**4. Validation and Verification**

The accuracy of the free surface is an important factor that needs to be considered before the CFD interpretation of the motion properties and flow of an FSP subjected to waves.

Previously, it was found that the importance of the variables such as time step, damping, and forcing constants can prevent wave reflection under boundary conditions, and can minimize waves [18,19]. In general, the computations that apply to the wave take a long computational time. Kim et al. [20] performed a numerical sensitivity simulation for two-dimensional waves and completed a numerical error distribution map based on the grid and CFL (Courant–Friedrichs–Lewy) conditions. Through those results obtained from the two-dimensional problem, a FPSO (floating production storage and offloading) motion

simulation was conducted under wave conditions. The tested two-dimensional wave conditions showed a relatively good agreement compared to potential results, which shows that the verification of the waves through the two-dimensional test is valid. In this study, two-dimensional sensitivity analysis for waves was performed, and this was applied to the FSP to perform grid convergence analysis.

#### *4.1. Two-Dimensional Wave Generation Sensitivity*

This section presents the results of an examination of the computational conditions to accurately simulate waves under the conditions given in the experiment. Considered as a 2D problem, the target waves are two types of stokes waves (*λ* = 1.35 m, *H* = 0.0405 m; and *λ* = 1.80 m, *H* = 0.054 m). The total simulation time corresponded to eight cycles of the wave. The data were extracted at three points (*x* = +1.650 m, +0.675 m, and −0.675 m from the origin) to verify the correctness of the implementation for the target wave (see Figure 10).

**Figure 10.** Computation domain in waves through grid sensitivity analysis.

The medium grid was set to approximately 135 cells per wavelength and 20 cells per wave height with √ *λ* = 1.35 m and *H* = 0.0405 m. The basic grid size was varied by 2 times. In total, three types of grid systems were constructed with coarse and fine grids. The coarse grid was set to approximately 95 cells per wavelength and 12 cells per wave height. Meanwhile, the fine grid was set to approximately 190 cells per wavelength and 26 cells per wave height. In all the grid systems, the time step was set to a value at which the CFL was lower than one.

The wave profiles of the numerical and theoretical solutions obtained at each point by the grid systems were compared as shown in Figures 11 and 12, where the black solid line is the theoretical solution. The coarse, medium, and fine grids are the green-triangle, blue-square, and red-circle symbols, respectively.

**Figure 11.** Comparison of wave profiles at *λ* = 1.350 m: (**a**) *x* = +1.650 m from the origin; (**b**) *x* = +0.675 m from the origin; (**c**) *x* = −0.675 m from the origin.

**Figure 12.** Comparison of wave profiles at *λ* = 1.800 m: (**a**) *x* = +1.650 m from the origin; (**b**) *x* = +0.675 m from the origin; (**c**) *x* = −0.675 m from the origin.

Table 4 shows the wave amplitude difference between the theoretical and numerical solutions according to the variation in the grid system at each position. The fine grid shows the best results. However, it requires an excessively long calculation time because of the number of grids and time steps. The medium grid appears to be suitable considering efficiency. In this study, we selected a medium-grid system to express the free surface.


**Table 4.** Comparison of wave amplitude for each position according to the grid system.

#### *4.2. Grid Convergence Test for the Floating Solar Power Farm under Wave Conditions*

In this section, the grid convergence tests for the FSP computation are presented. The test subject was set to one unit, and the free surface was set to three stages, as described in Section 4.1. The grid convergence index (GCI) [21–23] of the test conditions are listed in Table 5. The GCI in this study represents the difference in the result value due to the grid variation represented by Grid-1, Grid-2, and Grid-3. The grid numbers of Grid-1, Grid-2, and Grid-3 were 0.96 million, 1.36 million, and 1.92 million, respectively. The number of grids was changed by changing the basic grid. In the grid convergence test, the difference of motion RAOs was reduced with the increase of grid size. The heave RAO GCI is 0.00048, 0.00031, and 0.00055 at wavelength ratios of 1.6, 2.4, and 3.2, and the pitch RAO GCI is 0.00057, 0.00183, and 0.00180 at the same wavelength ratios.

**Table 5.** Comparison of the motions under the wave conditions *(H*/*λ* = 0.03).


The results of FSP RAOs according to the variation in the grid systems are shown in Figure 13. The coarse, medium, and fine grids and experiment results are the greentriangle, blue-square, red-circular, and black-circular symbols, respectively. Unlike other grid systems, Grid-1 indicates lower RAOs in heave and pitch motions. At short wavelengths, the number of grids was insufficient for the wave height to express the free surface. Even when Grid-3 was expressed effectively for motion RAOs of the FSP and free surface, it required a computation-time approximately two times that for Grid-2. In addition, Grid-2 indicated that the accuracy of the free surface and the results of motion RAOs did not differ significantly from those for Grid-3. The heave and pitch RAOs show GCI values below 1.0% for Grid-3 and Grid-2. In addition, the motion RAOs for Grid-2 show satisfactory agreement with the experiment results with an error of about 3.0%. The computations in this study were performed using Grid-2.

**Figure 13.** Comparison of motion RAOs according to grid conditions: (**a**) heave; (**b**) pitch.

#### **5. Results**

A model test was conducted using the IUTT to investigate the motion and load characteristics of the FSP under wave conditions. The results were compared with the computation results obtained with identical conditions. It was attempted to ensure the reliability and validity of the calculation based on the motions generated in each unit by the connection and the load applied to the mooring lines.

#### *5.1. Head Sea Conditions at H*/*λ = 0.03*

The experimental results of the motion RAOs of each unit of the FSP under waves at *H*/*λ* = 0.03 according to the wavelength are shown in Figure 14. Here, the circular and square symbols represent an odd-numbered and even-numbered unit, respectively. Black and blue denote the first and second groups, respectively.

In the case of surge motion, the surge motion generated in the wave direction in each unit was expressed. The difference between the groups was large at *λ*/*Lunit* = 2.0. Subsequently, the units tended to converge as the wavelength increased.

In the case of heave motion, the heave RAO increased linearly with the increase in wavelength, and relative behavior occurred between the first and second groups at *λ*/*Lunit* = 2.0 and 2.4. It appeared that when the incident wave lost energy while contacting the first group, this effect was reduced by the motion of RAOs of the second group. In addition, it was considered that a similar heave RAO occurred at *λ*/*Lunit* = 1.6 owing to a small wave height and short period.

In the case of roll motion, the hinge motion parallel to the wave travel direction was predominant because of the head sea conditions, the FSP was bilaterally symmetric, and almost no roll RAO was generated.

In the case of pitch motion, because the wavelength was longer than the FSP length for *λ*/*Lunit* = 2.8, a motion that rides on a wave appeared. Furthermore, the pitch RAO tended to converge as the wavelength increased. In addition, at *λ*/*Lunit* = 2.4, the length of the FSP became equal to the wavelength, and the maximum pitch RAO occurred.

**Figure 14.** Motion RAOs of the experiment according to the wavelength at *H*/*λ* = 0.03: (**a**) surge; (**b**) heave; (**c**) roll; (**d**) pitch.

The experimental and computation results were compared to verify the wave pattern around the FSP according to the wavelength (see Figure 15). Here, the images on the left and right represent the experiment and computation, respectively. It was observed that the waves were diffracted and reflected by floating bodies, and wave run-up was also observed. When the waves passed through the floating bodies, the bodies were significantly affected by the periodic variation in the incident wave, which especially appeared at a relatively short period condition (*λ*/*Lunit* = 1.6). Then, the nonlinear wave profile reveled a non-uniform load on the support structure coupled to the suspended bodies [24]. At *λ*/*Lunit* = 2.4, the deformation of the first and second groups were similar to the experiment. Also, the motions of riding on the wave were observed at long wavelengths.

The computation results of motion RAOs according to wavelength were compared with the experimental results (see Figures 16 and 17). Here, the black-square and red-circular symbols represent the experimental and computation results, respectively. The heave RAOs are shown in Figure 16. Similar results were observed overall. The heave RAOs increased linearly with the wavelength. The pitch RAOs are shown in Figure 17. The computation results matched a maximum value at *λ*/*Lunit* = 2.4, which is the same as the experiment, which was influenced by the wave conditions. However, the heave and pitch RAOs showed a difference from motion RAOs.

At *H*/*λ* = 0.03, *λ*/*Lunit* = 2.4, the time series data of the total tension was measured on the mooring lines connected to the FSP (see Figure 18). Here, the black-solid and red-dotted lines represent the experimental and computation results, respectively. The tension time series shows two points. One is the maximum load point where the FSP was farthest from the initial position. The second point is the average load point where the position of the FSP converged owing to the restoration of the mooring lines. Compared to the experiment, it showed a similar trend that the period in which the load was generated and the two tension change points. The total tension of the mooring lines is expressed as the maximum

and average loads (see Table 6). The maximum tension was observed at *λ*/*Lunit* = 2.0, where the difference of surge RAO between the units was large. We observed a tendency to converge even at tensions with *λ*/*Lunit* ≥ 2.4.

**Figure 15.** Wave pattern of experiment (**left**) and computation (**right**) according to the wavelength for *H*/*λ* = 0.03 under head sea: (**a**) *λ*/*Lunit* = 1.6; (**b**) *λ*/*Lunit* = 2.0; (**c**) *λ*/*Lunit* = 2.4; (**d**) *λ*/*Lunit* = 2.8; (**e**) *λ*/*Lunit* = 3.2.

**Figure 16.** Heave RAOs for *H*/*λ* = 0.03 under head sea: (**a**) Unit 1; (**b**) Unit 2; (**c**) Unit 3; (**d**) Unit 4.

**Figure 17.** Pitch RAOs for *H*/*λ* = 0.03 under head sea: (**a**) Unit 1; (**b**) Unit 2; (**c**) Unit 3; (**d**) Unit 4.



**Figure 18.** Time series data of total tension at *λ*/*Lunit* = 2.4 and *H*/*λ* = 0.03.

#### *5.2. Head Sea Condition at Different Wave Steepness*

The time series motion data for the heave and pitch of Unit 1 in the experiment according to the wave steepness for *λ*/*Lunit* = 1.6 and 2.4 are shown in Figure 19. Here, the pink and blue colors represent *λ*/*Lunit* = 1.6 and 2.4, respectively. The solid and dotted lines represent *H*/*λ* = 0.03 and 0.05, respectively. When the wave steepness is constant, the variation at *λ*/*Lunit* = 2.4 is larger than that for 1.6, because the height of the wave varies significantly.

**Figure 19.** Time series motion data according to wave steepness at Unit 1: (**a**) heave; (**b**) pitch.

The experimental results of the motion RAOs of each unit of the FSP under waves according to wave steepness are shown in Figure 20. In the motion RAOs, the results for *H*/*λ* = 0.03 were higher than those for 0.05. This tendency was converse to that of the time series motion data. It was thought that the nonlinearity of the motion increased with respect to the rate of increase in wave steepens. In addition, the difference appeared to be smaller in the second group than in the first. The wave lost energy while contacting the first group, and the second group in contact with the losing wave had reduced motion.

**Figure 20.** Motion RAOs of the experiment according to wave steepness: (**a**) heave (first group); (**b**) heave (second group); (**c**) pitch (first group); (**d**) pitch (second group).

The experiment and computation results were compared to verify the wave pattern around the FSP according to the wavelength (see Figure 21). In *H*/*λ* = 0.03, a large motion characteristic appeared at the center of the hinge. The bigger wave run-up and diffraction were observed, which showed similar results in computation. Also, the green water observed at *λ*/*Lunit* = 2.4 in the experiment was also observed in the computation results.

The computation results of motion RAOs according to the wavelength at *H*/*λ* = 0.03 were compared with the experimental results (see Figures 22 and 23). Similarly, as in the previous section, the amplitude of the RAOs increased as the wavelength increased. The total tension of the mooring lines is expressed as the maximum and average loads (see Table 7).

The tension was larger at *λ*/*Lunit* = 2.0, and the maximum load occurred at a wavelength ratio equal to that mentioned in Section 5.2. In addition, as the wave height increased, the wave energy increased with the load. It is expressed non-dimensionally to analyze the tension by excluding the wave conditions (see Figure 24). The tension coefficient (*Ctension*) is identical to that in Equation (5):

$$C\_{Tension} = \frac{F\_{Total\\_tension}}{\frac{\rho g A^2 B\_T^2}{L\_{avrit}}} \tag{5}$$

where *ρ* is the density, *g* is the gravitational acceleration, *A* is the wave amplitude, and *BT* is the total breadth of the floating bodies at one unit. Regarding the tension coefficient, *H*/*λ* = 0.03 appeared to be larger than 0.05. This was because the tension increased with the increase in wave height. However, the non-dimensional tension coefficient tended to decrease because of the nonlinear decrease in the pitch RAO.

**Figure 21.** Wave pattern of experiment (**left**) and computation (**right**) according to the wavelength for *H*/*λ* = 0.05 under head sea: (**a**) *λ*/*Lunit* = 1.6; (**b**) *λ*/*Lunit* = 2.0; (**c**) *λ*/*Lunit* = 2.4.

**Figure 22.** Heave RAOs for *H*/*λ* = 0.05 under head sea: (**a**) Unit 1; (**b**) Unit 2; (**c**) Unit 3; (**d**) Unit 4.

**Figure 23.** Pitch RAOs for *H*/*λ* = 0.05 under head sea: (**a**) Unit 1; (**b**) Unit 2; (**c**) Unit 3; (**d**) Unit 4.



**Figure 24.** Comparison of tension coefficient according to wave steepness.

#### *5.3. Oblique Sea Conditions at H*/*λ = 0.03*

In this study, we conducted a model test under oblique wave conditions by rotating the FSP, which leads the waves to enter diagonally. Units 2 and 4 of the FSP were arranged in a straight line, and the FSP was rotated by 41◦. The experimental results of the motion RAOs of each unit of the FSP under oblique waves are shown in Figure 25.

**Figure 25.** Motion RAOs of the experiement according to wavelength under oblique sea: (**a**) heave; (**b**) roll; (**c**) pitch.

In the case of heave motion, similar motion characteristics occurred between units (Units 1 and 4, and Units 2 and 3) located at the same position. In Unit 3, the waves that passed between Units 1 and 4 overlapped, thereby causing excessive motion.

Roll motion varies significantly depending on the wave direction, and this increases gradually according to the wavelength. In the uniaxial hinge system, the units in a column exhibited similar movements. In the case of pitch motion, the motion was lower than that under head sea conditions. When the wavelength was increased, the relative movements between the units were getting bigger. On the other hand, in the experiment, it was observed that the diffracted waves spread widely, and the waves would re-enter the FSP through the walls. In addition, the overlapping and dissipation of waves between floating bodies (which were difficult to observe in the experiment) were inspected.

The computation under oblique sea conditions was performed using the numerical method verified for head sea conditions. The results were compared with the experimental results for *λ*/*Lunit* = 2.4 and 2.8. The experimental and computation results were compared to verify the wave pattern around the FSP according to the wavelength (see Figure 26). The generated wave encountered the floating body at the front of Unit 2. The effect of wave (e.g., diffraction, upwelling, run-up, and reflection) appeared to be larger than those under the head sea conditions because the geometry of the FSP.

The computation results of the motion RAOs according to wavelength were compared with the experimental results (see Figure 27). The oblique sea conditions showed similarities to head sea conditions. The translational heave motion showed a tendency to match. However, the roll and pitch motions differed, and it is thought that the difference of the hinge causes errors between experiments and simulations. This was determined to be an error that occurred in the hinge gap, as mentioned in the previous section. Theoretically, the uniaxial hinge does not cause simultaneous rotational motion of the horizontal and vertical axes. However, in the experiment, it was determined that the rotational motion occurred simultaneously owing to the hinge gap and that additional rotational motion was generated during overlapping rotation. However, the motion occurred only in the axial direction in the computation. This was observed a difference in the rotational components.

The time series data of the total tension measured on the mooring lines connected to the FSP under oblique sea are shown in Figure 28. The total tension of the mooring lines is expressed as the maximum and average loads in the oblique sea (see Table 8). The periods in which the load was generated, and the load exhibited similar tendencies. There was a difference in the load pattern after the maximum tension. This is unlike the head sea condition. Furthermore, it was determined that the oblique sea condition did not achieve precise geometrical similarity for the mooring line system. It was then determined that the external force applied to the FSP appeared to be relatively low tension because the

waves were dispersed and escaped from the configuration of FSP and the movements were smaller than those under the head sea.

**Figure 26.** Wave pattern of experiment (**left**) and computation (**right**) according to the wavelength under oblique sea: (**a**) *λ*/*Lunit* = 2.4; (**b**) *λ*/*Lunit* = 2.8.

(**b**)

**Figure 27.** *Cont.*

**Figure 27.** Motion RAOs of heave (**left**), roll (**center**) and pitch (**right**) under oblique sea: (**a**) Unit 1; (**b**) Unit 2; (**c**) Unit 3; (**d**) Unit 4.

**Figure 28.** Time series data of total tension under oblique sea: (**a**) *λ*/*Lunit* = 2.4; (**b**) *λ*/*Lunit* = 2.8.

**Table 8.** Comparison of total tension determined by experiment and computation under oblique sea.


According to these results, it is necessary to develop an accurate computational technique for a connection similar to that in the experiment. When applying multi-directional articulations to the FSP, substantial research needs to be carried out on articulations for a more accurate interpretation.
