Numerical study on the 6-DOF motions of ship turning in waves

Abstract

Ships are often disturbed by waves when they are sailing on sea, and the influence is usually not negligible. When a ship is manoeuvering in waves, the wave will seriously threaten the ship’s navigation safety. Therefore, it is of great significance to study the manoeuverability of a ship in waves, and the study is a hot research topic in recent years. In this paper, the viscous CFD method is used to directly simulate the 6-DOF manoeuvering-seakeeping coupling motions of the series 175 (S-175) container ship model in regular waves. The turning motions of the ship in different wave cases are simulated in the time domain. The Unsteady Reynolds-Averaged Navier-Stokes (URANS) method is adopted to solve the viscous flow around the ship, and the overset grid method is adopted to deal with the rotation of the rudder. Based on the open water characteristic curves, the body force propeller method is used to mimic the propeller behaviour to reduce the calculating cost. The time history of the ship’s force, the characteristic parameters related to manoeuvrability, and some details of the flow field in the typical manoeuvring motion are obtained. At the same time, the model tests of the S-175 container ship turning in waves were carried out in the ship seakeeping/manoeuvrability tank, and the numerical results were compared with the experimental results. The comparison indicates the direct numerical simulation method adopted is a reliable method for predicting ship manoeuvrability in waves.

Introduction

As modern ships become larger and more specialized, maritime safety issues related to manoeuvrability are becoming increasingly prominent. In 2002, the International Maritime Organization (IMO) issued the ‘Standards for Ship Manoeuvrability’ (2002), which put forward a clear quantitative requirement for ship manoeuvrability¹. The International Towing Tank Conference (ITTC) listed ship manoeuvrability in waves as a research topic (2014)². These have greatly promoted the research on ship manoeuvrability prediction. The focus on manoeuvrability has also gradually shifted from calm water to that in waves. However, the validity and representativeness of this prediction method require further study, especially for navigation in severe sea conditions. As a coupled manoeuvring/seakeeping problem, wave forces force the ship motion to drift and generate high-frequency oscillatory motions. It is difficult to predict the manoeuvrability of ships in waves due to the complex interaction among the hull, propeller, rudder, and waves. Therefore, ship manoeuvrability prediction in waves still requires extensive and in-depth research.

Free-running model tests are the most reliable method for simulating ships sailing in waves. They can help us observe the real ship motion phenomenon and wave action law, and at the same time, they can help researchers to verify the accuracy of numerical simulation and theoretical calculation. Iwashita et al. (1993) carried out the wave-induced test of the VLCC model under captive motion³. Ueno (2003) experimented with a typical manoeuvring motion test in a regular wave for the VLCC self-propulsion model and analyzed the test results of changing the wavelength, encounter angle, etc⁴. Yasukawa et al. (2006) measured the wave-induced force under the towing test of the S-175 ship model⁵. Lee et al. (2009) carried out the turning and zig-zag tests in regular waves with different wave parameters for the KVLCC ship model⁶. Yasukawa et al. (2021) experimented with self-propulsion model tests in calm water and regular waves for the KCS container ship model⁷.

In addition to the experimental method, the widely used method is to predict the manoeuvrability in waves by introducing the wave force into the motion equation of the ship in still water based on the potential flow theory. At present, the unified theory method and the two-time scale method are mainly adopted. In the unified theory, the 1-order and 2-order wave forces and the manoeuvering forces are both put into the hydrodynamic equations of the 6-DOF ship motion; the unified solving model of the manoeuvering motion and the seakeeping motion is constructed, and the manoeuvring motions and the seakeeping motions are solved simultaneously. The two-time scale method divides ship motions into manoeuvring motions in waves and seakeeping motions with manoeuvring motion, and the two types of motions are solved independently with different time scales in separate motion equations.

The numerical simulation with a fully viscous CFD method has always been the international frontier method of ship manoeuvrability prediction, but in the past, it could not be widely applied due to a large amount of computational cost. In recent years, direct numerical simulation method has been applied for the prediction of manoeuvrability in waves. According to the description of the propeller’s action, it can be divided into the detailed propeller model and the body force propeller method. The detailed propeller model can clearly express the influence of the mutual coupling among the ship, propeller and rudder. Due to the huge number of grids and the extremely small time step, there are strict requirements on computational effort, so this method is difficult to be applied in engineering. Carrica (2008) used the detached eddy simulation (DES) method to numerically simulate the zig-zag motion of the KVLCC1 ship model⁸. The level set method was used to capture the free surface, the PI controller was used to control the propeller revolution rate, and the overset grid method was used to handle the rotation of the propeller and the rudder. The results show that it is feasible to use the DES method to simulate the 6-DOF manoeuvring motion of the propeller and rudder, but its disadvantage is that it requires a huge amount of CPU time and memory. Castro et al. (2011) used a dynamic overset grid method to simulate the propeller model and numerically simulated the self-propelled motion of the full-size KCS ship⁹. The calculation result shows the feasibility of the dynamic overset grid method in full-scale self-propulsion motion numerical simulation. Mofidi and Carrica (2014) used the overset grid method to simulate the zig-zag motion of the KCS ship model based on their previous work¹⁰. Through the comparison of the numerical results of the 6-DOF motions and the experimental results, the reliable accuracy shows that it is feasible for CFD to directly simulate the manoeuvring motion, but there is a problem of high computational cost. Therefore, it is necessary to explore and study the simplified propeller processing method. For the self-propelled model simulation using the real propeller method, Carrica et al. (2016) conducted an experiment and a numerical simulation of the zig-zag motion of the KCS ship in calm shallow water¹¹. The grid research shows that reliable results can be obtained by using a coarser grid (8.7 M) to predict forces, moments and motions. However, for obtaining accurate flow field details, a fine grid (24.6 M) needs to be adopted.

As an alternative to the real propeller, the body force propeller method is no longer trapped by the extremely small time-step required for propeller rotation, while maintaining a certain accuracy. Carrica et al. (2013) adopted the body force propeller method for manoeuvring motion, and the rudder was rotated by overset grids¹². Compared with the experimental results, the numerical simulation of the manoeuvring motion in calm water shows that the error is within 10%, which is satisfactory for complex manoeuvring motions. Naing et al. (2016) used the body force propeller method to simulate the viscous flow of KVLCC2 and analyzed the wake field information in detail¹³. The calculation results were compared with other body force models and the real propeller method, and the comparison results show that the used method can provide a complex wake field close to the experiment. Jin et al. (2019) carried out numerical simulation of ship manoeuvring trajectory, with the overset grid method to deal with the relative motion of hull and rudder, and with the body force method to deal with the rotational motion of propeller¹⁴. At the same time, exploratory work was carried out on the direct simulation of manoeuvring motion in waves, but the comparison with experimental data was lacking. Kim et al. (2021) succeed in numerically simulating the manoeuvring motion of the KCS ship in waves, and used the body force method to describe the action of the propeller¹⁵. Sanijo Đurasević et al. (2024) Numerically researched the ONRT ship manoeuvring in head waves with the partially rotating grid method¹⁶. In recent years, research on improving the efficiency of ship maneuverability prediction based on CFD simulation has become an important development hotspot. Changzhe Chen et al. (2024) proposed an improved reduced-order model (ROM) based on higher order dynamic mode decomposition (HODMD) to real-time predict ship maneuvering motion in waves¹⁷. Xiao Zhou et al. (2025) developed a novel unified physics-data-driven modeling method with integrated time-series identification framework and motion data, the trajectory-drift features and variable fluctuations in the ship maneuvering motions under wave impacts are captured with good accuracy¹⁸. Xiangrui Zhang et al. (2025) introduced artificial neural networks (ANN) into the field of trimaran maneuverability modeling, established an ANN surrogate model Basing on simulation dataset covering multiple hull configurations and operating conditions, to provide the efficiency of predicting the maneuverability of a trimaran in waves¹⁹.

In this paper, the model test in tank and the direct numerical simulation of manoeuvring motions in waves are both carried out for the S-175 container ship model. The model test and numerical simulation both considered the ship’s turning in cases of different rudder angles and different initial wave directions. The time history of the drift motions and sway motions of the ship in the process of manoeuvring is studied and analyzed. At the same time, the feasibility and effectiveness of the direct numerical method, which is based on the body force propeller method and the overset grid method, for the study of the ship’s manoeuvrability in waves are further verified.

Model experiment

Ship model

This paper takes the S-175 container ship model as the research object. The geometry of the hull is shown in Fig. 1. The rudder is suspended, and the B-4-55 series propeller is adopted. The main parameters of the hull, rudder, and propeller are listed in Tables 1, 2 and 3. The scale of the ship model is 1:42.8.

Fig. 1.

3-D geometry of the hull, propeller and rudder model.

Table 1.

Hull parameters.

Parameter		Full scale	Model scale
Length between perpendiculars	Lpp (m)	175.0	4.088
Maximum beam of the waterline	B (m)	25.4	0.593
Draft	d (m)	9.5	0.222
Displacement	▽ (m3)	24367.5	0.3108
LCB (%LPP), fwd +	-	-1.412	-1.412
Vertical Center of Gravity (from keel)	KG(m)	9.5	0.222
Yaw Radius of Inertia	Kzz/LPP	0.24	0.24

Table 2.

Rudder parameters.

Parameter		Full scale	Model scale
Rudder area	AR (m2)	32.44	1.771 × 10− 2
Rudder area ratio	μ R	0.0195	0.0195
Rudder height	hR (m)	7.70	0.1799
aspect ratio	λ R	1.827	1.827

Table 3.

Propeller parameters.

Parameter		Full scale	Model scale
Diameter (m)	Dp (m)	6.5	0.152
No. of blades	z	4	4
Rotation	-	Right hand	Right hand

Experimental tank and test cases

All relevant experiments in this paper are carried out in the seakeeping/manoeuvrability tank in the China Ship Scientific Research Center. As shown in Fig. 2, the tank is 69 m long, 46 m wide, and 4 m deep. It is an indoor windless tank. The tank is equipped with a 3-D flap-type wave generator (a total of 188 wave-making boards), which can simulate regular waves, long-peak irregular waves, and short-peak waves. The regular waves created are head wave and beam wave. The wave height H is 82 mm (H/L_pp = 0.02), and the regular wave wavelength λ is 4.088 m (λ/L_pp = 1.0). The test cases are shown in Table 4. The parameters of the ship’s roll, pitch, yaw, yaw rate, rudder angle, and the ship’s trajectory are measured. First, the remote controller and the onboard power supply were turned on, and started the test after obtaining the feedback on the rudder angle. In the sailing of the ship model, the movement of the propeller and the steering gear was controlled. The collected data is processed by the data acquisition module and is sent back to the computer through the wireless network to monitor the time history of the ship model motion parameters in real time.

Fig. 2.

Seakeeping/maneuverability experimental tank. Photograph taken by one of the co-authors. The co-author give his full consent for publication of this photograph.

Table 4.

Propeller parameters.

Experimental schemes				Experimental cases
	Speed		Fr.	Turning motion tests
	Full-scale kn	Model-scale (m/s)	Fr
Calm water	12.08	0.95	0.15	± 35°
Wave	12.08	0.95	0.15	± 35°

Mathematical and numerical model

Numerical method

In this paper, the numerical simulation of 6-DOF self-propulsion manoeuvring motion in waves is carried out using the commercial software Star-CCM+. The unsteady Reynolds-averaged Navier-Stokes (URANS) numerical simulation method is adapted. For unsteady incompressible flows, the mean continuity and momentum equations are expressed as follows:

1

2

where is the mean velocity component, is the Reynolds stress, is the mean pressure, is the mean viscous stress tensor component, and is defined in the following formula:

3

In this paper, a realizable two-layer turbulence model (Rodi, 1991) is adopted²⁰. The model is applied to the viscous-affected layer, and the calculation is carried out in two layers. In the near-wall layer, the turbulent dissipation rate and the turbulent viscosity are as functions of wall distance. For the solution of the near-wall flow, the all-y+ wall processing method is adopted to achieve 30 < y + < 200 along the hull surface to predict the near-wall flow.

The free surface capture is achieved using the Volume of Fluids (VOF) method. When both water and air coexist in the computational domain, the volume fraction of the i-th phase fluid satisfies 0 < c_i < 1, . For incompressible fluids, the governing equation of the fluid volume fraction is:

4

The finite volume method is used to discretize the governing equations. The second-order upwind scheme is used for the convection term, the central difference scheme is used for the diffusion term, and the second-order time accuracy is used for the time transient term. The solving of the coupling of velocity field and pressure field is conducted by the SIMPLE algorithm. For the regular wave case, ITTC (2011) gives a guideline to set the time-step less than 1/100 wave period. Compared to seakeeping calculations, the maneuvering motion is a slow motion with a long period, so the time step required is relatively large. Therefore, the time step of seakeeping calculation should be dominant in direct simulation. In this paper, the regular wave period of the working case is about 1.6 s, so the time step of the numerical simulation of the manoeuvring motion in all the regular waves in this paper is selected as 0.01 s.

To ensure the grid quality and numerical solution accuracy when dealing with the 6-DOF motion in waves, the overset grid method is used to deal with the multi-body motions. The overset grid which is adopted in the multi-body motion of ship, propeller and rudder, has the following advantages. The overset grid allows the sub-domain grid to be generated separately for refinement. The single sub-domain grid is refined independently without affecting the grids of other sub-domains. The motion of the sub-domain grid does not affect the whole grid. Therefore, overset grid method can ensure high grid quality in dealing with complex multi-body motion. Linear interpolation method is used to exchange flow field information between domains.

In this paper, the numerical simulation of manoeuvring motion in waves adopts the fifth-order VOF regular wave that was proposed by Fenton (Fenton, 1985)²¹. According to the work of Kim et al. (2012), the wave force method can combine the numerical solution of fluid with the theoretical solution of the VOF wave, and it forces the simulated solution to the theoretical solution in the wave force zone²². In this way, it not only reduces the wave-absorbing domain but also reduces the computational cost. The problem of surface wave reflection at the computational boundary when the ship model rotates in the computational domain can also be eliminated in the wave force zone. The layout diagram of the virtual disk of the body force propeller method is shown in Fig. 3. The open water characteristic curve of the propeller used in this paper is also shown in Fig. 3, and it can be obtained in the relevant design manual (China State Shipbuilding Corporation, 2013)²³.

Fig. 3.

Virtual disk layout and performance curve of propeller B-4-55.

In this paper, four coordinate systems are defined for numerical simulation. They are the Earth-fixed coordinate, the Ship-fixed coordinate, the Propeller-fixed coordinate and the Rudder-fixed coordinate. The Earth-fixed coordinate system is a fixed coordinate system on the shore. The Ship-fixed coordinate system is fixed at the centre of mass and moves with six degrees of freedom in its coordinate system. The Propeller-fixed coordinate system is used to locate the position and direction of the propeller. The Rudder-fixed coordinate system is fixed on the rudder stock to control the rotation angle of the rudder. The Propeller-fixed and Rudder-fixed coordinate systems are subsidiary coordinate systems of the Ship-fixed coordinate system. The four coordinate systems used are shown in Fig. 4.

Fig. 4.

Coordinate systems.

The computational domain is divided into three parts: the background domain, the overset domain around the hull, and the overset domain around the rudder. The rudder motion as DFBI superimposed rotation motion is attached to the hull of the DFBI 6-DOF motion.

The background grids are forced to move with 3 DOF (surge, sway, and yaw) to allow a relatively static motion between the ship overset grids and background grids. The velocity components of the background domain are set to the velocities of the ship surge, sway and yaw. At this point, the background grid can maintain the same 3-DOF motion as the ship, instead of making a background domain that can contain the ship’s motion trajectory. Thus it can effectively reduce the grid range of the background domain and computational effort.

Numerical simulation

The computational domain is shown in Fig. 5. The incoming and outgoing flow boundaries are 2.5L_pp and 3.5L_pp from the ship’s center of gravity. The sides are 2.5L_pp from the ship’s mid-longitudinal plane. The top and bottom boundaries are 1.0L_pp and 2.0L_pp from the base plane respectively. The boundary conditions are set as: the non-slip boundary condition for the hull surfaces, pressure outlet condition for the top boundary, and the velocity inlet condition for the other boundary surfaces.

Fig. 5.

Calculation of domain and boundary conditions.

The whole calculation domain is grided by the trimmed cell mesh. The mesh at the free surface is refined, and the overset grid around the hull is added to realize the 6-DOF movement of the hull. The layout of the overset grid is shown in Fig. 6.

Fig. 6.

The scheme of the computational domain for the free-running CFD model.

In order to ensure the reliability of the calculation results, the uncertainty analysis in grid was carried out on the basis of the comparison with the experimental results. Considering that the uncertainty analysis in grid of the long-period maneuvering motion conditions will bring a great computational burden, the sailing straight motion is selected, and the pitch and heave motions are used as the main parameters for comparative analysis. Comparisons are made to ensure reasonable validity of the numerical model. Considering computational resources and flow field details, based on the calm water grid scheme 2 (Shang and Zhan, 2021), three different grid schemes are used to predict the motion parameters of the S-175 in sailing straight with head waves²⁴. Compared with the experimental results, the grid structure and foundation size remain unchanged, only the grid size of the free surface is changed to 25, 50, and 100 grids in wavelength direction, and 10 grids in the wave height direction. Uncertainty analysis in grid is shown in Table 5; Fig. 7. After comprehensive consideration, the number of grids is selected as 632 M. At the same time, the wave height at 0.5L_pp in front of the ship is monitored in the calculation. The wave height time history in histories in three Schemes are shown in Fig. 8.

Table 5.

Uncertainty analysis in grid.

Grid scheme	Total grid number /million	Pitch /°	Error /%	Heave /m
1	4.92	2.58	4.4	0.022
2	6.32	2.60	3.7	0.022
3	16.00	2.59	4.1	0.023
Experiment	/	2.70	/	/

Fig. 7.

Pitch motion amplitude obtained from three sets of grid calculations.

Fig. 8.

Wave height time history in front of ship.

The numerical simulation cases of maneuvering motion in waves are as follows. Starboard 35° turning motion simulations with initial wave directions as head waves and beam waves are carried out in a parallel computer. The CPU is IntelR XeonR E5-2630 v4@2.20 GHz, parallel computing with 30 processes. A total of 22,841 core hours were spent to complete a single turning motion in waves. It took a total of 5,839 core hours to complete a single zig-zag motion. Port 35° turning motion tests simulations in head waves and beam waves, which are carried out in a computing cluster. The CPU is AMD EPYC 7502 4 × 128-core 256 G 2.5 GHz. It took 13,486 core hours to complete a single turning motion in waves. With the development of computer technology and the popularization of computing clusters, direct simulation can quickly predict the maneuverability of ships in waves with high accuracy at a lower cost. This is of great significance for the accurate and efficient overall evaluation of the ship’s manoeuvrability in waves in the ship design stage without carrying out tank experiments.

The results of experiment and simulation

Turning motion

In order to ensure the reliability and consistency of the comparison situation, all experiments and numerical simulations commence with the ship having steady speed and heading, and with the initiation of rudder deflection. Figure 9(a) shows the comparison of the turning trajectories between the numerical results and the experimental results in calm water cases. The current numerical prediction results accurately predict the turning trajectory of the ship in calm water. Figure 9(b, c) show the comparison of the turning trajectories between the numerical results and the experimental results with the initial wave direction as the head wave and the beam wave. From the comparison in wave cases, the numerical prediction of the turning motion trajectory and the ship motion drift directions is similar to the experimental results, but the numerical drift distance is larger than the experimental results. From the comparison of the turning motion trajectory, it can be seen that the current numerical simulation can well predict the turning motion in calm water and in waves.

Fig. 9.

Turning trajectory.

Hereinafter referred to as ‘the initial wave direction is the head wave’ as the head wave.

Take the example of a starboard 35° turning motion in the head wave. Figure 10 shows the 6-DOF motion parameter time history. Due to the influence of different wave directions and encounter frequencies in the turning motion, the time history of the ship’s 6-DOF motion shows obvious frequency oscillation. In the time history, the typical encountered wave direction stages are head wave, bowing wave, beam wave, quartering wave and following wave stages, which occur at 0–6.3 s-20.3–28 s-43.3–47.2 s respectively.

Fig. 10.

Time history of 6-DOF motion parameter (starboard 35° turning motion of the head wave).

From Fig. 10(a and b), it can be seen that after turning the rudder, the surge velocity of the ship gradually decreases. At this point, it is the stage of the head wave and the bow wave. After 20 s, it stabilizes at around 0.5 m/s, and the lateral velocity sway velocity gradually increases and gradually stabilizes around 0.1 m/s after 10 s. In the beam wave, quartering wave and following wave stages, surge velocity slightly increases, while the amplitude of sway velocity oscillation increases significantly in the beam wave stage, and the amplitude of oscillation is small in the following wave stage. From 80 s to 110 s, the surge velocity appears to be lower. At t = 109 s, after the hull completed a full 360-degree rotation in the waves, it entered the second rotation stage of head waves. The slow drift force in the longitudinal direction of the waves caused a significant surge velocity decrease. After this stage, the surge velocity will gradually recover.

Figure 10(c) shows the comparison of the yaw rate between the numerical and the experimental results. Before 80 s, there is almost the same regular distribution, and compared with the numerical prediction, the experimental yaw angular velocity amplitude is slightly larger. From 80 s to 110 s, there is a phase difference between numerical and experimental results, which is also the reason for the partial deviation of the trajectory when entering the second turning circle stage.

Figure 10(d) shows the time history of the numerical prediction of the heave of the ship. The heave is between − 0.05 m and 0.04 m, and the maximum amplitude is 0.046 m. From the bow wave stage to the beam wave stage, the frequency of heave increases and the amplitude of heave becomes larger, while in the quartering wave to following wave stages it shows smaller changes in frequency and amplitude.

Figure 10(e and f) show the time history of roll and pitch in the turning motion respectively. The roll angle varies from − 3° to 3°, and the pitch angle varies from − 3° to 2.5°. On the whole, the numerical prediction results show high accuracy, but only the rolling amplitude of the experimental results is slightly larger than that of the numerical simulation. The pitch motion is similar to the experimental results. Similarly, after 80 s, the roll and pitch time histories show a phase difference, and the numerical prediction trend is obvious, which indicates the movement changes in different wave directions and encounter frequencies.

Figure 11 shows the force time history of the ship model in the turning motion. Figure 11(a, b, c) show the longitudinal force changes of the whole ship, bare hull and rudder in the turning motion in different encountered wave stages. When the ship enters the bow wave stage from the head wave stage, the oscillation of the longitudinal bare hull force is weakened, and the oscillation amplitude is the smallest in the beam wave stage. The longitudinal rudder force gradually increases after initial time, maintains a certain amplitude oscillation in the bow wave stage, weakens in the beam wave stage, gradually increases in the quartering wave stage, and gradually decreases in the following wave stage. Figure 11(d, e, and f) show the calculation results of the lateral force changes of the whole ship, bare hull and rudder in the turning motion in different encountered wave stages. Before the end of the beam wave stage, the lateral bare hull force gradually increases, while in the quartering wave stage and the following wave stage it gradually decrease. The lateral rudder sway force increases in the opposite direction. Its oscillation weakens in the beam wave stage, then gradually increases in the quartering wave stage, and then gradually decreases in the following wave stage. At 80–110 s, the ship enters the bow wave stage, and the amplitude of the longitudinal force further increases, which also explains the further decrease of surge velocity at this stage. With the change of encountered wave directions, the yaw, roll and pitch motions are consistent with the change law of the corresponding moment.

Fig. 11.

6-DOF forces time-history of CFD results (starboard 35° turning motion in head wave).

Figure 12 shows the free surface changes in turning motion. When sailing straight in the head wave, the wave free surface around the ship is symmetric. The superposition of the ship making wave and the incident wave is obvious. In the beam wave stage, the disturbance of the ship making wave to the incident wave is minimal, and the ship trim is the smallest, because the forward speed and turning speed of the ship are small. It can be seen from the change of the ship’s attitude in the figure that the ship will have a relatively large 6-DOF motion when maneuvering in the waves.

Fig. 12.

Free surface changes in turning motion (starboard 35° turning motion in head wave).

Figure 13 shows the time history of the motions in different cases which are listed in Table 4. It can be seen that CFD can predict the trend of ship motion changes, and the accuracy of motion amplitude is also satisfactory. At the same time, the CFD and EFD results show a cumulative phase difference of motions. Through comprehensive analysis, there are two main possible reasons for it. The first of that, with the accumulation of time, wave attenuation occurs. The second is that the body force propeller method cannot accurately describe the lateral force generated when the real propeller rotates. The experimental test also has some errors, but the comparison between the two results shows that the numerical simulation can predict the trend of ship motion in waves. Therefore, the CFD method can effectively predict the maneuverability of S-175 in waves.

Fig. 13.

Yaw, roll and pitch compared EFD with CFD in waves (turning motion).

Comparison of ship motions in different cases

Tables 6 and 7; Fig. 13 show the parameters of turning motion in different cases listed in Table 4. It can be seen from Fig. 14(a and b) that the numerical simulation shows results similar to the experimental results for the parameters of the turning motion. The tactical diameter is relatively small in the cases of starboard and port turning motion in head wave and starboard turning motion in beam wave, and the tactical diameter is relatively large in the case of port turning motion in beam wave. The tactical diameter of the starboard and port turning motion of the calm water is the largest. The advance and transfer of the port and starboard turning motion in head wave are the smallest. Compared with the experiment, the numerical prediction of the advance and transfer in the beam wave cases has relative errors. From Fig. 14(c), it can be seen that the prediction accuracy of the turning motion is reliable, although there are relative errors in the cases of the port turning motion in head wave and beam wave. In the simulation of maneuvering motion in waves, the accuracy of the slow-drift force is of vital importance. Compared with the turning maneuvering in head waves, the lateral and rotational slow-drift force/torque in the initial stage of turning in beam wave have a more significant impact on the turning trans and advance. In this paper, in the numerical model, to ensure the efficiency of step-by-step solution during turning in wave, a trade-off decision was made in the setting of grid density and time step considering the existing computing hardware conditions. Therefore, the calculation accuracy of the slow-drift force in the initial stage of beam wave turning was compromised, which might be the main reason for the relative deviation between the numerical results and the experimental results. In future research, with the further improvement of computing conditions, the accuracy of numerical simulation can be enhanced.

Table 6.

Parameters of starboard 35° turning motion in different cases.

Main parameters	Calm water (CFD)	Calm water (EFD)	Head wave (CFD)	Head wave (EFD)	Beam wave (CFD)	Beam wave (EFD)
Tactical diameter /LPP	4.48	4.30	3.53	3.57	3.53	3.52
Advance /LPP	3.56	3.50	2.60	2.72	3.17	3.62
Transfer /LPP	1.97	1.84	1.43	1.36	1.67	1.90
Time to turn 90°/s	23.20	23.06	24.12	23.24	24.08	25.92
Time to turn 180°/s	46.80	44.6	46.78	44.42	47.22	46.40

Table 7.

Parameters of port 35° turning motion in different cases.

Main parameters	Calm water (CFD)	Calm water (EFD)	Head wave (CFD)	Head wave (EFD)	Beam wave (CFD)	Beam wave (EFD)
Tactical diameter /LPP	4.49	4.38	3.67	3.43	4.33	4.09
Advance /LPP	3.74	3.90	2.69	2.55	3.75	3.34
Transfer /LPP	1.97	1.84	1.50	1.35	1.98	1.67
Time to turn 90°/s	23.60	24.56	24.80	22.24	25.83	20.64
Time to turn 180°/s	47.00	46.56	48.22	42.64	48.59	41.84

Fig. 14.

Parameters of turning motion.

Conclusions

In this paper, the model experiment and the numerical simulation of the turning in waves of the S-175 container ship were carried out.

On the one hand, the feasibility and effectiveness of the direct numerical simulation method based on the body force propeller method and the overset grid method in the prediction of ship maneuverability in waves were discussed. Generally speaking, the numerical prediction results are in agreement with the experimental results. Except for the case of port 35° in beam wave case, the numerical errors of the three parameters of transfer, time to turn 90°, and time to turn 180° in the rest cases are less than 15%. Although the direct simulation of maneuvering motion in waves based on body force propeller model and overset grid method is still extremely time-consuming heavy, the problem can be solved with the development of computers and the application of computer clusters. Meanwhile, further improving the body force model to adapt to the actual wake field of the propeller during the maneuvering motion can further enhance the accuracy of the maneuvering motion prediction in waves by the method proposed in this paper. The current research in this paper is limited to the ship turning motion in small-amplitude regular waves. For more complex large-amplitude and irregular waves, further studies are needed in aspects such as the non-regular wave numerical wave generation method, overlapping grid setting, time step convergence, and computational efficiency, in order to make the applicability of this method more extensive.

On the other hand, the 6-DOF ship maneuvering/seakeeping coupling motions in waves with different initial wave directions were analyzed, and the numerical simulation of the maneuvering motion in calm water and waves and the change of the parameters of the maneuverability in the experiment are comprehensively analyzed, and the changes of the parameters of the maneuverability in different cases are given.

Author contributions

J.Z., Y.M., C.Z. and H.S.; research methodology: Y.M.; experiment: C.Z.; numerical simulation: J.Z., H.S.; data analysis: J.Z.; writing: J.Z.; All authors have read and agreed to the published version of the manuscript.

Funding

This research was supported by the National Natural Science Foundation of China [grant numbers 52271331], Joint supported by Hubei Provincial Natural Science Foundation and Sport Innovative Development of China [grant numbers 2025AFD620] and supported by the East Lake Scholars Sponsorship Program of Wuhan Sports University in China.

Data availability

The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.