**3. Results**

#### *3.1. Verification Study*

Spatial and temporal verification studies were performed to estimate the numerical uncertainties of the CFD model. The Wigley hull CFD simulations were conducted using three different resolutions of grids and time steps (i.e., Fine, Medium and Coarse), at *Fr* = 0.3 with the smooth hull condition. The Grid Convergence Index (GCI) method [33] was used to determine the spatial and temporal uncertainties (*U*Grid and *U*Δt) in the total resistance coefficient, *C*T, predictions as similarly used by other recent studies.

Table 2 shows the *U*Grid and *U*Δ<sup>t</sup> values estimated from the convergence studies. As shown in the Table 2, *U*Grid and *U*Δ<sup>t</sup> for the Wigley hull simulation are 0.053% and 0.022%, respectively, resulting in *U*Total of 0.057%. It is of note that the following simulation results were obtained using the fine mesh and fine time step.


**Table 2.** Spatial and temporal convergence study of the Wigley hull simulation, *Fr* = 0.3, smooth hull.

#### *3.2. Effect of Heterogeneous Roughness on Ship Resistance*

The Wigley hull CFD simulations were performed with various hull conditions at the speed range of *Fr* = 0.2 − 0.4, with the corresponding Reynolds numbers of *ReL* = 2.6 − 5.3 × 106. Figures 7–9 compare the total resistance coefficient, *CT*, of the Wigley hull with the different hull roughness conditions obtained from the current CFD simulations and the Experimental Fluid Dynamics (EFD) results of Song et al. [27]. The *CT* values were calculated by as

$$C\_T = \frac{R\_T}{\frac{1}{2}\rho S V^2} \tag{8}$$

where *RT* is total resistance, *ρ* is the density of water, *S* is the wetted surface area, and *V* is the towing speed (i.e., inlet velocity). As can be seen in Figure 7, the current CFD result agrees well with the experimental data of Song et al. [27]. This confirms the validity of the modified wall-function approach as previously demonstrated by Song et al. [25].

Figures 8 and 9 show the *CT* values of the Wigley hull with the heterogeneous hull roughness conditions (i.e., <sup>1</sup> <sup>4</sup> -bow-rough, <sup>1</sup> <sup>4</sup> -aft-rough, <sup>1</sup> <sup>2</sup> -bow-rough and <sup>1</sup> <sup>2</sup> -aft-rough). As observed from the physical towing tests of Song et al. [27], the current CFD simulations predicted larger *CT* values for the bow-rough conditions ( <sup>1</sup> <sup>4</sup> -bow-rough and <sup>1</sup> <sup>2</sup> -bow-rough) than the aft-rough conditions ( <sup>1</sup> <sup>4</sup> -aft-rough and <sup>1</sup> <sup>2</sup> -aft-rough). The percentage differences between the CFD and EFD results can be found from Table A1 in Appendix A.

Figures 10 and 11 compare the frictional and residuary resistance coefficients, *CF* and *CR*, with the different hull conditions, respectively. The frictional and residuary resistance were calculated by simply decomposing the total drag into the shear and pressure components. The *CF* and *CR* were calculated as

$$C\_F = \frac{R\_F}{\frac{1}{2}\rho S V^2} \tag{9}$$

$$C\_R = \frac{R\_R}{\frac{1}{2}\rho S V^2} \tag{10}$$

where, *RF* and *RR* are the frictional (shear) and residuary (pressure) resistance, respectively.

**Figure 7.** *CT* of the Wigley hull with smooth and full-rough conditions obtained from the current Computational Fluid Dynamics (CFD) simulations and the Experimental Fluid Dynamics (EFD) result [27].

**Figure 8.** *CT* of the Wigley hull with <sup>1</sup> <sup>4</sup> -bow-rough and <sup>1</sup> <sup>4</sup> -aft-rough conditions obtained from the current CFD simulations and the EFD result [27].

**Figure 9.** *CT* of the Wigley hull with <sup>1</sup> <sup>2</sup> -bow-rough and <sup>1</sup> <sup>2</sup> -aft-rough conditions obtained from the current CFD simulations and the EFD result [27].

**Figure 10.** *CF* of the Wigley hull with different hull conditions predicted from the current CFD simulations.

**Figure 11.** *CR* of the Wigley hull with different hull conditions predicted from the current CFD simulations.

As shown in Figures 10 and 11, the effect of different hull conditions on the *CF* values is apparent as expected, while the effect on the *CR* value is negligible. As expected, in Figure 10, the *CF* values of the smooth case show a descending trend while the *CF* values of the full-rough case show an ascending trend, which implies that the flow of each case is within the hydraulically smooth and transitionally rough flow regimes, respectively.

The bow-rough conditions show larger *CF* values than the aft-rough conditions with the same area of the rough surface. Accordingly, the differences in the added resistance between the bow-rough and aft-rough conditions can be mainly attributed to the different effects on the frictional resistance of the ship. Therefore, it is worthwhile to examine the effect of heterogeneous hull roughness on the distributions of the local skin friction on the hull.

Furthermore, in Figure 10, the *CF* trends of the different hull conditions show different transition behaviours in terms of roughness flow regimes. In other words, the bow-rough cases show more developed flow features at the same speed range than the aft-rough cases. For example, the *CF* values of the <sup>1</sup> <sup>4</sup> -bow-rough case converge when *Fr* > 0.3 (i.e., the fully rough regime is reached), while those of the <sup>1</sup> <sup>4</sup> -aft-rough case keep increasing (i.e., still within transitionally rough regime). For a similar reason, *CF* values of the full-rough case

keep increasing, although its forepart is expected to reach the fully rough regime because its aft part is still within the transitionally rough regime. The locally different flow regimes on the hull can be further correlated with the roughness Reynolds number, *k*+, on the hull.

#### *3.3. Rationale behind the Effect of Heterogeneous Roughness*

As discussed in the previous section, the effect of heterogeneous hull roughness on ship resistance is believed to be closely related to the distributions of the local skin friction and the roughness Reynolds number. Therefore, this section discusses and compares the local skin friction, and the roughness Reynolds numbers with different hull conditions.

Figure 12 compares the local skin friction, *Cf* , values on the Wigley hull with different hull conditions. The local skin friction was obtained by dividing the wall shear stress, *τw*, by the dynamic pressure, <sup>1</sup> <sup>2</sup> *<sup>ρ</sup>V*2, where *<sup>ρ</sup>* is water density and *<sup>V</sup>* is the towing speed (i.e., inlet velocity). As shown in Figure 12, significant increases in the local *Cf* due to the roughness effect were observed. In the case of the homogeneous conditions (smooth and full-rough), the highest local *Cf* values are observed in the first quarter of the hull. The heterogeneous hull conditions (Figure 12b–e) showed blended *Cf* distributions, where the smooth surfaces show similar *Cf* distributions as the smooth condition, while the rough surfaces show those similar to the full-rough condition. For example, the first quartile of the <sup>1</sup> <sup>4</sup> -bow-rough case (Figure 12b) has a similar *Cf* distribution as that of the full-rough case, while the rest of the hull has a similar *Cf* distribution as that of the smooth case. As the full-rough condition has higher values in the bow region, the increase in the *Cf* values of the bow-rough cases (Figure 12b,d) are more apparent compared to the aft-rough cases.

**Figure 12.** *Cf* distribution on the Wigley hull with different hull conditions, *Fr* = 0.3.

Figure 13 clearly shows the increase in the *Cf* values due to the presented hull roughness (i.e., Δ*Cf* = *Cf* , rough − *Cf* , smooth). The full-rough case shows greater Δ*Cf* in the bow region, and thus the bow-rough conditions show larger Δ*Cf* values compared to the aft-rough conditions. The locally different Δ*Cf* values suggest different roughness effects in the different regions, and it can be best attributed to the different roughness Reynolds numbers, *k*+, in the local regions.

**Figure 13.** Increase in the *Cf* on the Wigley hull (Δ*Cf* = *Cf* , rough − *Cf* , smooth), *Fr* = 0.3.

Figure 14 shows the distributions of the roughness Reynolds number, *k*+, on the Wigley hull with different hull roughness conditions. As expected, the full-rough case shows larger *k*<sup>+</sup> values in the bow region due to higher local skin friction (i.e., *k*<sup>+</sup> = *kτw*/*ν*). For a similar reason, the bow-rough cases show larger *k*<sup>+</sup> values compared to the aft-rough cases, and these differences result in different Δ*Cf* values. This observation supports the hypothesis of Song et al. [27].

**Figure 14.** *k*<sup>+</sup> distribution on the Wigley hull with different hull conditions, *Fr* = 0.3.

The observations in Figures 12–14, with regards to *Cf* and *k*+, are in correspondence with the effect of different heterogeneous hull roughness on ship resistance shown in Figures 7–10. In other words, it can be seen that the greater increases in the *Cf* and *k*<sup>+</sup> of the bow-rough cases resulted in the greater added resistances compared to the aft-rough cases as shown in Figures 7–10.

Furthermore, the *k*<sup>+</sup> values in Figure 14 can be also correlated with the different trends of the *CF* with different hull conditions in Figure 10. As shown in Equation (6), when *k*<sup>+</sup> value is higher than 25, it is considered that the fully rough flow regime is reached. Therefore, for example, it can be seen that the fully rough flow regime is reached for most of the rough wetted surface of the <sup>1</sup> <sup>4</sup> -bow-rough condition (Figure 10), while the transitionally rough flow regime is expected for most of the wetted surface of the <sup>1</sup> <sup>4</sup> -aft-rough condition (Figure 10).

Figure 15 shows the boundary layers represented by the axial velocity contours limited to *Vx*/*Vship* = 0.9. When it comes to the homogeneous hull conditions, the results were as expected. The full-rough case (Figure 15b) shows a thicker boundary layer compared to the smooth case (Figure 15a) and the difference becomes more apparent along with the flow, as similarly observed from previous studies, e.g., [17,18,25]. On the other hand, differences were observed with the heterogeneous hull conditions. As shown in Figures 14f and 15d, the boundary layer thicknesses around the forward part of the aft-rough conditions (where the surface is smooth) showed almost no differences compared to that of the smooth case (Figure 15a). In contrast, the bow-rough conditions (Figure 15c,e) showed increases in the boundary layer thickness not only around the forward parts (where the surface is rough) but also around the aft parts (where the surface is smooth), compared to the smooth case (Figure 15a). Interestingly, the bow-rough conditions showed thicker boundary layers on the aft parts compared to the aft-rough conditions. For example, the <sup>1</sup> <sup>2</sup> -bow-rough condition (Figure 15e) shows a thicker boundary layer than the <sup>1</sup> <sup>2</sup> -aft-rough condition (Figure 15f) even around the aft part.

**Figure 15.** Boundary layer represented by slices limited to axial velocity (*Vx*/*Vship* = 0.9), *Fr* = 0.3.

#### **4. Concluding Remarks**

A numerical investigation was completed on the effect of heterogeneous hull roughness on ship resistance. A URANS-based CFD model was developed to investigate the effect of heterogeneous hull roughness using the modified wall-function approach. The predicted total resistance coefficients with different hull conditions were compared with the experiment of Song et al. [27] and showed a good agreement. As similarly observed

by Song et al. [27], the bow-rough conditions showed larger added resistance compared to the aft-rough conditions with the same wetted surface area of the roughness region, confirming that the hull roughness of the fore part of the ship has a greater impact on the results than the hull roughness in other regions.

The observations on the effects of heterogeneous hull roughness were correlated with the distributions of the local wall shear stress and the roughness Reynolds number. The results showed that the local differences in the wall shear stress result in different roughness Reynolds numbers and thus different roughness effects depending on the locations of the hull roughness. Therefore, the hypothesis of Song et al. [27] was confirmed in this study.

This study provides a numerical investigation into the effect of heterogeneous hull roughness using the modified wall-function approach. The results can be useful from an industrial point of view, since they give insight into different priorities of partial hull cleaning depending on the impact of the roughness in different hull regions.

The investigation was carried out in model-scale using idealised surface conditions. However, the same methodology can be extended to incorporate real hull conditions of ships where heterogeneous biofouling accumulations are present. Furthermore, the numerical approach presented in this study can also be adopted for predicting the effect of heterogeneous roughness on propellers.

**Author Contributions:** Conceptualisation, S.S. and Y.K.D. methodology, S.S. and Y.K.D.; software, S.S.; validation, S.S.; formal analysis, S.S. and Y.K.D.; investigation, S.S. and Y.K.D.; resources, C.D.M.M.-F. and A.I.; data curation, S.S.; writing—original draft preparation, S.S.; writing—review and editing, S.S., Y.K.D., C.D.M.M.-F., T.S., D.V., T.T., A.I.; visualisation, S.S.; supervision, Y.K.D., C.D.M.M.-F., T.S., D.V., T.T. and A.I.; project administration, C.D.M.M.-F.; funding acquisition, C.D.M.M.-F. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors gratefully acknowledge that the research presented in this paper was carried out as part of the EU funded H2020 project, VENTuRE (grant no. 856887).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Acknowledgments:** Results were obtained using ARCHIE-WeSt High Performance Computer (www.archie-west.ac.uk).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**


