*2.1. Details of the 8000 TEU-Class Container Ship*

The ship used for the analysis of resistance performance was an 8000 TEU-class container ship with 322.6-m L.B.P. (length between perpendiculars), 45.6-m breadth, and 24.6-m depth. The details are as provided in Table 1. The model ship for numerical simulations was set to 7.279 m, which was the same size as the KCS (3600 TEU KRISO Container ship).

In order to consider the superstructure, the ship was modeled with the containers loaded, as shown in Figure 1a, and the container was designed in a simple rectangular shape. In addition, breakwater, hatch cover, and accommodation were included in the modeling, whereas the lashing structures for the containers were omitted. Figure 1b shows a ship without superstructures, generally used for experiment in basin and numerical simulations.

**Table 1.** Principal dimensions of the 8000 TEU-class container ship (center of gravity means, from A.P. (Aft Perpendicular) to F.P. (Forward Perpendicular), centerline, from baseline to upward).


**Figure 1.** Modeling of the 8000 TEU-class container ship. (**a**) Design of model with superstructure. (**b**) Design of model without superstructure.

#### *2.2. Full-Scale Prediction Method*

In the full-scale prediction method, the total resistance coefficient (*CTS*) was calculated by a two-dimensional method, as the sum of the frictional resistance coefficient (*CF*), residuary resistance coefficient (*CR*), correlation allowance (*CA*), and air resistance coefficient (*CAA*), as shown in Equation (1). *CF* is calculated according to the ITTC-1957 (International Towing Tank Conference-1957) frictional correlation line, *CA* is calculated by the Harvald formulation, and *CAA* is calculated by the ITTC method [6].

$$\mathbf{C}\_{TS} = \mathbf{C}\_{F} + \mathbf{C}\_{R} + \mathbf{C}\_{A} + \mathbf{C}\_{AA} \tag{1}$$

$$C\_F = \frac{0.075}{\left(\log R\_N - 2\right)^2} \tag{2}$$

$$\mathbb{C}\_{R} = \mathbb{C}\_{TM} - \mathbb{C}\_{FM} \tag{3}$$

$$C\_A = \frac{0.5\log(\Delta) - 0.1(\log(\Delta))^2}{10^3} \tag{4}$$

$$\mathbb{C}\_{AA} = \mathbb{C}\_{DA} \frac{\rho\_A \cdot A\_{VS}}{\rho\_S \cdot \mathbb{S}\_S} \tag{5}$$

where Δ is the displacement in ton, *RN* is the Reynolds number, and *CDA* is the air drag coefficient of the ship above the water line that can be determined through the wind tunnel testing or calculations. Typically, 0.8 can be used as the default value of *CDA* in the range 0.5–1.0 if the specific value is not known [6]. ρ*<sup>A</sup>* is the density of air, ρ*<sup>S</sup>* is the density of seawater, *AVS* is the projected area of the ship above the water line to the transverse plane, and *SS* is the wetted surface area of the ship. The subscript *M* signifies the model and *S* signifies the full-scale ship.

#### **3. Numerical Simulation**

In this study, the commercial software Star-CCM+ was used to perform the numerical simulation. The governing equations were the continuity equation and momentum equation for three-dimensional unsteady incompressible viscous flow, shown in Equations (6) and (7) [10].

$$\frac{\partial \mathcal{U}\_i}{\partial \mathbf{x}\_i} = 0 \tag{6}$$

$$\frac{\partial \mathcal{U}\_i}{\partial t} + \mathcal{U}\_j \frac{\partial (\mathcal{U}\_i)}{\partial \mathbf{x}\_j} = -\frac{1}{\rho} \frac{\partial p}{\partial \mathbf{x}\_i} + \frac{1}{\rho} \frac{\partial}{\partial \mathbf{x}\_j} \Big(\mu \frac{\partial \mathcal{U}\_i}{\partial \mathbf{x}\_j} - \rho \overline{u\_i' u\_j'}\Big) + B \tag{7}$$

where *U* is the average velocity vector, *x* is the coordinate system, *t* is the time, ρ is the density, *p* is the pressure, and μ is the coefficient of viscosity. ρ*u i u <sup>j</sup>* is the turbulent shear stress that is determined using a turbulence model, and *B* is the body force. In this study, a realizable k- model was used for the turbulence model.

The governing equations mentioned above were discretized using the finite volume method (FVM). The convection and diffusion terms were discretized with the second-order upwind scheme. The second-order implicit scheme was used for temporal discretization.

The semi-implicit method for a pressure-linked equations (SIMPLE) algorithm was used for velocity-pressure coupling. The volume of fluid (VOF) method with a high-resolution interface capturing (HRIC) algorithm was used to define the water and air area of the free surface.

Equation (8) related to the translation of the center of mass of the body, and Equation (9) related to the rotation of the body, formulated with the origin at the center of mass of the body.

$$m\frac{dv}{dt} = f\tag{8}$$

$$M\frac{d\overrightarrow{\omega\nu}}{dt} + \overrightarrow{\omega\nu} \times M\overrightarrow{\omega\nu} = \mathbf{n} \tag{9}$$

where *m* is the mass of the body, *f* is the force acting on the body, *v* is velocity of the center of mass, *M* is the tensor of the inertia moments, <sup>→</sup> ω is the angular velocity of the rigid body, and n is the moment acting on the body.

### *3.1. Initial Conditon and Boundary Condition*

In the numerical simulation, the length, breadth, and height directions were set as 4.0 L, 1.5 L, and 2.5 L, as shown in Figure 2a. Here, L is the L.B.P. of the ship.

As shown in Figure 2b, velocity inlet, pressure outlet, symmetry, no-slip wall of the ship, and free-slip wall conditions were used for each boundary. To limit the calculation time, only half the breadth of the ship was modeled and the symmetry boundary condition was applied. Heave and pitch motion were considered by using the dynamic body fluid interaction (DFBI) method for the translation and rotation of the entire domain. The total calculation time of the numerical simulation was 90 s and the time increment was 0.02 s.

**Figure 2.** Computational domain and boundary conditions for numerical simulation; (**a**) computational domain and (**b**) boundary condition.

The above conditions were verified by conducting numerical simulations using the KCS hull form. KCS is a popular hull form like KVLCC (KRISO Very Large Crude-Oil Carrier) and DTMB (David Taylor Model Basin) it is often used to verify the conditions of numerical simulation through comparisons with experimental data [11–14]. Table 2 shows the main particulars of the KCS hull form; numerical simulation was performed for the model scale.

**Table 2.** Principal dimensions of KCS (center of gravity means (from A.P. to F.P., centerline, from baseline to upward)).


### *3.2. Grid System*

The grid system for the numerical simulation consisted of approximately 1.5 million cells, as shown in Figure 3. It was created using surface re-mesher, prism layer, and trimmer grid, which are auto-meshing methods provided by Star-CCM+. Five layers were generated in the normal direction to the hull, to consider the viscous flow field. In addition, we arranged the grid more closely around the free surface, to consider the wave generated by the hull. The minimum size of a cell was set to 1.0 <sup>×</sup> <sup>10</sup>−<sup>2</sup> <sup>m</sup> and *<sup>Y</sup>*<sup>+</sup> was less than 100 for the entire area of the hull, as shown in Figure 4. Additional numerical simulation was performed to validate the grid sensitivity of the 8000 TEU container ship, with the superstructure, as shown in Table 3.

**Figure 3.** Grid system for numerical simulation. (**a**) KCS, (**b**) 8000 TEU-class container ship without superstructure, and (**c**) 8000 TEU-class container ship with superstructure.

**Figure 4.** *Y*<sup>+</sup> entire area of the hull. (**a**) KCS, (**b**) 8000 TEU-class container ship without superstructure, and (**c**) 8000 TEU-class container ship with superstructure.



As the number of grids increased from coarse to fine, *CTM* tended to converge. In particular, since the difference in *CTM* between the medium and the fine grid system was less than 1%, so the medium grid was applied to reduce the calculation time in numerical simulation.

#### **4. Results of the Numerical Simulation**

#### *4.1. Validation Study*

Numerical simulations were conducted under six different speed conditions (Froude number (*FN*) of 0.108, 0.152, 0.195, 0.227, 0.260, 0.282) for validation of the simulation conditions. The results are as shown in Figure 5.

As shown in Figure 5a, the sinkage tended to increase as the speed increased. As shown in Figure 5b, the trim by stern tended to increase as speed increased, when *FN* was at or below 0.269 and decreased when *FN* exceeded 0.269. Overall, under the six speed conditions, the results for trim and sinkage were quantitatively similar to the experimental simulation results [11], when compared with the numerical simulation results of Villa et al. [14]. However, a quantitative difference from the experiment results was observed for the trim when *FN* was less than 0.15 or more than 0.28, and for the sinkage when *FN* was 0.16 or below. A difference of approximately 3% was observed from the experimental value of the total resistance coefficient, at the low speed of *FN* = 0.108. Overall, the results were quantitatively similar to the experimental results under all the six speed conditions. It was also relatively more consistent with the experimental results than the numerical simulation results of Villa et al. [14], as shown in Figure 5c.

Therefore, as the accuracy of the numerical simulations for the ship's attitude appeared to be relatively low in the low-speed range (*FN* < 0.16) or in the high-speed range (*FN* > 0.28), the numerical simulations of the 8000 TEU-class container ship were conducted in the *FN* range of 0.16–0.27.

**Figure 5.** Comparison of KCS simulation results between EFD (Experimental Fluid Dynamics) and computational fluid dynamics (CFD) (a positive trim value was defined bow up and positive sinkage value was defined upward). (**a**) Sinkage; (**b**) trim; and (**c**) total resistance coefficient.
