*4.2. 8000 TEU-Class Container Ship*

Numerical simulations were conducted under five different speed conditions (Froude number (*FN*) 0.165, 0.192, 0.219, 0.247, and 0.274) for validation of the simulation conditions in a model scale. The results are as shown in Table 4 and Figure 6.

Similar to the KCS hull form, the sinkage tended to increase as the speed increased, and trim by head tended to increase as speed increased, when *FN* was 0.247 or below and decreased when *FN* was above 0.247. Sinkage was observed at a significant level in all cases where the superstructures were absent, and varied by approximately 3% to 9% between cases with superstructures. However, even as the speed increased, the quantitative difference remained consistent at approximately 0.0004. Trim was

observed at a significant level at *FN* of 0.2 or below when the superstructures were considered and at *FN* of 0.2 or above when the superstructures were not considered.


**Table 4.** Comparison of the simulation results with and without a superstructure (RD—relative difference).

**Figure 6.** Comparison of simulation results with and without superstructures (a positive trim value was defined bow up and positive sinkage value was defined upwards). (**a**) Sinkage and (**b**) trim.

Wave patterns tended to become similar as the speed increased, with the biggest difference observed at the lowest speed (*FN* 0.165) in Figure 7. Here, the vector distribution around the ship according to the presence or absence of the superstructure is shown in Figure 8.

*CTM* differed by a maximum of approximately 2%, under five different speed conditions, with and without the superstructures, as shown in Figure 9a. When *FN* was 0.2 or less, the resistance was higher in the case without superstructure and when *FN* was 0.2 or above it showed opposite results. This showed a typical tendency where the trim by head had a relatively lower resistance, compared to the even conditions, or the trim by stern [15,16].

To analyze the effects of the presence or absence of superstructures on the resistance, the resistance performance of the full-scale ship was estimated using Equations (1)–(5). Here, *CAA* obtained from Equation (5) was calculated using the coefficient in Table 5, for the case without the superstructure and *CAA* was set to 0, when the superstructure was considered.

Table 6 and Figure 9b show that depending on whether *CAA* is considered or not, *CTS* differs by approximately 1% to 5%, under the six speed conditions and the difference was significant at approximately 5% when *FN* was relatively low at 0.192 or below.

This indicated that calculation using an empirical formula could lead to over-estimation of the resistance performance of a full-scale ship, compared to a direct numerical interpretation, when considering the superstructures.

*J. Mar. Sci. Eng.* **2020**, *8*, 267

**Figure 7.** Comparison of wave pattern between the 8000 TEU container ship with superstructure and without superstructure. (**a**) *FN* 0.165, (**b**) *FN* 0.192, (**c**) *FN* 0.219, (**d**) *FN* 0.247, and (**e**) *FN* 0.274.

**Figure 8.** Velocity vector around the 8000 TEU-class container ship (velocity coefficient was defined as velocity divided by the inlet velocity). (**a**) *FN* 0.165 and (**b**) *FN* 0.274.

**Figure 9.** Comparison of total resistance coefficient for ships, with and without a superstructure. (**a**) Model scale and (**b**) full scale.

To analyze the effects of overestimating the *CAA*, the default value of *CDA* by ITTC was compared with the *CAA* for a container ship, calculated by Kristensen and Lützen [17] and the result of the equations proposed by Fujiwara et al. [18].

The *CAA* proposed by Kristensen and Lützen [17] estimates the air resistance coefficient according to the loading capacity of a container ship, as shown in Equation (10), and is not more than 0.09.

$$C\_{AA} \cdot 1000 = 0.28 \cdot TEI^{-0.126} \text{ less than } 0.09 \tag{10}$$

**Table 5.** Factors for calculating the air resistance coefficient.



**Table 6.** Comparison of resistance coefficients with and without superstructure.

The Fujiwara formula [18], which is mainly used for resistance correction in sea trials, is shown in Equations (11)–(14). The value of each parameter used in the calculation is provided in Table 7; Table 8. Figure 10 shows the profile of the 8000 TEU-class container ship used to calculate the coefficient values.

$$\mathbf{C}\_{DA} = \mathbf{C}\_{LF}\cos\varphi\_{WR} + \mathbf{C}\_{XLI}\left(\sin\varphi\_{WR} - \frac{1}{2}\sin\varphi\_{WR}\cos^2\varphi\_{WR}\right)\sin\varphi\_{WR}\cos\varphi\_{WR} + \\ \mathbf{C}\_{ALF}\sin\varphi\_{WR}\cos^3\varphi\_{WR} \\ \tag{11}$$

$$\mathcal{C}\_{LF} = \beta\_{10} + \beta\_{11} \frac{A\_{YV}}{L\_{OA}B} + \beta\_{12} \frac{\mathcal{C}\_{M\mathcal{C}}}{L\_{OA}} \tag{12}$$

$$C\_{XLI} = \delta\_{10} + \delta\_{11} \frac{A\_{YV}}{L\_{OA} h\_{BR}} + \delta\_{12} \frac{A\_{XV}}{B h\_{BR}} \tag{13}$$

$$\mathbb{C}\_{ALF} = \varepsilon\_{10} + \varepsilon\_{11}\frac{A\_{OD}}{A\_{YV}} + \varepsilon\_{12}\frac{B}{L\_{OA}}\tag{14}$$

Here, *AOD* is the lateral projected area of the superstructures, *AXV* is the area of the maximum transverse section exposed to the wind, *AYV* is the projected lateral area above the waterline, *B* is the ship breadth, *LOA* is the overall length, *CMC* is the horizontal distance from the mid-ship section to the center of the lateral projected area, *hBR* is the height of the top of the superstructure, and ϕ*WR* is the relative wind direction (0 indicates the wind heading). The values of the non-dimensional parameters (β*ij*, δ*ij*, ε*ij*) are listed in Table 8.

The calculation results of *CDA* are as shown in Table 9. Here, *CAA* was calculated using the method proposed by Kristensen and Lützen [17], which is shown in Equation (10). The ITTC value was the counter-calculated value of *CDA*, using Equation (5). The value of 0.67 calculated by the Fujiwara formula was the same result as the *CDA* value of the 6800 TEU-class container ship, with containers in the laden condition, provided by ITTC [19]. The result indicated that ships with typical forms, such as a container ship, would show similar results.

The *CDA* value was 16% lesser with the Fujiwara formula and 10% lesser with the method proposed by Kristensen and Lützen [17] than the ITTC value of 0.8, which was the default value of *CDA*.

The results of estimating the total resistance coefficient by applying the *CDA* calculated by the respective methods are shown in Table 10 and Figure 11. All three methods over-estimated the resistance values when compared with the numerical simulations in the case where the superstructures were considered, but the quantitative differences were reduced by using a *CDA* value lower than the default value. For the Fujiwara formula, which used the lowest *CDA* value, the difference was approximately at a 4%lower speed of *FN* at 0.192 or below, but decreased to 2% or below at higher *FN*.

**Table 7.** Parameters for calculating the Fujiwara formula.


**Parameter** *<sup>i</sup> <sup>j</sup>* **01 2** β*ij* 1 0.922 −0.507 −1.162 2 −0.018 5.091 −10.367 δ*ij* 1 −0.458 −3.245 2.313 2 1.901 −12.727 −24.407 <sup>ε</sup>*ij* 1 0.585 0.906 <sup>−</sup>3.239 2 0.314 1.117 -

**Table 8.** Non-dimensional parameters for calculating the Fujiwara formula.

**Figure 10.** Schematic profile above the waterline for calculating the Fujiwara formula.



**Table 10.** The total resistance coefficient according to *CDA*.


**Figure 11.** Comparison of total resistance coefficient by *CDA*.

#### **5. Conclusions**

In this study, a numerical simulation was conducted on the 8000 TEU-class container ship to study the variation in resistance performance, according to the presence or absence of superstructures on a ship. Prior to the numerical simulation for the 8000 TEU-class container ship, numerical simulations using the KCS hull form were conducted to verify the numerical simulation conditions. The numerical simulation results of the KCS hull form for total resistance acting on the ship, showed a similar tendency as that observed for the experimental results, with a quantitative difference of approximately less than 3%. However, in the case of trim and sinkage, as excessive quantitative differences were observed at low and high speeds, numerical simulations for the 8000 TEU-class container ship was conducted at the *FN* range of 0.16–0.27. The results of the study are summarized below:
