Hydrodynamics and Wake Flow Analysis of a Floating Twin-Rotor Horizontal Axis Tidal Current Turbine in Roll Motion

Abstract

The study of hydrodynamic characteristics of floating double-rotor horizontal axis tidal current turbines (FDHATTs) is of great significance for the development of tidal current energy. In this paper, the effect of roll motion on a FDHATT is investigated using the Computational Fluid Dynamics (CFD) method. The analysis was conducted in the CFD software STAR-CCM+ using the Reynolds-averaged Navier–Stokes method. The effects of different roll periods and tip speed ratios on the power coefficient and thrust coefficient of FDHATT were studied, and then the changes in the vorticity field and velocity field of the turbine wake were analyzed by two-dimensional cross-section and Q criterion. The study indicates that roll motion results in a maximum decrease of 30.76% in the average power coefficient and introduces fluctuations in the instantaneous load. Furthermore, roll motion significantly accelerates the recovery of wake velocity. Different combinations of roll periods and tip speed ratios lead to varying degrees of wake velocity recovery. In the optimal combination, at a position 12 times the rotor diameter downstream, roll motion can recover the wake velocity to 92% of the incoming flow velocity. This represents a 23% improvement compared to the case with no roll motion.

1. Introduction

With the depletion of coal, oil, and other energy sources, as well as environmental issues such as global warming, people have realized the importance of developing and utilizing renewable energy. Compared with land-based renewable energy, ocean-based renewable energy has greater development potential, and the global renewable tidal current energy is about 3 TW [1]. Tidal current energy, as a marine renewable energy, has the advantages of high predictability, high energy density, and no occupation of land area [2]. Tidal current turbines are devices with mature technology that generate electricity using tidal currents, which convert the kinetic energy of water into mechanical energy of rotor rotation and then generate electricity. Due to the limited resources in nearshore areas, floating tidal current turbines (FHATTs) are used in deep-sea areas, and the loads on FHATTs are more complex. Among them, wave loads can greatly affect the performance of FHATTs, and it is important to study the operating characteristics of tidal current turbines in wave environments.

Barltrop [3] analyzed the hydrodynamic performance of tidal current turbines under wave–current interactions using blade element momentum theory and linear wave theory. The experiments were conducted in a towing tank with a length of 77 m, width of 4.6 m, and depth of 2.4 m. The rotor had a diameter of 0.4 m and a three-bladed structure. The study found that at lower flow velocities, the presence of waves slightly increased the average torque while the average thrust was almost unaffected. However, the instantaneous load produced significant fluctuations. Galloway [4] conducted experiments on a three-bladed tidal current turbine in a towing tank and obtained similar conclusions, with wave-induced thrust and torque fluctuations of 37% and 35%, respectively. Luznik [5] conducted model experiments on a three-bladed tidal current turbine in the U.S. Navy’s mechanics laboratory. The rotor had a diameter of 0.46 m and used an E387 airfoil. The towing tank had a length of 37 m, width of 2.4 m, and depth of 1.5 m. The selected relative wave depth was λ/h = 0.335 (λ represents the wavelength, and h represents the water depth). The study found that the difference in the average performance of the turbine under wave and no-wave conditions was very small, and the horizontal and vertical wave velocities were strongly correlated with the rotor torque. Guo [6] further investigated the variations in load on a tidal current turbine operating in a wave environment. The experiments were conducted in a towing tank at Zhejiang Ocean University with a length, width, and depth of 130 m, 6 m, and 3 m, respectively. The rotor was a three-bladed horizontal axis structure, and to reduce machining difficulties and blockage effects, the rotor diameter was proportionally reduced to 800 mm. Similar conclusions were drawn that waves do not affect the average load on the tidal current turbine but have a significant impact on the instantaneous load. The instantaneous torque and thrust reached a maximum at the wave crest and a minimum at the wave trough, with the load fluctuation frequency matching the wave frequency. The following conclusions were drawn: load fluctuations have a linear relationship with wave height, with an increase in wave height resulting in an increase in load fluctuation. Load fluctuations decrease with an increase in H/D (H represents the water depth at which the rotor is located, and D represents the rotor radius), but increase with an increase in the tip speed ratio (TSR) and λ/h.

The floating platform is subject to wave influences, which cause the tidal current turbine and floating platform to move in six degrees of freedom (6-DOF). These movements include roll, sway, pitch, heave, yaw, and surge, and may alter the fluid dynamic performance of the turbine [7]. Currently, there is limited research on the hydrodynamic performance of tidal current turbines under 6-DOF motion. Zhang [8] found that the turbulence intensity distribution in the wake region becomes asymmetrical when the turbine is subjected to yaw inflow. The thrust decreases as the yaw angle increases and reaches a minimum value at a yaw angle of 40°. The range of thrust fluctuations increases with increasing yaw angle but begins to decrease when the yaw angle exceeds 30%. Ma [9] analyzed the characteristics of a vertical-axis tidal current turbine under heave motion and found that different heave frequencies and periods only affect the instantaneous load fluctuations, but not the average load of the turbine. The shedding of vortices in the turbine is affected by the heave motion, and the vortices are broken down into smaller ones. The shedding frequency is independent of the heave amplitude, but the increase in heave frequency accelerates the vortex shedding frequency and also leads to greater load fluctuations of the turbine. Pranav [10] conducted a CFD simulation of a three-bladed horizontal axis tidal current turbine under yawed inflow conditions and found that at a yaw angle of 15 degrees, the power coefficient (C_P) and thrust coefficient (C_T) of the turbine decreased by about 8.7% and 9%, respectively. The yawed condition generated an asymmetric vortex structure near the axis of the turbine. Meanwhile, the recovery velocity of the wake was faster due to the higher momentum flux entering the turbine under yawed conditions, which led to the disruption of the wake structure and reduced the load fluctuations on the downstream turbine caused by the wake of the upstream turbine, thus increasing the turbine’s lifespan, reducing maintenance costs, and improving energy yield and power generation efficiency. Wang [11] studied the hydrodynamic performance of a horizontal axis tidal current turbine under roll motion. It was found that the load of the turbine would produce corresponding fluctuations under roll motion, with the frequency of load fluctuations consistent with that of the roll motion. The amplitude of load fluctuations increased with the frequency and amplitude of the roll motion. When the direction of roll motion was the same as the direction of turbine rotation, it was equivalent to increasing the rotational speed of the turbine. When the direction of roll motion was opposite to the direction of turbine rotation, it was equivalent to reducing the rotational speed of the turbine. Therefore, the hydrodynamic performance of the tidal current turbine would change with the direction of the roll angular velocity. Zhang [12] discovered that surge motion could cause fluctuations in the axial force coefficient and power coefficient of the turbine, which were positively correlated with the surge frequency, amplitude, and TSR. The axial damping coefficient of the turbine was derived by using the least squares method and was found to have no significant correlation with the amplitude and frequency of the surge motion. Osman [13] obtained similar results in their study. Jiang [14] studied the load variation of a vertical-axis tidal current turbine under the coupling effect of roll and yaw. The study found that the influence of roll motion on the load is greater than that of yaw motion, and the instantaneous load fluctuation amplitude increases with the increase in roll frequency.

Extensive research has been conducted on tidal current turbines, but most studies have focused on single-rotor structures. However, a type of tidal current turbine with a double rotor has a greater advantage in capturing tidal energy. Ma [15] conducted a study on the load of a double-rotor vertical-axis tidal current turbine, and the results showed that the double-rotor system can increase power generation. Compared to a single-rotor system, the average output power of a double rotor can be increased by 15.3%, and the interaction between the two rotors can increase power output at high TSRs. Liu [16] conducted a comparative experiment on coaxial and parallel double-rotor tidal current turbines in a tank with a width of 0.8 m and a depth of 0.65 m. When coaxially arranged, the distance between the upstream and downstream rotors had a significant impact on the performance of the tidal current turbine. The efficiency of the upstream rotor decreased significantly when the distance was narrow, while the load fluctuation of the downstream rotor gradually increased as the distance increased. The double-rotor system had the best overall performance when the distance between the two rotors was 0.4 D (rotor diameter), with a performance improvement of nearly 10% compared to a single-rotor tidal current turbine. When the rotors are arranged in parallel, the highest power generation efficiency was achieved at a distance of 0.3 D between the two rotors. As the distance increased, the power generation efficiency decreased, and when the distance increased to 1 D, there was almost no interaction between the two rotors. Wei [17] summarized the relationship between the power coefficient of the double-rotor tidal current turbine and the axial distance between the rotors through experiments, showing that the performance of the front rotor improved significantly with an increase in axial distance, while the performance of the rear rotor did not show significant improvement. Lee [18] demonstrated through experiments that the performance of the double-rotor turbine is significantly affected by the difference in rotor speed ratio and radius.

In tidal current turbine arrays, the performance of downstream turbines is mainly influenced by upstream wake. The loss of wake velocity results in decreased energy extraction by downstream turbines, which in turn affects the overall efficiency. Divett [19] conducted a wake analysis of tidal current turbine arrays and found that the wake of upstream turbines significantly impacts the power generation of downstream turbines. An alternate arrangement of turbines can help the downstream turbines extract more energy. Thus, a comprehensive understanding of the wake structure of individual turbines is crucial for optimizing the layout of tidal current turbine arrays. Stallard [20] conducted a wake analysis on a single tidal current turbine and a tidal current turbine array. The study found that when only one turbine was operating, the wake velocity decreased to 20% of the ambient flow velocity at 2 D downstream of the rotor, and recovered to 80% within 10 D and 90% at 20 D. When the turbines were arranged in an array, the wakes merged within 4 D downstream of the rotor when the lateral spacing between rotors was 2 D, and there was almost no interaction between wakes when the spacing was 6 D. Mycek [21,22] showed that turbulence intensity has a significant impact on the wake of a turbine. The higher the turbulence intensity, the faster the wake dissipates. This means that the wake generated by an upstream turbine recovers faster under high turbulence intensity. Therefore, the interaction between turbines is also severely affected by turbulence intensity. Mycek [22] demonstrated that the power coefficient of downstream turbines is higher when turbulence intensity is high. Tahani [23] investigated the wake characteristics of a turbine with turbulence intensities of 1% and 8% using blade element momentum theory (BEM). The results showed that as turbulence intensity increased, the turbulent kinetic energy in the wake increased, resulting in faster wake recovery.

Faizan [24] analyzed the accuracy of Improved Delayed Detached Eddy Simulation (IDDES) and k-omega Shear Stress Transport (k-ω SST) turbulence models in predicting wake velocity and power coefficient through CFD methods. The study found that the error in predicting wake velocity and power coefficient using the IDDES model was smaller than that of the k-ω SST model. Leroux [25] compared steady-state and transient numerical methods using Reynolds Averaged Navier–Stokes (RANS) methods. The study showed that both methods could accurately predict the power coefficient and thrust coefficient of tidal current turbines. The steady-state method had a shorter time and lower cost, while the transient method could better predict wake changes, flow rate, and velocity distribution.

Experimental methods offer advantages in monitoring numerical values, but they struggle to capture wake structures. Numerical simulation methods can address this issue. The commonly used numerical methods for turbine studies are BEM and CFD. To accurately simulate turbine wakes, transient studies are required, and BEM has been proven inadequate for unsteady load calculations [26]. Tian [27] utilized the CFD approach for three-dimensional transient calculations, and the computational results match the experimental values closely. Additionally, CFD has proven its ability to finely resolve turbulence in both near-field and far-field regions [28]. Therefore, this study uses the CFD method for transient simulations, selecting the k-omega Shear Stress Transport (SST) and Reynolds averaged Navier–Stokes (RANS) turbulence models to precisely predict wakes. A detailed explanation of the turbulence model selection is provided in Section 2.2.

The literature above presents numerous studies on single-rotor tidal current turbines, including analyses of load and wake under various degrees of freedom (DOF) motions and in complex environments. However, research on double-rotor tidal current turbines under DOF motions is almost non-existent, and current studies have not explained the recovery patterns of wake. This significant research gap has motivated the authors to explore further in this potential research direction. Therefore, this paper focuses on investigating the load and wake variations of double-rotor tidal current turbines under roll motion. The innovation of this study is in its establishment of a link between vortex structures and the recovery of wake velocity for the first time, offering a significant method for wake research. First, variations in power coefficient and thrust coefficient of double-rotor tidal current turbines under roll motion are analyzed. Then, it is found that different combinations of roll periods and tip speed ratios have varying effects on the recovery of wake velocity. Finally, the differences in wake velocity recovery are elucidated using the vortex structures.

2. Methods

2.1. Basic Theory

Tidal current turbines have practical dimensions that are relatively large. In order to reduce computational costs, numerical simulations often employ scaled models for research. Therefore, for the purpose of facilitating performance comparisons among turbines of different sizes, dimensionless treatment was conducted on the crucial parameters of the water turbine addressed in this article.

The Tip Speed Ratio (TSR) is employed to illustrate the relationship between the rotor’s rotational speed and the flow velocity. It is defined as follows:

$$TSR=\frac{\omega R}{U}$$

(1)

The Power Coefficient (C_P) is the most crucial parameter for energy conversion in tidal current turbines. The power coefficient is defined as the ratio of the power extracted by the turbine (P) to the maximum available power of the incoming flow through the rotor area:

$$C_P=\frac{P}{0.5\rho S{U}^3}=\frac{Q\omega }{0.5\rho \pi R^2{U}^3}=\frac{Q}{0.5\rho \pi R^3{U}^2}\times \mathrm{TSR}$$

(2)

The Thrust Coefficient (C_T) is a fundamental indicator for evaluating the hydrodynamic performance of a turbine. The thrust coefficient is defined as the ratio of the thrust force acting on the turbine (T) to the kinetic energy of the incoming flow through the rotor area:

$$C_T=\frac{T}{0.5\rho S{U}^2}=\frac{T}{0.5\rho \pi R^2{U}^2}$$

(3)

where U is the water flow velocity (unit: m/s), ω is the angular velocity of the turbine blades (unit: rad/s), R is the blade Radius, ρ represents the density of water, S = πR² is the swept area of the turbine, and Q represents the torque of the turbine.

2.2. Numerical Model Setup

2.2.1. Geometrical Model

The rotor diameter of the tidal current turbine is 0.7 m, and the rotor is composed of two blades. The hub diameter is 0.1 D (0.1 times the rotor diameter). The blade profile selected in this study was designed by the Institute of Ocean Renewable Energy System (IORES) at Harbin Engineering University in China, and the profile model is S809.

2.2.2. Boundary Conditions Setting

The computational domain shown in Figure 1 is a rectangular structure using the Cartesian coordinate system, where the directions of length, width, and height correspond to the Z, X, and Y axes of the Cartesian coordinate system, respectively. The length, width, and height of the computational domain are 14 D, 5.7 D, and 4.28 D, respectively. The entire computational domain is divided into a stationary domain and a rotating domain, where the shape of the rotating domain is a cylinder with a diameter of 1.29 D. To ensure data exchange between the two domains during the calculation, the contact boundary between the rotating and stationary domains is set to overset mesh. The inlet boundary is set as velocity inlet to specify a uniform flow velocity and turbulence intensity. The outlet boundary is set as pressure outlet with an average pressure of 0, and the water flow direction is parallel to the Z-axis. The surface of the turbine blades is set as a wall. To avoid wall effects, the top, bottom, and two side faces of the computational domain are set as symmetry planes. Wu [29] demonstrated that when the distance between the rotor and the inlet and outlet of the computational domain is 3 D and 5 D, respectively, computational accuracy is ensured. Therefore, in this study, the distance between the rotor and the velocity inlet is set at 3 D. For ease of wake observation, the distance between the rotor and the pressure outlet is increased to 14 D, with a horizontal spacing of 1.5 D between the centers of the two rotors. Ultimately, the overall length of the entire computational domain is 14 D, the width is 5.7 D, and the height is 4.28 D.

2.2.3. Mesh

As shown in Figure 2b, a structured mesh was used for the calculation, and the mesh was refined near the blade edges with a target size of 5 × 10⁻⁴ m controlled by the feature curve. As shown in Figure 2c, a prismatic layer mesh is employed near the blade surface. The total thickness of the prismatic layer mesh is 4 × 10⁻³ m, consisting of 8 layers with a growth ratio of 1.2. The value of y⁺ is approximately 10. Stringer [30] used the same turbulence model as this study and found that when y+ is around 10, it can better predict the lift and drag coefficients at low angles of attack, while also providing more stable computational results in stall conditions. As shown in Figure 2a, in order to better analyze the details of the wake structure and facilitate the convergence of the calculation results, the mesh was refined in the wake region of the turbine, with a target mesh size of 0.03125 m. The mesh size of the rotating domain was also set to 0.03125 m to improve the calculation accuracy and stability. The mesh size in other areas of the stationary domain was set to 0.125 m. The total number of mesh cells was 5 million.

2.2.4. Roll Motion

The focus of this study is not on the motion response of the floating platform in waves, and therefore the floating structure has been omitted. Instead, user-defined motion functions are used. As shown in Figure 3, the tidal current turbine is forced to undergo a roll motion, and the angular velocity of the roll motion can be controlled using a harmonic function.

$${\omega }_r=A\frac{2\pi }{T}\cos \left(\frac{2\pi }{T}t\right)$$

(4)

In this context, the roll angular velocity of the horizontal axis turbine is denoted as ω_r (unit: rad/s), where T is the period of roll motion and A is the amplitude of roll motion. The distance in the Y direction between the roll motion axis and the water turbine axis is 0.78 m. Additionally, the tidal current turbine rotates around its own axis with an angular velocity of ω during the roll motion.

2.2.5. Solver and Turbulence Model Settings

The time solver chosen is an implicit unsteady method, with a second-order time discretization. The time step is the time taken for the blade to rotate 3 degrees, and the maximum number of iterations is set to 5. Considering the complex dynamic mesh motion involved in this study, the convergence criteria are appropriately relaxed to residuals less than 1 × 10⁻³. Simultaneously, the inlet and outlet flow rates are monitored, and a flow rate difference less than 1 × 10⁻⁴ kg/s is used as a crucial criterion for convergence. The total computation time is 50 s, by which point the wake behind the rotor has fully developed. The k-omega Shear Stress Transport (SST) and Reynolds averaged Navier–Stokes (RANS) turbulence models are selected. The SST model is a two-equation eddy viscosity model that combines the advantages of the k-omega and k-epsilon models. The standard k-omega model is used within the boundary layer, while the model gradually transitions to the k-epsilon model outside the boundary layer. The reliability of this model has been confirmed in similar studies [31]. Moreover, Wang’s [32] comparison of numerical simulations with experimental data demonstrated the accuracy of this turbulence model in predicting outcomes at a Reynolds number of 1.2 × 10⁻⁵. The Reynolds number of the model employed in this study is approximately 1 × 10⁻⁵, which can be determined from the equation where Re represents the Reynolds number and v denotes the kinematic viscosity of water. In conclusion, it can be argued that the turbulence model utilized in this research is reasonable. A workstation equipped with an Intel(R) Xeon(R) C_PU E5-2680 v2 (20 cores, 40 threads) and 64 GB of RAM is used. It takes three days to complete a simulation with a duration of 50 s.

$$Re=\frac{UD}{v}$$

(5)

3. Results and Discussion

3.1. Feasibility Validation

3.1.1. Feasibility Validation of CFD Method

To validate the accuracy of the numerical simulation. In Figure 4a,b, a single-rotor turbine with S809 airfoil is numerically simulated using the aforementioned mesh and boundary condition settings, with a water flow velocity of 1.5 m/s. In Figure 4a, the C_P corresponding to different TSR is calculated and compared with experimental values. It can be seen that there is a significant difference between the CFD method and experimental values at TSR = 3. This is because a large amount of boundary layer separation occurs on the blade surface in the low TSR range, resulting in CFD results higher than the experimental values. Both CFD and experimental values increase first and then decrease with increasing TSR, and C_P reaches its peak at around TSR = 5. In Figure 4b, the C_T corresponding to different TSRs is calculated using the CFD method. The CFD results are slightly higher than the experimental results for TSRs between 3 and 7, and slightly lower for TSRs between 7 and 8. Overall, it is believed that the CFD method’s predictions of C_P and C_T are reliable.

Figure 4c demonstrates the accuracy of CFD predictions for wake trajectories, using a vertical-axis tidal current turbine with a NACA0012 blade profile, a turbine diameter of 1.22 m, a blade chord length of 0.0914 m, 2 blades, and a flow velocity of 0.091 m/s. In the CFD approach, the Lagrangian particle method is employed to trace wake trajectories, and these results are compared to Strickland’s experiments [33], revealing good agreement between CFD and experimental values. Thus, it is deemed that the CFD method’s predictions of tidal current turbine wakes are reliable.

3.1.2. Verification of Time Step and Mesh Independence

Using the geometric model, mesh, and boundary condition parameters described in Section 2.2, a verification of time step and mesh independence is conducted for a single-rotor tidal current turbine under the conditions of U = 0.3 m/s and TSR = 5. The power coefficient (C_P) and thrust coefficient (C_T) are chosen as comparative parameters. First, an independence verification of the time step is performed, where three selected time steps (0.006 s, 0.012 s, 0.018 s) correspond to rotor rotations of 1.5°, 3°, and 4.5°, respectively. The verification results in the

Table 1. Results of the time step independence test.

	Time Step (0.006 s)	Time Step (0.012 s)	Time Step (0.018 s)
CP	0.7533	0.7549	0.7555
CT	0.3519	0.3525	0.3530
Relative error of CT (%)	−0.2119	0	0.0794
Relative error of CP (%)	−0.1702	0	0.1418

indicate that the relative errors are all below 1%, ensuring that changes in the time step do not affect the accuracy of the computed results and confirming the independence of the time step. Subsequently, an independence verification is conducted for the mesh. As shown in the

Table 2. Mesh size and number.

	Mesh 1	Mesh 2	Mesh 3
Mesh size in rotational domain (m)	0.035	0.03125	0.025
Mesh size in wake region (m)	0.035	0.03125	0.025
Mesh size the stationary domain (m)	0.14	0.125	0.1
Total mesh number (million)	2.78	3.68	6.16

, the mesh sizes and total mesh number are presented for three different mesh schemes. The verification results in the

Table 3. Results of the mesh independence test.

	Mesh 1	Mesh 2	Mesh 3
CP	0.7559	0.7549	0.7528
CT	0.3521	0.3525	0.3543
Relative error of CT (%)	0.1324	0	−0.2781
Relative error of CP (%)	−0.1134	0	0.5106

reveal relative errors all below 1%, ensuring that the number of mesh does not influence the computed results and successfully passing the mesh independence verification.

3.2. Effects of Roll Period and TSR on the Average Load of the Turbine

This study conducted numerical simulations for various combinations of roll periods and tip speed ratios (TSR). The roll amplitude was 8°. Assuming that the rolling period is the same as the wave period, and then scaling the real water field data [34] using the Froude number similarity principle, As shown in the

Table 4. Scaling models.

	Real Environment/Scale Model
Water velocity U (m/s)	0.49/0.3
Roll period T (s)	2/0.75	4/1.5	8/3

, the rolling periods of this study are obtained as 0.75 s, 1.5 s, and 3 s, with a water flow velocity of 0.3 m/s. Additionally, the scaled roll periods are close to the rotor’s rotation periods (0.916 s, 1.46 s, 2.44 s). The rotor’s rotation period can be determined by TSR and flow velocity (U). Some conclusions in the subsequent analysis rely on this relationship; this is also another reason for choosing this proportion scaling. Figure 5 shows the variations of the average C_P and C_T under different tip speed ratios and roll periods. A double-rotor structure was used in this study, and to facilitate comparison with other studies, the C_P and C_T in Figure 5 and Figure 6 are half of the total provided by the two rotors. From Figure 5a, it can be observed that the C_P at T = 0.75 s is lower than that at the other two roll periods, but it exceeds the C_P without roll motion at high TSRs (TSR = 8). The optimal TSRs for the three roll motions are between 5 and 6, while the optimal TSR without roll motion is between 4 and 5, where the optimal TSR refers to the TSR at which the C_P reaches its peak. It can be seen that the C_P curve peaks at T = 1.5 s, T = 3 s, and without roll motion are relatively close, at 0.36, 0.38, and 0.39, respectively. However, the peak value of the C_P curve decreases to 0.27 at T = 0.75 s. The C_P at T = 1.5 s and T = 3 s gradually decreases after reaching the peak, while the C_P at T = 0.75 s remains relatively unchanged after reaching the peak. Figure 5b reflects the variation of C_T with TSR. It can be seen that the C_T values for the three roll motions are similar, and all increase with increasing TSR. The growth rate of C_T is relatively uniform, and the shape of the curve is roughly a straight line. The C_T curve without roll motion increases more gradually and is slightly lower than the other operating conditions at high TSRs (TSR = 8).

3.3. Effects of Roll Period and TSR on the Instantaneous Load of the Turbine

Figure 6 investigates the changes in the instantaneous C_P and C_T for different combinations of roll period and TSR in tidal current turbines. Previous studies have shown that C_P and C_T basically do not fluctuate after the tidal current turbine operates stably in a uniform flow. This study found that the instantaneous load of the tidal current turbine has strong fluctuations after adding the roll motion. Because the angular velocity of the roll motion changes continuously in a cosine law, this means that the resultant velocity acting on the blades is also constantly changing, resulting in the fluctuation of the instantaneous load. The total simulation time is 50 s, and the middle section is analyzed. It can be seen that C_P and C_T exhibit periodic fluctuations in the overall trend, but the periodicity is weak. As shown in Figure 6a (TSR = 3), as the roll period increases, the amplitude of C_P fluctuation gradually decreases. When the roll period doubles, the amplitude of C_P fluctuation is reduced by about half. Similar conclusions are drawn from Figure 6b,c (TSR = 5.8): as the roll period increases, the amplitude of C_P fluctuation decreases. Figure 6d–f show that C_T has similar conclusions with C_P.

Figure 7 compares the instantaneous loads of rotor-1 and rotor-2 and explains the weak periodicity of load fluctuations shown in Figure 6. In Figure 7a,c (TSR = 3), it can be seen that the instantaneous loads of the two rotors differ significantly, and the variation has weak periodicity. In contrast, in Figure 7b,d (TSR = 2.44), the loads exhibit regular periodic fluctuations. When TSR = 2.44, the rotor speed is 2.09 rad/s, and since it has a two-bladed structure, the blade position is restored every 180° rotation of the rotor, and the time it takes for the rotor to rotate 180° is 1.5 s. Therefore, the roll period is equal to the blade rotation period at this time. This means that the rotor position is exactly the same every 1.5 s, resulting in periodic load fluctuations with a period of 1.5 s. Figure 8 shows the changes in rotor position and velocity field with a time interval of 1.5 s. In Figure 8b (TSR = 2.44), the rotor position and velocity field near the blade remain unchanged, while in Figure 8a (TSR = 3), the rotor position and nearby velocity field show significant differences. This is the main reason for the weak periodicity at TSR = 3.

3.4. Effects of Roll Period and TSR on the Wake of the Turbine

The tidal current turbine extracts energy from high-velocity water to drive its blades for power generation. However, the velocity loss caused by the water flow passing through the turbine will significantly affect the downstream turbine’s ability to extract energy from the water flow, resulting in power loss. Therefore, the study of the wake is crucial.

Figure 9 shows the wake velocity field at an axial distance of 4 D behind the turbine for different roll periods and TSRs. The wake velocity distribution and recovery are reflected, where blue represents the low velocity area, yellow represents the medium velocity area, and red represents the high velocity area. In Figure 9a, it can be observed that when there is no roll motion, the wakes behind the two rotors do not interfere with each other, and the distribution of the blue area is continuous and symmetrical. As the TSR increases, the velocity behind the blade tip decreases. Figure 9b–d show the velocity fields with roll motion. It can be seen that within the 4 D range, there is no direct correlation between roll period and the recovery of wake velocity. In Figure 9c,d (T = 1.5, T = 3), it can be observed that within the 4 D range, as the TSR increases, the blue area increases, indicating a slower recovery of wake velocity. This phenomenon is not observed in Figure 9b (T = 0.75 s).

Figure 10 shows the vorticity field of the wake within a 4 D distance behind the tidal current turbine for different roll periods and TSRs. In Figure 10a (no roll motion), the wake of the two rotors is symmetrical and they do not interfere with each other. In Figure 10a1 (TSR = 3), symmetrical “vortex pairs” are formed by the tip vortices, which persist until about 2 D and then form continuous elongated vortices. As the TSR increases, the vortex pairs gradually disappear, and the intensity of the tip vortices increases. Figure 10b–d (with roll motion) clearly change the vortex structures of the tidal current turbine. The interaction between the tip vortices and central vortices of the same rotor, as well as the interaction between the tip vortices of the two rotors, destroys the originally neat vortex structures and generates more small vortices. Only in Figure 10c2, d2 and d3, obvious “vortex pairs” are found. The central vortices are also basically disappearing, and only in Figure 10c2 (T = 1.5 s, TSR = 5), fragmented central vortices are found. In Figure 10c,d (T = 1.5, T = 3), it can be seen that the vorticity intensity increases as the TSR increases within a 4 D range. Comparing Figure 9 and Figure 10, it can be observed that there is no clear correlation between the intensity of vorticity and the velocity.

Figure 11 illustrates the distribution of wake velocity recovery (U_m/U) along the x-direction. Sixty detection points are selected to monitor the instantaneous velocity Um, with U = 0.3 m/s. The dashed area behind the rotor represents the wake region, and the average values within this dashed region reflect the average recovery of wake velocity. The Figure 12 shows the average recovery of wake velocity at axial distances of 4 D and 12 D. As shown in Figure 11a, at an axial distance of 4 D, the recovery of wake velocity depends on the TSR, with slower recovery observed for higher TSRs. However, there is no significant correlation between the recovery of wake velocity and the roll period. The slowest recovery of wake velocity (48%) was observed for T = 1.5 s and TSR = 8, while the fastest recovery (76%) was observed for T = 1.5 s and TSR = 3. At an axial distance of 12 D, the recovery of wake velocity depends on the combination of different roll periods and TSRs. The fastest recovery (92%) was observed for T = 1.5 s and TSR = 3. For the three cases with no roll motion (TSR = 3,5,8), the recovery of wake velocity was slowest, with recoveries of 69%, 54%, and 53%, respectively, indicating that roll motion accelerates the recovery of wake velocity. It can be seen that in the near-wake region (4 D), the recovery of wake velocity is mainly determined by the TSR, while in the far-wake region (12 D), it is determined by the combination of different roll periods and TSRs. The subsequent study found that the vortex structures varied greatly under different combinations, and there is a strong correlation between the recovery of wake velocity and the vortex structures.

Figure 13 analyzed the three-dimensional vortex structure under different roll periods and TSRs using the Q criterion (Q = 5 × 10⁻³ s⁻²), which enables a clearer visualization of the external structure of the vortex. In Figure 13a, the flow structure is clearly symmetric under no roll motion. The wake is composed of tip vortices and central vortices. The red dashed line annotates the tip vortices. Tip vortices are spiral-shape vortex structures formed at the tip of the tidal current turbine rotor, and the seven spiral-shape vortices produced continue until they break down at 2 D and disappear at 3 D. The distance between two spiral-shape vortices is called the pitch, which remains constant under no roll motion. Central vortices form behind the rotor hub and extend as a long, thin, columnar structure until they start to diverge at 7 D.

As shown in Figure 13b–j, with the introduction of roll motion, under some cases, the spiral-shaped tip vortices transform into semicircular ring structures. In Figure 13c,f,i (TSR = 5), as indicated by the red dashed lines, when semicircular ring or spiral-shaped vortex structures appear regularly and repetitively, it is considered that the tip vortices persist. As depicted in Figure 13c,f,i (TSR = 5), for the same tip speed ratio, a larger roll period leads to a greater distance of persistence for the tip vortices. At T = 0.75 s, the tip vortices persist up to 1 D. At T = 1.5 s, the tip vortices persist up to 3 D. At T = 3, the tip vortices nearly persist throughout the entire wake region. The distance of persistence for the tip vortices is not influenced by the tip speed ratio.

It can be observed from Figure 12b (at an axial distance of 12 D) that the recovery of wake velocity depends on the combination of different roll periods and TSRs. The two sets with the fastest wake velocity recovery are T = 0.75 s, TSR = 5, and T = 1.5 s, TSR = 3. The corresponding vortex structures for these two sets are illustrated in Figure 13c,e, where it can be observed that the central and tip vortices are disrupted and merged, and the wake flow path gradually widens as the axial distance increases. Different combinations of roll periods and TSRs result in distinct wake structures, where the wake velocity recovers faster when the tip and central vortices are disrupted. On the other hand, when the wake flow structure is relatively intact, the wake velocity recovery is slower. The experiments conducted below further validate this view.

Figure 14 corresponds to Figure 15, where Figure 14 shows the velocity field and Figure 15 shows the three-dimensional vortex structure. Figure 16 depicts the vorticity field, corresponding to Figure 15c,d. Figure 15b shows the wake structure at a TSR = 2.44, where the roll period is equal to the time taken for the blade to rotate 180°. At this TSR, the wake structure is not disrupted, the wake flow path is narrower, and the tip vortices persist further downstream, while a columnar central vortex is generated behind the hub. In Figure 15a (TSR = 3), the wake structure oscillates and starts to diverge at 2 D. The tip and central vortices are disrupted and the wake of the two rotors merge with each other between 10 D and 12 D. As shown in Figure 17, the wake velocities at a distance of 4 D recover by approximately 76% for TSR = 2.44 and TSR = 3. However, there is a 9% difference in the wake velocity recovery at 12 D, where the wake velocity recovers by 89% for TSR = 3 and by 81% for TSR = 2.44. Figure 15d and Figure 16b show the vortical structures at TSR = 4.88, where the roll period equals the time for the blade to rotate 360°, and the vortical structures are similar to those at a TSR of 2.44. The vortical structures are complete, and the tip vortices and central vortices do not diverge at this TSR. As shown in Figure 17, the wake recovery reaches 66% for TSR = 5 at 4 D downstream, 59% for TSR = 4.88, 75% for TSR = 5 at 12 D downstream, and 60% for TSR = 4.88. The analysis above indicates that there is a direct relationship between the vortical structures and wake recovery, which also explains why the wake recovery is different for different combinations of roll period and TSR, because the vortical structures change under different combinations. When the vortical structures are neat and the tip vortices and central vortices are not destroyed, the wake recovery is slower, and when the tip vortices and central vortices are destroyed, the wake recovery is faster. This also explains why the wake recovery in the far wake region without roll motion is slower, because the vortical structures are not destroyed by roll motion. Roll motion can promote the destruction and fusion of vortical structures between the two rotors, which is the main reason for the acceleration of wake recovery caused by roll motion.

4. Conclusions

This study investigated the hydrodynamic performance of a double-rotor horizontal axis tidal current turbine under roll motion using CFD simulations. Three different roll periods (0.75 s, 1.5 s, and 3 s) and three TSRs (3, 5, and 8) were considered, and the instantaneous and average loads as well as the wake characteristics were analyzed.

(1)

The study found that for the average load, the peak value of C_P was similar for no roll motion and roll periods of 1.5 s and 3 s, with values of 0.39, 0.36, and 0.38, respectively. However, for a roll period of 0.75 s, the peak value of C_P decreased significantly to 0.27, and the roll motion shifted the C_P curve to the right, whereas the effect on the C_T curve was minimal.

(2)

For the instantaneous load, roll motion caused C_P and C_T to fluctuate, but the periodicity of the fluctuations was weak, and the amplitude of the fluctuations decreased with increasing roll period. When the roll period was equal to the time for the rotor to rotate 180 degrees, C_P and C_T showed strong periodic fluctuations.

(3)

When analyzing the three-dimensional vortical structure using the Q-criterion, it was found that the vortex structure was not disrupted in no roll motion conditions, and the wake consisted of spiral-shaped tip vortices and slender central vortices with a fixed pitch between the tip vortices. However, under roll motion, the spiral-shape tip vortices transformed into semi-circular ring structures that were more easily disrupted, and the central vortices almost disappeared.

(4)

At a near-wake location (axial distance of 4 D), wake recovery was mainly determined by the TSR, with no dependence on the roll period. As the TSR increased, the wake velocity recovery slowed down. However, at a far-wake location (axial distance of 12 D), wake recovery was mainly determined by the vortical structure. When the tip and central vortices were not disrupted, the wake velocity recovery was slower. When the roll period was equal to the time for the rotor to rotate 180 or 360 degrees, the vortical structure was less susceptible to disruption, resulting in slower wake recovery. Overall, roll motion can enhance wake recovery.