The equation of motion below can be used to determine the free floating RAOs of the structure [1]:

Where,

M(S) = mass matrix,
M(a) = added mass matrix,
C = linear damping matrix,
K(s) = total system stiffness matrix,
F = wave force on system (per unit wave amplitude), and
X = response amplitude operators.

When X = X₀e^−iwt and F = F₀e^−iwt , the solution of the equation of motion has the form

Where,

and H is the transfer function that relates input forces to output responses.

The effects of mooring lines on a body alter the external forces and stiffness matrix for the floating body and change the calculation of the equation of motion:

M(s) = mass matrix,
M(a) = hydrodynamic added mass matrix,
K(h) = hydrostatic stiffness matrix, and
F(t) = external harmonic forces on the structure.

The system damping C(s) and stiffness matrix K(s) are

C(s) = C(h) +C(w) and
K(s) = K(h) + K(m) + K(d) + K(a),

where

C(h) = hydrodynamic damping matrix
C(w) = damping due to wind = 0,
K(m) = stiffness due to mooring lines,
K(d) = hydrodynamic stiffness due to environmental effects (wind, waves, and currents), and
K(a) = stiffness due to articulations.

From the characteristics of low pressure and turbulent flow, the k-epsilon model is used to describe the flow of fluid in internal devices. K-epsilon is a model with two equations to describe turbulence of flows. The first convective variable is turbulent kinetic energy k, which determines the energy in a turbulent flow, and the second convective variable is dissipative turbulence ϵ, which determines the scale of turbulence. The convection equations used to model the Kepsilon standard [13] are as follows:

Turbulent kinetic energy k

Dissipative turbulence ϵ

Viscous turbulence is modeled as follows:

where

S is the modulus of ratio average stress tensor:

The influence of push force is given by:

The Pradtl number is Prt = 0.85.

Thermal expansion coefficients are

Figure 4 shows the geometry of the FWEC and buoy used in ANSYS AQWA. Figure 5 and Table 1 detail the schematic and dimensions of the mooring line used for station keeping with the FWEC device. This system is modeled in ANSYS AQWA as shown in Figure 6 and the naming convention for device motions corresponding to the coordinate system in ANSYS AQWA is listed in Table 3. WEC is modeled as a rigid body with six free DOFs for motion. To determine the best mooring line configuration, we solved many cases of mooring lines previously. In this research, we choose the most effective configuration used for sea trials to determine the relationship between pitch motion of a WEC and rotational speed of the device. Additionally, the model mooring configuration and parameters in Figure 5 are the same as in the sea trail.

Figure 4: 
Geometry of a WEC and buoy

Figure 5: 
Mooring model

Figure 6: 
Model in ANSYS AQWA

The load applied in ANSYS AQWA is shown in Table 2, which includes a wind load of 20 m/s and irregular Pierson- Moskowitz-generated waves with significant wave heights of 0.48 m and a wave period of 4.46 s as show in Figure 7. The Pierson-Moskowitz spectrum is a special case for a fully developed long-crested sea. The spectral ordinate at a given frequency (in rad/s) is given by:

where

H_s: significant wave height (m) and
T_Z: average (mean zero-crossing) wave period(s).

Figure 7: 
Wave elevation vs. time based on the Pierson- Moskowitz spectrum

To save time and obtain the best results, the model was divided into multiple parts. The part near the blades and the part containing the blades must have fine-quality meshes and small elements for accurate flow visualization, as shown in Figure 8. In addition, the interfaces of the parts must have the same element sizes.

Figure 8: 
Turbine model in ANSYS Fluent

User-defined functions (UDF) are written as C-functions, which can be implemented dynamically into ANSYS FLUENT to enhance the standard features of the software. Source files containing UDFs can be either interpreted or compiled in ANSYS Fluent [14].

For interpreted UDFs, source files are interpreted and loaded directly at run time in a single-step process.

For compiled UDFs, the process involves two separate steps. A shared object code library is built and then loaded into ANSYS Fluent.

In this study, the sliding mesh technique is used to simulate the motion of the blades. Motion data for the device are imported from the results of ANSYS AQWA. ANSYS Fluent uses the data for the sliding mesh. The function DEFINE_ZONE_MOTION macro is defined in UDF. DEFINE_ZONE_MOTIONmacro specifies the cell zone motion components in a moving reference frame or moving mesh simulation. In the DEFINE_ZONE_MOTION macro, “For” loops are used to set the angular velocity (rad/s) dependent on time.

The DEFINE_ZONE_MOTION macro is used to describe the motion of the blades. The motion of the blades depends on the water pressure on the blades and ANSYS Fluent can export force and moment on the boundary zone. So we write the function in DEFINE_ZONE_MOTION macro to get force and moment and calculate torque on blades, area of blades by the code Lookup Thread (d, 10), where 10 is ID of area blade. By this way, we can take the angular velocity of blade against time. Angular velocity is calculated by multiplying the torque and moment of inertia of a blade. The velocity is used to update the node position in blade zone motion.

Being a closed domain, all solid surfaces interfacing with the fluid are treated as wall boundaries. This relates to the fact that fluid at the wall has the same velocity as the wall. For example, fluid on the surface of the turbine has the same velocity as the rotating blades, whereas fluid at the case surface has zero velocity.

Studies on the strength of FWEC structures have mainly focused on numerical work due to the complexity of the structures involved. It is important to note that, while experimental studies provide necessary physical insight, predictive tasks such as design, analysis, and evaluation of structures are often done by computational methods. Hence, numerical methods have advantages over experimental methods, especially in terms of studying stress distributions and fracture mechanisms.

Figure 9 shows the static structural geometry of the hull and the blade. The geometry of the hull includes the shell of the box and a reinforced link for the shell. The parts of the device that do not affect the strength of the box are removed to reduce the computational time required.

Figure 9: 
Geometry of the blades and outer hull

The static structural mesh used is composed of tetrahedral elements with finer meshing where appropriate. The boundary conditions of the model used in static structural analysis include maximum pressure on blades taken from ANSYS Fluent, i.e., 2,700 Pa in the Z direction. This pressure is applied to the blades to check stress and deformation.

For the outer hull, the directional forces of cable connections taken from ANSYS AQWA are applied as follows: fx=69,002 N, fy= -61,221 N, and fz= -25,780 N.

AQWA can generate a time history of the simulated motions of floating structures arbitrarily connected by articulations or mooring lines under the action of wind, waves, and current forces. The positions and velocities of the structures are determined at each time step by integrating the accelerations due to these forces in the time domain using a two-stage predictor-corrector numerical integration scheme. This analysis is used to simulate the real-time motion of a floating body while operating in irregular waves.

Figure 10(a) shows the resulting output of the simulation in ANSYS AQWA, which includes X direction position (m) (surge motion of the FWEC), Z directional position (m) (heave motion of the FWEC), and Y directional angular velocity RY (0/s) (pitch motion of the FWEC). Results from ANSYS AQWA include other parameters such as sway, yaw, and rotation motion as well as cable force (Figure 11) and structural stress. We discuss these results below.

Between 0 s to 25 s, the heave and pitch motions had large amplitudes, as seen in the enlarged Figure 10(b). After 25 s, the motions were slightly damped. Pitch motion (0/s) after 25 s had a maximum velocity of 27 (0/s) and a minimum velocity of -27 (0/s). Since the applied wave motion function is irregular, the resulting motion is not regular. To obtain an exact evaluation of the motion, results from ANSYS AQWA were considered from 130 s onwards. Input parameters of phase 2 and phase 3 were also used from 130 s onwards.

Figure 10(a): 
ANSYS AQWA simulations results

Figure 10(b): 
Motion of device in ANSYS AQWA

Figure 11: 
Cable forces in ANSYS AQWA

Results evaluated in ANSYS Fluent include the velocity distribution, pressure distribution inside the device, change of velocity at the blades over time, change of pressure on the blades over time, and torque values on the blades. A comparison is also made for the change in blade velocity with pitch motion and Y direction velocity of the devices.

Figures 12 and 13 show velocity and pressure distributions of fluid in an internal device at three different times, 10 s, 12.1 s, and 18.8 s. General velocity distribution and pressure distribution show the general fluid flow and general pressure of the fluid. Velocity and pressure distributions on the blades show local distributions of fluid near the turbine blades. At 10 s, the device had a small rotational angle. The difference between water levels in the device is very small, and therefore water velocity through the blades was small. The maximum velocity observed was 1.78 m/s, which was concentrated at less important locations. At 12.1 s and 18.8 s, the difference between water levels in the device was larger, so the velocity of the fluid was larger and the velocity of 1.78 m/s appeared at many locations. The direction of water movement depends on the tilt angle of the device. The inversion of tilt angles leads to a change of direction for the moving water, as illustrated in Figures 12 and 13. In these figures, we observe the difference of tilt angles for the device at 12.1 s and 18.8 s; the direction of fluid motion was opposite at these times. Moreover, the distribution of fluid velocity also depends on the rotational velocity of the device, pitch motion, and rotational velocity of the blades. The effect on the velocity of water due to rotation of the blades is smaller than the effect on the velocity of the water due to rotation of the device.

Figure 12: 
Distribution of velocity

Figure 13: 
Distribution of pressure

Figure 14 shows the relationship between pitch motion and blade velocity, linear-angular velocity (pitch motion). For the Y directional angular velocity RY (0/s) (pitch motion of the FWEC), the units of velocity (0/s) were changed into (rad/s) between 130–150 s. Angular velocity (pitch motion) has negative or positive values, and as the blade is rotated in one direction, it therefore has negative or positive values, as well. It can be seen that similar changes occur between velocity lines, and depends on blade velocity and angular velocity (pitch motion). The peaks (maximum velocity values) of pitch motion were the same as for peaks of blade velocity. This shows that the difference of phases between velocity lines is very small. A small difference in the phases means that the shapes of turbine blades, their moments, and materials (blade materials affect the inertial moments of the blades) are suitable for satisfying the design of the turbine.

Figure 14 also shows the difference between velocity values. At earlier time points, the difference between the two velocities was small. The difference was close to zero from 5.5–7 s; during 0–4.5 s, pitch motion had a velocity of (-0.2)–(-0.18) rad/s, and blade velocity was 0.2–0.3 rad/s. In this case, the difference during 0–4.5 s had a value of 0.1 rad/s. From 9.9–20 s, the maximum value of the difference was 0.1 rad/s during this period. The maximum approximate value of the difference in velocities was 0.3 rad/s at 12.2 s and at 12.2 s the blade velocity and pitch motion had maximum values. Therefore, when pitch motion has a large value, the blade is in a high position, and the difference between the two values is high.

Figure 14: 
Relationship between pitch motion and blade velocity

In Figure 15, the graph shows that the maximum pressure on the blade is 2,000 Pa and the minimum is approximately -2,800 Pa.

Figure 15: 
Pressure on blade over time

From results for velocity and pressure, the relationship between pitch motion and angular velocity is shown. This is useful in judging the power of a device. The sliding meshing technique used in ANSYS Fluent provides a pathway to improving the design of the system and its efficiency.

The water column velocity distribution and pressure distribution show general fluid flow and pressure; at the column for the velocity distribution on the blade, and distribution of pressure on the blade shows the local distribution of fluid at a location near turbine blade. In general, it is expected that, when the difference between water levels in the device is small, the effective water velocity and thus blade velocity are small, and when the water level difference is relatively larger, a higher fluid flow velocity results.

Table 4 results were obtained from a real sea test. The tabulated results are maximum peak values obtained instantaneously for peak efficiency. As observed from the torque graph in Figure 16, the nominal peak torque value is close to the obtained value. The torque sign changed from negative to positive approximately when the water stream hit the turbine blades. Factors that could lead to uncertainties include friction losses in the bearings and generator, which were not considered in the simulation. Limitations still exist in applying the problems in this study with realistic loading and examining how FWECs behave in various changing loading conditions. These aspects have an effect on the solutions obtained for the generated energy, which will be considered in future studies.

Figure 16: 
Torque from the simulation

Static structural analysis was presented for the equivalent stress and deflection on the outer hull and deflection of the turbine blades of the modeled FWEC. The resultant structural forces and maximum torque were considered as input for static structural analysis of the outer hull and turbine blades of the FWEC.

Figures 17 and 18 show the results of total deformation and stress distribution on the blades and outer hull. The maximum deformation value of the blades was 0.000276 mm. The position of maximum deformation was the middle of the blades. The maximum stress (von Mises stress) was 2.61220 MPa. The positions of maximum stresses were at the edges of the blades. The maximum stress ratio of the blades was 0.09 (<1), and hence the blades were safe under the effect of water pressure. In addition, the maximum stress and deformation, as expected, were located at the contact joint between the cable and the device. The stress ratio of the outer box was 0.432, which meant that the stress was within a safe margin. However, under the effects of severe wave conditions, stress values can be high and would be nearly equal to the strength of the hull, leading to potential damage. Consequently, the device is suitable to work in the following conditions: a wave height of 0.48 m and wave period of 4.8 s with a wind load of 20 m/s. The simulation results provided great insight into modifying and upgrading structural strength while maintaining device efficiency. Areas in need of further consideration to improve structural integrity are the mooring connection point, the blades, and stiffeners.

Figure 17: 
Total deformation and stress distributions on the turbine blades

Figure 18: 
Deformation and stress of the outer hull at the mooring connection point