# Comparative Study of Potential Flow and CFD in the Assessment of Seakeeping and Added Resistance of Ships

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## Abstract

**:**

^{®}, a viscous flow solver with a turbulence model; it is based on the finite volume method (FVM) combined with a volume-of-fluid (VOF) technique for sea-surface evolution. The study is carried out for two ship seakeeping cases in head-sea regular waves, respectively, without and with ship forward speed. The first case refers to a 6750 TEU containership scale model developed at the LHEEA laboratory in Nantes for a benchmark study, providing experimental data for all test cases. Pitch and heave response is calculated and compared with the experimental values. The second case refers to a KRISO container ship, an extensively researched hull model in ship hydrodynamics. In addition to the pitch and heave, added resistance is also calculated and compared with the experimental values. Hence, it provides a comprehensive basis for a comparative analysis between the selected solvers. The results are systematically analyzed and discussed in detail. For both cases, deterioration of the PF solution with increasing wave steepness is observed, thus suggesting limitations in the modeled nonlinear effects as a possible reason. The accuracy of the CFD solver greatly depends on the spatial discretization characteristics, thus suggesting the need for grid independence studies, as such tools are crucial for accurate results of the examined wave–body interaction scenarios.

## 1. Introduction

^{®}[10]. Furthermore, appropriate turbulence modeling needs to be addressed for wave generation if turbulence is expected to matter for the examined wave–body interaction problems. The most used turbulence models in marine applications are k-ε and k-ω SST [11]. However, with both models, numerical instability occurs with free-surface waves through the non-physical build-up of turbulent viscosity, thus creating high and unrealistic damping of the waves [12]. The overall influence of turbulence modeling in seakeeping simulations within CFD is speculative, since the pressure and velocity gradients in the flow are very high near the hull, making single-phase turbulence models rogue [13]. Moctar et al. [14] even suggest omitting fine boundary layers along the hull if viscous effects are to be neglected. However, further investigations are needed in this area. In the present paper, a seakeeping response assessment, with added resistance included, is performed for two ships for which model-test results are available. The first case concerns a 6750 TEU containership model with zero speed in head-sea regular waves, in which pitch and heave responses are analyzed. Preliminarily, a thorough investigation was performed regarding the mesh quality within CFD for the propagation of waves with a two-dimensional (2D) numerical wave tank; an assessment was performed for five different wave heights and keeping the wavelength fixed and equal to the ship length. The second case refers to the KRISO containership model advancing in head-sea waves. Both cases are investigated with two numerical tools for marine hydrodynamics: (A) the commercial code Wasim from DNV classification society, based on potential flow theory, i.e., a Rankine panel method, and including second-order nonlinear effects within a perturbation approach, hereafter briefly indicated as PF; and (B) the open source CFD toolkit OpenFOAM

^{®}based on the co-located finite volume method (FVM), hereafter briefly indicated as CFD. This comprehensive study provides additional insights into the performance of these two hydrodynamic tools for estimating ship response and added resistance in waves, their shortcomings, and their advantages. The rest of this paper is structured as follows. In Section 2, the experimental setups and test matrices for the two studied cases are outlined. The two selected numerical tools are described in Section 3. In the same section, the incident-wave parameters for the first experimental case are used to carry out a systematic analysis to identify the grid size and time-step needed to limit the numerical errors of the CFD solver. The results from all seakeeping simulations are documented in Section 4, together with their comparison against the corresponding physical data, and the main conclusions are drawn in Section 5.

## 2. Benchmark Tests

#### 2.1. 6750 TEU Containership

_{1}, k

_{2}, k

_{3}and, k

_{4}, all of which have the same stiffness of 56 N/m. The angle α between pairs of springs at the bow and stern is equal to 45°. The mooring arrangement is shown in Figure 2.

#### 2.2. KRISO Container Ship

## 3. Numerical Methods

^{®}is an open-source type of software; hence, detailed information about the numerical set-up ensures the reproducibility of this study. Turbulence modeling is summarized for such applications with appropriate boundary conditions used.

#### 3.1. Numerical Schemes—Potential Flow

_{h}, Equation (1).

_{x}represents the smallest panel length in the longitudinal ship direction. For more detailed information on this, the reader is referred to [24]. Computational meshes are shown in Figure 4 for the 6750 TEU containership and KRISO containership, respectively. HydroMesh utility within Wasim offers fast, automatic meshing of both the hull and free surface.

#### 3.2. Numerical Schemes—CFD

^{®}, in which the co-located finite volume method is applied to solve fluid mechanics equations. Fluid flow is governed by Navier–Stokes (NS) equations for incompressible flows, the continuity Equation (2) and momentum Equation (3):

**u**is the local fluid velocity. The LHS of the NS equation contains the total derivative, i.e., the change of velocity in time, convective term and viscous term, respectively. $\nabla p$ is the pressure gradient,

**g**is the gravitational acceleration and $\nu $ stands for effective kinematic viscosity. For solving these equations, the interFoam solver is engaged for two incompressible, isothermal immiscible fluids (water and air). The air–water interface is modeled using the Volume of Fluid method (VOF), in which indicator function α is introduced into the partial differential equations. The α function represents a scalar field that enters the NS equations through density ρ and effective kinematic viscosity $\nu $, as depicted in Equations (4) and (5), respectively.

#### 3.2.1. Two-Dimensional Numerical Wave Tank

^{®}, setting the breadth (i.e., the domain size normal to the flow motion) equal to one-cell size and using proper boundary conditions to ensure two-dimensional wave conditions. Due to the reduced number of cells, this procedure provides results within minutes. Relaxation zones [9] are employed to tackle the problem of wave reflections at the outlet and artificial velocities at the inlet during wave build-up. The length of the relaxation zones is determined as a function of the wavelength λ of the targeted generated waves, as proposed in [26]. The second-order Stokes wave model is used as input for the wave kinematics. Five different wave heights are tested at a given λ, corresponding to the experimental waves listed in Table 1. To measure the simulated wave height, a virtual wave gauge is placed at a longitudinal distance from the wave-generation side that corresponds to the forward perpendicular of the ship model (when the seakeeping simulations are performed); this position is hereafter indicated as L

_{PP}. The main individual zones of the numerical wave tank are depicted in Figure 5.

_{i}is the local fluid velocity and ∆x

_{i}is the local cell size in the x direction. Here, to reduce the numerical dissipation at the free surface, the Courant number is kept below 0.2. As for the number of cells per wave height N, three grid densities are examined while keeping the Courant number at the free surface equal to 0.2, so the time-step is modified consistently. Figure 2 confirms the significant influence of N for the development of the free surface wave prescribed for case 3 (see parameters in Table 1) in terms of the wave elevation at the selected virtual gauge. Time histories appear very coarse due to the plotting time interval set to 0.2 s. Asymmetry between wave trough and crest also appears, which happens due to nonlinearities in free surface flows for the examined wave parameters. Turbulence modeling was not employed, since turbulence effects are expected to play a negligible role for these scenarios of wave propagation, as we have in mind sufficiently long gravity-driven waves. Nevertheless, it was also investigated, as seen in Figure 6. After 17 s of simulation time, the damping of the wave height becomes more exaggerated for the targeted wave height of 0.23 m.

_{max}, and minimum, η

_{min}, values of the wave elevation are estimated at the position of x = LPP in a time interval nT, equal to n = 7 wave periods T, when nearly steady-state conditions are established. This is used to estimate the wave height in each oscillation period kT, with $1\le k\le n$, as shown in Equation (7). The mean value of the wave height is calculated according to Equation (8), and the percentage error is estimated using Equation (9), where H is the targeted wave height given in Table 1 for each wave case. Corresponding results for N = 8 are given in Table 5. while the influence of N on the wave amplitude error is documented in Figure 7 for case 3.

^{®}. A computational grid with refinement zones within free surface for case 5 is shown in Figure 8.

#### 3.2.2. 6750 TEU Containership: Computational Grid, Linear Solvers and Boundary Conditions

^{®}are covered in [29]. The refined mesh in the vicinity of the hull can be seen in Figure 12.

^{−8}. Velocity components and turbulence terms are solved using a smooth solver, i.e., Gauss–Seidel smoother with a residual tolerance of 1

^{−8}.

#### 3.2.3. KRISO Containership: Computational Grid, Linear Solvers and Boundary Conditions

^{+}on the hull of approximately 40. These values are determined based on the author’s previous experience with such calculations. The K-ω SST turbulence model is applied for calm water conditions, where turbulent kinetic energy k for the far-field boundary conditions is specified as:

_{fs}stands for free stream velocity and I is the turbulence intensity, which is assumed at 3%. The specific dissipation rate is defined following the guideline from Eça and Hoekstra [32]:

_{PP}is the length between the ship perpendiculars. The realizable k-ε turbulence model is chosen for wave conditions, since the dissipation and dispersion errors in the wave propagation are sufficiently small for short-duration simulations including regular waves if compared to the k-ω SST model. It is assumed that using different turbulent models will not affect any of the physical quantities being investigated in the paper, i.e., rigid body motions and second-order longitudinal force (added resistance in waves). Computational grids are shown in Figure 14 and Figure 15 for calm water and wave conditions, respectively. For the wave application, the mesh in the free-surface zone is refined in horizontal and vertical directions, thus drastically increasing the number of cells. As for the linear solvers, for asymmetrical matrices (turbulence terms, velocity components and α), the setup is the same as depicted in Section 3.2, while for symmetrical matrices, i.e., pressure equation, a Generalized Geometric–Algebraic multigrid solver (GAMG) is chosen with a Gauss–Seidel smoother with a residual tolerance of 1

^{−7}. Computational grids for the KRISO containership are outlined in Figure 14 and Figure 15.

## 4. Results and Discussion

^{®}for visualization and python Spyder for Fourier transformations and plots.

#### 4.1. Heave and Pitch Response of the 6750 TEU Containership

#### 4.2. Calm Water Resistance with Free Sinkage and Trim of KRISO Containership

_{T}, Equation (12), in which R

_{T}stands for total resistance,

**r**being the fluid density, U the speed of the ship and S

_{0}the wetted surface area at rest:

^{7}for model scale.

_{T}is the resistance coefficient, θ

_{t}is the trim angle in degrees, and z

_{s}is the sinkage in meters, normalized with L

_{PP.}

#### 4.3. Heave, Pitch and Added Resistance of KRISO Containership

_{e}, see Figure 22, Figure 23, Figure 24, Figure 25 and Figure 26. Furthermore, transfer functions are graphically shown to satisfy the more conventional presentation. Regarding the fact that the PF tool, being a seakeeping solver, does not provide calm water resistance, the calm water component of the total resistance coefficient in Equation (12) for Wasim is taken from CFD results. Guidelines for the Fourier transform of the signals are given in [24].

#### 4.4. Discussion

^{®}. In Figure 22c, no report is made for the added resistance from Wasim, since the resonant behavior in the measurements was reported. Interestingly, resonance is also captured in CFD with two force peaks over one encounter period. Observing the total resistance coefficient, i.e., added resistance from Wasim, higher harmonics are not captured.

## 5. Conclusions

^{®}. In the study, two different cases are investigated, the first being a 6750 TEU containership scale model without forward speed, subjected to steep head waves. Heave and pitch response are analyzed, while only the latter is compared to the experiment. Regarding heave motion, a different trend of amplitudes is revealed from Wasim and OpenFOAM

^{®}. For Wasim, the amplitude is significantly smaller for the bigger wave heights compared to the CFD solver. For the pitch motion, the underestimation of the motion by the potential flow is clear. CFD shows better agreement with the experimental values, leading to the conclusion that for increasingly steep waves, heave can be underestimated with the potential flow tools with Wasim features. The use of CFD can also ensure better accuracy, but grid independence studies should be carried out. It is particularly evident in the second case of this work: a KRISO containership subjected to head waves with forward towing speed. Rigid body motions, i.e., heave and pitch, agree quite well for CFD, while the solution from PF differs for all cases. For the viscous flow method, differences in the experiment for lower wave heights are attributed to an insufficient number of computational cells per wave height, regardless of the accurate incident wave. Regarding second-order effects such as added resistance, CFD also shows better agreement. From a practical point of view, the constant towing speed and the restricted surge motion strongly affect the forces acting on the hull. Hence, the solution is not entirely suitable for estimating the sea margin for a real ocean-going ship. Clearly, the CFD solver will yield a more accurate solution, in which careful modeling of the surge response should be taken care of. Comparing the solutions for the potential flow in both cases, it is clear that the percentage difference is of a very different order of magnitude. This indicates that the forward speed effect in the PF tool should be studied in detail using the double-body linearization method, free-surface nonlinearities, etc. This is the subject of further planned work in future studies. Investigations of seakeeping performance with stabilized turbulence models within CFD are scarce, making it a fruitful area for further research. As a general guideline for the choice of a potential flow tool, or a fully viscous CFD for seakeeping or added resistance derived from this work: If a very sharp accuracy of the solution is required, i.e., a difference of less than 3 or 5%, then a fully viscous CFD is an option, considering the expected high computational cost. If such high accuracy is not of utmost importance, then a potential flow tool with features such as those analyzed here is the right choice, taking care that the vessel is in a linear region of motion, i.e., the height of the incoming waves is relatively low with respect to their characteristic length.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Joakim Tveiten Vigsnes Seakeeping Analysis Comparison between Viscous and Inviscid CFD. Master Thesis. Available online: https://ntnuopen.ntnu.no/ntnu-xmlui/handle/11250/2566949 (accessed on 10 January 2023).
- Katsidoniotaki, E.; Göteman, M. Numerical modeling of extreme wave interaction with point-absorber using OpenFOAM. Ocean Eng.
**2022**, 245, 110268. [Google Scholar] [CrossRef] - Kim, S.-G.P. CFD as a seakeeping tool for ship design. Int. J. Nav. Archit. Ocean Eng.
**2011**, 3, 65–71. [Google Scholar] [CrossRef][Green Version] - Dorozhko, V.M.; Bugaev, V.G.; Kitaev, M.V. CFD Simulation of an Extreme Wave Impact on a Ship. In Proceedings of the Twenty-fifth International Ocean and Polar Engineering Conference, Kona, HI, USA, 21–26 June 2015. [Google Scholar] [CrossRef]
- Bi, X.; Zhuang, J.; Su, Y. Seakeeping Analysis of Planing Craft under Large Wave Height. Water
**2020**, 12, 1020. [Google Scholar] [CrossRef][Green Version] - Gao, Z.; Wang, Y.; Su, Y.; Chen, L. Validation of a combined dynamic mesh strategy for the simulation of body’s large amplitude motion in wave. Ocean Eng.
**2019**, 187, 106169. [Google Scholar] [CrossRef] - Galbraith, A.; Boulougouris, E. Parametric Rolling of the Tumblehome hull using CFD. In Proceedings of the 12th International Conference on the Stability of Ships and Ocean Vehicles, Scotland, UK, 14–19 June 2015. [Google Scholar]
- Shen, Z.; Wan, D.; Carrica, P.M. Dynamic overset grids in OpenFOAM with application to KCS self-propulsion and maneuvering. Ocean Eng.
**2015**, 108, 287–306. [Google Scholar] [CrossRef] - Jacobsen, N.G.; Fuhrman, D.R.; Fredsøe, J. A wave generation toolbox for the open-source CFD library: OpenFoam®: Wave Generation Toolbox. Int. J. Numer. Methods Fluids
**2012**, 70, 1073–1088. [Google Scholar] [CrossRef] - Weller, H.G.; Tabor, G.; Jasak, H.; Fureby, C. A tensorial approach to computational continuum mechanics using object-oriented techniques. Comput. Phys.
**1998**, 12, 620. [Google Scholar] [CrossRef] - Pena, B.; Huang, L. A review on the turbulence modelling strategy for ship hydrodynamic simulations. Ocean Eng.
**2021**, 241, 110082. [Google Scholar] [CrossRef] - Larsen, B.E.; Fuhrman, D.R. On the over-production of turbulence beneath surface waves in Reynolds-averaged Navier–Stokes models. J. Fluid Mech.
**2018**, 853, 419–460. [Google Scholar] [CrossRef][Green Version] - Gatin, I.; Vukčević, V.; Jasak, H.; Seo, J.; Rhee, S.H. CFD verification and validation of green sea loads. Ocean Eng.
**2018**, 148, 500–515. [Google Scholar] [CrossRef] - el Moctar, B.O.; Schellin, T.E.; Söding, H. Numerical Methods for Seakeeping Problems; Springer: Cham, Switzerland, 2021; ISBN 978-3-030-62561-0. [Google Scholar]
- Bouscasse, B.; Merrien, A.; Horel, B.; Hauteclocque, G.D. Experimental analysis of wave-induced vertical bending moment in steep regular waves. J. Fluids Struct.
**2022**, 1111, 103547. [Google Scholar] [CrossRef] - Kim, Y.; Kim, J.-H. Benchmark study on motions and loads of a 6750-TEU containership. Ocean Eng.
**2016**, 119, 262–273. [Google Scholar] [CrossRef][Green Version] - Kim, W.J.; Van, S.H.; Kim, D.H. Measurement of flows around modern commercial ship models. Exp. Fluids
**2001**, 31, 567–578. [Google Scholar] [CrossRef] - Larsson, L.; Stern, F.; Visonneau, M. Numerical Ship Hydrodynamics: An Assessment of the Gothenburg 2010 Workshop; Springer: Dordrecht, The Netherlands, 2014; ISBN 978-94-007-7189-5. [Google Scholar]
- Simonsen, C.D.; Otzen, J.F.; Joncquez, S.; Stern, F. EFD and CFD for KCS heaving and pitching in regular head waves. J. Mar. Sci. Technol.
**2013**, 18, 435–459. [Google Scholar] [CrossRef] - Tokyo 2015: A Workshop on CFD in Ship Hydrodynamics. Available online: https://www.t2015.nmri.go.jp/Instructions_KCS/Case_2.10/Case_2-10.html (accessed on 10 January 2023).
- Heo, J.; Park, D.; Berg-Jensen, J.H.; Pan, Z.; Vada, T.K. Hull Form Optimization to Fulfil Minimum Propulsion Power by Using Frequency and Time Domain Potential Flow Solvers. In Practical Design of Ships and Other Floating Structures; Okada, T., Suzuki, K., Kawamura, Y., Eds.; Lecture Notes in Civil Engineering; Springer: Singapore, 2021; Volume 65, pp. 220–236. ISBN 9789811546792. [Google Scholar]
- Raven, H.C. A Solution Method for the Nonlinear Ship Wave Resistance Problem; TU Delft: Delft, The Netherlands, 1996; ISBN 978-90-75757-03-3. [Google Scholar]
- Kim, K.-H.; Kim, Y.-H. Numerical Analysis of Added Resistance on Ships by a Time-domain Rankine Panel Method. J. Soc. Nav. Archit. Korea
**2010**, 47, 398–409. [Google Scholar] [CrossRef] - SESAM USER MANUAL—Wasim: Wave Loads on Vessels with Forward Speed. DNV, Norway. 2004. Available online: https://manualzz.com/doc/7270848/wasim-user-manual (accessed on 10 January 2023).
- Weller, H.G. Bounded Explicit and Implicit Second-Order Schemes for Scalar Transport; OpenCFD Ltd.: Bracknell, UK, 2006. [Google Scholar]
- Constance, C. Investigation of Floating Offshore Wind Turbine Hydrodynamics with Computational Fluid Dynamics. Ph.D. Thesis, University of Rouen Normandy, Rouen, France, 2021. [Google Scholar]
- Shen, Z.; Hsieh, Y.-F.; Ge, Z.; Korpus, R.; Huan, J. Slamming Load Prediction Using Overset CFD Methods. In Proceedings of the Offshore Technology Conference, Houston, TX, USA, 2–5 May 2016; p. D011S014R004. [Google Scholar]
- Windt, C.; Davidson, J.; Schmitt, P.; Ringwood, J. On the Assessment of Numerical Wave Makers in CFD Simulations. J. Mar. Sci. Eng.
**2019**, 7, 47. [Google Scholar] [CrossRef][Green Version] - Lemaire, S.; Vaz, G.; Deij-van Rijswijk, M.; Turnock, S.R. On the accuracy, robustness, and performance of high order interpolation schemes for the overset method on unstructured grids. Int. J. Numer. Methods Fluids
**2022**, 94, 152–187. [Google Scholar] [CrossRef] - OpenFOAM 10 C++ Source Code Guide. Available online: https://cpp.openfoam.org/v10/ (accessed on 10 January 2023).
- ITTC—Recommended Procedures and Guidelines (Practical Guidelines for Ship CFD Applications). Available online: https://ittc.info/media/1357/75-03-02-03.pdf (accessed on 15 January 2023).
- Eça, L.; Hoekstra, M. The numerical friction line. J. Mar. Sci. Technol.
**2008**, 13, 328–345. [Google Scholar] [CrossRef] - Roenby, J.; Bredmose, H.; Jasak, H. A computational method for sharp interface advection. R. Soc. Open Sci.
**2016**, 3, 160405. [Google Scholar] [CrossRef] [PubMed][Green Version]

**Figure 4.**Panel meshes for PF solution for both cases: (

**a**) 6750 TEU containership mesh; (

**b**) KRISO containership mesh.

**Figure 6.**Influence of realizable k-ε turbulence model on wave height dissipation at wave gauge at LPP, after 10 oscillation periods and for N = 8.

**Figure 7.**Influence of a number of cells per wave height, N, on wave dissipation for case 3, as defined in Table 1.

**Figure 14.**CFD computational domain of the KRISO containership in calm water conditions: (

**a**) Kelvin wake refinements; (

**b**) boundary layer.

**Figure 18.**Difference of numerical and experimental nondimensional pitch amplitudes the 6750 TEU containership.

**Figure 19.**6750 TEU containership in wave case 5. (

**a**) Experiment photography; (

**b**) OpenFOAM snapshot.

**Figure 20.**Time signals of pitch motion from experiment and numerics for H = 0.46 m for the 6750 TEU containership.

**Figure 21.**Time histories of horizontal forces on the KRISO containership in calm water and Re = 1047 × 10

^{7}.

**Figure 22.**Time histories for case 1 of the KRISO containership: nondimensional (

**a**) heave, (

**b**) pitch, (

**c**) total resistance.

**Figure 23.**Time histories for case 2 of the KRISO containership: nondimensional (

**a**) heave, (

**b**) pitch, (

**c**) total resistance.

**Figure 24.**Time histories for case 3 of the KRISO containership: nondimensional (

**a**) heave, (

**b**) pitch, (

**c**) total resistance.

**Figure 25.**Time histories for case C4: nondimensional (

**a**) heave, (

**b**) pitch, (

**c**) total resistance for the KRISO containership.

**Figure 26.**Time histories for case 5 of the KRISO containership: nondimensional (

**a**) heave, (

**b**) pitch, (

**c**) total resistance.

**Figure 30.**Difference of numerical and experimental heave transfer functions for the KRISO containership.

**Figure 31.**Difference of numerical and experimental pitch transfer functions for the KRISO containership.

**Figure 32.**Difference of numerical and experimental total resistance coefficients for the KRISO containership.

**Figure 33.**Top view snapshots from the simulations for case 3 of the KRISO containership: (

**a**) OpenFOAM; (

**b**) Wasim.

Units | Model | Full Scale | |
---|---|---|---|

LPP, length between perpendiculars | m | 4.41 | 286.6 |

B, breadth | m | 0.615 | 40 |

T, draught | m | 0.1843 | 11.98 |

∆, deadweight | kg | 312.61 | 85,849,972 |

LCG, longitudinal center of gravity | m | 2.13 | 143.7 |

VCG, vertical center of gravity | m | 0.256 | 16.66 |

Case Number | Wave Height H (m) | Wavelength λ (m) | Steepness H/λ % |
---|---|---|---|

1. | 0.09 | 4.41 | 2.1 |

2. | 0.17 | 4.41 | 3.8 |

3. | 0.23 | 4.41 | 5.2 |

4. | 0.38 | 4.41 | 8.7 |

5. | 0.45 | 4.41 | 10.5 |

Case Number | Wave Height H (m) | Wavelength λ (m) |
---|---|---|

1. | 0.062 | 3.949 |

2. | 0.078 | 5.164 |

3. | 0.123 | 6.979 |

4. | 0.149 | 8.321 |

5. | 0.196 | 11.840 |

Units | Model | |
---|---|---|

L_{PP} | m | 6.0702 |

B | m | 0.8498 |

T | m | 0.2850 |

∆ | kg | 956 |

LCB(%L_{PP}), fwd+ | m | −1.48 |

VCG | m | 0.378 |

S | m^{2} | 6.697 |

U | m/s | 2.017 |

Case Number | $\overline{{\mathit{H}}_{\mathit{L}\mathit{P}\mathit{P},\mathit{n}}}$ (m) | e (%) |
---|---|---|

1. | 0.086 | 4.28 |

2. | 0.162 | 4.37 |

3. | 0.22 | 3.12 |

4. | 0.37 | 2.81 |

5. | 0.45 | 2.52 |

Case Number | $\overline{{\mathit{H}}_{\mathit{L}\mathit{P}\mathit{P},\mathit{n}}}$ (m) | e (%) |
---|---|---|

T/200. | 0.222 | 3.32 |

T/400. | 0.222 | 3.17 |

T/600. | 0.222 | 3.07 |

**Table 7.**Difference of nondimensional numerical and experimental pitch amplitudes for the 6750 TEU containership.

H, m | 0.09 | 0.17 | 0.23 | 0.38 | 0.46 |
---|---|---|---|---|---|

OpenFOAM, % | 3.06 | 5.08 | 0.05 | 1.60 | 1.96 |

Wasim, % | 0.82 | 1.46 | 3.9 | 2.03 | 3.33 |

R_{e} = 1047 × 10^{7} | C_{T} | θ_{T} | z_{s}/L_{PP} |
---|---|---|---|

CFD | 3.835 | −0.152 | −0.001 |

EFD | 4.096 | −0.165 | −0.002 |

(EFD/CFD)∙100% | 6.372 | 7.87 | 50.771 |

**Table 9.**Difference of numerical and experimental heave transfer functions for the KRISO containership.

H (m) | 0.062 | 0.078 | 0.123 | 0.149 | 0.196 |
---|---|---|---|---|---|

OpenFOAM, % | −14.33 | −26.83 | −3.76 | −0.42 | +0.62 |

Wasim, % | −33.81 | +25.12 | −31.59 | −9.75 | −0.348 |

**Table 10.**Difference of numerical and experimental pitch transfer functions for the KRISO containership.

H (m) | 0.062 | 0.078 | 0.123 | 0.149 | 0.196 |
---|---|---|---|---|---|

OpenFOAM, % | −30.41 | +26.46 | −10.21 | +2.40 | +2.09 |

Wasim, % | −36.20 | +56.85 | −38.62 | −36.42 | −31.76 |

**Table 11.**Difference of numerical and experimental total resistance coefficients for the KRISO containership.

H (m) | 0.062 | 0.078 | 0.123 | 0.149 | 0.196 |
---|---|---|---|---|---|

OpenFOAM, % | −40.51 | −38.82 | +19.61 | −16.20 | −0.79 |

Wasim, % | −44.36 | −40.69 | n.a.* | +22.64 | +12.76 |

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## Share and Cite

**MDPI and ACS Style**

Sulovsky, I.; Hauteclocque, G.d.; Greco, M.; Prpić-Oršić, J. Comparative Study of Potential Flow and CFD in the Assessment of Seakeeping and Added Resistance of Ships. *J. Mar. Sci. Eng.* **2023**, *11*, 641.
https://doi.org/10.3390/jmse11030641

**AMA Style**

Sulovsky I, Hauteclocque Gd, Greco M, Prpić-Oršić J. Comparative Study of Potential Flow and CFD in the Assessment of Seakeeping and Added Resistance of Ships. *Journal of Marine Science and Engineering*. 2023; 11(3):641.
https://doi.org/10.3390/jmse11030641

**Chicago/Turabian Style**

Sulovsky, Ivan, Guillaume de Hauteclocque, Marilena Greco, and Jasna Prpić-Oršić. 2023. "Comparative Study of Potential Flow and CFD in the Assessment of Seakeeping and Added Resistance of Ships" *Journal of Marine Science and Engineering* 11, no. 3: 641.
https://doi.org/10.3390/jmse11030641