Numerical and Experimental Analysis of A Vertical-Axis Eccentric-disc Variable-Pitch Turbine (VEVT)

A combined experimental and numerical investigation is carried out to study the performance of a vertical-axis eccentric-disc variable-pitch turbine (VEVT). A scheme of eccentric disc pitch control mechanism based on double-block mechanism is proposed. The eccentric control mechanism and the deflection angle control mechanism in the pitch control structure are designed and optimized according to the functional requirements of the turbine, and the three-dimensional model of the turbine is established. Kinematics analysis of the eccentric disc pitch control mechanism is carried out. Kinematics parameters and kinematics equations which can characterize its motion characteristics are derived. Kinematics analysis and simulation are carried out, and the motion law of the corresponding mechanical system is obtained. By analyzing the force and motion of blade of VEVT, the expressions of the important parameters such as deflection angle, attack angle and energy utilization coefficient are obtained. The lateral induced velocity coefficient is acquired by momentum theorem, the hydrodynamic parameters such as energy utilization coefficient are derived, and the hydrodynamic characteristics of VEVT are also obtained. The experimental results show that the turbine has good energy capture capability at different inflow velocities of different sizes and directions, which verifies that VEVT has good self-startup performance and high energy capture efficiency.


Introduction
Tidal current turbine is the device that converts tidal current energy into electric energy. Variable pitch refers to that the blade mounted on the hub can be controlled to change the pitch angle. For hydraulic turbines, the active control of pitch angle can overcome many disadvantages of pitch fixing and passive stall regulation. Kirke and Lazauskas (2008) put forward the design of variable pitch, which can make the vertical axis turbine have higher starting torque and peak efficiency. Experiments verify that the variable pitch turbine is easy to start-up and has less vibration (Kirke, 2011).
The active pitch control mechanism mainly has two types: direct drive mechanism and mechanical structure ad-justment mechanism (Kirke and Lazauskas, 2011). Directdriven variable-pitch turbine drives each blade directly by hydraulic cylinder or stepper motor. Through hydraulic control or motor driving each blade to swing in the process of turning the hub for a cycle, the periodic variation of blade deflection angle is realized. The mechanism of adjustable pitch of mechanical structure realizes the periodic variation of blade deflection angle by the motion synthesis of linkage mechanism, angle pendulum mechanism, slider mechanism and wheel hub rotation (Zhang et al., 2012).
At present, many researchers have done a lot of researches on the variable pitch technology. Zeiner-Gundersen (2014) proposed a variable pitch turbine, which uses passive spring variable pitch mechanism. However, the spring pitch control system requires high spring stiffness standard, and the variation of deflection angle can only improve the energy utilization of the system to a limited extent. Schönborn and Chantzidakis (2007) introduced a hydraulic pitch control system. The pitch control of a hydraulic turbine is realized by a swashplate mechanism and a hydraulic circuit for each blade. This pitch control system based on hydraulic control can better realize the law of deflection angle change, but it requires high precision for hydraulic device, and it is easy to fail due to sealing problems. Zhang et al. (2015) designed an active collective blade pitch control device, which consists of four driving wheels attached to the blade and a servo motor. The experimental results show that the power coefficient of the prototype can be between 0.5 and 0.7. The mechanism part of this active pitch control system is relatively simple, but the control requirements are relatively high.
In view of the above problems in the design and application of pitch control device, this paper presents an innovative mechanical scheme of VEVT, gives the working principle of the active pitch control device, and designs an eccentric-disc variable-pitch mechanism based on the principle of double-rotating block mechanism, which makes the blade realize the cycloidal deflection angle variation law with controllable eccentricity, and its mechanical kinematics is analyzed. Finally, the hydrodynamic theory analysis and experimental study on VEVT based on this principle of eccentric-disc variable-pitch are carried out, and the influence of some important parameters on the energy utilization ratio of the turbine is discussed. The theoretical calculation results of VEVT are compared with the experimental results. The three-dimensional model and experimental prototype of VEVT are shown in Fig. 1 and Fig. 2.

Motion equation of the mechanism of variable pitch system and hydrodynamic characteristic of VEVT
The overall schematic diagram of VEVT is shown in Fig. 3. The eccentric control mechanism and the eccentric angle control mechanism in the variable pitch structure are   connected by the connecting shaft between the translating disc and the eccentric disc. When the incoming flow impacts the blades of the turbine and makes hub rotate, the blade deflection angle will change periodically, which improves the energy utilization rate.
When the eccentric disc of a hydraulic turbine coincides with the rotation centre of the hub, the pitch angle of each blade of VEVT is zero. As shown in Fig. 4a, the energy capture efficiency of the blade is quite limited. When the eccentric disc and hub do not coincide, different blade pitch angles are formed at different positions of the blade, thus improving the distribution of the attack angle of the blade, as shown in Fig. 4b.
The linear motor in the eccentric control mechanism of the hydraulic turbine controls the movement of the double cross slideway to drive the translation of the translation disc, and the translation disc controls the position transformation of the eccentric disc, thus controlling the variation of the blade deflection angle. The output rotation of the hub achieves speed change through the increaser, and then achieves the rated generator speed, thus realizing the power generation function.

Eccentric control mechanism
The two-dimensional diagram of the eccentric control mechanism of VEVT is shown in Fig. 5a. A part of the eccentric control mechanism is a translational mechanism composed of the linear motor 6, ball screw 4, double cross slideway 5, linear motor 9, ball screw 7 and translational disc 8. It can adjust the relative position between the centre of the eccentric disc and the centre of the impeller, that is, the eccentricity e. When the adjustment of the translational mechanism is finished, the hub rotation drives the rotational mechanism composed of the generator 1, idler gear 2 and big gear 3 to rotate.
The translational disc of the eccentric control mechan- ism and the eccentric disc of the deflection control mechanism are coupled by a connecting shaft. During the movement of the translational disc, the eccentric disc moves translationally. The centre of the eccentric disk and the actual control point N of the blade deflection are two different position points. Therefore, it is necessary to establish the mathematical model of the control point N of the deflection angle of the blade of the eccentric control mechanism, as shown in Fig. 5b.
When the centre point of eccentric disc is moved from to by the movement of double cross slides in eccentric control mechanism, the position of control point N of pitch control mechanism is shown in the figure. Define diameter and eccentric disc diameter . Establish a rectangular coordinate system as shown in  CHEN Hai-long et al. China Ocean Eng., 2020, Vol. 34, No. 3, P. 411-420 413 Fig. 5b. Eccentricity can be deduced as: (1) −50 mm≤x≤50 mm −52 mm≤y≤ 52 mm In the equation, , .
Considering the length of the swing link and the length parameters of the double cross slideway, the eccentricity of the VEVT ranges from 0 to 0.7.
By adjusting the moving distance x and y of the double cross slideway, the eccentricity of the VEVT changes. After the eccentric disc deflects to , the blade rotates around the rotating centre in the process of rotating with the hub, and the change of the deflection of the blade is realized.

Deflection control mechanism
The sketch of the deflection control mechanism is shown in Fig. 6a. In order to make the eccentric disc bear uniform force, the mechanism of double turning blocks is symmetrically arranged. The double cross slideway can move in a straight line in the fixed block connected to the eccentric disc and hub. Fig. 6b is the kinematics analysis diagram of the deflection control mechanism. When the connecting rod 1 rotates at an angular velocity , the angular velocity of the connecting rod 4 can be obtained by the transmission of doubleblock mechanism. and drive the double crank slider mechanism to produce specific motion, and output the angular displacement of connecting rod 2, i.e. the blade deflection angle. Finally, some parameters, such as angular velocity and angular acceleration, can be obtained to characterize the motion characteristics of eccentric control mechanism.
Set up the coordinate system as shown in the figure, the angular displacement of the rod 2 (θ 2 ) can be obtained: where, l 1 and l 4 are the length of rod 1 and rod 4 respectively; d is the length of Point D from the origin of coordinate; θ 1 is the angle between rod 1 and the horizontal direction, and . Then, the range of deflection angle is: . (3)

Motion
As shown in Fig. 7, the overall coordinate system Oxy of the vertical axis turbine is established with the hub rota- . (4) Attack angle equation is shown as: Equation for the coefficient of resultant velocity is shown as: β Blade inclination is shown as: Ω Then, the expression of angular velocity of blade rotating around its own axis is as follows: After some computation, the expressions of thrust coefficient , lateral force coefficient , torque coefficient and energy utilization coefficient are respectively as follows: (9) (10) The induced velocity of VEVT is determined by streamtube model method (Li et al., 2018). The streamtube model analysis schematic diagram is shown in Fig. 8. Among them, and are the upstream area on both sides of the disk, is the upstream area on the disk, and are the inflow velocity on both sides of the disk, is the fluid velocity on the disk, is the pressure at infinity on both sides of the disk, and is the pressure at the disk.
Define the induced velocity at the disk: (13) Then the fluid velocity: (14) Change the rate of flow momentum: (15) According to the momentum theorem (Kosaku et al., 2002), the thrust acting on the turbine along the x-axis direction is as follows: (16) The thrust acting on the disk is: The simplification is as follows: .
Therefore, in order to obtain the lateral induced velocity coefficients, the above two equations must be solved.

Development of experimental prototype
As shown in Fig. 2, the experimental prototype is composed of pitch control mechanism, speed-increasing power generation system and control system. CHEN Hai-long et al. China Ocean Eng., 2020, Vol. 34, No. 3, P. 411-420 415 3.2 Experimental principle The control scheme is as follows: the system automatically resets after starting up, uses metal proximity switch and rotary encoder to make two linear motors in x and y directions return to their respective zeros, so that the eccentricity of the turbine is zero. Later, the rotary encoder is used to detect and control the operation of the linear motors. The system block diagram for controlling the linear motors is shown in Fig. 9.
The voltage and current at both ends of generators are measured. By changing the resistance value of the digital scribed rheostat to change the speed of the turbine, the speed ratio of the turbine can be changed.

Experimental parameters
(1) Turbine parameters The hub diameter is 400 mm, the number of blades is 4, the blade airfoil is NACA 0018, the eccentric disc diameter is 270 mm, and the eccentricity . can be obtained from the eccentricity formula. As the energy utilization rate decreases when is calculated theoretically, the values are 0, 0.1, 0.2, 0.3, 0.4, 0.5, and the corresponding values of L are 0, 5, 11, 16.5, 22, 27.5, respectively.
(2) Experimentally environmental parameters The experimental flume is a conventional circulating flume of Harbin Engineering University. The total length of the flume is 17.3 m, the height is 2.88 m and the maximum width is 6.2 m. The working section is 7 m in length, 1.7 m in width, 1.5 m in depth and 2.3 m/s in velocity (Yu, 2009).

Experimental content
For the purpose of the research, the experiments on VEVT mainly include the following contents: (1) When the incoming flow velocity is the same and the eccentricity is different, the difference of the hydrodynamic performance of the turbine is analyzed.
(2) When the incoming flow velocity is different and the eccentricity is the same, the difference of the hydrodynamic performance of the turbine is analyzed.
(3) When the incoming flow velocity is different and the eccentricity is different, the influence of starting torque on the turbine is analyzed.
The prototype experiment of VEVT is shown in Fig. 10a, and the experimental measurement scheme is shown in Fig. 10b.

Governing characteristics of VEVT
By introducing the kinematics equation of pitch control mechanism, we can obtain the variation law of blade deflection angle, the angular velocity and angular acceleration of blade rotating around its own axis, and make the kinematics simulation by MATLAB/SIMULINK software. By comparing with the mechanism simulation in Pro/E software, we can verify the rationality of the design of eccentric disc variable pitch mechanism.
As shown in Fig. 11, the kinematics analysis block diagram of pitch control mechanism is established. According to the "measurement" function of the kinematics simulation module in Pro/E, the ends kinematics parameters, such as the variation law of blade deflection angle, can be obtained and compared with those obtained from kinematics analysis.
The initial conditions in the program are the size parameters of each component of the pitch control mechanism, as shown in Table 1. When the hub rotates at angular velocity , it is necessary to find out the variation  The Simulink simulation model of pitch control mechanism is established under MATLAB/SIMULINK module (Zhao et al., 2018). In the model, when the eccentricity and , the instantaneous displacement and velocity of the corresponding components are the initial values of each integration module when the hub rotates counterclockwise with velocity . According to the motion characteristics of the system, the function module is set, and the geometric dimensions and initial position angles of each mechanical component are defined in the constant module. After completing the above definition, the running time of the simulation model is set up for 8 s. After running, the simulation results are output to simout, which is the workspace variable, and the corresponding data are obtained.
By running the above mechanism simulation model and Simulink simulation model, the curves of angular displacement (i.e. the change of blade deflection angle), angular velocity and angular acceleration of the blade rotating around its own axis are obtained respectively. As shown in Fig. 12, the variation curve of the actual measured parameters of the mechanism simulation model is consistent with that of the Simulink simulation model.
The curves of blade deflection angle, angular velocity and angular acceleration obtained by mechanism simulation and kinematics simulation coincide, which proves that the results of the two simulation analysis are consistent and the kinematics analysis is correct.

Hydrodynamic characteristics of VEVT
Based on the hydrodynamic analysis and calculation of VEVT, combined with the geometric parameters and related experimental parameters of the turbine, the hydrodynamic numerical results of variable pitch turbine are ob-tained by using MATLAB software.
Before calculating the parameters that characterize the hydrodynamic performance of the turbine, the relationship between the inflow size and the Reynolds number of the blade must be taken into account (He et al., 2014). For the same inflow, its velocity affects the Reynolds number. The hydrodynamic calculation of VEVT has been carried out since the flow velocity was taken as an example. When , the Reynolds number is . Figs. 13a and 13b are the curves of blade deflection angle, blade inclination angle and blade position angle. Fig. 13c is curves of lateral induced velocity coefficient and blade position angle. Fig. 13d is curves of blade attack angle and blade position angle when speed ratio is and eccentricity e is different. Figs. 13e and 13f are the relationship curves of thrust, side force and torque curves of single blade under eccentricity and speed ratio . Figs. 13g and 13h are the relationship curves of the instantaneous hydrodynamic coefficients of a single  Fig. 11. Verification of kinematics analysis of pitch control mechanism. From the above figures, it can be seen that the maximum power obtained by the turbine is about 30W when the inflow velocity is , and the maximum energy utilization coefficient of the turbine is when eccentricity and speed ratio .

Comparison of numerical and experimental results
λ − C P 4.3.1 curve with the same inflow velocity and different eccentricities Fig. 15a shows the relationship between energy utilization coefficient and velocity ratio when and are different in the prototype experiment. For the curve with the same velocity and eccentricity, the energy utilization factor increases gradually with the increase of velocity ratio until it reaches . When the velocity ratio exceeds a certain value, the energy utilization factor decreases gradually until zero, and then the electricity generated by the turbine is zero. With the increase of eccentricity , the coefficient of energy utilization in- creases gradually. When eccentricity , the coefficient of energy utilization of the turbine has a maximum value , which shows that VEVT has a high energy capture efficiency.
Removal gear secondary transmission efficiency is , ball bearing efficiency , and double cross slideway loss efficiency . The other loss efficiency is . It can be seen that the mechanical transmission efficiency of VEVT transmission system: In addition to the mechanical transmission efficiency, the generating efficiency of the generator is 85%, so the total efficiency can be expressed as: η=η m η e = 0.646; (21) C Pj C Ps where is the experimental energy utilization coefficient obtained by calculation and is the energy utilization coefficient obtained by experiment.
As shown in Fig. 15b, when the incoming flow velocity and eccentricity , the comparison of experimental curve and theoretical curve shows that η = 75% C P max the experimental value is lower than the theoretical value. This is mainly because the output power of the generator is measured in the experiment, while the generating efficiency of the generator is about (Batten et al., 2007), and there is power consumption in the circuit. The experimental value at peak of the curve is about 70% of the theoretical value, which proves that the analysis is correct.
curve with different inflow velocities but the same eccentricity λ − C P e = 0.3 λ As shown in Fig. 16b, the experimental curves of VEVT with three incoming velocities and the same eccentricity can be seen from the figure that the energy utilization coefficient increases gradually with the increase of , and then decreases gradually after the peak value.
Comparing the experimental curve with the theoretical curve in Fig. 16a, it can be seen that the experimental values are relatively low, but the law of the curve under the same conditions is roughly the same, which can accurately reflect the changing law of the relationship between inflow velocity and experimental curve. and eccentricity are selected. is the energy utilization coefficient obtained by experiment, is the energy utilization coefficient obtained by theoretical calculation, and is the experimental energy utilization coefficient obtained by calculation.
As shown in Fig. 17, it can be seen that the deviation between the calculated experimental data and the theoretical data after considering the mechanical transmission efficiency and generator generation efficiency is small, and the trend is consistent. Therefore, the experiment proves the rationality of the theoretical analysis.

Conclusions
In summary, it can be concluded that VEVT has good starting performance and high energy capture efficiency because of its unique mechanism scheme. Through the kinematic analysis and numerical simulation of the eccentric disc mechanism, the feasibility of the eccentric disc mechanism based on the double slider mechanism is verified. According to the kinematic results and the hydrodynamic performance theoretical results of VEVT, the hydrodynamic parameters of VEVT are derived. Through the flume experiment of VEVT prototype, it is found that the pitch system of VEVT can achieve the desired pitch action of turbine blade. Through the analysis of the theoretical data and experimental data, it is found that the theoretical calculation results are basically consistent with the experimental results, which verifies that the VEVT has good self-starting performance and high energy capture efficiency relative to the fixed pitch turbine.
In the actual marine environment, the VEVT studied in this paper often has reliability problems. In order to ensure the normal operation of VEVT under the actual sea conditions, the more effective scheme is to rely on the tidal turbine on the offshore platform. In the aspect of turbine control, this paper presents the optimal blade deflection law of vertical axis turbine under any sea condition by means of theoretical research. However, in practical application, it is unrealistic to change the blade deflection angle of VEVT in real time according to the sea conditions. The more practical scheme is that according to the optimal blade deflection law found in this paper and the tidal current in the target sea area, through sampling analysis and probability statistics, the optimal range of VEVT deflection angle is found in different quarters (or months), and the optimal blade deflection angle of VEVT is changed in different quarters (or months) to achieve higher energy capture efficiency.