*2.3. Covegence of Mesh Using for Computation*

In this research, the effect of mesh numbers generated in the computed domain on aerodynamic performance and wind drag acting on the original container ship has been investigated. The computed domain is shown in Figure 2, where twelve different mesh numbers have been used. The mesh numbers are from 0.05 to 21.52 million elements and value of y+ is taken from 1.08 to 502.49, respectively. The meshes have been used to compute the aerodynamic performances of the ship under the same conditions. Analyzing the obtained CFD results, the effect of mesh numbers on aerodynamic performances and wind drag as well as the state of independence of mesh numbers have been found. The computation was done in head wind condition at a wind velocity of 14 knot instead of a Reynolds number of 6.73 <sup>×</sup> 107. Table <sup>2</sup> shows a detailed mesh of the computed domain.




**Table 2.** *Cont.*

Tables 3 and 4 show detailed results of wind drag, and the interaction effect of the mesh number on wind drag acting on the ship. Figure 3 shows the curve of the effect of the mesh number on wind drag, as well as the state of the mesh number becoming independent of the wind drag acting on the ship obtained in the computation.

**Table 3.** Wind drag acting on the original container ship in the different mesh numbers.


**Table 4.** <sup>E</sup>ffect of mesh number on wind drag acting on the ship, at Re <sup>=</sup> 6.73 <sup>×</sup> <sup>10</sup>7, in head wind.


**Figure 3.** <sup>E</sup>ffect of mesh number on wind drag acting on the ship in head wind, at Re <sup>=</sup> 6.73 <sup>×</sup> <sup>10</sup>7.

Tables 3 and 4 clearly show wind drag acting on the original ship in the different computed mesh numbers, where the wind drag acting on the ship is defined by the following equation [27,28].

$$\mathbf{C}\_{\rm d} = \frac{\mathbf{R}\_{\rm x}}{0.5 \rho \mathbf{A}\_{\rm F(x)} \mathbf{V}^2} \tag{1}$$

where:

Cd is the wind drag coefficient. Rx is the wind drag acting on the hull, N. AF(x) is the frontal projected area, m2. V is the ship velocity, m/s.

The results given in the Tables 3 and 4 show that the wind drag acting on the ship decreases slowly with increasing mesh number, although the wind drag is slightly different when the y+ is less than 50 and retains the same value when the y+ less than 7. The effect of mesh number on wind drag acting on the ship decreases with the increasing of the mesh number. The difference of wind drag is around 5% when the y+ is less than 50 and the same value when y+ is less than 7. When the y+ increases to over 50 but continues to be less than 500, the difference in wind drag increases by up to 42%, as shown. This is important in studying aerodynamic performances using the CFD tool. Apart from paying attention to calculating the wind drag acting on the ship, it is very useful to get a clear pattern of pressure distribution around the hull to understand how the flow has an effect on the hull, as well as factors affecting the wind drag, and so on.

Figure 3 shows the effect of mesh number on wind drag acting on the ship as well as the mesh convergence curve obtained in computation. From the results shown, it can be seen that when the mesh number increases to over two million instead of a y+ of less than 50, the effect of the mesh number on total wind drag acting on the ship drastically reduces and becomes zero when the y+ is less than 7, as shown. The obtained result is useful in the application of CFD to investigate aerodynamic performances of the ship.

#### **3. Interaction E**ff**ect between Hull and Accommodation**

In this section, aerodynamic performances of the ship have been computed in two different cases, namely the hull with and without accommodation on its deck. Analyzing the obtained CFD results for the two cases, the interaction effect on aerodynamic performances of the hull and accommodation in

head wind has been obtained. Figure 4 shows the pressure distribution around the ship in the two computed cases at a Reynolds number of 6.73 <sup>×</sup> 107 in head wind.

**Figure 4.** Pressure coefficient distribution over the hull surface of the model in the two computed cases at a Reynolds number of 6.73 <sup>×</sup> 107, in head wind.

Analyzing the results as shown in Figure 4, the red and yellow regions show high pressure and the blue region shows lower pressure acting on the model. Hence, the effect of hull and accommodation on pressure distribution over the hull surface of the ship could be evaluated. The interaction effect between hull and accommodation may also be determined by comparison of the wind drag acting on the hull and accommodation in both of the computed cases, as shown.

In this research, interaction effect between hull and accommodation on wind drag acting on the ship has been determined by the following Equation (2) [12–14]:

$$
\Delta \text{C}\_{\text{d} \prime} \,\%= \frac{\text{C}\_{\text{d}}(\text{Hull with Acc}) - \text{C}\_{\text{d}}(\text{Independence Hull and Acc})}{\text{C}\_{\text{d}}(\text{Independence Hull and Acc})} \,\,100\% \tag{2}
$$

where:

ΔCd, % is the interaction effect between hull and accommodation.

Cd (Hull with Acc) is wind drag coefficient acting on the hull with accommodation.

Cd (Independent Hull and Acc) is total wind drag coefficient acting on hull without accommodation and independent accommodation.

Tables 5–7 show, in detail, the wind drag acting on the hull, the accommodation, and the interaction effect between the hull and accommodation on the wind drag of the ship in head wind.

Analyzing the results shown in the Tables above, the wind drag acting on hull part and accommodation part in the case of the hull with accommodation is less than that of the case of accommodation independent of hull, by up to 42%. The total wind drag acting on the ship in the case of hull with accommodation is also less than that of the independent hull by up to 10%. For the wind drag acting on the hull part and the accommodation part, the interaction effect is about 42% for the hull part and 4% for the accommodation part. The interaction effect between the hull and accommodation on wind drag is around 10%, as shown.


**Table 5.** Wind drag acting on the hull with accommodation.

**Table 6.** Wind drag acting on an independent hull and independent accommodation.


**Table 7.** Interaction effect between hull and accommodation on wind drag.


#### **4. Reduced Interaction E**ff**ect on Wind Drag Acting on the Container Ship**

#### *4.1. Proposed Accommodation Shape for the Ship*

The original container ship has been designed with an accommodation located at the aft of the ship. In this section, three new hull forms with an accommodation located at the frontal part of ship have been proposed. Three models with frontal accommodation, named N1, N2 and N3, have been used for computation to determine the interaction effect on aerodynamic performances of the proposed streamlined hull shapes. The dimensions of the models are the same ones as those of the original ship shown in Table 1. The computation for all ships have also been done under the same conditions as shown in the Section 2. Figure 5 shows the proposed new models N1, N2 and N3.

**Figure 5.** Newly proposed shapes for the ship with frontal accommodation N1, N2 and N3.

In the proposed models, the N1 has the same accommodation shape of the original ship but is located at frontal hull, the N2 has a streamlined bow cover, and the N3 has a streamlined accommodation shape and it is located at frontal hull. Computation has been done for all the models under the same conditions to investigate the aerodynamic performances of the ships.

#### *4.2. Reduced Interaction E*ff*ect of the New Hull Shapes on Pressure Distribution*

In this section, the effect of the streamlined hull shapes on aerodynamic performances of the proposed container ships have been investigated by the CFD. Figures 6–11 show pressure distribution around and over the hull surface of the ships. Analyzing the obtained CFD results of pressure distribution around and over hull surfaces of the ships, the reasons for the increasing interaction effect between hull and accommodation have been clearly found.

**Figure 6.** Dynamic pressure coefficient distribution around the ships at the center plane (y/L = 0) in head wind, at Re <sup>=</sup> 6.73 <sup>×</sup> 107.

In the above results, the blue region shows low dynamic pressure and, consequently, high static pressure acting on the ship. The results, as shown in Figure 6, clearly show reduced separation flow at the lower dynamic pressure region (blue color region) around the streamlined hull shape N3. Figure 7 shows dynamic pressure distribution around the ships at several horizontal planes of the computed domain.

The results, as shown in Figure 7, clearly show the effects of the accommodation shapes on the dynamic pressure distribution around the ships. The separation flow regions at the low dynamic pressure regions (blue color regions) at the aft of the accommodation N2 and N3 have been drastically reduced, as shown. Figure 8 shows the dynamic pressure distribution around several cross sections of the computed domain of the ships.

The results obtained in the above figures clearly show differences of dynamic pressure distribution around the ships. A drastic reduction in low dynamic pressure region (blue color) could be seen around the hull shapes of N2 and N3, as shown. These results clearly show the reduced interaction effect on ship aerodynamic performance of streamlined accommodation shapes which are located at frontal hull of the ship. There is a reason why a suitable frontal accommodation should be installed to reduce wind drag acting on the ship hull.

**Figure 7.** Dynamic pressure coefficient distribution around the ships at the horizontal plane in head wind, at Re <sup>=</sup> 6.73 <sup>×</sup> <sup>10</sup>7. (**a**) z/<sup>L</sup> <sup>=</sup> 0.08 and (**b**) z/<sup>L</sup> <sup>=</sup> 0.11.

**Figure 8.** Dynamic pressure coefficient distribution around the ships at various cross sections of the computed domain in head wind, at Re <sup>=</sup> 6.73 <sup>×</sup> <sup>10</sup>7.

**Figure 9.** Wind drag acting on the ships with different frontal accommodation shapes in head wind.

**Figure 10.** Different wind drag coefficients acting on the ships due to the developed frontal streamlined accommodations.

**Figure 11.** Comparison of interaction effect on wind drag acting on the ship with different frontal accommodation shapes.
