The Maritime Safety Committee (MSC) under the International Maritime Organization (IMO) has established regulations for the safe maneuvering performance of ships since the 1960s by enacting standards on ship maneuvering performance evaluation (IMO, 2002a; 2002b). The evaluation of ship maneuvering performance is essential in the design stage because the maneuvering performance should satisfy the global standards in the sea trial.

Shipyards have mostly evaluated the maneuvering performance of the ship hull based on the data of previously built ships and empirical formulas. This method enables a simple evaluation compared with other methods, but its application is difficult when an innovative hull is considered(Kim et al., 2020; Yeo et al., 2020).

The model test method can be used to evaluate maneuvering performance of the hull that the basic design is already completed. The maneuvering performance can be evaluated for a single and/or multiple hull(s) since the model ship should be fabricated. In the model test, captive and free running model test techniques can be used (Kim et al., 2009; Im and Seo, 2010; Shin and Choi, 2011; Choe and Im, 2016). In a captive model test, the maneuvering coefficients of a ship are estimated by measuring loads with constrained motion, which limits the degrees of freedom of the ship. Oblique tests can be conducted in a towing tank, the maneuvering coefficients, however related to the turning motion of the ship are difficult to estimate. Instead, the rotating arm (RA) and computerized planar motion carriage (CPMC) tests can be conducted in a large rectangular/circular tank for maneuvering coefficients related to turning motion. These model tests require large facility and equipment, test scheduling and model fabrication. Furthermore, there are still an issue on scale effects on the maneuvering performance (Yun et al., 2021; ITTC, 2021).

Recent advances in computing resources have enabled the evaluation of ship maneuvering performance using computational fluid dynamics (CFD) (Hajivand and Mousavizadegan, 2015; Sung and Park, 2015; He et al., 2016). CFD can reflect changes in hull design relatively freely compared with model tests; however, the robustness and accuracy of the applied numerical methodology and the numerical solver related to the mesh must be verified.

This study aimed to examine the applicability of foamStar, an OpenFOAM-based wave-structure coupled analysis software program, to ship maneuvering performance evaluation. foamStar is an in-house code jointly developed by Bureau Veritas, École Centrale de Nantes (ECN), and Pusan National University. It is primarily aimed at ship motion in waves, numerical simulation techniques for the global performance of marine structures and their verification, and analyzing the behavior of rigid/elastic bodies in waves (Seng, 2012; Li et al., 2021; Aliyar et al., 2022; Engel et al., 2023). To verify the applicability of foamStar to maneuvering performance evaluation, we conducted a numerical analysis to estimate the maneuvering coefficients of the hull of the Korea Research Institute of Ships & Ocean Engineering (KRISO) container ship (KCS), which is an open hull form. The same meshes and numerical techniques used in a study on the numerical analysis of the seakeeping performance of the KCS hull in waves using foamStar (Descamps, 2022) were utilized. Descamps (2022) simulated the ship’s resistance with a uniform stream flow incident to the hull, but this study conducted resistance simulation in calm water using the moving mesh method. The ship’s resistance are compared with previous numerical and experimental results for the validation (Descamps, 2022; Hino et al., 2020). Then, the adopted moving mesh method was also used to conduct numerical analysis for oblique and rotating arm tests to evaluate the hydrodynamic coefficients of KCS hull, and the obtained hydrodynamic coefficients are compared with other numerical and experimental results.

In this study, the Reynolds-averaged Navier–Stokes equation (RANSE) was applied for the viscous flow analysis. The continuity equation, which is the governing equation for a fluid, is expressed by Eq. (1), and the Reynolds-averaged Navier–Stokes momentum equation in Eq. (2). In addition, the volume of fluid (VOF) was introduced to calculate the two-phase problem to consider free-surface effects. The VOF transport equation is expressed in Eq. (3). The free-surface SSTk – ω model that based on the SSTk – ω turbulence model considering the existence of free surface was utilized (Menter, 1994; Larsen and Fuhrman, 2018; Kim, 2021).

where u, ρ, p_d, μ_eff, and g are the fluid velocity, fluid density, dynamic pressure, effective viscosity, and gravitational acceleration, respectively. α is the VOF value, which has α ∈ [0.1]. α = 0 and α = 1 represent cases in which the computational mesh is filled with air and water, respectively, and values between 0 and 1 represent an interface between water and air. The Mathematical Modeling Group (MMG) model (Okuda, 2023) was utilized as the formula to estimate the ship maneuvering coefficients. The coordinate system used in this study is shown in Fig. 1. u is the surge speed, υ_m is the sway speed at midship, r is the yaw rate, ψ is the yaw, U=u2+vm2, and the drift angle is defined as β = tan⁻¹ (−υ_m/u), respectively.

In this study, the KCS hull was used for the numerical analysis. The geometry with a scale ratio of λ = 37.98 was used, and it is shown in Fig. 2. Information on the full-scale KCS and the model scale used is listed in Table 1. The geometries of the bow and stern in the coarse, medium, and fine meshes used for the numerical analysis are shown in Fig. 3. The schematic view on the entire mesh system with numeric setting are given in Figs. 4 and 5 and Table 2. The relaxation method applied to waves2Foam was utilized to eliminate the generated waves with an improved dynamic polynomial weight technique demonstrated in previous studies (Jacobsen et al., 2012; Seng, 2012; Choi, 2019). The VOF and flow velocity were gradually mixed with the target values in the relaxation zone to absorb waves, as described below.

where φ is physical quantities such as VOF and fluid velocity, in the present study ω is weight function, and φ^Target is the physical quantity targeted in relaxation zones. Table 3 lists the number of computational cells used during the numerical analysis and the maximum non-orthogonality of the meshes.

In this study, the moving mesh method was considered to evaluate the maneuvering performance. The resistance simulation of KCS in the calm water was performed to verify the moving mesh method prior to the OTT and RA simulations. In the resistance simulation, the KCS hull, which included the bare hull and rudder, was used in the same manner as in Case 2.10 of the CFD Tokyo 2015 Workshop (Hino et al., 2020). Tables 4 and 5 list the speed conditions and constraints of the resistance simulation. The resistance simulation in the calm water was conducted under four conditions, each described in Table 6. REL-C was the condition with a stationary mesh, and a relative flow velocity corresponding to the forward speed was applied. For MOV-C, MOV-M, and MOV-F, the mesh advanced according to the forward speed of the ship. Fig. 6 shows the mesh movement of MOV-C. Fig. 7 shows the total resistance time-series data, forward speed of the mesh, and pitch time-series data for each case. The total resistance converged as the mesh advanced along with the ship, but its periodic perturbations were observed. This was owing to the ship’s pitch and heave motions caused by the initial flow disturbance. The wave elevation at the final time step for each case is shown in Fig. 8. Fig. 9 shows the wave elevation profile in the nondimensionalized y/L_pp and x/L_pp directions presented at the CFD Tokyo 2015 Workshop (Hino et al., 2020). x/L = 0 is forward perpendicular (FP), whereas x/L= 1 is aft perpendicular (AP). When the stationary mesh (REL-C) was compared with the moving mesh, the overall wave patterns are similar. The small discrepancy of wave profile is understood that the heave and pitch motions of ship for moving mesh simulation.

Table 7 presents the results of the resistance simulation for calm water. Compared with the EFD results, REL-C exhibited an error of +1.19%, MOV-C an error of +3.40%, MOV-M an error of +1.24%, and MOV-F an error of −1.71%. These results are presented in Fig. 10 along with those of other studies that conducted resistance analyses under the same scale and Froude number conditions (Wang et al., 2020; Mandru et al., 2024; Filip et al., 2017). The results of all the meshes used in this study showed an error of less than 4% compared with the test results, and the medium and fine meshes exhibited an error of ±2% or less. This was within the error range of other research results, confirming that the meshes and numerical method used can be applied to the KCS hull flow analysis. In other words, a numerical technique that uses a moving mesh can be used in OTT and RA simulations to estimate the hydrodynamic coefficients of a ship.

The OTT simulation for the KCS hull in calm water was compared with the experimental and other CFD results (Franceschi et al., 2023). The KCS hull with the same scale ratio (λ = 37.98) used in the resistance simulation for calm water was utilized. Same meshes as those used in the resistance simulation for calm water were used. These constraints are listed in Table 8. “Forced” in the table means that the conditions were given for forced motion under the test conditions. The OTT simulation setup is listed in Table 9. The positions of the computational mesh at 0 and 30 s under the β = 12° condition are shown in Fig. 11. The sway force time-series data and yaw moment time-series data at β = 0°, 3°, 6°, 9°, 12° and 18° for each mesh are shown in Fig. 12. The wave elevations of the coarse, medium, and fine meshes at β = 12° are shown in Fig. 13.

The OTT simulation results for calm water are shown in Fig. 14 along with the experimental and other CFD results (Franceschi et al., 2023). Figs. 14(a) and 14(b) show the analysis results of the nondimensionalized sway force and yaw moment with respect to β when the coarse mesh was used, Figs. 14(c) and 14(d) show the results when the medium mesh was used, and Figs. 14(e) and 14(f) show the results when the fine mesh was used. These analysis results were similar to the experimental and other CFD results. Among the maneuvering coefficients calculated using only the OTT simulation results for still water, the results of Yv′′ and Nv′′, which are linear hydrodynamic coefficients, are listed in Table 10. In addition, the grid convergence index was calculated using the Nv′′ coefficient for each grid based on the recommended procedure of the International Towing Tank Conference (ITTC, 2024), and the results are listed in Table 11.

The RA simulation for the KCS hull in still water was compared with the experimental and CFD results of the OTT simulation (Franceschi et al., 2023). The KCS hull with the same scale ratio (λ = 37.98) and meshes was utilized. The ship motion was also constrained in the RA for forced turning. The RA simulation setup is provided in Table 12. The positions of the computational mesh at 0, 10, 20, and 30 s under the r’ = 0.4, β = 12° condition are shown in Fig. 15. In addition, the wave elevations of the coarse, medium, and fine meshes at r’ = 0.4 and β = 12° are shown in Fig. 16.

The RA simulation results for calm water are shown along with the experimental and other CFD results (Franceschi et al., 2023) in Figs. 17 and 18, respectively. Figs. 17 and 18 show the nondimensionalized sway force and yaw moment according to the computational mesh for the nondimensional yaw rates r’ = 0.2 and r’ = 0.4, respectively.

We found that all of the analysis results were similar to the experimental and other CFD results for β ≤ 12°. The sway force analysis results for β ≥ 12° in particular, were significantly different from the EFD analysis results for β = 20°. Because other CFD results exhibited a similar discrepancy in the drift angle, the analysis results were considered acceptable to some extent. Table 13 shows the results of the numerical analysis conducted and the errors of the linear hydrodynamic coefficients estimated through other CFD results and those estimated through experimental results. for the damping coefficients Yv′′ and Nr′′, which correspond to the linear hydrodynamic coefficients related to the sway and yaw motions, respectively, the error from the experimental results decreased when a fine mesh was used. These values were similar to the CFD results obtained by Franceschi et al. (2023). For the coupled damping items Nv′′ and Yr′′ which are linear hydrodynamic coefficients with coupled sway–yaw motions, the relative error was large, but the absolute value of the coupled hydrodynamic coefficient term was considered to be small. Tables 14 and 15 show the results of the numerical analysis, the errors of the sway and yaw nonlinear hydrodynamic coefficients estimated through other CFD results, and the nonlinear hydrodynamic coefficients estimated through the experimental results. The errors in the hydrodynamic coefficients estimated through other CFD results and the results of this study were found to be significant. Based on these findings, we concluded that the results of this study are acceptable. Figs. 19 and 20 show the estimated values of the linear hydrodynamic coefficients Yv′′and Nv′′ for each mesh. The grid refinement index h is the value obtained by dividing the entire mesh volume by the number of cells. When it approaches zero, it indicates a finer mesh. All the values estimated using the fine mesh exhibited an error of less than 10% from the values estimated from the experimental results. The coefficient changes from medium to fine was larger than that from coarse to medium, indicating that a mesh with a higher resolution than the fine mesh used in this study must be considered. This problem appears to have occurred because the Courant number was not identical to Δt = 0.005 was applied to all of the coarse, medium, and fine meshes to reduce the computational cost. For the RA, the medium mesh consumed 2940 core-hours and the fine mesh 7946 core-hours. The RA results confirmed that a computational mesh as dense as the fine mesh used in this study was required to derive the most similar results to the experiment under all calculation conditions.

In this study, the maneuvering coefficients of the KCS hull were estimated to determine and validate the applicability of the OpenFOAM-based in-house code (foamStar) for evaluating maneuvering performance. The linear hydrodynamic coefficients were compared with those of previous model tests and CFD analysis results. The RANSE model was used for the viscous flow analysis, and the finite volume method was applied as a numerical technique for the RANSE analysis. The VOF transport equation was applied to consider the free-surface effects.

Based on the meshes and setup of a previous study on the verification of the seakeeping performance of ships in waves (Descamps, 2022), the numerical analysis results of the resistance simulations with fixed and moving meshes in still water were compared with the experimental values from Case 2.10 of the CFD Tokyo 2015 Workshop (Hino et al., 2020) to verify the application of the moving mesh. All the analysis results of coarse, medium, and fine meshes exhibited an error of less than 4% from the experimental results. Therefore, the remaining OTT and RA simulations were conducted using a moving mesh.

The OTT simulation for calm water was performed under the same motion conditions as those used in a previous experiment and other CFD analyses (Franceschi et al., 2023). While the resistance simulation for calm water, meshes, and numerical techniques were the same, a numerical analysis was conducted by providing different conditions only for the forward speed and motion. When the numerical analysis was conducted for 11 drift angles, we found that the analysis results were similar to the experimental and other CFD results for the coarse, medium, and fine meshes.

The RA simulation for calm water was performed under the same motion conditions as those of the previous experiment and other CFD analyses (Franceschi et al., 2023) as with the OTT simulation for calm water. The same meshes and numerical technique as the resistance simulation for still water were used, and an analysis was conducted by combining the yaw rates and with drift angles. All the coarse, medium, and fine meshes were analyzed under the motion conditions, and the maneuvering coefficients were estimated using the analysis results. Among the estimated maneuvering coefficients, the estimated values of Yv′′,Yr′′,Nv′′, and Nr′′, which are linear hydrodynamic coefficients, were compared with the values estimated based on the experimental and other CFD results. The results obtained using medium and fine meshes were similar to the experimental and other CFD results. In particular, Yr′′, a linear hydrodynamic coefficient with the most significant impact on the turning performance of the ship, exhibited excellent results in the fine mesh, whereas Nv′′ showed excellent results for the medium and fine meshes.

The linear hydrodynamic coefficients Yv′′ and Nv′′ estimated using only the OTT simulation results for calm water and the linear hydrodynamic coefficients Yv′′ and Nv′′ estimated using the RA simulation results for still water exhibited similar values. However, Yv′′ estimated using the fine mesh, had a difference of approximately 9.3%. This is because the additional flow caused by the rotational motion that occurs in an actual ship maneuvering scenario is considered in the RA test, unlike in the OTT. When a numerical analysis was conducted on the same hull form in another study, a difference was observed between the hydrodynamic coefficient estimated through the OTT simulation alone and that estimated through other test techniques (Ziabari and Mousavizadegan, 2024).

In conclusion, when the maneuvering model simulation of the KCS hull was performed using the OpenFOAM-based in-house code (foamStar) with moving mesh technique, results similar to those of the model experiment and other CFD results were obtained. This confirmed that foamStar can be used to estimate the maneuvering coefficients of ships. However, we confirmed that a fine mesh compared with the seakeeping analysis must be used to obtain good results in the maneuvering simulation.