5.1.2. Variation of Refinement Region

Using the overlapping grid technique, the single block grid describing the flow domain can be effectively refined as explained in Section 3. As shown in Figure 5, three geometrically similar refinement regions (R1, R2, and R3) are determined for all test cases. The refinements are applied to the g2 grid of each test case presented in Section 5.1. The starting and ending boundaries of the refinement regions are positioned at the same longitudinal position (with respect to *L*PP of each hull) relative to the aft perpendicular. The smallest refinement limits in the circumferential and normal directions are determined with respect to the propeller diameter to be able to encapsulate the *nominal wake*, i.e., flow field at the propeller plane without the presence of the propeller.

**Figure 4.** The histogram of the acquired y<sup>+</sup> values for the H3 hull for the average (**a**) y<sup>+</sup> = 0.11, (**b**) y<sup>+</sup> = 0.26, (**c**) y<sup>+</sup> = 0.50, (**d**) y<sup>+</sup> = 0.73, (**e**) y<sup>+</sup> = 0.95 and (**f**) y<sup>+</sup> = 1.36.

**Figure 5.** Systematic refinement variation in three steps for R1, R2, and R3 regions.

As observed in Figure 6, the comparison error of the form factor remains relatively unchanged between the grids without refinement (computations with 7M grid cells) and the grids with different refinement regions (shown in Figure 5). The regression lines are fitted to computations of each test case. As seen in Figure 6, the steepest line belongs to the H4 hull where a mild flow separation at the bilge of the gondola (i.e., lower part of the stern bulb) is observed. In this case, the varying refinement regions had a somewhat noticeable effect on the local flow as well as the computed resistance components. However, the variations in *C*F, *C*PV and *C*<sup>V</sup> are well within the grid uncertainties reported in Table 1 for all hulls. Therefore, it is concluded that, for the purpose of obtaining CFD based form factors, g2 (the baseline) is fine enough for all test cases, and no further grid refinements are required.

**Figure 6.** Comparison error of all test cases for the refinement region variation.

#### 5.1.3. Domain Size Variation

The size of the domain was varied from the default settings where the distance between inlet and fore-perpendicular (FP) is 0.5LPP (denoted as U/LPP), outlet plane is located at 0.8LPP behind the aft-perpendicular (denoted as W/LPP), and the radius of the cylindrical outer boundary is 3LPP (denoted as N/LPP). The default domain size is increased in two steps by changing the distance of only one part (either U/LPP, W/LPP or N/LPP) and keeping the rest the same. When a part of the domain is changed (for example U/LPP), the number of cells are also changed with the same ratio in that region (e.g., number of cells are doubled when the distance is doubled), while the rest of the domain (in this example W/LPP or N/LPP) parameters and the grid cells within the coverage of the unchanged domain are kept identical. In this way, errors due to the discretization is aimed to be kept similar to the baseline grid (g2).

The distance between the outlet plane and the aft-perpendicular (W/LPP) is varied between 0.8 and 2.4. The variations of CF due to changing W/LPP are smaller than 0.03% which is more than one order smaller than the numerical uncertainties shown in Table 1. The calculated viscous pressure resistance component increased up to 2% with respect to the base line. Investigations in the modified part of the flow domain showed that the traces of the wake of the hulls are reaching the outlet boundary in most of the test cases. However, when the outlet boundary moved further away from the hull (W/LPP = 1.6 and 2.4), the wake almost completely dissipated and diffused into the free stream. The difference in CV for all test cases varied between 0.1 to 0.4%, and, therefore, it is concluded that the Neumann boundary condition at the outlet boundary worked satisfactorily to handle the wake reaching the outlet plane. The resulting changes can be seen in Figure 7b where the predicted form factors are rather insensitive to the W/LPP change regardless of the turbulence model used.

The radius of the cylindrical domain varied between N/LPP = 3 to 5. Similar to the W/LPP variations, the change in the frictional resistance component is limited (up to 0.2%) and approximately one order smaller than the numerical uncertainties on Cf. The change in the CPV and CV is also comparable for N/LPP and W/LPP variations. Considering the insignificant variation in form factor predictions as seen in Figure 7c and already having much smaller blockage effect (ratio between mid-sectional area of the hulls and the towing tank section area) in CFD compared to towing tank, the domain size of the baseline grid (N/LPP = 3) is found to be large enough.

**Figure 7.** Comparison error of all test cases for the domain size variations: (**a**) U/LPP, (**b**) W/LPP, (**c**) N/LPP, (**d**) in all directions.

The distance between the inlet and the fore-perpendicular (U/LPP) is varied between 0.5 and 1.5. Contrary to the previously mentioned parts of the domain (N/LPP and W/LPP), the choice of the inlet plane location had prominent implications both on the local flow and the resistance components. The analysis performed on all test cases indicated that the significant changes between different U/LPP values are mainly due to the differing turbulence intensity reaching the hull. As explained in Section 3, the specific turbulent dissipation rate and turbulent kinetic energy (TKE) are set at the inlet plane for each hull regardless of the distance between inlet boundary and the FP (see Equations (6) and (7)). As a result, the TKE reaching the hulls decreased since the TKE is dissipated more by traveling greater distances with the increasing U/LPP. The reduced turbulence intensity in the free stream arriving to the hulls caused a significant drop in *C*<sup>V</sup> varying between 1.3% to 3.5%. The contribution of the frictional and viscous pressure resistance to the reduced viscous resistance is nearly equal for all cases. As seen in Figure 7a, the comparison errors in form factors are highly dependent on the U/LPP.

In addition to commonly known modeling errors such as turbulence modeling, the transition from laminar to turbulent flow is also considered for this study. Contrary to the common perception, the flow in a typical model scale RANS computation with a wall resolved approach is *not* fully turbulent in the boundary layer even with the commonly used turbulence models (i.e., without transition models). The analysis performed on infinitely thin 2D flat plates by Eça and Hoekstra [39] and Korkmaz et al. [30] indicated that, even though the correct position of the transition could not be predicted using ordinary turbulence models such as k-omega SST [33] and EASM [32], the transition occurs qualitatively rather accurately. In order to show if this is the case also for the hulls, local skin friction coefficient, *Cf* is calculated for baseline grid of each test case as

$$C\_f = \frac{\mu \left(\frac{\partial \mu\_x}{\partial y}\right)\_{y=0}}{\frac{1}{2} \rho l L\_{\infty} A} \, , \tag{10}$$

where *∂ux ∂y y*=0 is the velocity gradient at the wall and *A* is the wetted surface of the hull. The velocity gradients of the cells in the same x-position were summed, and all no-slip cells are included in the calculation of the local skin friction coefficient.

In Figure 8, the local skin friction coefficient of all test cases are presented together with the Blasius solution [41] for the reference of the friction coefficient of the laminar flow, Prandtl–Schlichting formula [41] for the fully turbulent flat plate skin friction coefficient, and the numerical friction line [30] for the EASM turbulence model derived using SHIPFLOW.

**Figure 8.** Local skin friction resistance coefficients for all hulls versus Reynolds number.

As seen in Figure 8, *Cf* of all test cases are even below the Blasius line before *Re* <sup>&</sup>lt; <sup>4</sup> <sup>×</sup> <sup>10</sup>4. However, *Cf* values are steeply increasing approximately between *Re* <sup>=</sup> <sup>7</sup> <sup>×</sup> 104 and *Re* = 1.5 × <sup>10</sup><sup>5</sup> where the numerical friction line is also showing remarkably similar increase from laminar regime towards turbulent. The *Cf* values of each test case are expected to have some quantitatively and qualitatively differences both among themselves and the friction lines (Blasius, Prandtl–Schlichting, and NFL) since each hull shape is unique and the flow is not under zero-pressure-gradient. However, it can be argued that the flow is not fully turbulent in all parts of the hull in the computations as it may be also the case for the towing tank tests. To make sure that turbulent flow is achieved in the model tests, turbulence simulators are attached to the hull. The model tests used for all the test cases in this study also utilized turbulence stimulators placed at 5% of the *L*PP from the fore perpendicular. In Figure 8, these locations are marked with circles. It is ensuring to observe that the flow in CFD transitioned into turbulent flow before the position of the turbulence stimulators. In the previous studies with flat plates [30,39], the Reynolds number where transition occurred was found to be dependent on the choice of turbulence model and turbulence intensities. Additionally, different CFD solvers (SHIPFLOW and FINEMARINE) using the same grid also led to not only different transition behaviors but also significantly different *Cf* values in the turbulent region [30]. Therefore, it is recommended to adjust the turbulence intensities for each code and the turbulence model when the wall resolved approach is used for making sure the flow characteristics are similar in CFD to the experiments.

In order to investigate the effect of varying turbulence intensities, the local skin fiction coefficient of the H1 hull is presented in Figure 9a for U/LPP = 0.5, 1 and 1.5. *Cf* of all U/LPP values seem to be nearly identical in Figure 9a except where the *Cf* is steeply increasing, i.e., transition of the flow. In order to visualize the differences in more detail, *Cf* of the U/LPP = 1 and 1.5 are plotted relative to *Cf* of the baseline domain. As seen in Figure 9b, the *Cf* [%] values are differing the most at the position where transition from laminar to turbulent flow occurs. In the laminar and turbulent regions, *Cf* values are nearly uniformly 0.7% and 0.8% less for U/LPP = 1 and 1.5 compared to U/LPP=0.5. The local viscous pressure resistance component also decreased with the lower turbulence intensity but not as uniformly as is the case for *Cf* . The local viscous pressure resistance coefficient, *Cpv*, of U/LPP = 1 and 1.5 was predominantly lower in the laminar parts of the hull and at the stern region that is covered by the boundary layer.

**Figure 9.** (**a**) Comparison of the local frictional resistance coefficients of H1 hull for the domain size variation in U/LPP and (**b**) comparison of *Cf* of H1 hull relative to U/LPP = 1.

In the final step, the domain size is varied in all directions (U/LPP, N/LPP and W/LPP) at the same time in two steps. Starting from the baseline grid (U/LPP = 0.5, W/LPP = 0.8 and N/LPP = 3), the boundaries in all directions are first moved to U/LPP = 1.0, W/LPP = 1.6 and N/LPP = 4 and then U/LPP = 1.5, W/LPP = 2.4 and N/LPP = 5 for all test cases. The number of cells are also increased proportionally in the region where the domain size is enlarged. As seen in Figure 7d, the resulting form factors due to varying the domain size in all directions are nearly the same with the domain size variation only in U/LPP as seen in Figure 7a. As a result, the turbulence intensities are playing a more significant role in terms of modeling errors than the wake reaching the outlet plane and the blockage effect. The default domain size gives the smallest comparison error on average, and it is concluded that further enlarging the domain is not necessary.

#### 5.1.4. Variation of the Model Scale Speed

The previous CFD studies presented by Raven et al. [42], Wang et al. [25], Dogrul et al. [26], Korkmaz et al. [27], Terziev et al. [24], Van et al. [43], and Korkmaz et al. [28] supported the existence of speed dependency for the form factors even though this should not be the case according to the hypothesis of Hughes [6]. Therefore, regardless of the choice of the CFD code, numerical methods, and settings, the choice of speed that the double body computations are performed for will have a significant impact on the CFD based form factors when the ITTC-57 model to ship correlation line is used. The towing tank tests are also not immune to the variation of form factors with changing the scale factor of the model as shown by García Gómez [21], Toki [22] and Van et al. [43].

The baseline grid, g2, of each test case is simulated in three different speeds: the lowest speed tested in the towing tank, the design speed of the vessel, and an interim speed between the two speeds. The average non-dimensional first cell height, y+, and other CFD settings are kept the same for all speeds. As can be seen in Figure 10a, all test cases indicate a definitive trend for the comparison error of factors which is based on the ITTC-57 line. As also explained in the earlier studies [22,28,42], the ITTC-57 line is the main reason for

form factors increasing with increasing Reynolds number due to its excessive steepness in model scale Reynolds numbers.

**Figure 10.** Comparison error of all test cases for the speed variations, (**a**) using the ITTC-57 line and (**b**) using the NFL.

Using the results of the same simulations but changing the friction line from the ITTC-57 line to a numerical friction line (NFL) [30] of the same CFD code and turbulence model reduced, and, in some cases nearly eliminated, the speed dependency of CFD based form factors, as can be seen in Figure 10b. As there is no experimental comparison point with the NFL, the form factors are directly plotted in Figure 10b. Contrary to other test cases, the H4 hull shows a decreasing trend with Reynolds number. The investigation in the local flow highlighted that the trend observed in H4 is due to the existence of mild flow separation at the stern. The separation is noticeably larger in the computations with the EASM than with the *k* − *ω* SST turbulence model. As Reynolds number is increased, the separation diminishes in intensity; therefore, the form factor is reduced as expected. Note that, according to the form factor hypothesis of Hughes [6], there should not be any flow separation in the model tests nor in the CFD. In such cases, the CFD simulations are recommended to be performed at higher Reynolds number until the separation is vanished. Except for the case with flow separation, the smallest average comparison error is observed at the slowest model tow speed. Therefore, it is recommended to perform the CFD simulations at the lower end of the model speed interval. However, such a conclusion may as well be different for other codes, numerical methods, or the size of the model used in towing tank tests.

#### 5.1.5. Turbulence Model

The systematic variations applied to the CFD set-ups have been performed with the *k* − *ω* SST and EASM turbulence models. The conclusions regarding the other CFD settings: the non-dimensional cell height normal to the wall, additional grid refinement at the stern, domain size, and model scale speed, are valid for both turbulent models. The form factor predictions of all test cases using the *k* − *ω* SST model are approximately 10% higher than the computations with EASM using the same grid. As a result of this consistent difference between the form factors from different turbulence models, the full scale viscous resistance (*C*VS = (1 + *k*)*C*FS) predictions using the *k* − *ω* SST model will be higher while the residual resistance (see Equation (2)) predictions will be lower than the predictions from EASM when the ITTC-57 line is used.

When the comparison error of the form factors is presented for each turbulence model with respect to the loading conditions, a certain prediction pattern is observed as presented in Figure 11. Computations at ballast, design, and scantling loading conditions are stacked in separate columns where box-and-whisker plots are placed with markers. The box plot can be identified with the gray color and sized with the lower and upper quartiles. Lines extending from the boxes (whiskers) extend to the data within 1.5 times the interquartile range (IQR). The markers are colored with the test cases and sized according to the number of cells. The computational results from the finest two grids (original g1 and g2 from Section 5.1), from average y<sup>+</sup> <sup>≤</sup> 0.5 presented in Section 5.1.1 and all grid refinements presented in Section 5.1.2 are presented in Figure 11. The form factor predictions from the *k* − *ω* SST model at ballast loading condition corresponds better to the experimentally determined form factors than the EASM turbulence model. However, the opposite trend is true for the design and scantling loading conditions. When the results are considered regardless of the loading condition, the average comparison error is 0.75% and −0.9% for the EASM and the *k* − *ω* SST turbulence models, respectively. Therefore, the absolute mean comparison error of the two turbulence models is similar.

**Figure 11.** Comparison error of form factors for different loading conditions for the EASM and the *k* − *ω* SST turbulence models.

#### 5.1.6. Validation in Model Scale

In order to complete the verification and validation study, experimental uncertainty needs to be determined [44]. As the experimental data used in this study were obtained through routine towing tank testing, thorough uncertainty analyses according to ITTC [45] could not be performed but instead the simplified implementation of this procedure as presented in ITTC [46] was used.

Using the standard uncertainty of calibration (SEE) for routine tests and the SSPA database of the repeatability of resistance tests, the uncertainties in resistance measurement [46] (without repeat tests) are calculated for each measurement point of each test case. However, the combined uncertainty of measured resistance, *u*c(*R*T), cannot be used as direct indicatives of the uncertainties related to the form factors. Therefore, an additional step is required to consider the uncertainties due to the data reduction process of the form factor, i.e., the linear regression in the Prohaska method. The regression lines in Figures 12 and 13 are obtained by using the measurement uncertainties for the 95% confidence interval (*k*<sup>p</sup> = 2) and applying the method explained in [47], which considers the experimental uncertainties in the regression progress and predicts the uncertainties in the form factor as well. The resulting regression line is indicated as York's method in Figures 12 and 13, where the uncertainty on the form factor is illustrated with magenta colored error bar at *Fn*4/*C*<sup>F</sup> = 0, and the measurement uncertainties are shown as the blue colored error bars. The uncertainty of the form factors, UD, for the 95% confidence interval are varying between 0.015 and 0.023, which corresponds to 1.3% and 2.0% of the (1 + *k*).

**Figure 12.** Prohaska plot of test cases in ballast loading condition: (**a**) H1, (**b**) H2-b, (**c**) H3.

**Figure 13.** Prohaska plot of test cases in design and scantling loading conditions: (**a**) H2-d, (**b**) H2-d, (**c**) H4.

The numerical uncertainty, USN, of the CFD based form factors is calculated similar to form factor calculation in Equation (3),

$$\mathcal{U}\_{\rm SN}(k) = \frac{\mathcal{U}\_{\rm SN}(\mathcal{C}\_{\rm V})}{\mathcal{C}\_{\rm FM}},\tag{11}$$

where *C*FM is the equivalent flat plate resistance in two-dimensional flow obtained from the same Reynolds number as the computations and obtained from the ITTC-57 line [4].

The validation uncertainty is calculated as UV = *U*SN<sup>2</sup> + *U*D2.

The numerical and experimental uncertainties, absolute comparison error, and the validation uncertainties for the baseline grids (g2) of all test cases are presented in Table 2 in percent of (1 + *k*) where the form factor from the Prohaska method is used. The validation uncertainty, UV, is bigger than the absolute comparison error for all test cases with the *k* − *ω* SST turbulence model. Except the H2-b test case with the EASM, all other test cases are also |E| <UV; validation is achieved at UV level, i.e., the comparison error is below the noise level. However, it should be noted that the UV of H1 and H2-d test cases are exceptionally high due to very large numerical uncertainties as explained in Section 5.1. When only UD and E are compared, the form factor predictions made with the *k* − *ω* SST are within the experimental uncertainty for the same number of test cases as the EASM turbulence model.

**Table 2.** Validation of the CFD based form factors.


The required uncertainty, U*reqd*, is determined based on the typical USN%*D* and UD%*D* values observed in Table 2. The numerical uncertainties of 2.5% to 3.5% and the experimental uncertainties of 1.3% to 1.8% were considered satisfactory levels in consideration of the U*reqd*. The combination of USN%*D* and UD%*D* indicates that U*reqd*%*D* should be approximately 4%. This required uncertainty level results in approximately ±4% variation in the full scale power predictions. It can be seen in Table 2 that U*reqd*%*D* = 4% is larger than the comparison error of all test cases with a considerable margin. The comparison of U*reqd*%*D* and U*V*%*D* for H2-b, H3, and H4 (only with the *k* − *ω* SST) shows that required uncertainty is larger than the validation uncertainty, and, therefore, the validation of these cases is successful for a programmatic standpoint [44]. The rest of the test cases U*reqd*%*D* is smaller than U*V*%*D* due to substantial numerical uncertainties.

It should be noted that the experimental determination of the form factor, i.e., the Prohaska method [5], is not a direct measurement but obtained as a result of data reduction. The Prohaska method is solely an approximation to obtain the form factor described by Hughes [6]. Therefore, the comparison error of the form factor should be interpreted with care since the experimental form factors may not always represent the true value.

#### **6. Demonstration of Quality by Comparison of Full Scale Predictions**

In the final step of the proposed quality assurance procedure, full scale speed-powerrpm relations between speed trials and full scale predictions based on model tests carried out at SSPA are compared. In order for such comparisons to be meaningful, a large number of sea trials are required since the uncertainty of each trial is large. The combination of the precision and bias limits of single speed trial result in approximately 10% of total uncertainty as indicated by Werner and Gustafsson [48] and Insel [49].

Correlation of model test power predictions to the speed trials are quantified with the *correlation factors* which are also used as "correction for any systematic errors in model test and powering prediction procedure, including any facility bias" [50] in the 1978 Power Prediction method [20]. There are three different schemes of correlation factors that can be used: CA, CP − CN and ΔCFC − ΔwC. In this study, the correlation scheme of CP − CN coefficients are applied. In order to obtain these coefficients, the correlation of each individual speed trial, C <sup>P</sup> and C N, are calculated as

$$\mathbf{C}'\_{\rm P} = \frac{P\_{\rm D \ trial}}{P\_{\rm D \ tank}} \quad \text{and} \quad \mathbf{C}'\_{\rm N} = \frac{n\_{trial}}{n\_{tank}} \tag{12}$$

where the *P*<sup>D</sup> *trial* and *ntrial* are the power and propeller turning rate from a speed trial, while *P*<sup>D</sup> *tank* and *ntank* represent the corresponding predictions based on the model test. The power, *P*D, is derived from the faired speed-power curve at the design speed. After C <sup>P</sup> and C <sup>N</sup> are calculated for a large number of sea trials, an assembled correlation factor for CP and CN are determined by taking the median of C <sup>P</sup> and C <sup>N</sup> of all trials of sufficient quality [48]. In this study, assembled correlation factors are not disclosed, but the probability density functions (PDFs) of C <sup>P</sup> and C <sup>N</sup> are presented by shifting the median of PDFs to 1, i.e., normalizing the correlation factors.

In the determination of the ITTC 1978 Power Prediction Method, the standard deviation of normalized C <sup>P</sup> and C <sup>N</sup> were used as the main measure to rank different extrapolation methods. In this study, the same approach is adopted, and reduced scatter of the normalized C <sup>P</sup> and C <sup>N</sup> is interpreted as an improvement for the extrapolation methods explained in Sections 6.1 and 6.2.

#### *6.1. Comparison of the Standard ITTC-78 Method and CFD Based Form Factors Using the ITTC-57 Line*

Correlation of the speed trials to model test power predictions is quantified by using three different sources of form factors: the Prohaska method, CFD based form factors using EASM, and *k* − *ω* SST turbulence models. The ITTC-78 method [20] is used for all predictions with the ITTC-57 model to ship correlation line [4] and the correlation allowance stated in Equation (4). A special wake scaling suggested in the ITTC 1999 method [51] is applied to the vessels with a pre-swirl stator type of device ahead of the propellers. All predictions used the same model test data, but only the source of the form factor is changed. The difference in the form factors among predictions leads to a change in the residual resistance as calculated in Equation (2) and the viscous resistance of the full scale ship ((1 + *k*) × *C*FS) as calculated by using Equation (1). As a result of the change in CTS, the predicted delivered power and propeller rate of revolution vary.

The probability density functions (PDFs) of the normalized correlation factors, C P and C <sup>N</sup>, are calculated for the speed trials that have an uncertainty index less than eight using the different sources of the form factors. The uncertainty index, *u*, is an in-house developed index that quantifies the trustworthiness of each speed trial by summarizing the largest error sources and weighting them according to their impact on the results. In addition to the PDF curves, the standard deviations (*σ*) of C <sup>P</sup> and C <sup>N</sup> are also presented in Figure 14. The comparison of the standard deviations for the power predictions (CP) indicates that the scatter is reduced considerably when the CFD based form factors from the EASM turbulence model are used compared to the Prohaska method. The PDF curve of CFD based form factors from EASM suggests that the frequency of predictions that are within ±5% of the sea trials is increased, while the predictions that are off more than 10% are slightly reduced. The PDF curve of CFD based form factors from *k* − *ω* SST for the power prediction remained nearly the same as the standard ITTC-78 method. The propeller rate of revolution predictions remained the same with EASM but slightly worsened by the predictions with CFD based form factors with *k* − *ω* SST model when the standard deviation is considered. It should be noted that the reduction of scatter in the power predictions is a more significant measure than the propeller turning rate since the scatter in power prediction is much larger than the prediction of rps. Hence, it can be concluded that the usage of CFD based form factors with ITTC-57 line improves the predictions in general or at least do not deteriorate them.

**Figure 14.** The probability density functions (PDFs) of the normalized correlation factors for the standard ITTC-78 method and using CFD-based form factors with two different turbulence models. (**a**) C <sup>P</sup> and (**b**) C N.

#### *6.2. Comparison of the CFD Based Form Factors Using the ITTC-57 Line and Numerical Friction Lines*

To investigate if the predictions can be further improved by modifying the standard ITTC-78 method, the ITTC-57 model to ship correlation line is replaced by the numerical friction lines [30]. The CFD based form factor of each hull is recalculated using the same simulation results as in Section 6.1 for EASM and *k* − *ω* SST turbulence models using the corresponding numerical frictional line as explained in Section 2. Similar to the previous section (Section 6.1), the correlation of the speed trials to model test power predictions is quantified using four different sources of form factors; CFD based form factors for EASM and *k* − *ω* SST using ITTC-57 line and NFLs.

The same population of the speed trials presented in the Section 6.1 are used for generating the PDFs of the normalized C <sup>P</sup> and C <sup>N</sup>. As can be seen in Figure 15, the standard deviation of the predictions with the CFD based form factors is lower with the application of NFL compared to using the ITTC-57 line, while the scatter of C <sup>N</sup> increased slightly. Another distinctive result of using the NFLs is the reduced difference between the turbulence models. The previous studies [27,28] indicated that, when the numerical friction lines are used, the full scale viscous resistance (CVS = (1 + *k*)*C*FS) is nearly the same regardless of using EASM and *k* − *ω* SST turbulence models for the derivation of the form factor. However, the residual resistance varies with regard to the turbulence model, and, therefore, leading to the different full scale total resistance (see Equation (1)). As observed in Figure 15, the form factors from using the EASM turbulence model and its numerical friction line correlates better than when using the *k* − *ω* SST model. In addition, the CFD based form factors using numerical friction lines considerably reduced the frequency of the predictions that differ from the speed trials more than 10%, which is the level of the total uncertainty of a speed trial [48,49].

**Figure 15.** The probability density functions (PDFs) of the normalized correlation factors using the CFD based form factors with the ITTC-57 and the numerical friction lines. (**a**) C <sup>P</sup> and (**b**) C N.

#### *6.3. Analysis of the Extrapolation Methods and Speed Trials*

The full scale speed-power-rpm relations between speed trials and full scale predictions using different extrapolation methods have been presented in Sections 6.1 and 6.2. It is important to make sure that the conclusions are not biased for a specific population of speed trials and also the speed trials with large error sources are excluded when general conclusions are made. Therefore, the statistical analysis on the C <sup>P</sup> is repeated for the speed trials with varying maximum uncertainty indexes. As presented in Table 3, three different levels of maximum uncertainty index, *u*, are used. In practice, the cut off uncertainty index should be as low as possible since the larger *u* index value of a speed trial indicates the existence of larger error or uncertainty sources such as large wave and wind corrections due to adverse weather conditions. However, the number of sea trials with very low uncertainty is limited and, as a result, the danger of drawing conclusions from small number of speed trials arise.

In Table 3, the statistics of different populations of speed trials with maximum cut off values of *u* varying between 4 and 8 are presented. The number of speed trials is 46 for the cut off value of *u* = 4. A lower *u* cut off value for the uncertainty index is not preferred since the size of the population decreases significantly. The comparison of the standard deviations between the corresponding *u* index shows that *σ* increases slightly with the increasing *u* index as there are more speed trials with lower quality in the larger

populations. However, the ranking of the magnitude of the standard deviations among each extrapolation method remains consistent. The scatter of the standard ITTC-78 method, where the form factors are obtained from the Prohaska method, is higher than all other extrapolation methods where the form factor is obtained from CFD except when *k* − *ω* SST is used with the ITTC-57 line. The standard deviations when using CFD based form factors with the ITTC-57 line are slightly in favor of the EASM turbulence model and also the percentage of the predictions with less than a 5% error are consistently higher when the EASM is used, while the predictions that differ from the speed trials more than 10% remained similar to the *k* − *ω* SST with the ITTC-57 line. The replacement of the ITTC-57 line with the numerical friction lines in combination with the CFD based form factors shows promising and consistent improvements towards not only reduction in the scatter but also decline in the number of speed trials where the prediction error is larger than 10% for both turbulence models. The correlation between the predictions and the speed trials is improved the most when the EASM turbulence model is used in combination with NFL.

**Table 3.** Statistics of the power predictions compared to speed trial.


The relatively large standard deviations observed in Table 3 are mainly due to the scatter in the speed trials of the sister ships. In order to illustrate this, an ideal prediction scenario has been prepared. Ideal case means if the model test prediction is fully correct compared to the speed trial. In theory, it would mean that the mean C <sup>P</sup> of a series of sisters would be 1. The resulting C <sup>P</sup> values are presented in Figure 16 together with the standard deviations of each ship series. As observed in Figure 16, the standard deviations among the sister ship series are varying between 2% to 6%. The standard deviation of all the C <sup>P</sup> values in the ideal prediction case is 3.6%, which is only marginally smaller than the predictions made with the CFD based form factor with the EASM turbulence model and NFL as shown in Table 3. Additionally, the percentage of predictions within 5% and the predictions that differ from the speed trials more than 10% are also nearly the same with the ideal prediction case and the predictions made with the CFD based form factors with the NFL. This indicates that it is hardly possible to achieve a better accuracy than this, unless the uncertainty of the speed trials become lower.

**Figure 16.** Illustration of the speed trial spread between sister ships for 14 ship series. C <sup>P</sup> values normalized to the mean of each series. Dashed black lines indicates the standard deviation.

#### *6.4. Discussion*

The analysis of the model test power predictions is further deepened by grouping the vessels based on their main dimensions, operational conditions, and general characters. Four main groups have been identified:


The comparison error of the predictions to the speed trials on a group level is calculated as

$$P\% T = \left(\overline{\mathbf{C}\_P} - 1\right) \times 100\tag{13}$$

where *P* represents predictions, *T* is the speed trials, and C <sup>P</sup> is the average normalized C <sup>P</sup> of each group. *P*%*T* values are calculated for the five different extrapolation methods presented in Sections 6.1 and 6.2. The *P*%*T* values from CFD based form factor methods (Section 6.2) are compared to the *P*%*T* of the standard ITTC-78 method. This comparison, Δ*P*%*T*, is calculated by subtracting the absolute *P*%*T* values of the CFD based form factor methods from the absolute *P*%*T* values of the standard ITTC-78 method. As a result, the Δ*P*%*T* = 0 indicates that the prediction accuracy with the CFD based form factors remained the same as the standard ITTC-78 method, positive values Δ*P*%*T* indicate improvement and negative values point out that the predictions are worsened for the corresponding prediction method relative to the predictions from the standard ITTC-78 method. Δ*P*%*T* of CFD based form factor methods are presented in Figure 17 for the four groups. In order to give an indication of the uncertainty of the speed trials for each group, the standard deviation of the *C* <sup>P</sup> values from the standard ITTC-78 method is combined with the bias limit of 4% as estimated by Insel [49]. The resulting total uncertainties ( *σ*<sup>2</sup> + *U*<sup>2</sup> *bias*) are indicated in Figure 17 as error bars for each group. It should be noted that the total uncertainties of trials for the each group are larger than the corresponding comparison error from the standard ITTC-78 method, i.e., *P*%*T* is below the noise level. Therefore, the improvements (if any) as a result of using the CFD based form factors will be within the noise levels caused by the uncertainty of the trials. However, the changes in the accuracy of predictions, Δ*P*%*T*, are considered statistically valid as the number of speed trials are rather large.

**Figure 17.** Relative change of the accuracy of the predictions, Δ*P*%*T* = 0, made with the CFD based form factors compared to the predictions with the standard ITTC-78 method, based on 18 test cases and 78 trials.

Starting with the extrapolations based on the CFD based form factors using the ITTC-57 line, there is no significant change in the accuracy of predictions for the full block and slow speed (*Fn* < 0.17) vessels. This outcome is expected since the form factor determination with the Prohaska method often functions well for such vessels and the CFD based form factors predict similar form factors to the Prohaska method as discussed in Section 5.1.6. As a result of obtaining nearly the same form factors from EFD and CFD while using the same friction line, the correlation between the predictions and speed trials remained nearly the same. However, it should be noted that the usage of *k* − *ω* SST turbulence model with the ITTC-57 line led to negative or too small CR values in some test cases as a result of over-prediction of form factors which was also the observation in validation in the model scale (see Figure 11).

The accuracy of predictions for the low block vessels also remain nearly the same when the CFD based form factors are used in combination with the ITTC-57 line. This group of vessels are equipped with large protruding bulbous bows where the Prohaska method does not work in most cases. Therefore, the CFD based form factors are expected to introduce improvement to the predictions. The analysis performed on each vessel in this group showed that the CFD based form factors were up to 60% lower than the EFD based form factors. This caused the full resistance predictions to increase just enough to be predicted with similar accuracy but making it an over prediction on average instead of under prediction as it was the case for the standard ITTC-78 method.

The medium block and medium speed group are another group of vessels where a significant improvement is expected because the bulbous bow designs of these ships and typical loading condition for which the speed trials are performed for these vessels are not ideal for the Prohaska method. The CFD based form factors with the ITTC-57 line are predicted 25% to 50% lower than the form factors based on the Prohaska method. As can be seen in Figure 17, the predictions are in fact improved by implementation of the CFD based form factor for the both turbulence models.

The analysis is also performed for the CFD based form factors with the numerical friction lines. As can be seen in Figure 17, nearly all groups of vessels are predicted better than the standard ITTC-78 method and also the CFD form factors with the ITTC-57 line. The accuracy of the predictions for the full block vessels with LWL > 320 m remained nearly the same as the predictions from the standard ITTC-78 method were already in good agreement with the speed trials. The improvement in the predictions for the medium block and medium speed vessels are doubled on average and the low block vessel indicates a gain in accuracy when the CFD based from factors with NFL are used compared to using the ITTC-57 line.

The full block vessels with 220 m < LWL < 280 m are also predicted better on average with the numerical friction line than other extrapolation methods. Considering that the form factors for the vessels in this group are nearly the same for EFD and CFD when the ITTC-57 line is used, it is significant to investigate why such an improvement is observed. When the CFD based form factors are obtained for the ITTC-57 line and the NFL, the same computation is used for a test case and turbulence model. This leads to variations in the full scale resistance predictions even when the same turbulence model is used. The origin of such variations is explained by the difference


The full scale resistance components explained above in the items 3 to 5 have varying effects on the final predictions as the main dimensions of each hull and its operational conditions such as *Re* and *Fn* are different. In Figure 18a, the difference between the full scale resistance predictions using the CFD based form factors with the EASM turbulence model but with different friction lines are presented. This difference is calculated as ((CTS(*NFL*)/CTS(*ITTC*57)−1)\*100) where the friction line used for the prediction of CTS is indicated in brackets. In order to simplify the visualization and evaluation, the speed CTS curve of each test case is averaged. As seen in Figure 18a, the difference between the average total resistance of the predictions made with the NFL and the ITTC-57 shows a linear trend when plotted against LWL. At around LWL = 240 m, the total resistance predictions of both friction lines intersect, the vessels shorter than 240 m are under-predicted, and the ships longer than 240 m are over predicted up to 6% by the application of the numerical friction lines compared to using the ITTC-57 line.

Among the possible sources of the trend observed in the total resistance predictions, the full scale viscous resistance (the roughness and correlation allowances summed with (1 + k)CFS) is identified as the main contributor. The comparison of CVS for the rough hulls are calculated in a similar fashion to the CTS comparison and presented in Figure 18b. It can be noticed that the trends for the difference between the predictions made with the NFL and the ITTC-57 line are highly similar for CTS and CVS, which is by far the biggest resistance component when combined with roughness and correlation allowances. The comparison between Figure 18b,c also suggests that the difference in the CR when NFL and ITTC-57 line are used indeed has a noticeable effect on the total resistance, but it is limited when compared to the contribution of the viscous resistance.

The full scale viscous resistance for the smooth hull (CVS = (1 + k)CFS) could not have been the reason behind the trend observed in Figure 18b,c because the slope of CFS curves for NFL and ITTC-57 line are highly similar at the Reynolds numbers that most of the conventional ships operate. Following this statement and the previous arguments, no other part of the extrapolation is left but the roughness allowance [10] and the correlation allowance as in Equation (4). In Figure 18c, the proportion of the Δ*C*<sup>F</sup> + *C*<sup>A</sup> and Δ*C*<sup>F</sup> in the full scale total resistance of the each test case are presented for the predictions with the ITTC-57 line and the NFL, respectively. The contribution of the roughness allowance to the total resistance varies between 4% to 7% when NFL is used. However, the usage of the ITTC-57 line led 7% to 11.5% of the total resistance to be constituted by *C*<sup>F</sup> + *C*A. The formulation of the *C*<sup>A</sup> in Equation (4) is dependent on *Re* and clearly explains the trends observed in Figure 18c. The relationship observed in Figure 18b for the full scale viscous resistance including the roughness and correlation allowance is a direct result of the contribution of the *C*<sup>A</sup> in Equation (4), which propagated into the full scale total resistance seen in Figure 18c.

**Figure 18.** Difference between the (**a**) CTS, (**b**) CVS (including Δ*C*<sup>F</sup> + *C*A), and (**c**) Δ*C*<sup>F</sup> + *C*<sup>A</sup> predictions using the CFD based form factors with the EASM turbulence model and different friction lines.

Going back to the discussion on correlation of the predictions and the speed trials for the full block vessels with 220 m < LWL < 280 m, it can be stated that the full scale resistance predictions vary 0 to 2% as a result of using different friction lines with CFD based form factors. Therefore, the improvement due to using NFL as presented in Figure 17 is arguably due to the combination of the changes in resistance predictions and also as a result of the overall changes in the whole population (shift in the median of the C <sup>P</sup> of all speed trials).

As explained in this section, the full scale speed-power-rpm relations between speed trials and full scale predictions using CFD based form factors can be improved compared to the standard ITTC-78 prediction method. This conclusion is confirmed for the different populations of speed trials where the trials were filtered through varying uncertainty indices as shown in Table 3. Using the numerical friction lines derived by Korkmaz et al. [30], correlation between the predictions and the speed trials are further improved in general. The difference in the predictions when the NFL and the ITTC-57 line is used with the CFD based form factors largely originates from the correlation allowance recommended by ITTC [20]. The correlation allowance was omitted when the numerical friction lines were used in the extrapolations since it is only fit to be used in combination with the ITTC-57 line. As a result of omitting the CA term, the predictions are significantly improved for the 18 test cases. However, this may not be the case for the ships that are shorter than the test cases considered in this study because of the way the CA term [20] is formulated. Since the CA is a logarithmic function of the Reynolds number, the contribution of CA increases rapidly with decreasing *Re*, i.e., the shorter vessels. At lower Reynolds numbers than the ones investigated in this study, it may be desirable to have new CA in combination with the roughness allowance derived for the numerical friction lines to sustain the improvements. Therefore, the conclusions regarding the CFD based form factor method explained in this study is limited to the ships with 1.05 × <sup>10</sup><sup>9</sup> < *Re* and 200 < LWL.

#### **7. Conclusions**

In order to respond to the need for a practical procedure for the organizations that regularly perform CFD predictions on similar cases, a new procedure of quality assurance has been proposed by the ITTC Specialist Committee of Combined CFD and EFD Methods. This study serves as an example of how the procedure can be applied in practice to a problem: CFD based form factors. The quality assurance of this practical problem is

demonstrated in three parts: the content of the Best Practice guidelines of the specific CFD code used in this study as explained in Sections 3 and 5.1.1–5.1.5, the quality Assessment of the BPG methodology through verification and validation studies presented in Section 5.1.6, and finally the demonstration of quality by the comparisons of 78 speed trials to the predictions made by combined CFD/EFD methods explained in Section 6.

In order to investigate and derive a best practice guideline for CFD based form factors, systematic variations have been applied to the CFD set-ups. The non-dimensional cell height normal to the wall, additional grid refinement at the stern, domain size, and model scale speed were analyzed in Sections 5.1.1–5.1.4 and the following observations and recommendations were made:


to the form factor approach of Hughes [6], there shall be no flow separation to ensure its validity. Therefore, for the cases where flow separation is observed, higher *Re* numbers should be used for the simulations regardless of the friction line used.


In the second step of the proposed quality assurance procedure, verification and validation of the CFD based form factor method were performed. Experimental uncertainties of the six test cases were determined and the uncertainties on the form factors were derived. The following conclusions were made for the verification and validation studies:


The last step of the of the proposed quality assurance procedure, demonstration of quality, was performed by investigating the full scale speed-power-rpm relations between the speed trials and the full scale predictions based on different extrapolation methods but using the same model test data. In total, 18 test cases were simulated using the best practice guidelines presented in this study and CFD based form factors were determined. The conclusions regarding the comparison of the full scale predictions and speed trials are that:


Following the proposed quality assurance procedure, the CFD based form factor method has been investigated. As a result of the systematic variations applied to the CFD settings, a best practices guideline was derived for the CFD based form factor method. Using the BPG, form factor predictions have been made for a variety of ship types. The investigation on the correlations between the full scale power predictions and speed trials showed that the CFD based form factors can be considered as an alternative or supplementary method to the Prohaska method. As explained, the selection of friction lines in combination with CFD based form factors played a prominent role towards reducing the scatter between the predictions and trials. However, the change of the friction line would also require altering the well accustomed correlation factors.

It is worth remembering the statement of Gilbert Dyne who was the mastermind behind the ITTC 1978 Power Prediction Method: "It is ... relatively easy to criticise the different details of the...ITTC method. It is much more difficult to find an alternative which gives better power and RPM predictions" Lindgren and Dyne [52] (p. 14).

As stated by Dr. Dyne, there are still shortcomings with full scale resistance predictions. However, the combination of EFD and CFD is expected to provide immediate improvements to the 1978 ITTC Performance Prediction Method. Further studies should be performed with many more test cases to be used for the comparison of speed trials and power predictions with the CFD based form factors as the uncertainty of the speed trials are rather large.

**Author Contributions:** Conceptualization, K.B.K., S.W., and R.B.; methodology, K.B.K., S.W., and R.B.; validation, K.B.K.; formal analysis, K.B.K.; investigation, K.B.K., S.W., and R.B.; data curation, K.B.K.; writing—original draft preparation, K.B.K.; writing—review and editing, K.B.K., S.W., and R.B.; visualization, K.B.K.; supervision, S.W. and R.B.; project administration, S.W.; funding acquisition, S.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by VINNOVA, the Swedish Governmental Agency for Innovation Systems, grant 2017-02953, and the computational resources provided by Chalmers Center for Computational Science and Engineering (C3SE).

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**

