*4.1. Verification Study*

A verification study is carried out in order to estimate sufficient grid spacings and adequate time steps. This study is carried out using three different meshes and three different time steps. Verification study for grid size is made with fine time step and verification study for time step is made with fine mesh. Thereafter, numerical uncertainty, which is consisted of both spatial and temporal uncertainties, is calculated using the grid convergence index (GCI) method. This method is recommended by the American Society of Mechanical Engineers, as well as by the American Institute of Aeronautics and Astronautics for the assessment of grid uncertainty (*UG*) [35], but can be used for the assessment of temporal uncertainty (*UT*) as well [35–37]. More details regarding the GCI method and numerical uncertainty can be found in [18].

For the purposes of verification study three meshes are generated for smooth surface condition and fouling condition R1. Since all mesh parameters, except prism layer mesh, are set to be relative to cell base size, mesh is refined by changing cell base size. It should be noted that all remaining CFD simulations, i.e., for the fouling conditions R2, R3, R4, R5 and R6 are performed using fine mesh. In Table 4, the number of cells used in the verification study is shown. Three different time steps, i.e., *T*/50, *T*/100 and *T*/200 are used in the verification study for time step.


**Table 4.** Number of cells within CFD simulations.

It should be noted that the verification study for CFD simulations of resistance tests of KCS and KVLCC2 is carried out in [16]. Numerical uncertainties in the prediction of *RF* and *RV* consisted of grid uncertainties solely, and *RT* consisted of grid and temporal uncertainties, which are calculated using the GCI method. The obtained numerical uncertainties in the prediction of *RF* were below 1.3% for both ships and for all analyzed fouling conditions (Table 1). Numerical uncertainties in the prediction of *RV* were slightly higher, however, the highest obtained numerical uncertainty was equal to 2.86%. Finally, the highest numerical uncertainties are obtained for the prediction of *RT*. Nevertheless, these grid and time step uncertainties were relatively low, i.e., the highest obtained grid uncertainty in the

prediction of *RT* was equal to 2.99%, while the highest time step uncertainty in the prediction of *RT* was equal to 0.1%. Within this paper, the numerical uncertainty in the prediction of *KTO* and 10*KQO* from CFD simulations of OWT are calculated for one *J* value and the obtained results are presented in Tables 5 and 6. Additionally, numerical uncertainty in the prediction of *PD*, *n*, *T* and *J* from CFD simulations of SPT are calculated.


**Table 5.** The verification study for *KTO*.

**Table 6.** The verification study for *KQO*.


As can be seen from Tables 5 and 6, relatively low numerical uncertainties are obtained, and are in line with numerical uncertainties of other CFD studies regarding open water tests [38,39]. Thus, the highest *UG* in the prediction of *KTO* and 10*KQO* is obtained for the WB propeller with smooth surface condition, and it is equal to 3.565% and 2.815%, respectively. It should be noted that numerical uncertainties obtained for smooth and fouled propellers are relatively close, i.e., numerical uncertainty has not raised due to the roughness effects.

From the results of verification study of SPT, Tables 7–9, it can be concluded that *UT* are lower than *UG*. Generally, the obtained *UG* related to the prediction of *PD* for smooth and fouled ships are slightly higher than for the other investigated key variables and the highest *GCI*<sup>21</sup> *fine* for KCS is equal to 3.123%, for KVLCC2 is equal to 1.174% and for BC is equal to 7.318%. The obtained *UT* related to the prediction of *PD* for smooth and fouled ships are lower and the highest *GCI*<sup>21</sup> *fine* for KCS is equal to 1.366%, for KVLCC2 is equal to 1.502% and for BC is equal to 3.390%. The obtained *UG* related to the prediction of *n* for smooth and fouled ships are the lowest amongst investigated key variables and the highest *GCI*<sup>21</sup> *fine* for KCS is equal to 0.255%, for KVLCC2 is equal to 0.164% and for BC is equal to 1.661%. Interestingly, the obtained *UT* values related to the prediction of *n* for smooth and fouled ships are higher than *UG* values and the highest *UT* for KCS is equal to 0.401%, for KVLCC2 is equal to 0.701% and for BC is equal to 2.909%. The obtained *UG* values related to the prediction of *T* for smooth and fouled ships are low and the highest *GCI*<sup>21</sup> *fine* for KCS is equal to 3.273%, for KVLCC2 is equal to 1.478% and for BC is equal to 4.717%. The obtained *UT* values related to the prediction of *T* for smooth and fouled ships are lower or similar to *UG* and the highest *GCI*<sup>21</sup> *fine* for KCS is equal to 0.807%, for KVLCC2 is equal to 1.529% and for BC is equal to 3.499%. Finally, the obtained *UG* values related to the prediction of *J* for smooth and fouled ships are low and the highest *GCI*<sup>21</sup> *fine* for KCS is equal to 0.452%, for KVLCC2 is equal to 1.257% and for BC is equal to 2.041%. The obtained *UT* values related to the prediction of *J* for smooth and fouled ships are low as well, and the highest *GCI*<sup>21</sup> *fine* for KCS is equal to 0.451%, for KVLCC2 is equal to 0.703% and for BC is equal to 2.719%.


**Table 7.** The obtained grid uncertainties in the prediction of *PD*, *n*, *T* and *J*.

The obtained *UT*, *UQ*, *Un*, *UPD* and *UJ*, which consist of both *UG* and *UT*, are shown in Table 9. As can be seen from Table 9, the lowest *USN* values for smooth and fouled ships are obtained for KCS, which was expected, since *UG* values are higher than *UT* values and the mesh for KCS had more cells than for KVLCC2 and BC. The highest *USN* is obtained for the prediction of *UPD* for BC fouled with R1 and it is equal to 7.421% and other obtained *USN* values are lower than 5.5%. Higher *UPD* were expected, since, for the prediction of *PD*, both *n* and the propeller torque should be determined. It should be noted that the obtained *UPD* are in line with the previously published studies [8,25]. From Table 9, it can be seen that higher numerical uncertainties are obtained for the prediction of *PD* and *T*, than for *n* and *J*, which was also obtained in [8]. Additionally, it can be seen that *USN* in the prediction of key variables for R1 are mostly below *USN* for smooth surface condition. Higher *USN* obtained for R1 than for smooth surface condition can be ascribed to the lower cell number used in CFD simulations of SPT for rough surface condition (Table 4). Therefore, it can be concluded that the implementation of Δ*U*<sup>+</sup> within the wall function did not cause higher uncertainties in the prediction of the key variables.


**Table 8.** The obtained temporal uncertainties in the prediction of *PD*, *n*, *T* and *J*.

**Table 9.** The obtained simulation uncertainties (*USN*) in the prediction of *PD* (*UPD* ), *n* (*Un*), *T* (*UT*) and *J UJ* .


### *4.2. Validation Study*

Relative deviations between numerically obtained and extrapolated results are calculated using the following equation:

$$RD = \frac{\phi\_{\text{CFD}} - \phi\_{\text{EX}}}{\phi\_{\text{EX}}} \cdot 100\% \tag{15}$$

where φCFD is the certain hydrodynamic characteristic obtained using CFD and φEX is the certain hydrodynamic characteristic obtained using the ITTC 1978 Performance Prediction Method (PPM) and experimental results [30].

The obtained *CT* for full-scale KCS and KVLCC2 is validated within [16] through comparison of the obtained numerical results with extrapolated values using original ITTC 1978 PPM, based on Equation (9). Within ITTC 1978 PPM, *CF* is determined using the ITTC 1957 model-ship correlation line. In Table 10, the validation of the numerically obtained *CT* for the smooth surface condition is presented. As can be seen from Table 10, the obtained results are in satisfactory agreement with the extrapolated results, i.e., the highest *RD* is obtained for BC and it is equal to −4.338%.

**Table 10.** The validation study for *CT*.


The numerically obtained open water characteristics for all three propellers have been validated, with the towing tank results published in the literature [29,31,32]. It should be noted that CFD simulations of OWT are performed in full-scale, while experimental OWT are performed in model scale. Towing tank tests for all three investigated propellers are performed at *Rn* above *Rn* = 2 · 105, as prescribed by ITTC [30]. In Figure 7, the comparison between the numerically and experimentally obtained open water characteristics is presented. From this figure, it can be seen that numerically obtained *KTO*, 10*KQO* and η*<sup>O</sup>* are in satisfactory agreement with the experimentally obtained ones. Slightly higher *RD* between numerically and experimentally obtained *KTO* and especially 10*KQO* is obtained at lower *J* values, however, at higher *J* values, these *RD* are significantly lower.

The obtained results of the validation study for *PD* and *n* are presented in Table 11, from which it can be concluded that satisfactory agreement is obtained. The highest obtained *RD* between numerical and extrapolated *PD* is obtained for KVLCC2 and it is equal to −5.701%, while the highest obtained *RD* for *n* is obtained for BC and it is equal to −1.786%. The validation study for ship propulsion characteristics is presented in Table 12. From Table 12, it can be seen that the obtained *RD* for 1 − *t* are lower than 3.7%, for 1 − *w* are lower than 7.4% and for η*<sup>H</sup>* are lower than 5.6% for all analyzed ships. It should be noted that slightly higher *RD* for 1 − *w* is obtained only for BC, and this can be attributed to the application of body force method. However, this *RD* is in line with previously published studies dealing with CFD simulations of SPT where the virtual disk model is applied [40,41]. The obtained *RD* for η*<sup>O</sup>* are lower than 3.1%, for propeller efficiency behind ship (η*B*) are lower than 3.8%, for η*<sup>R</sup>* are lower than 2.9% and for η*<sup>D</sup>* is lower than 6.2%. It should be noted that slightly higher *RD* for η*<sup>D</sup>* is obtained only for KCS. However, in [42] where the authors carried out full-scale SPT for KCS using discretized propeller, η*<sup>D</sup>* was equal to 0.766, which is also lower than the extrapolated result. From this result, the obtained η*<sup>D</sup>* in this paper has *RD* equal to −3.394%. In Table 12, the validation for the obtained *J*, *KT* and *KQ* for self-propulsion point is shown as well. It can be seen that the obtained *RD* for *J* are lower than 5.7%, for *KT* are lower than 4.1% and for *KQ* are lower than 3.4% for all analyzed ships. Generally, the obtained *RD* presented in Tables 11 and 12 can be ascribed to different reasons. For example, insufficiently precise assessment of the nominal wake, as well as the

propeller performance in OWT can be related to the inaccurate assessment of *J* for self-propulsion point, which then leads to inaccurate assessment of other propulsion characteristics. In addition to this, the modelling error should also be taken into account, as, in the body force method, the effect of propeller is modelled, rather than propeller itself. Furthermore, there is a numerical error as well, which is related to the applied mesh and time step. Lastly, there are also aspects regarding the applied PPM for the extrapolation of towing tank results. Namely, in [25] four different PPM are compared, and it was shown that extrapolated values can significantly vary with respect to the applied PPM. Thus, it was shown that for BC, extrapolated value of *PD* can vary up to 1.5%, for *n* up to 0.4%, for 1 − *t* up to 0.5%, for 1 − *w* up to 6.3%, for η*<sup>R</sup>* up to 1.1% and for η*<sup>B</sup>* up to 2.6%. In addition to these variations, experimental uncertainty should also be considered. Considering all above mentioned aspects, it can be concluded that satisfactory agreement is achieved for *PD*, *n* and all propulsion characteristics.

**Figure 7.** The validation study for open water characteristics of KP505 (**upper**), KP458 (**middle**) and WB (**lower**).

**Table 11.** The validation study for self-propulsion point.



**Table 12.** The validation study for propulsion characteristics.

#### **5. The Impact of Hard Fouling on the Ship Performance**

Within this section, the impact of hard fouling on the resistance, open water and propulsion characteristics is presented for three investigated ships. While detail investigation of the impact of hard fouling on resistance characteristics for KCS and KVLCC2 is presented in [16], within this study this impact is only briefly mentioned as emphasis is given to the impact of hard fouling on the ship performance, which is defined by propeller operating point.

#### *5.1. The Impact of Hard Fouling on Resistance Characteristics*

As demonstrated within [16,18] the impact of biofouling on each resistance component is different. Thus, the presence of biofouling causes the increase in *CF*, decrease in *CW*, while the impact of biofouling on 1 + *k* value is almost negligible. Consequently, it is valuable to study the increase in *RT*, due to the presence of hard fouling through analysis of decomposed *RT* and the portion of each resistance component in *RT* for certain fouling condition. In Figure 8, decomposition of *RT* for three investigated ships and fouling conditions is presented. Additionally, within Figure 8 the portions of *RF*, *RVP* and *RW* in *RT* are given. From Figure 8, it is clear that, for all analyzed ships, the portion of *RF* in *RT* increases, due to the presence of hard fouling, and this increase is the highest for KCS, which can be attributed to the ship speed. Namely, KCS is investigated at the highest speed and therefore *u*τ values along the KCS hull are higher than *u*<sup>τ</sup> values along the KVLCC2 and BC hulls. Since *k*<sup>+</sup> values and consequently Δ*U*<sup>+</sup> values for given fouling condition and fluid properties depend only on *u*τ values, those values are higher for KCS than for KVLCC2 and BC resulting in higher increases in *CF* [16]. Additionally, *CF* for rough surface condition at high *Rn* value depends solely on *k*/*L* value, i.e., relative roughness [16]. The portion of *RVP* in *RT* due to the presence of hard fouling has increased for KCS and BC, while for KVLCC2 this portion has decreased. Regardless of this, from Figure 8, it is clear that the absolute value of *RVP*, due to the presence of hard fouling, has increased, which is expected, since the impact

of biofouling on 1 + *k* value is minimal [16]. Finally, the portion of *RW* in *RT* due to the presence of hard fouling decreases for all analyzed ships and this decrease is the highest for KCS, which can be also attributed to ship speed. What is more, from Figure 8 it is clear that absolute values of *RW* due to the presence of hard fouling have decreased for all analyzed ships [16]. Generally, KVLCC2 is the most affected, due to the presence of hard fouling in terms of the increase in *RT*, which can be seen from Figure 9. Thus, the increase in *RT* due to the presence of hard fouling for KVLCC2 ranges from 63.8% (R6) to 120.9% (R1), for BC ranges from 59.5% (R6) to 114.6% (R1) and for KCS ranges from 49.9% (R6) to 95.8% (R1). This can be mostly attributed to the portion of *RV* in *RT*, since, due to the presence of biofouling *RV*, significantly increases. The portion of *RV* in *RT* is the highest for KVLCC2 and for smooth surface condition this portion is equal to 99.46%, as *RW* of KVLCC2 is negligible [28]. However, beside the portion of *RV* in *RT*, the ship speed also affects the increase in *RT*, as already explained. Thus, the increase in *RT* due to the presence of hard fouling is only slightly lower for BC than for KVLCC2 and the portion of *RV* in *RT* for smooth surface condition is equal to 83.6%. It should be noted that the significantly lower increase in *RT* is obtained for KCS, as KCS has relatively large portion of *RW* in *RT* (for smooth surface condition this portion is equal to 24.7%). Due to the presence of hard fouling, *RW* decreases, and, therefore, the increase in *RT* for KCS is lower.

**Figure 8.** Decomposition of *RT* for KCS (**upper**), KVLCC2 (**middle**) and BC (**lower**) for smooth and fouled surface condition.

**Figure 9.** The impact of hard fouling on *CT*.
