular shape at 7$D$ behind the stationary turbine. In contrast, under surging and pitching conditions, wake deformation initiates earlier—at approximately 3D and 2D, respectively—due to the dynamic motion of the platform. The low-speed region induced by rotor rotation is observed downstream in the direction opposite to the rotor’s rotational movement. However, the prominence of this region diminishes due to the turbine’s dynamic motions. For the stationary turbine, the low-speed region is still observed at 7$D$ behind the turbine. In contrast, under the surging and pitching motions, the low-speed region disappears after 4$D$ and 2$D$ behind the turbine, respectively.

To further analyze the wake characteristics, this study considers the swept area of the turbine rotor along the downstream as the wake region. With the swept area denoted by $A$, the normalized average axial velocity of rotor surface $V^{*}$ is defined as follows:

$$V^{*}={\displaystyle \frac{1}{A}}\int V_x^{*}dA.$$

(12)

With the velocity standard deviation of the rotor surface, representing its spatial non-uniformity, denoted by $\sigma$, the normalized velocity standard deviation of the rotor surface is defined as the following:

$${\sigma }^{*}={\displaystyle \frac{\sigma }{V_w}}.$$

(13)

The normalized time-averaged velocity ${\overline{V}}_t^{*}$ and normalized time-averaged standard deviation ${\overline{\sigma }}_t^{*}$ are defined as follows:

$${\overline{V}}_t^{*}={\displaystyle \frac{1}{t}}\int V^{*}dt,$$

(14)

$${\overline{\sigma }}_t^{*}={\displaystyle \frac{1}{t}}\int {\sigma }^{*}dt.$$

(15)

To discuss the influence of the surging and pitching behaviors on the wake characteristics, the differences in ${\overline{V}}_t^{*}$ and ${\overline{\sigma }}_t^{*}$ between the surging and stationary turbines, as well as between the pitching and stationary turbines, are defined as $∆{\overline{V}}_t^{*}$ and $∆{\overline{\sigma }}_t^{*}$, respectively.

Figure 10a illustrates the variation in the difference in the normalized time-averaged velocity between surging and stationary turbines at various downstream locations. The results indicate that the surging behavior causes almost no difference in the velocity at the beginning of the simulation. At $t/t_p=40$, the largest $\left|∆{\overline{V}}_t^{*}\right|$ is 1.53% of $V_w$, occurring at 5$D$ behind the turbine. At $t/t_p=80$, the largest $\left|∆{\overline{V}}_t^{*}\right|$ is 4.30% of $V_w$, occurring at 6$D$ behind the turbine. At $t/t_p=120$, $160$, and $200$, the largest $\left|∆{\overline{V}}_t^{*}\right|$ are 5.73%, 6.34%, and 6.90% of $V_w$, respectively, all located at 7$D$ behind the turbine. Figure 10b presents the differences in the normalized time-averaged velocity between pitching and stationary turbines at various downstream locations. Similar to the surging case, the pitching motion causes almost no difference in the velocity at the beginning of the simulation. At $t/t_p=40$, the largest $\left|∆{\overline{V}}_t^{*}\right|$ is 5.75% of $V_w$, occurring at 4$D$ behind the turbine. At $t/t_p=80$, the largest $\left|∆{\overline{V}}_t^{*}\right|$ is 10.08% of $V_w$, occurring at 5$D$ behind the turbine. At $t/t_p=120$, $160$, and $200$, the largest $\left|∆{\overline{V}}_t^{*}\right|$ are 12.01%, 12.94%, and 13.72% of $V_w$, respectively, all occurring at 6$D$ behind the turbine. The results indicate that $∆{\overline{V}}_t^{*}$ between both the surging and stationary turbines exhibits a downstream trend of initially increasing and then decreasing. The largest $\left|∆{\overline{V}}_t^{*}\right|$ increases with the simulation time, and the corresponding location gradually shifts farther downstream. Noticeable differences in $∆{\overline{V}}_t^{*}$ are observed between $t/t_p=0$ to $160$, with the variations diminishing after $t/t_p=120$. This suggests that the influence of the surging and pitching behaviors on the wake characteristics is not fully propagated throughout the flow field before $t/t_p=160$. Beyond $t/t_p=120$, the location of the largest $\left|∆{\overline{V}}_t^{*}\right|$ tends to stabilize, and its value gradually converges.

The comparison of the difference in the normalized time-averaged standard deviation between surging and stationary turbines at various downstream distances is shown in Figure 11a. The results indicate that the surging behavior causes barely any difference in the velocity at the beginning of the simulation. At $t/t_p=40$, the largest $\left|∆{\overline{\sigma }}_t^{*}\right|$ is 1.26% of $V_w$, located 12$D$ behind the turbine. At $t/t_p=80$, $120$, $160$, and $200$, the largest $\left|∆{\overline{\sigma }}_t^{*}\right|$ are 1.64%, 2.05%, 2.27%, and 2.37% of $V_w$, respectively, located at 1$D$ behind the turbine. The comparison of the difference in the normalized time-averaged standard deviation between pitching and stationary turbines at different downstream distances is shown in Figure 11b. The results show that the pitching behavior also causes barely any difference in the velocity at the beginning of the simulation. At $t/t_p=40$ and $80$, the largest $\left|∆{\overline{\sigma }}_t^{*}\right|$ are 2.07% and 2.81% of $V_w$, respectively, located at 4$D$ behind the turbine. At $t/t_p=120$, $160$, and $200$, the largest $\left|∆{\overline{\sigma }}_t^{*}\right|$ are 3.14%, 3.63%, and 3.89% of $V_w$, respectively, located at 1$D$ behind the turbine. The results also show that $∆{\overline{\sigma }}_t^{*}$ between surging and stationary turbines, as well as pitching and stationary turbines, exhibits a fluctuating trend along the downstream direction. The largest $\left|∆{\overline{\sigma }}_t^{*}\right|$ increases with the solution time. The highest values of $\left|∆{\overline{\sigma }}_t^{*}\right|$ mainly appear at 1$D$ and 4$D$. Similar to the results of $∆{\overline{V}}_t^{*}$, significant differences in$∆{\overline{\sigma }}_t^{*}$ are found between $t/t_p=0$ to $160$, and the differences become smaller at $t/t_p=160$. This further indicates that the influence of the surging and pitching behaviors on the wake characteristics is not fully propagated throughout the flow field before $t/t_p=160$. After $t/t_p=160$, the value of $\left|∆{\overline{\sigma }}_t^{*}\right|$ gradually converges.

From the above explanation, the wind turbine wake evolves over time and space. Figure 12a visualizes the contour of the difference in the normalized time-averaged velocity between surging and stationary turbines. In the near-wake region within 5$D$ behind the turbine, it is clear that the influence of surging behaviors on the normalized time-averaged velocity is always less than 1% of $V_w$ within 2$D$ behind the turbine. The influence is less than 1% of $V_w$ in the whole downstream region when the solution time is less than $t/t_p=$ 40. The influence increases from less than 1% to 5% as the solution time increases from $t/t_p=$ 40 to 120, indicating that it takes at least 120 rotor revolutions for the influence to converge in the near-wake region. After 120 rotor revolutions, the influence becomes more evident, increasing from less than 1% to 5% of $V_w$ along 2$D$ to 5$D$ behind the turbine. In the far-wake region, which is after 5$D$ behind the turbine, the influence increases from less than 1% to 6% after 40 rotor revolutions. The largest propagation distance of the surging behavior extends from 5$D$ to 12$D$ with a linear trend of 0.128$D$ farther per revolution. Figure 12b shows the contour of the difference in the normalized time-averaged velocity between pitching and stationary turbines. In the near-wake region, which is within 5$D$ behind the turbine, the influence of pitching behaviors on the normalized time-averaged velocity is less than 1% of $V_w$ when the simulation time is less than $t/t_p=$ 10. The influence increases from less than 1% to 12% as the simulation time increases from $t/t_p=$ 10 to 140. This indicates that it takes at least 160 rotor revolutions for the influence to converge in the near-wake region. After 160 rotor revolutions, the influence becomes stable and increases from less than 2% to 12% of $V_w$ along 2$D$ to 5$D$ behind the turbine. In the far-wake region, which is beyond 5$D$ behind the turbine, the influence increases from less than 1% to 12% after 10 rotor revolutions. The largest propagation distance of the pitching behavior increases from 5$D$ to 12$D$ with a linear trend of 0.25$D$ farther per revolution. The difference in the normalized time-averaged velocity appears before 10 rotor revolutions at the mid-wake regions within 6$D$ to 8$D$, as well as from 10$D$ to 12$D$ behind the turbine, and the error caused by the stationary rotor during the 200 rotor revolutions is negligible.

In terms of wake homogeneity, the pitching wind turbine exhibits greater fluctuations than the surging motion in the near-wake region. However, there appears to be a sweet spot around 7D where the wake is well mixed under wind turbine motions. In other words, if floating wind turbines are spaced 7D distance or more apart, the dynamic wake effect can positively enhance the average inflow and reduce fluctuations compared to the stationary baseline.

5. Conclusions

The dynamic wake characteristics of a semi-submersible floating wind turbine were investigated through numerical simulations, incorporating surging and pitching motions and comparing them with the stationary condition. The wind turbine under surging and pitching motions exhibited reductions in average power output of 2.2% and 3.5%, respectively, compared to the stationary condition. The near-wake region exhibited significant unsteady fluctuations induced by the turbine’s motions. The fluctuations also enhanced mixing with the undisturbed flow field, leading to an increased average velocity within the wake region. In the present case, the surging and pitching motions contribute to increases of 6.9% and 13.7%, respectively, in the average flow velocity at a downstream distance of seven rotor diameters. Moreover, the motion-induced mixing effect contributed to a reduction in the spatial inhomogeneity of the average flow velocity beyond the downstream distance of 7D. Nevertheless, the simulation assumed steady wind, and future works on wind spectrum and wind–wave correlated conditions are expected to reproduce more realistic operations.