**3. Numerical Modeling**

#### *3.1. Governing Equations*

In this study, the flow around the FPS was assumed to be an incompressible turbulent flow. The continuity equation and Reynolds average Navier–Stokes (RANS) equations were used as the governing equations. These can be expressed as shown in Equations (1) and (2):

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \overline{u\_i}) = 0 \tag{1}$$

$$
\rho \overline{u\_{\dot{j}}} \frac{\partial \overline{u\_{\dot{i}}}}{\partial \mathbf{x}\_{\dot{j}}} = \rho \overline{f}\_{\dot{i}} + \frac{\partial}{\partial \mathbf{x}\_{\dot{j}}} \left[ -\overline{\mathbf{p}} \delta\_{\dot{i}\dot{j}} + \mu \left( \frac{\partial \overline{u\_{\dot{i}}}}{\partial \mathbf{x}\_{\dot{j}}} + \frac{\partial \overline{u\_{\dot{j}}}}{\partial \mathbf{x}\_{\dot{i}}} \right) - \rho \overline{u\_{\dot{i}}^{\prime} \dot{u}\_{\dot{j}}^{\prime}} \right] \tag{2}
$$

where *ui* is the three-dimensional velocity vector in the *x, y*, and *z* directions. *u* is the mean component and *p*, *ρ*, *μ*, and *f* are the pressure, density, dynamic viscosity, and body-force per mass, respectively. The left-hand side of this equation represents the variation in the mean momentum of a fluid element owing to the unsteadiness in the mean flow and convection by the mean flow −*ρu i u <sup>j</sup>* owing to the fluctuating velocity field. It is generally referred to as the Reynolds stress. The nonlinear Reynolds stress term requires additional modeling to close the RANS equation for solving. In this study, the Realizable *k* − turbulence model was used.

#### *3.2. Numerical Method and Setup*

The commercial CFD program STAR-CCM + v15.06 [14] was used to determine the motion and load characteristics of the FSP under wave conditions. The Dynamic fluid–body interaction (DFBI) model was used for the movement of the FSP. Furthermore, an overset mesh was used considering the deformation of the grid system owing to the wave, which leads to larger movement of the FSP. Six-DOF analyses were performed, and the free surface was generated by a volume-of-fluid (VOF) model showing non-mixed fluid. The wave was generated as the fifth-order Stokes waves [15].

The domain and boundary conditions used in the computation under head sea and oblique sea conditions are shown in Figures 7 and 8, respectively. The boundary conditions were set to a velocity inlet, and the forcing method supported by STAR-CCM + was utilized to minimize the dissipation of generated waves. The size of the forcing zone is shown in the red area.

**Figure 7.** Computation domain and boundary conditions for head sea conditions.

**Figure 8.** Computation domain and boundary conditions for oblique sea conditions.

In a previous study, the effects of the wave (e.g., wave run-up, reflections, and diffractions) occurred owing to the blunt geometry and spacing of the floating bodies [16,17]. Due to the forcing boundary absorbed the effects of the wave, the domain size was set as shown in Figures 7 and 8, respectively. The computation was an implicit unsteady condition, and the time step was set approximately *Te*/250 s. Here, *Te* is the wave period. Approximately 5 million and 6 million grids were used in the computations for the head sea and oblique sea conditions, respectively.

A catenary coupling system was used to express a mooring line in the computation. Catenary coupling models an elastic, quasi-stationary catenary such as a chain or towing rope, which hangs between two endpoints and is subject to its weight in the gravity field. In the local cartesian coordinate system, the catenary shape is given by Equations (3) and (4):

$$x = au + b\sinh + a$$

$$y = a\cosh(u) + \frac{b}{2}\sinh^2(u) + \beta$$

$$a = \frac{c}{\lambda\_{0\overline{\xi}}}, \ b = \frac{ca}{DL\_{eq}}, \ c = \frac{\lambda\_0 L\_{eq}\overline{\xi}}{\sinh(u\_2) - \sinh(u\_1)}$$

$$\tan\phi = \sinh(u)\tag{4}$$

where the curve parameter *ui* is related to the inclination angle *φ* of the catenary curve by Equation (4). *g* is the gravitational acceleration. *λ*<sup>0</sup> and *Leq* are the mass per unit length and relaxation length, respectively, of the catenary under force-free conditions. *D* is the stiffness of the catenary, and *α* and *β* are integration constants that depend on the positions of the two endpoints and the total mass of the catenary.

Also, a revolute joint coupling system was applied to express a hinge system in the computation. A revolute joint connects two 6-DOF bodies, each of which has its own local-body coordinate system. The position and axis direction of each body must coincide. However, the position and axis of the revolute joint can vary over time.

The overall forms in which the mooring lines and hinges were applied to the FSP are shown in Figure 9. The same geometry and physical property values of the model used in the experiment were applied in the computation simulations. The test conditions used to verify the numerical method are listed in Table 3.

**Figure 9.** Computation setup of mooring lines (catenary) and the hinge system.


**Table 3.** Test conditions of computation.
