**1. Introduction**

During the design process of ships, power predictions are of utmost importance because the speed attained at a certain power consumption in a trial run, so called the contract speed, is specified at the contract of a new ship order. If the speed does not meet the specifications, the yard is forced to pay a penalty depending on the terms in the contract. Therefore, designers are under a pressure of being just in the limits. The dilemma for the designer and the yard as stated by Larsson and Raven [1] is that too optimistic predictions might end up in a big burden for the yard while too conservative predictions will be a lost order. In addition to increasing competitiveness of the market, legal authorities have also been taking steps towards further improvement of the energy efficiency of ships due to environmental concerns IMO [2]. Therefore, the importance of the power predictions and the required accuracy are increasing ever more.

The towing tank testing and extrapolation procedures have been used for more than a century to predict the performance of a ship in deep and calm water. The efforts to standardize and improve the early towing tank testing and extrapolation practices resulted in the foundation of the International Towing Tank Committee (ITTC) in 1933. The extrapolation of full scale ship resistance evolved in time, starting from a rather simple William Froude's method and going through several major revisions: adoption of using both Froude and Schoenherr lines in ITTC [3]; adoption of ITTC 1957 model to ship correlation lines [4]; recommendation of the Prohaska method [5] following the introduction of the form factor concept of Hughes [6]; adoption of the Bowden and Davison [7] formula in ITTC [8]; and

confirmation and integration of the previous concepts and formulas [4–7] via comparison of approximately one thousand sea trials to model test predictions in the formulation of the ITTC 1978 Power Prediction Method [9]. The roughness allowance was updated by replacing the previous formulation with Townsin and Dey [10] and introducing a new correlation allowance formulation in the ITTC Report of Power Performance Committee [11], and the calculation of air resistance was modified in the Revision 03 of the 1978 ITTC Performance Prediction Method [12]. The extrapolation procedures and towing tank tests are still considered as the last step of the performance prediction considering the current commercial tendencies and the evaluations such as EEDI calculations as enforced by IMO [2] where the applicable ships must go through the pre-verification by model testing during the design phase of a new ship.

Even though towing tank testing and extrapolation methods have been improved over many decades, there are inherent and well known shortcomings mainly because the Froude and Reynolds similarities cannot be fulfilled simultaneously, i.e., scale effects. Therefore, towing tank facilities must rely on experience obtained from large databases of model tests and sea trials. Computational fluid dynamics (CFD), however, can handle these scale effects and have been under development for more than a century as the advent of computational hydrodynamics dates back to 1898 by the work of Michell [13]. However, it was in 1980s when the "numerical methods to started to become really useful in ship design" Larsson [14] (p. 2). The development of the Reynolds Averaged Navier–Stokes (RANS) methods have been evaluated since 1980 [15] and the verification and validation (V&V) of CFD methods in model scale is now a well established practice. According to the statistics presented in Hino et al. [16], the mean absolute comparison error of JBC is around 2% for the towed cases (resistance) and 3% for the self-propulsion, while the standard deviation is approximately 2% and 4%, respectively. It is noticeable that no further reductions neither in comparison error nor in scatter were obtained at the 2015 Workshop [16] compared to the 2010 Workshop [15]. Unlike in model scale, the accuracy of CFD on prediction of full scale performance is still under concern mainly due to scarcity of the full scale validation data in the literature and limited CFD studies. Recently, studies presented by Sun et al. [17] and Niklas and Pruszko [18] showed that full scale simulations can predict the power similar to or better than the towing tank tests. On the other hand, the results of Lloyd's Register workshop on ship scale hydrodynamics [19] indicated that differences between the numerical setups can lead to very diverse predictions on both power and propeller turning rate.

Instead of choosing between towing tank testing (EFD) and computational hydrodynamics (CFD), a combination of the two methods can be a feasible solution to increase the accuracy of the power predictions. As identified by the ITTC Specialist Committee on the Combined CFD/EFD Methods, if a part of the model testing or extrapolation procedure causes higher uncertainty than the numerical uncertainty and modeling errors of the CFD applications, the accuracy is expected to increase. In the 1978 ITTC Performance Prediction method [20], the form factor approach is identified as a major uncertainty source due to its determination method, i.e., the Prohaska Method [5], and the scale effects when the ITTC-57 model to ship correlation line is used [21–28]. The study performed by Wang et al. [29] showed that when the Prohaska Method is replaced by CFD based form factors in the ITTC-78 Power Prediction Method, the sea trials correlated better for a ship. As it is the case for the direct full scale CFD predictions, the CFD/EFD combined methods were applied to only a limited number of cases. Therefore, this paper aims to address this issue by investigating the verification and validation of the CFD based form factor approach in model scale and by comparing large number predictions using combined CFD/EFD methods to sea trials.

Verification and validation of CFD codes and methods are essential measures not only for the improvement of the CFD methods, but also the quality assurance of the CFD applications. It requires significant computational resources and validation data with an experimental uncertainty at hand. Since the derived uncertainty levels are only valid for a unique case and condition, each test case should be subjected to V&V studies in order to ensure the quality of the CFD application. However, thorough verification studies cannot be performed for each commercial application for practical reasons and experimental uncertainty analysis is not available in advance. This raises the question on whether a V&V result is valid for a similar case and also if it is required to be done only once. Therefore, a practical procedure is needed for the organizations that regularly perform CFD predictions for similar cases. To respond to this need, two ITTC committees jointly proposed a new procedure for the quality assurance comprised of the following parts: (1) requirement of an organization's Best Practice Guideline (BPG), (2) Quality Assessment (QA) of the BPG methodology, and (3) demonstration of quality. This paper will follow the proposed Quality Assurance Recommended Procedures when demonstrating the application of the CFD based form factors. For the first time in the literature, the practicality and usefulness of the proposed QA procedure will be tested for the quality assurance of CFD based form factor method and are presented in the following steps:


Through these steps, the following research questions are aimed to be answered by this study:


#### **2. Extrapolation of Model Tests to Full Scale**

In this study, the procedure recommended by ITTC [20] is used to extrapolate the towing tank test results to full scale. According to the 1978 ITTC Performance Prediction Method [20], the resistance of the full scale ship is calculated as

$$\mathbf{C}\_{\rm T} = (1+k)\mathbf{C}\_{\rm FS} + \Delta \mathbf{C}\_{\rm F} + \mathbf{C}\_{\rm A} + \mathbf{C}\_{\rm R} + \mathbf{C}\_{\rm A \& S},\tag{1}$$

where *k* is the form factor, *C*FS is the frictional resistance coefficient in full scale (the subscript 'S' signifies the full scale ship), *C*<sup>R</sup> is the residual resistance coefficient, Δ*C*<sup>F</sup> represents the roughness allowance, *C*<sup>A</sup> is the correlation allowance, and *C*AAS is the air resistance coefficient.

According to the recommended procedure [20], the form factor is obtained by the Prohaska method [5] in model scale. The residual resistance, which is assumed to be the same in model and full scale, is then obtained as

$$\mathbf{C}\_{\rm R} = \mathbf{C}\_{\rm TM} - (1 + k)\mathbf{C}\_{\rm FM} \tag{2}$$

where *C*TM is the total resistance coefficient (the subscript 'M' signifies the model). *C*TM is measured at each speed in the towing tank, and *C*FM is obtained from the friction lines. In the study, two form factors were obtained from the Prohaska method and CFD based

form factor determination methods. The latter method follows the assumptions of [6] and is derived using the relation:

$$\mathbf{C}(1+k) = \frac{\mathbf{C}\_{\text{F}} + \mathbf{C}\_{\text{PV}}}{\mathbf{C}\_{\text{FM}}} = \frac{\mathbf{C}\_{\text{V}}}{\mathbf{C}\_{\text{FM}}},\tag{3}$$

where the frictional resistance coefficient (*C*F) and viscous pressure coefficient (*C*PV) are obtained by the double body CFD simulation. *C*FM in the denominator of Equation (3) is the equivalent flat plate resistance in two-dimensional flow obtained from the same Reynolds number as the computations. When the CFD based form factor determination is used, *C*FM in Equation (3), *C*FM in Equation (2) and *C*FS in Equation (1) are derived from the same friction line.

The frictional resistance coefficients of the ship, *C*FM and *C*FS, are obtained by using three different friction lines: the ITTC-57 model-ship correlation line [4], and two numerical friction lines for EASM and *k* − *ω* SST turbulence models, respectively, proposed by Korkmaz et al. [30].

Correlation factors for the extrapolation were separated from the roughness allowance by ITTC [11] and the formulation of Bowden and Davison [7] is replaced by Townsin and Dey [10]. The correlation allowance recommended by the 19th ITTC is

$$\rm C\_A \approx (\Delta C\_F)\_{\rm Bowden} - (\Delta C\_F)\_{\rm Townsin} = 5.68 - 0.6 \log(Re) \times 10^{-3} \,\text{s} \tag{4}$$

where (Δ*C*F)Bowden is the roughness allowance proposed by Bowden and Davison [7], and (Δ*C*F)Townsin [10] is the recommended roughness allowance in the present recommended procedures [20]. Note that, if the recommended *C*<sup>A</sup> in Equation (4) is used for the extrapolation, summation of (Δ*C*F)Townsin and *C*<sup>A</sup> is essentially the same as using the old formulation of roughness allowance proposed by Bowden and Davison [7], i.e., the original 1978 ITTC method. As an option, it was recommended in the ITTC Report of Power Performance Committee [11] that each institution maintains their own *C*<sup>A</sup> formulations. However, a certain reluctance can be expected from towing tanks to change the *C*<sup>A</sup> value since it would require a substantial amount of work and risk-taking to derive new model-full scale correlations that are derived from a consistent model testing and extrapolation practices. Therefore, in the context of this study, *C*<sup>A</sup> used for the extrapolations is the recommended correlation factor in Equation (4). As explained in Section 6, *C*<sup>A</sup> is omitted in calculation of the full scale resistance when the numerical friction lines are used as (Δ*C*F)Bowden "is suitable when extrapolating resistance using the 1957 ITTC line on a form factor basis..." Bowden and Davison [7].

For the cases when a flow separation is observed in model scale CFD computations, an additional computation has been performed also in full scale. The extrapolation method is slightly altered to mitigate the adverse effect of the flow separation which causes an overestimation of the full scale viscous resistance. In the altered method, the residual resistance is obtained by using the model scale form factor as in Equation 2, but the viscous resistance of the full scale ship ((1 + *k*) × *C*FS) is calculated by the form factor obtained from the full scale double body computations.

#### **3. Flow Solver, Grid Generation, Computational Domain, and Boundary Conditions**

The XCHAP module of SHIPFLOW 6.5 is used for solving the steady state viscous flow [31]. Reynolds Averaged Navier–Stokes (RANS) equations are solved with a finite volume method. The first order accurate Roe scheme is used for discretization of the convective terms and a flux correction is applied explicitly in order to increase the order of accuracy. The equations are solved with a Krylov solver (adopted from PETSc) which implements the Generalized Minimal Residual method (KSPGMRES). Two turbulence models are available in XCHAP solver: EASM as described by Deng and Visonneau [32] and *k* − *ω* SST as described by Menter [33]. Both turbulence models are used in this study.

The viscous flow solver XCHAP can only handle structured grids, which can be in H-H, H-O, or O-O topologies. Although it is possible to import external grids to the

solver, the grid generator of SHIPFLOW, XGRID, is used for the study. The coarsest body fitted grid used for the study is presented in Figure 1. The parametrized nature of grid generation with XGRID ensures almost identical grid distribution in the longitudinal and circumferential directions for the most conventional hulls simulated. However, the grid distribution in the normal direction to the hull varies between different hulls as the *Re* differs; therefore, different first cell sizes in the normal direction to the wall and cell growth ratios are obtained to achieve (nearly) the same *y*<sup>+</sup> values. The overlapping grid technique is used to model the flow around the rudders [34] and to apply refinement on the single block of structured grid. As can be seen in Figure 1b, a refinement is applied to the region within the black line boundaries. The refinement does not improve the geometry resolution but only divides the existing cells into two or more pieces in desired directions [31]. In this study, the refinements are applied in all directions and the cells are divided in two. All simulations in this study were performed as double body computations with rudders that are integrated into the flow domain with an overlapping grid technique as seen in Figure 1b.

The computational domain is shaped as a quarter of a cylinder that consists of six boundaries where the center plane of the ship is set as the symmetry boundary condition. By default, the distance between inlet and fore-perpendicular (FP) is 0.5LPP, outlet plane is located at 0.8LPP behind the aft-perpendicular, and the radius of the cylindrical outer boundary is 3LPP.

Two boundary conditions are used: Dirichlet and Neumann conditions. Boundary types employed in XCHAP are noslip, slip, inflow, outflow, and interior. Inlet boundary condition sets a fixed uniform velocity (*U*∞), the estimated turbulent quantities and a zero pressure gradient normal to the inlet boundary. The turbulent quantities, specific turbulent dissipation rate, and turbulent kinetic energy at the inlet are estimated as

$$
\omega\_{inlet} = \lambda \mathcal{U}\_{\infty} / \mathcal{L} \tag{5}
$$

$$k\_{inlet} = \mu \omega\_{inlet} \mathbf{C\_i} / \rho \; , \tag{6}$$

where the factor of proportionality, *λ*, is set to *λ* = 10, *L* is the reference length, *μ* represents the dynamic viscosity, and *<sup>ρ</sup>* is the water density and constant *Ci* = <sup>1</sup> × <sup>10</sup>−4. Outflow condition only consists of a Neumann boundary condition that sets the gradient of velocity, turbulent kinetic energy, and pressure to zero, normal to the outflow plane. Slip condition is similar to a symmetry condition where the normal velocity and normal gradient of other variables are set to zero. No-slip condition specifies the velocities components, turbulent

kinetic energy, and normal pressure component as zero at the wall. The *ω* on the wall for a smooth surface is specified as described by Hellsten [35]:

$$
\omega\_w = \frac{\mu\_\tau^2}{\nu} \times \left(\frac{50}{4.3(y^+)^{0.85}}\right)^2,\tag{7}
$$

where *u<sup>τ</sup>* is the friction velocity and *ν* is the kinematic viscosity. Since no wall-functions are used in XCHAP, all simulations were performed with the wall resolved approach.

#### **4. Test Cases and Computational Conditions**

Fourteen common cargo vessels having a model test and full scale speed trial results are used as test cases. As the speed trials of some vessels were carried out at more than one loading condition, the total number of tests cases are 18. The LPP of the vessels are ranging from 200 m to 355 m, block coefficients (CB) vary between 0.5 and 0.84, and the Froude numbers (the achieved speed at 75% MCR) are covering the range of 0.14 to 0.23. The vessels are built in series and speed trials were performed for each sister ship. The data set consists of 78 sea trials in total. The trial measurements were conducted by the yards and analyzed by SSPA with an in-house software according to ITTC Recommended Procedures and Guidelines for Preparation, Conduct and Analysis of Speed/Power Trials [36] and ISO Ships and marine technology—Guidelines for the assessment of speed and power performance by analysis of speed trial data [37]. The trials fulfill the ISO 15016/ITTC limits on weather condition. The corresponding model tests were conducted at SSPA.

The model tests corresponding to the speed trials were performed at SSPA's towing tank (260 m long, 10 m wide and 5 m deep). The models were made of the plastic foam material Divinycell, and they were manufactured with a 5-axis CNC milling machine at SSPA. A trip wire is mounted at 5% of *L*PP aft from the fore perpendicular for the turbulence stimulation. All hull models are equipped with a dummy propeller hub and a rudder (two rudders for twin skeg hulls) for the resistance tests. The computational conditions for each test case are replicating the same conditions as in the corresponding towing tank tests such as the non-dimensional quantities, *Re* and *Fn*, loading condition, and geometrical features.

#### **5. Best Practice Guidelines and the Quality Assurance of the CFD Based Form Factor Methodology**

The proposed ITTC QA procedure consists of three steps, the first one being derivation of a Best Practice Guideline. In this section, the derivation of a best practice guideline for the CFD based form factors will be presented. Considering the plethora of numerical methods and possible CFD set-ups, it is not possible to formulate a general standard procedure for CFD-work that is generally applicable to all codes and cases [28]. Instead, each organization is required to formulate their own process (BPG) and assess its suitability for a specific application.

In order to derive the best practice guidelines for the determination of form factor using the SHIPFLOW code, CFD setups were varied systematically and verification and validation of the predicted form factors were performed. Since the validation is a key factor for the evaluations, the hulls were selected on the basis that experimentally determined form factors are suitable for the Prohaska method (i.e., *C*T/*C*FM values are fairly linear as presented in Section 5.1.6). The analysis is based on 300 double body simulations of the six test cases consisting of four hulls (H1, H2, H3, and H4) out of which one is in three different loading conditions (indicated as H2-b, H2-d, H2-s). The variations applied to the CFD set-ups are explained in Sections 5.1.1–5.1.6.

#### *5.1. Grid Dependence Study*

Grid dependence studies were performed to quantify the numerical uncertainty (USN). Four geometrically similar grids were generated for each test case. The simulations were performed in double precision in order to eliminate the round-off errors. The iterative uncertainties were quantified by the standard deviation of the force in percent of the

average force over the last 10% of the iterations. Iterative uncertainty for CF and CPV were kept below 0.01% and 0.20% for all simulations except two computations where mild separation is observed at the stern, and standard deviations are 3 to 4 times higher than the rest of the simulations. Considering the small standard deviations in both resistance components, it was assumed that the numerical errors are dominated by the discretization errors and both iterative errors and round-off errors are neglected. The procedure proposed by Eça and Hoekstra [38] was used to predict the grid uncertainties which are presented for the finest grid as a ratio of the computed value (USN%S1) in Table 1.


**Table 1.** Estimated numerical uncertainties of SHIPFLOW in model scale for EASM and *k* − *ω* SST turbulence models, in a percentage of the computed result of the finest grid, S1.

Numerical uncertainty of the viscous resistance is predicted by a combination of the frictional and viscous pressure resistance components (*UCV* = *UCF* + *UCPV* ) as *C*<sup>V</sup> is not directly computed. As seen in Table 1, grid uncertainties on CF mostly vary between 0.6 to 1.5 percent of the computed result of S1. The grid uncertainty on CPV varies greatly between different hulls. As a result of the large fluctuations in the CPV, the grid uncertainty on the viscous resistance coefficient varies between 1.1% and 10.2%. The reason for the large fluctuation in the grid uncertainties is explained by the scatter in the computed values which strongly penalizes the estimated uncertainties [38]. Computed values for CF, CPV and CV are presented in Figure 2 in a percentage of the result of the finest grid which has approximately 10M cells (*Ngrid*). As seen in Figure 2, a majority of the CF and CPV values shows somewhat oscillatory behavior, which is observed significantly more for the latter. The fluctuations stems mostly from the grid generation strategy, which is a structured grid with a stair-step profile in the stem and stern profiles. As the curvature around the bulb changes rapidly, the structured grid that captures the profile of the bulb changes abruptly with changing grid. As a result, the computed quantities are influenced by the such variations. Considering the tip of the bulb where the stagnation point is often situated and followed by a steep pressure gradient, it is expected that CPV will be influenced more than CF as observed in Figure 2.

The estimated numerical uncertainties shown in Table 1 do not indicate a particular trend for a specific ship type or loading condition. Even though USN varies significantly between the test cases due to the drawbacks of the grid generation, its reflection on CV is limited and the variation on the predicted form factors is rather small, especially between the finest two grids. Therefore, the g2 grid settings have been chosen as a baseline for the rest of the BPG investigation since the grid cell count were reduced to approximately 7 million, and computational time was shortened compared to the finest grid.

**Figure 2.** Computed values for (**a**) CF, (**b**) CPV and (**c**) CV with the EASM model and (**d**) CF, (**e**) CPV and (**f**) CV with the *k* − *ω* SST model vs. grid refinement ratio, *h*1/*hi* = <sup>3</sup> *Ngrid*1/*Ngridi* .
