Numerical Experiments on Hydrodynamic Performance and the Wake of a Self-Starting Vertical Axis Tidal Turbine Array

Abstract

In this paper, based on the CFD software ANSYS-Fluent, two-dimensional numerical models are established to investigate the hydrodynamic performance of a self-starting H-Darrius vertical axis tidal turbine (VATT) array of three turbines in a triangular layout with 3D in axial and radial distance. Three main aspects are explored in this study: (1) the self-starting performance, power coefficient, flow fields, and blade force of the double-row VATT array, which are compared with a stand-alone turbine, (2) the wake development of the front and rear displacement turbines, and (3) the feasibility of the double-row self-starting VATT array in practical applications. It is found that the power coefficients of the three turbines in the array all improved compared with that of the stand-alone turbine, and as the load increased, the difference between the averaged power coefficient of the array and a stand-alone turbine was more obvious, with a maximum difference of 3%. The main effects of the front turbines on the rear turbine are energy utilization and turbine vibration. Due to the beam effect between the front turbines, the incident flow rate of the rear turbine increased to approximately 1.2 times the free flow rate. However, the greater rotational fluctuations of the rear turbine mean that although it had a higher power factor, it was more susceptible to fatigue damage. The wake of the rear turbine in the array had a much larger area of influence on both the length and width, but the velocity deficit recovered more quickly to over 95% at a distance of 10D behind it. The rate of wake velocity recovery is load-dependent for a stand-alone self-starting turbine, but this was not evident in the arrays. The positive torque of the turbine is mainly generated when the blade rotates through an azimuth angle from 45° to 160° and mainly benefits from the inner side of the blade. For the double-row three-turbine array, the axial and radial spacing of 3D is reasonable in practical applications.

1. Introduction

With the rapid development of technologies, renewable energies have attracted more attention for utilization as a promising alternative solution to the fossil fuel crisis. The ocean energy in China has abundant reserves, and the market is on the ascent [1]. Among all types of marine renewable energies, tidal current energy has advantages such as high predictability and stability over other types [2]. Exploring tidal current energy also has accelerated the fast track of development [3]. In the recent two decades, a variety of tidal devices has been widely studied, such as the horizontal axis tidal turbine (HATT), vertical axis tidal turbine (VATT), oscillating hydrofoils, and tidal kites, among which the first two kinds are most widely studied [4]. Compared with other kinds, the rotation of the VATT is not affected by the incident waterflow direction [5], and manufacturing costs are relatively low due to the simple structures of the blades [6]. Moreover, for HATT, the velocity recovers to 65–80% of the inflow velocity at a distance of 6–10D behind the turbine [7], while the recovery of the VATT is mostly accomplished at 5D [8]. This suggests that the wake length of the VATT is shorter than the HATT, allowing more devices to be placed in a limited space to increase power generation per unit of water area. Obviously, the VATT is an alternative choice for large-scale applications in limited sea areas with complex flow directions and abundant tidal energy resources.

Active control and passive flow-driven control are two modes of VATT rotation. The former controls its speed by applying a motor, while the latter causes the turbine to realize self-starting by the hydrodynamic action of the incoming flow. The concept originated from the study of wind turbines. Much of our technical understanding of wind turbines was well known by the mid-1920s, and their key characteristics were certainly known by the mid-1950s [9]. Active control requires an external power supply and therefore introduces additional complexity and energy costs [10]. This makes it necessary to study self-starting turbines. The definition of self-starting varies in wind turbine research. Elbert and Wood defined the self-starting process as a turbine going from a standstill to significant power extraction [11]. Celik et al. considered that self-starting is only achieved when the turbine accelerates from rest to its final operating tip speed ratio [12]. We hereby consider that the VATT has reached a stable rotational state from a standstill under hydrodynamic action, and the turbine has successfully self-started. The aerodynamic performance of a self-starting vertical axis wind turbine (VAWT) has been extensively studied [13,14]. There has also been a lot of research into the factors affecting VAWT self-starting performance, such as the moment of inertia, number of blades, airfoil type, and pitch angle [15,16,17]. Calautit et al. summarized the various types of numerical calculation models and analytical methods used in wind turbine research [18]. These studies provide good guidance for exploring self-starting VATTs.

Active-controlled VATT arrays have been extensively studied in recent years. Due to the stand-alone turbine system constantly being subjected to lateral force during operation, which is not beneficial to the system carrier to keep stable [19], many twin-rotor arrays studies have been proposed, such as on the relative rotation direction [20,21,22] of rotors and their relative positions [22,23]. For double-row turbine arrays, many studies are focused on the arrangement of arrays to minimize the influence of the wake on the rear turbines. Blade and wake interaction is remarkable during VATT operation [24], and the wake pattern varies greatly depending on the rotational speed of the turbine [25]. The wake not only affects the average power output of the VATT array but also increases structural fatigue [21,26] and affects the power quality by imposing fluctuations in the power [27]. Therefore, it is very important to arrange the spacing of the turbines reasonably. Sun et al. [21] introduced a symmetrical simplified model to study the interaction and wake characteristics of each turbine under different VATT array layout schemes. Sébastien et al. [28] introduced the effective performance turbine model (EPTM) based on high-fidelity CFD simulations to simplify turbine models and computational costs for planning VATT array deployment and predicting performance. Tremblay et al. [29] used the EPTM to predict the accurate downstream turbine performance of a VATT array. The team in [30] then used the EPTM to test the effect of local blockages, lateral and longitudinal spacing, and array staggering on VATT arrays in a turbulent flow environment. None of the rotors in the above studies were passive flow-driven controlled rotors, and there are currently no reports on the performance of VATT arrays under self-starting conditions.

For the passive flow-driven control, several studies have been conducted to improve the self-starting performance and power efficiency. Numerical methods such as the sinusoidal pitching method [31] and the combination of the sinusoidal and fixed pitching methods [32] have been proposed using experimental methods such as proposing a modified inclined-blade turbine [33]. Saini et al. [34] evaluated two different hydrokinetic turbine rotors (hybrid and Darrieus) by using numerical and experimental methods and found that the hybrid rotor is more efficient. However, all of these studies were carried out on a stand-alone turbine, and no studies have been reported on the self-starting of VATT arrays.

To the limit of our literature review, most previous studies focused on turbines under active control, which is different from the actual operation circumstances, and self-starting studies of VATT arrays have not been carried out. Therefore, it will be valuable to investigate the performance of a VATT array in the passive flow-driven mode, in which the rotor rotation is governed by Newton’s second law. Several crucial aspects have been analyzed, including (1) the self-starting performance and power coefficient of a stand-alone turbine and double-row VATT array, (2) the wake characteristics of the turbines in the array and the force changes of the blades during the stable rotation of the turbines, and (3) the difference in power efficiency, self-starting performance, flow fields between the VATT array and a stand-alone turbine, and the feasibility of the double-row self-starting VATT array in practical applications. This study provides theoretical guidance for practical applications of self-starting VATT arrays.

This paper is organized as follows. Section 2 describes the details of the numerical models, including the rotor selection and parameters, computational domain and grids, solver and turbulence model, self-starting equations, dimensionless parameters, and mesh and time step convergence study. Section 3 gives the numerical model validation. Section 4 presents the results, including the hydrodynamic performance and wake characteristics on a stand-alone turbine and a VATT array. Section 5 gives our conclusions.

2. Numerical Model

2.1. Selection and Parameters

In this study, the H-Darrieus VATT was considered, as shown in Figure 1. The NACA series asymmetrical airfoil of the turbine is now widely studied [35,36]. Among of them, the NACA0018 airfoil, which is widely applied in wind and tidal turbines, was used here. The VATT consisted of three blades spaced at an angel of 120° and a round shaft at the rotor’s center. The pivots of the blades were located on the cord line of the foil at a distance of 0.03 m from the leading edge. φ is the blade pitch angle. Following the right-hand rule, the positive and negative values of φ are defined to represent the inward and outward angles, respectively. An azimuth angle θ of 0° is defined to coincide with the positive Y-axis, with a positive azimuth in the counter-clockwise direction and the turbine rotating counter-clockwise in all calculating cases. According to previous research, the energy utilization rate is optimal for the NACA0018 three-blade VATT when the leading edge of the blades has an outward offset of 7° [37]. Therefore, we adopted VATTs with a preset pitch angle (φ = −7°) for the formal calculation, including the stand-alone turbine and the VATT array. The stand-alone turbine without a preset pitch angle (φ = 0°) was only used for mesh model validation calculations. The moment of inertia (I) was measured by drawing the three-dimensional (3D) model of the turbine in Solidworks. Detailed information on the VATT is presented in

Table 1. Detailed parameters of VATT.

Symbol	Description	Value
R	Rotor Radius	0.5 m
D0	Rotor Shaft Diameter	0.04 m
S	Blade Span	1.0 m
C	Blade Chord	0.12 m
N	Number of Blades	3
σ, σ = NC/πD	Solidity	0.1146
I	Moment of Inertia	7.0 kg·m2

.

2.2. Computational Domain and Grid

It has been shown that with appropriate corrections, two-dimensional (2D) model methods can be used to effectively simulate tidal current turbines [38]. Considering the size of turbines with the computational costs and the accuracy, a 2D numerical model was employed. Meshing was carried out using the ICEM. The computational domains of a stand-alone turbine and corresponding mesh structures are shown in Figure 2. A structured grid was used for the entire computational domain. The rotating circular domain of the turbine used a sliding mesh, and the red border is the sliding and static mesh boundary. By using the sliding mesh approach combined with loading user-defined functions (UDFs), the turbines realized self-starting rotation in the simulations. The UDFs were written in C. The velocity inlet and pressure outlet boundary conditions were set toward the left and right sides of the outer rectangular main domain, which were 5D and 10D away from the rotor shaft, respectively. The incident flow velocity was fixed at 1.5 m/s for all testing cases. The gauge pressure on the outlet was set to 0 Pascals. The symmetry boundaries of the no-slip condition were used for the upper and bottom sides, which were 5D away from the rotor shaft.

For the VATT array, it was necessary to consider minimizing the influence of the wake on the rear turbines. A triangular arrangement and side-by-side arrangement are generally used in double-row turbine arrays. The efficiency of the VATT can be improved when the radial spacing of the twin turbines is at a certain optimum distance [23]. According to the research, the triangular arrangement of the double-row turbine array performs better, and the optimal distance for the axial and radial spacing of the VATT is 3D [39]. Thus, a double-row triangular arrangement was adopted in this study, with axial (D_x) and radial spacings (D_y) of 3D. The basic layout scheme is shown in Figure 3. In the following, the three turbines are represented by T1, T2, and T3. The outlet boundary is 12D away from T3′s shaft in order to obtain a fully developed wake. The grid size and basic structure around the blades and at the interface in the double-row VATT array were the same as in Figure 2.

2.3. Solver and Turbulence Model

The numerical model was established based on the continuous equations and Reynolds-averaged Navier–Stokes equations, which were solved using the finite volume method. The commercial computational fluid dynamics software ANSYS-Fluent 16.0 was employed as the computing platform. For the low-velocity incompressible flow in this study, a pressure-based solver was used. The scheme for the semi-implicit method of pressure-linked equations (SIMPLE) was applied for velocity-pressure coupling. The Green–Gauss node-based method was used for gradient interpolation. Second-order upwind and implicit schemes were employed for spatial discretization and transient formulation, respectively. For the turbulence model, it has been pointed out that the SST k-ω model is more suitable for analyzing the performance and flow characteristics of a VATT [40] and tends to predict higher levels of isotropy in a turbulent wake [41]. Therefore, the SST k-ω model was utilized in unsteady numerical analysis.

2.4. Rotor Dynamics and Dimensionless Parameters

Different from the forced rotation mode with a constant speed, the VATT arrays rotate under hydrodynamic force. Self-starting of the VATTs in the numerical simulation was realized by loading UDFs which contained three main parts: solving for the blade torque, controlling the turbine rotation, and outputting the values of the monitored variables. These three parts were defined in the UDFs by means of the macros DEFINE_ADJUST, DEFINE_ZONE_MOTION, DEFINE_EXECUTE_AT_END, and DEFINE_ON_DEMAND.

Figure 4 shows the blade force of the turbine in the rotation process. The blade was subjected to the drag force F_D parallel to the direction of the combined velocity v_r and the lift force F_L perpendicular to the direction v_r. By decomposing F_D and F_L along the local coordinate system oxy and setting f_x as the component force along the x axis and f_y as the component force along the y axis, we obtain:

$$\mathit{f}=\left({\mathit{f}}_{\mathit{x}} {\mathit{f}}_{\mathit{y}}\right)=\left({\mathit{F}}_{\mathit{L}} {\mathit{F}}_{\mathit{D}}\right)\left(\begin{array}{c}-sin\alpha \\ cos\alpha \end{array} \begin{array}{c}cos\alpha \\ sin\alpha \end{array}\right)$$

(1)

In the study of the overall force during the rotation of the turbine, the drag force F_D and lift force F_L on a single blade are decomposed into the overall coordinate system, and the component forces in the X and Y directions are denoted as F_x and F_y, respectively. From the triangular geometric relationship, the angle between the drag force F_D and the X axis is 2π − (α + φ + θ). This gives an expression for the force on a single blade in the overall coordinate system as follows:

$$\mathit{F}=\left({\mathit{F}}_{\mathit{x}} {\mathit{F}}_{\mathit{y}}\right)=\left({\mathit{F}}_{\mathit{L}} {\mathit{F}}_{\mathit{D}}\right)\left(\begin{array}{c}-sin\left(\alpha +\phi +\theta \right)\\ cos\left(\alpha +\phi +\theta \right)\end{array} \begin{array}{c}cos\left(\alpha +\phi +\theta \right)\\ sin\left(\alpha +\phi +\theta \right)\end{array}\right)$$

(2)

The combined X and Y forces on the turbine in the overall coordinate system are the vector sum of the forces on the three blades in the X and Y directions, respectively. The turbine resultant moment versus the rotational speed is

$$\mathit{I}\frac{d\mathit{\omega }\left(t\right)}{dt}={\mathit{T}}_{\mathit{R}}\left(t\right)={\mathit{T}}_{\mathit{H}}\left(t\right)-{\mathit{T}}_{\mathit{L}}\left(t\right)-{\mathit{T}}_{\mathit{F}}$$

(3)

where ω(t) is the instantaneous angular velocity and T_R(t), T_H(t), T_L(t), and T_F represent the resultant torques of the three blades, hydrodynamic torque, loading torque, and friction torque, respectively. In this study, we ignore the friction effects, so ${\mathit{T}}_{\mathit{F}}$ is constant during the whole rotation process, as 0 N·m. T_L(t) is proportional to the rotation speed and defined as T_L(t) = Lω(t), where L is the loading coefficient and its unit is N·m·s/rad, which will be omitted in the following context.

The instantaneous acceleration a(t)of a VATT can be expressed as

$$\mathit{a}\left(t\right)=\frac{{\mathit{T}}_{\mathit{R}}\left(t\right)}{\mathit{I}}$$

(4)

The instantaneous angular velocities ω(t) at the times t and t + δt can be expressed as

$$\mathit{\omega }\left(t\right)={{\displaystyle \int }}_t^{t+\delta t}\mathit{a}\left(t\right)dt$$

(5)

By looping Equations (3)–(5), the passive self-starting mode can be realized for the VATT.

In order to compare the calculation results of different models, the following dimensionless parameters are used in this paper.

The instantaneous torque coefficient C_TI and power coefficient C_PI for the rotor or the individual blade are evaluated from the following equations:

$$C_{TI}\left(t\right)=\frac{{\mathit{T}}_{\mathit{R}}\left(t\right)}{\frac{1}{2}\rho {\mathit{v}}^{2}DSR}$$

(6)

$$C_{PI}\left(t\right)=\frac{{\mathit{T}}_{\mathit{R}}\left(t\right)\mathit{\omega }\left(t\right)}{\frac{1}{2}\rho {\mathit{v}}^{3}DS}$$

(7)

where v is the incident current velocity, ρ is the water density, D and R are the diameter and radius of the rotor, respectively, and S is the blade span. Furthermore, the averaged torque coefficient C_TA and the averaged power coefficient C_PA during the stable stage can be taken as the mean value integrated from the instantaneous coefficients for 10 rotation periods:

$$C_{TA}\left(t\right)=\frac{{{\displaystyle \int }}_t^{t+10T}{C}_{TI}\left(t\right)dt}{10T}$$

(8)

$$C_{PA}\left(t\right)=\frac{{{\displaystyle \int }}_t^{t+10T}{C}_{PI}\left(t\right)dt}{10T}$$

(9)

where T is the rotation period for the rotor in the stable stage.

The tip speed ratio λ is defined as

$$\lambda =\frac{\mathit{\omega }R}{\mathit{v}}$$

(10)

The wake recovery ratio η is defined as

$$\eta =\frac{{\mathit{v}}_x}{\mathit{v}}$$

(11)

where v_x is the component of the flow velocity along the X direction and v is the incident flow velocity.

2.5. Mesh and Time Step Selection

2.5.1. Mesh Convergence Study

Three different grids with varying accuracy were calculated to select a suitable grid for the calculation. The turbine speed was constant, and the tip speed ratio λ was 2.269.

Table 2. Detailed mesh information.

Mesh	Total Domain Mesh Quantity	Rotating Domain Mesh Quantity	Thickness of the Blade First Layer (×10−4 m)	Y+
Rough	50,018	28,329	3.5	18.2–44.6
Medium	93,715	58,230	0.8	2.79–5.11
Fine	313,258	225,258	0.3	0.75–1.71

gives the mesh quantities, thickness of the blade’s first layer, and Y plus (Y⁺), which in the CFD calculations is reflected by the calculation of the height of the first layer of grid nodes in the process of dividing the grid. During the rotation of the turbine, the viscous sublayer region of the blade wall had very high requirements on the grid, and the mesh fineness quality would directly affect the numerical simulation results. Theoretically, the Y⁺ value of the blade wall should be less than five.

Figure 5 shows a comparison of the instantaneous torque coefficients of a single blade in one cycle for three meshes. It can be seen that the forces on the blade of the rough mesh differed significantly from the other two meshes when the blade was rotated around an azimuth angle in the range of approximately 180~270°. There was little difference in the blade force between the medium mesh and fine mesh. Combined with the calculated Y⁺ values, the medium mesh achieved mesh convergence.

2.5.2. Time Convergence Study

The time step affects both the calculation precision and the solution time. Too small a time step wastes computing resources and increases the calculation time, while too large a time step can affect the accuracy of the results. In this study, four different time steps (Δt = 0.002 s, 0.004 s, 0.006 s, and 0.008 s) were compared to verify the time step convergence. The calculations using the grid model are shown in Figure 2. Figure 6 shows the resultant torque coefficient variation on the turbine as it rotated around two cycles in four different time steps. The turbine speed was constant at 65 rpm (ω = 6.807 rad/s). It can be seen that the curve variation trends were similar when the time steps adopted were 0.004 s and 0.006 s. When the time step was 0.008 s, the torque fluctuated significantly, and it could not meet the convergence requirements. Considering the reasonable choice of time step in relation to the mesh size of the interface and the turbine rotation speed, as well as the calculated operating conditions with rotation speeds greater than 65 rpm, the time step adopted was 0.004 s.

3. Numerical Model Validation

As experimental tests focusing on the self-starting VATT performance are still few in number, the laboratory results of a forced rotation turbine with the same model size and parameters were employed for the numerical model validation [37]. The test turbine blade profile was also that of an NACA0018, and the inlet velocity was 1.5 m/s. During the entire numerical simulation calculations, the incident flow was uniform (v = 1.5 m/s), and the flow direction was set to be parallel to the x-axis. A comparison of the numerical and experimental results of C_PA against the tip speed ratio λ is shown in Figure 7a. It can be seen that the numerical predictions exhibited similar tendencies toward the experimental results. Both the experimental and numerical data reached their peaks at λ = 2.28.

Table 3. Comparison of errors between numerical and experimental results.

Parameter	Value
λ	1.1971	1.4073	1.5898	1.7503	1.9328	2.0932	2.2790	2.4463	2.6223	2.8038
Numerical Results	0.1307	0.1569	0.2092	0.2478	0.2779	0.2965	0.3088	0.3027	0.2830	0.2546
Experimental Data	0.0892	0.1300	0.1647	0.1981	0.2291	0.2600	0.2860	0.2600	0.2191	0.1758
μ	0.0415	0.0269	0.0445	0.0497	0.0488	0.0365	0.0228	0.0427	0.0639	0.0788
$\overline{\mu }$	0.0456

gives the specific errors μ of the numerical and experimental results under different λ values. The maximum error was approximately 0.0788, corresponding to λ = 2.8038. The minimum error was 0.0228, corresponding to λ = 2.2790. The average error was 0.0456. The numerical averaged power coefficients were overestimated, evidently because of the lower accuracy of the 2D model in the complicated fluid-structure interactions and turbulent behavior of the faster rotating blades. Moreover, turbine friction was ignored in the calculations.

A comparison of the torque coefficients of the turbine between the numerical and experimental results at λ = 2.269 is given in Figure 7b. The experimental data were from [37]. The curve contains two complete rotation cycles of the turbine. It is clear that the two curve tendencies had good agreement. The peaks and valleys of the numerical and experimental results reached their extreme values at about the same time. Therefore, the numerical model could accurately simulate the force process of the turbine rotation.

Since the turbine with a preset pitch angle φ = −7° was used in the formal calculations, in order to further verify the reliability of the numerical model, the power coefficients of the forced rotation turbine with φ = −3° and −7° were compared with the experimental and numerical results. The turbine parameters and flow rate conditions were the same in all calculations. As shown in Figure 8, when φ = −3°, the calculation data had the same trend as the experimental data, and the C_PA reach its peak at around λ = 2.1. When φ = −7°, it can be seen from the trends of the two curves that when the λ was higher than 2, the mistakes decreased (around 4%). Overall, the numerical model could reflect to some extent the performance of the VATT under real operating conditions.

4. Results

4.1. Stand-Alone Turbine

In order to better explore the hydrodynamic performance and wake development of the turbine array and compare the work differences with a stand-alone turbine, we will first discuss a single turbine. Seven loads were applied for all calculations, and the load coefficient L = 2, 4, 6, 8, 10, 12, 14. The turbine was driven by hydrodynamic power and reached stable rotation.

Figure 9 shows the averaged power coefficient of a stand-alone turbine under seven different loads. Specific values of C_PA corresponding to λ are illustrated in

Table 4. Specific values of C_PA corresponding to λ.

Parameter	Value
L	14	12	10	8	6	4	2
λ	2.313	2.448	2.611	2.822	3.071	3.403	3.838
CPA	0.435	0.418	0.396	0.370	0.328	0.269	0.171

. The rotation speed decreased as the load increased. As the loads increased by an equal amount, the C_PA growth rate decreased. Under the promise that the turbine could realize its self-starting and achieve stable rotation, it could obtain a higher power output by choosing a large load as much as possible.

The instantaneous wake velocity distribution of the turbine during the steady rotation phase for four different loads (L = 2, 6, 10, 14) is examined in Figure 10. In order to better study the velocity of the turbine wake, the flow profiles in the vertical direction at axial distances of 2.5D, 5D, 7.5D, and 10D behind the turbine rotation axis were selected for analysis, as shown in Figure 10. The shaft center of the rotor was located at x = 0D.

From Figure 10, it can be seen that as the load increased, the wake acceleration region grew longer. As shown in Figure 10a, velocity deficit was apparently within 2.5D from the rotor shaft, and alternately, shedding wake vortices appeared beyond 2.5D. As shown in Figure 10b, when the L increased to 6, the downstream area with a severe velocity deficit extended to 3D, and this length grew to around 4D and 5D when the L increased to 10 and 14, as shown in Figure 10c,d, respectively. Although the wake acceleration region grew longer, the velocity deficit weakened. It can be noted in Figure 11 that the wake velocity recovered quickly when the load was high. This is because the turbine rotation speed was low when the load was high, and there was less disturbance to the water flow. Therefore, as given in Figure 10, there was significant vortex shedding over a relatively short distance behind the turbine at low loads. Figure 12 shows the velocity radial deficit along y = 0. The wake velocity recovered to approximately 95.49% and 92.80% at x = 10D for loads L = 10 and 14, respectively, and only to approximately 89.32% and 66.66% at L = 2 and 6, respectively. In summary, for a stand-alone self-starting turbine, the recovery of the turbine wake was load-dependent.

4.2. Double-Row Turbine Array

Figure 13 includes the averaged power efficiency results for a stand-alone turbine and the three turbines in the array under seven different loads. It is obvious that the averaged power coefficient of the turbines in the array was higher than that of the stand-alone turbine. This difference became significant as the load increased. When L = 2, the averaged power coefficient of the three turbines in the array was 0.181, a value 0.010 higher than that of the stand-alone turbine. When L = 14, the averaged power coefficient of the three turbines in the array was 0.465, a value 0.030 higher than that of the stand-alone turbine. For the turbines in the array, the three turbines showed different performances. The power output of T3 was higher than that of the other two turbines, and this difference also became more significant as the load increased. When L = 2, the C_PA of T3 was 0.193, which was 1.067 times that of T1 and 1.144 times that of T2. When L = 14, the C_PA of T3 was 0.510, which was 1.128 times that of T1 and 1.179 times that of T2. The power conversion efficiency of T2 was the lowest. In summary, the self-starting turbines in the triangle arrangement array showed more of an advantage in power acquisition than the stand-alone turbine.

In Figure 14, the time histories of the rotation speed of the turbines at different loads are compared. There was little difference in the self-starting times of the three turbines. It can be seen that the rotation speed of T3 fluctuated obviously under different loads. Although T3 had the fastest average rotation speed and the highest power efficiency, high vibration would make the turbine more vulnerable to fatigue damage.

In Figure 15, the instantaneous flow velocities of the double-row turbines at the stable rotation stage under four different loads are compared. The wake developments of T1 and T2 were different due to the rotation of T3 perturbing the flow field. As given in Figure 13 and Figure 14, although the incident velocities of T1 and T2 were the same, there were still slight differences in power efficiency and self-starting performance. This suggests that the rear-row turbine exerts an effect upon the performance of the front-row turbine by influencing its wake development. Furthermore, it can be clearly seen that T3 influenced the direction of wake development in the front turbines (T1 and T2). By comparison with the stand-alone turbine (Figure 10), the wake acceleration regions of both T1 and T2 were clearly contracted around 3D from the shaft and then deflected outward, with small shedding vortices developing downstream. In addition, the flow velocity between T1 and T2 increased significantly. Figure 16 shows the horizontal flow rate deficit at x = 0. It can be derived that the lower the turbine load, the faster the flow velocity between T1 and T2. The incident flow rate of T3 increased to approximately 1.2v. This is because when the turbine load is low, the turbine speed is faster. Thus, the turbulence intensity increases, and the beam effect between the two turbines becomes stronger. For T3, the incidence velocity increased, and its rotation speed was significantly faster (see Figure 14). The shedding vortices of the T1 and T2 wakes were superimposed onto the T3 wake. As a result, the wake of T3 developed more unevenly, and the wake’s influenced length and width increased (see Figure 15).

Figure 17 demonstrates the flow profiles in the vertical direction behind T3 at axial distances of 2.5D, 5D, 7.5D, and 10D from its rotation shaft, with the four positions corresponding to the x-axis coordinates 4D, 6.5D, 9D and 11.5D. The shaft center of T3 was located at x = 1.5D. The wake of T3 in the array was significantly more turbulent, and the wake velocity was more heterogeneous than in the stand-alone turbine. As can be seen in Figure 17a,b, the velocity deficit was severe in the area within 5D behind the rotating shaft of T3. In the region 7.5D off the shaft, the velocity deficit was much improved (Figure 17c), and at 10D, the wake velocity was largely recovered (Figure 17d). The velocity radial deficit behind T3 along y = 0 is presented in Figure 18. At an axial distance of approximately 10D (x = 11.5D) behind the T3 shaft, the wake velocity corresponding to the load L = 2, 6, 10, 14 returned to 98.32%, 102.07%, 98.56%, and 95.17% of the incident velocity, respectively. The wake of the array was more turbulent, and the velocity deficit was more severe than that of a stand-alone turbine, but it recovered significantly faster. The wake velocity recovery in the array was not affected by the loads. The above analysis also led to the conclusion that for the double-row turbines, the axial and radial spacings of 3D were reasonable.

The hysteresis curves of the surface pressure coefficient over a blade at four certain azimuth angles in a cycle for L = 2 are illustrated in Figure 19. The solid and dashed lines represent the outside and inside of the hydrofoil, respectively. It can be seen that, as a whole, there was a significant difference in the forces on the leading and trailing edges during the stable rotation of the turbine. The forces near the leading edge of the blade were larger, and when the blade rotated to azimuths θ = 0°, 180°, 270°, the forces on the outside and inside of the blade were basically the same magnitude and in opposite directions. Figure 20 shows a comparison of the instantaneous torque coefficients over a blade in two stable cycles for L = 2. From this diagram, the resultant force on the blade was approximately zero at these three azimuths. We can also see that the instantaneous torque of the blade was positive at an azimuth of 90° (corresponding to t/T = 0.25, 1.25). As highlighted in Figure 19b, the positive moment of the blade at this position was mainly caused by the inner side of the blade.

As seen in Figure 20, the positive torque was mostly generated during the first half-cycle (θ ≈ 0~180°) of blade rotation, corresponding to an azimuth angle of about 45–160°. Within this azimuth range, the force on the T3 blade increased due to the increase in the incident velocity, especially at an azimuth angle of around 100°. Due to the severe velocity deficit behind T3 (see Figure 15), the torque on the T3 blade was lower in the second half-cycle (θ ≈ 180~360°). Compared with T1 and T2, the force difference between the first and second half-cycle was greater due to the greater disturbances and differences upstream and downstream of T3, resulting in greater rotational fluctuations of the turbine (see Figure 14).

5. Conclusions

In this study, numerical models based on the CFD software ANSYS-Fluent 16.0 (ANSYS, Inc., Pittsburgh, USA) were established to investigate the hydrodynamic performance and wake development of the H-Darrius vertical axis tidal turbine array under self-starting conditions and compare the performance between the array and a stand-alone turbine. The array adopted double-row turbines in a triangular arrangement, and the axial and radial distances between the turbines were 3D. The fixed pitch angle of the VATTs was −7°. The calculated turbines could achieve self-starting to stable rotation under seven loads. The following conclusions were drawn by analyzing and discussing factors such as the power coefficients, flow velocity contours, and hydrodynamic torques:

(1)

Compared with a stand-alone turbine, the power coefficients of the three turbines in the array all improved, and as the load increased, the difference between the average power coefficients of the array and the stand-alone turbine became more obvious. When L = 2, the averaged power coefficient of three turbines in the array was 0.181, and the value was 0.010 higher than that of the stand-alone turbine. When L = 14, the average power coefficient of the three turbines in the array was 0.465, and this difference increased to 0.030.

(2)

In the array, the main effects of the front turbines T1 and T2 on the rear turbine T3 were energy utilization and turbine vibration. The fluid flow rate increased due to the beam effect between T1 and T2. The incident flow rate of T3 increased to approximately 1.2v, its rotational speed of T3 accelerated, and its power efficiency was enhanced. The superposition of the front turbine wake and other factors caused the velocity deficit area behind T3 to be larger, and the increase in the incident velocity made the force difference between the first and second half-cycles larger, resulting in greater rotational fluctuations in T3, meaning that although T3 had higher power coefficients, it was more susceptible to fatigue damage. There was little difference in the self-starting times between the three turbines. The rear turbine could affect the wake development and thus the performance of the front turbine, so there was a slight difference in performance between T1 and T2 despite using the same incident flow.

(3)

In the array, the wake behind T3 had a much larger area of influence and more intense turbulence, with a greater velocity deficit than for the stand-alone turbine at the same load, but it recovered more quickly, with the wake recovering to over 95% at a distance of 10D behind it. For a stand-alone self-starting turbine, the rate of wake velocity recovery is load-dependent, but this was not evident in the arrays. In practice, axial and radial spacings of 3D are reasonable for a double-row turbine array.

(4)

The positive torque of the turbine was mainly generated when the blade rotated through the azimuth angle of 45–160° and mainly benefited from the inner side of the blade.

In summary, this paper has discussed the basic characteristics of self-starting turbine arrays with specific arrangements. However, the effect of structures such as the turbine support arm was ignored in the study, and the array only considered one arrangement of double rows. These factors need to be improved in future research.