1   Introduction

Ships inevitably encounter extreme waves while sailing. In severe sea conditions, a large amount of seawater rushes onto the deck, and huge impact loads cause damage to the equipment and superstructures on the deck. Moreover, green water can even cause hull capsizing.

For a tumblehome vessel, such as the DDG1000 in the USA, the distinct attributes of this hull form are a low-tumblehome freeboard and a wave-piercing bow. These novel shape designs can effectively reduce ship resistance in waves and improve stealth performance. However, an inversely inclined bow makes the hull motion violent, which easily causes green water. Therefore, it is essential to accurately predict and study the tumblehome vessel green water during strong nonlinear regular waves.

In previous studies, green water has predominantly been studied using experimental and theoretical methods. Tasaki (1960) concluded from experiments that green water in regular waves is caused by the relative motion of bows and waves. Berhault et al. (1998) found that the water height on the deck is related to the bow shape and drift angle. Stansberg (2001) measured the relative motion of a floating production storage and offloading (FPSO) unit and the green water impact pressure.

With the improvement of the test technology, to be closer to extreme sea waves in the real ocean, researchers have paid great attention to green water under high-sea conditions. Duan et al. (2013) studied the relationship between the drift and impact pressure of an S-175 ship under high-sea conditions. Gao et al. (2016) evaluated the trim attitude influence on green water under different wave steepness values ([H/λ]_max = 0.13). Rosetti et al. (2019) verified the applicability of the computational fluid dynamics (CFD) method for predicting FPSO green water in beam waves with large wave steepness ([H/λ]_max = 0.08). Deng et al. (2021) evaluated the evolution of "plunging" green water of an FPSO through experiments ([H/λ]_max = 0.10). The green water model test is the most reliable method and provides an important reference for numerical calculations.

In terms of the theory method, Ochi (1964) used linear hydrodynamic theory and probability statistics to evaluate green water. He et al. (1996) and Wang and Wu (1998) improved this method and quantitatively evaluated the severity of green water pollution. Ogawa et al. (1997) evaluated the impact loads and water height on a deck during irregular waves using probability statistics. However, accurately evaluating impact loads utilizing probability statistics is challenging. The linear theory cannot deal with green water in strong nonlinear waves.

Based on the potential flow, dam-break (Goda et al. 1976), and flood wave theories (Ogawa et al. 1998), and fully nonlinear boundary element method (Greco 2001) have been proposed to numerically solve green water. However, the potential flow theory overlooks fluid viscosity. It is difficult to simulate strong nonlinear phenomena, including wave breaking, plunging jets, and splashing around ship hulls in extreme waves.

In recent years, CFD methods have been vastly utilized for green water numerical simulation under large wave steepness.

Liang et al. (2010) and Gong et al. (2014) studied green water using a dynamic grid and the volume of fluid (VOF) method in Fluent. Liu (2017) and He et al. (2018) evaluated the influence of wave amplitude and wavelength on the green water of ships. Liu (2017) focused on the "wave run-up" and the total volume of water on the tumblehome hull DTMB5613's deck under a large wave steepness ([H/λ]_max = 0.05). He et al. (2018) used the FINE/Marine software to evaluate the green water dynamic process of a wave-facing sailing Wigley ship ([H/λ]_max = 0.027). Xu and Tang (2018) evaluated the influence of air entrainment on green water. Liu and Li (2020) studied the use of green water for freak waves. Yang et al. (2021) evaluated the influence of velocity field change on the point pressure at the FPSO deck and the overall stress of the structure during green water.

Currently, most CFD numerical studies have investigated the green water of conventional ship types with wave steepness less than 0.05, and few studies have focused on the tumblehome vessels' green water attributes in extreme sea waves. A large wave steepness can affect the ship motion and green water severity. Therefore, it is essential to study the effect of large wave steepness > 0.05 on the tumblehome vessel green water.

In this study, the commercial CFD software Star-CCM+ was used to simulate the green water of wave-facing sailing tumblehome vessels in strong nonlinear regular waves. The predicted water height on the deck and slamming pressure were compared with experimental data to verify the reliability of the CFD method. When the wave steepness exceeded 0.05, the influence of the wave height on green water was evaluated. This study focuses on the dynamic process of green water and the "wave run-up" phenomenon to further evaluate the green water mechanism of tumblehome vessels under large wave steepness.

2   Numerical methods of green water

2.1   Governing equation

The numerical simulation of the tumblehome vessels' green water was based on the CFD commercial software Star-CCM+. The fluid was presumed viscous and incompressible. The governing equations are as follows.

where U is the velocity field vector, ρ is the fluid density, p is the pressure, and ν is the kinematic viscosity coefficient. Eq. (1) represents the continuity equation of incompressible viscous fluid. Eq. (2) represents the Navier–Stokes (N–S) equation of incompressible viscous fluid.

Each term in the N–S equation represents a force acting on the fluid of unit mass. $\frac{{{\rm{D}}\mathit{\boldsymbol{U}}}}{{{\rm{D}}t}}$ represents the inertial force; f, $\frac{1}{\rho }\nabla p$, and ν∇²U represent the mass force, resultant force of pressure, and viscous force, respectively.

In this study, the Reynolds-averaged Navier–Stokes (RANS) method was used to solve the N–S equation. The K-epsilon model was selected for viscous turbulence.

2.2   VOF method

The interface between water and air is captured using the VOF method. The definition of the ith phase volume fraction is given as follows:

where V_i is the volume of the ith phase in the cell, V is the total cell volume, and α_i is the relative proportion of the fluid in each cell, as depicted in Eq. (4).

where α_i = 0 means that the cell is full of air, α_i = 1 means that the cell is full of water, and 0 < α_i < 1 means that the cell is at the free surface.

2.3   Wave generation

To simulate regular head waves with large wave steepness, 5th-order Stokes waves were adopted at the inlet boundary to generate waves. According to the 5th-order Stokes wave theory in infinite deep water (Fenton 1985), the wave surface elevation η (x, t) is defined as

The encounter frequency ω_e in the regular head wave condition is defined by the dispersion relation:

where A = H/2, θ = kx - ω_et, H is the wave height, k is the wave number, ω is the frequency of the wave, and U_ship is the ship velocity.

2.4   Wave damping based on the Euler overlay method

The wave forcing function provided in STAR-CCM+ was used for wave damping based on the Euler overlay method (EOM) theory (Kim et al. 2012; Baquet et al. 2017).

The finite-volume formulation of the diffusion-transport equation of a scalar function ϕ within a volume V surrounded by the boundary surface A is given by

where ρ is the density, v denotes the fluid velocity vector, v_g denotes the fluid mesh velocity vector, a is the area vectoron the boundary surface A, Γ is the diffusion parameter, and S_ϕ is the source term for the scalar. The scalar function ϕ can be the velocity or volume fraction depending on whether Eq. (7) describes the conservation of momentum or phase.

In the EOM, an additional source term is added to each conservation equation to achieve forced wave absorption. The new source term S_ϕ^* in Eq. (7) is given by

where γ is the forcing function, ρ is the fluid density, ϕ is the current solution to the transport equation, and ϕ^* is the solution forced by the EOM.

The expression for the forcing function γ is given by

where γ_max is the maximum value of γ (x), and l is the ramping length of the forcing function.

Figure 1 shows the top view of the computational domain and the sketch of the forcing function. The green and red contour lines represent the inner and outer boundaries, respectively.

Figure  1  Schematic sketch for the forcing function definition in the EOM

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The N–S equation was solved in the inner zone, where the forcing function γ is 0. In the overlay zone with a width of l, γ (x) smoothly increases from 0 at the inner boundary to γ_max at the outer boundary, as depicted in Figure 1. Then, the discretized N–S equation solution is forced to move to the theoretical or a simplified numerical solution.

The wave forcing function based on the EOM can reduce the size of the computing domain to effectively shorten the solution time. In addition, it can eliminate the surface wave reflection problem at the boundary.

3   Model and simulation conditions

3.1   Geometric model

A tumblehome vessel model was used for all numerical simulations. The geometric model is depicted in Figure 2, and its principal dimensions are listed in Table 1. A wave baffle was inclined to the stern at an angle of 20° at the 8th station, as depicted in Figure 3.

Figure  2  Geometry of the tumblehome vessel

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Figure  3  Dimension of the wave baffle (superstructure)

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Wave probes PH1–PH6 and measuring points B1–B7 were used to measure the water height on the deck and slamming pressure on the hull surface, as depicted in Figures 4 and 5.

Figure  4  Wave probes of the water height on the deck

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Figure  5  Measuring points of the slamming pressure

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The origin of the Earth-fixed coordinate system is located at the keel in the midship. The X-axis is in the longitudinal direction from the stern to the bow, Y-axis is in the portside direction, and the Z-axis is vertically upward. The ship hull is divided into 20 stations. The 0th station is at the bow, and the 10th station is at the midship.

Table 2 enlists the detailed positions of PH1 – PH6 and B1–B7 in the Earth-fixed coordinate system. PH4 and PH6 are symmetrical about the midship plane.

The ship is free to heave and pitch during the simulation. The origin of the ship-fixed coordinate system is located at the ship's center of gravity. The directions of the ξ-axis, η-axis, and ζ-axis are consistent with those of the X-axis, Y-axis, and Z-axis of the Earth-fixed coordinate system.

The wave probes (PH1–PH6) were set in the ship-fixed coordinate system during the ship motion, as depicted in Figure 6, to ensure the relative position of the wave probes and so that the ship remains unchanged. The water height on the deck is defined as H_GW (height of the green water):

Figure  6  Schematic sketch for the water height on the deck

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where ζ_GW is the water height relative to the ζ-axis, ζ_Deck = Z_Deck - Z_CG = 0.18, and ζ_Deck is the height of the deck relative to the ζ-axis. Z_Deck and Z_CG are the height of the deck and center of gravity relative to the Z-axis, respectively.

3.2   Simulation conditions

The numerical simulations were performed in regular head waves with different wave heights for the tumblehome vessel advancing at Fr = 0.22. The wave steepness H/λ ranged from 0.033 to 0.067.

The simulation conditions are summarized in Table 3, where T is the period of the wave, Te is the encounter period based on the unlimited-depth 5th-Stokes wave theory (see Eq. (6)), λ is the wavelength, H is the wave height, and Fr is the Froude number, defined as:

where U_ship is the velocity of the ship, g is the acceleration of gravity, and L_wl is the waterline length of the ship.

4   Numerical validation

4.1   Numerical wave-generating validation

A numerical "forced wave tank" was established using the wave forcing function in the STAR-CCM+ software to simulate regular waves with large wave steepness.

The 5th-order Stokes wave was adopted to simulate regular waves with a 0.08 wave steepness. The water depth, wave height, and wavelength were 9, 0.36, and 4.5 m, respectively.

According to the EOM, the velocity inlet with wave forcing was used on all vertical boundaries. The ramping length of the forcing function was set to L_wl. The pressure outlet was used on the top, and the velocity inlet was used on the bottom.

To ensure the accuracy of the wave simulation, a 2^nd-order scheme was adopted for the time step. The CFL number is less than 0.5 on the free surface, defined as

where C_wave is the phase velocity of regular waves, dt is the time step, and dx is the horizontal cell dimension of the free surface region.

20–40 cells were within the wave height range. 80–160 cells were set within the wavelength range. The maximum aspect ratio of the grid element on the free surface Δz/Δx was 1/8. The grid refinement is depicted in Figure 7.

Figure  7  Boundary conditions and grid refinement of the "forced wave tank"

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The predicted numerical wave height was compared with the solution of the stream function theory, as depicted in Figure 8, where t is the simulation time, and T is the period of the wave.

Figure  8  Wave height in the whole domain

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The numerical wave height in the entire computational domain at t/T = 15 almost coincided with the theoretical solution of the stream function, as depicted in Figure 8.

Figure 8 depicts that the CFD strategy for wave generation can provide accurate information on regular waves with large wave steepness (H/λ ≤ 0.08).

4.2   Domain and boundary conditions

Half of the ship model was used for the numerical simulation. The computational domain size was set as follows: The inlet boundary was located at 2L_wl from the front perpendicular, and the outlet boundary was located at 3L_wl from the after perpendicular. The top and bottom boundaries were located at 1.5L_wl and 2L_wl, respectively. The lateral boundary was located 2L_wl away from the midship plane, as depicted in Figure 9.

Figure  9  Computational domain and boundary conditions

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The boundary conditions were set up as follows: The velocity inlet with wave forcing was used on the inlet, outlet, bottom, and side wall; there was no-slip wall condition on the ship hull; pressure outlet was utilized on the top; and the symmetry condition was located at the symmetry plane, as depicted in Figure 9.

4.3   Mesh convergence studies on green water

Trimmed meshes were generated using STAR-CCM+. Overset grids were adopted to accurately simulate large-amplitude ship motions in regular waves. A convergence study for the mesh size was conducted in Case 2. Coarse (Mesh 1), medium (Mesh 2), and fine (Mesh 3) grids correspond to the total cell numbers of 2.85, 3.94, and 5.66 million, respectively. The CFL number is 0.5.

The mesh refinement of the background, overlap, overset and free surface regions of Mesh 1 is depicted in Figure 10.

Figure  10  Side view of the mesh refinement of Mesh 1

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The mesh refinement strategy is the same for the three meshes. The percentage of the grid size relative to the base size in each region is summarized in Table 4. The total cell numbers are changed by resizing the size of the base size. Mesh 1, Mesh 2, and Mesh 3 correspond to the base sizes of L_wl/65, L_wl/80, and L_wl/100, respectively.

The grid size of each region is summarized in Table 4.

The sensitivity of the water height on the deck and slamming pressure to the grid scale was analyzed. The water height on the decks of PH2 and PH4 under different mesh sizes is depicted in Figure 11. H_GW is the water height on the deck (see Eq. (10)), and t is the simulation time.

Figure  11  Mesh convergence of the water height on the deck

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Figure 11(a) depicts that with an increase in the total cell number of meshes, the crest of the water height on the deck is basically unchanged at PH2. The three meshes had a slight phase difference. Figure 11(b) depicts that the predicted crest of the water height on the deck of Mesh 1 is lower than those of Mesh 2 and Mesh 3. The time history curves of Mesh 2 and Mesh 3 basically coincide at PH4.

The results show that the grid size mainly affects the simulation accuracy of the water height on the deck at the station, which is far from the ship bow in the green water.

The slamming pressures of B2 and B6 under different mesh sizes are depicted in Figure 12. P_GW is the dynamic pressure of slamming, and t is the simulation time.

Figure  12  Mesh convergence of the slamming pressure

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Figure 12 depicts that with an increase in the total cell number of the mesh, the crest of the slamming pressure is basically unchanged at B2 and B6. Mesh 1 has an obvious phase difference. The time histories of Mesh 2 and Mesh 3 basically coincide.

By comprehensively comparing the time history with different total cells, it is considered that Mesh 2 converged.

4.4   Time step convergence studies on the green water

A convergence study for the time step was conducted in Case 2 using the medium grid (Mesh 2). Coarse (∆t₁), medium (∆t₂), and fine (∆t₃) time steps correspond to Te/2⁸ (CFL = 1.0), Te/2⁹ (CFL = 0.5), and Te/2¹⁰ (CFL = 0.25), respectively.

The water height on the decks of PH2 and PH4 under different time steps is depicted in Figure 13. The slamming pressure of B2 and B6 under different time steps is depicted in Figure 14.

Figure  13  Time step convergence of the water height on the deck

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Figure  14  Time step convergence of the slamming pressure

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Figures 13 and 14 depict the predicted crest of the water height on the deck, and the slamming pressure of ∆t₁ is lower than that of ∆t₂ and ∆t₃. The time histories of ∆t₂ and ∆t₃ basically coincide.

The results show that when the CFL number is greater than 0.5, the predicted water height on the deck and slamming pressure will be lower. The comparison of the time history with different time steps shows that ∆t₂ converged.

The comparison of the time history with different time steps shows that ∆t₂ converged.

4.5   Green water validation

The convergent numerical results of the water height on the deck and slamming pressure in Case 2 (Fr = 0.22, H/λ = 0.050) were selected for comparison with the experimental data (Sun et al. 2017; Li et al. 2020).

The computed time history of the water height on the deck at PH1–PH5 and the slamming pressure at B1–B7 was compared with the measurements in Case 2, as shown in Figure 15.

Figure  15  Time history of the water height on the deck

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Figure 15 demonstrates that the predicted water heights on the deck at PH2– PH5 are in good agreement with the test results. The predicted crest value at PH1 after 32 s is much lower than the experimental results. This is due to the large movement of the bow during the experiment, resulting in a certain error in the measurement of the water height on the deck at PH1.

Figure 16 depicts a good agreement between the experimental results and the predicted slamming pressure at B1–B7.

Figure  16  Time history of the slamming pressure

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In general, the predicted water height on the deck and the slamming pressure at each measuring point on the bow agree with Case 2 test results. Hence, the CFD numerical strategy was accurate and reliable for simulating tumblehome vessel green water under large steepness waves.

5   Results and discussion

5.1   Influence of the wave height on ship motions

The four cases in Table 3 were selected to evaluate the influence of wave height on the ship motion response. The time history of the ship motion with different wave heights is depicted in Figure 17.

Figure  17  Time history of the ship motion

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Figure 17(a) depicts the heave motion at different wave heights. When H/λ changed from 0.050 to 0.056, the heave motion amplitude increased most significantly. Compared to Case 2, the trough and crest values increased by 31.25% and 30.37%, respectively, in Case 3. The sinkage in Case 4 increased by 11.23%, whereas the amplitude of the rise changed only by a little compared with Case 3.

Figure 17(b) depicts the pitch motion at different wave heights. Compared with Case 2, the trim amplitude by the bow increased by 14.57% in Case 3. Compared with Case 3, the trim amplitude by the bow in Case 4 increased by 10.27%.

Thus, for a wave-facing tumblehome vessel, when H/λ increases to 0.056, a large-amplitude heave motion will occur. When H/λ increased to 0.067, the increase in sinkage was significant. The influence of the wave height on the pitch motion is predominantly reflected in the trim change by the bow.

5.2   Influence of the wave height on the water height on the deck

The time history of the water height on the deck of the PH1 and PH3 wave height gauges at different wave heights is depicted in Figure 18.

Figure  18  Time history of the water height on the deck

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Figure 18(a) depicts that with an increase in the wave height, the water height on the deck gradually increases at PH1. Figure 18(b) depicts that the water height on the deck increased most obviously in Case 2 at PH3, and compared with Case 1, the crest value increased by 62 mm. However, the wave height had little effect on the green water duration at PH1 and PH3.

In Figure 18(b), an interesting phenomenon is observed. The water height on the deck of Case 1 has one significant peak at PH3. However, there are two peaks in Case 2, Case 3, and Case 4. The value of the first peak is larger than that of the second peak in Case 2 and Case 3. The value of the first peak is smaller than that of the second peak in Case 4.

This is because in the last encounter period, during the "wave run-up" process, a part of the water attached to the wave baffle returned to the deck, and accumulated water formed from the wave baffle to the bow. This part of the water body, "backwater, " participates in the green water process in the current encounter period.

When the wave steepness was small, due to the small amount of "backwater, " the peak value of the water height on the deck at PH3 was hardly affected in the current encounter period. Therefore, Case 1 formed one significant peak.

When the wave steepness was larger than 0.050, the total amount of water on the deck increased, and the amount of "backwater" returning from the top of the wave baffle to the bow also increased. The "backwater" formed in the last encounter period will meet the green water opposite the "backwater" direction at PH3 in the current encounter period, as depicted in Figures 20(a) and (b). The "backwater" increased the height of the green water H_GW. Therefore, Case 2 and Case 3 formed the second peak after the first peak of the water height on the deck was formed.

Figure  19  Time history of the slamming pressure

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Figure  20  Green water at t = 20.00Te

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However, in Case 4, the "backwater" in the last encounter period has a greater impact on the peak of H_GW. The "backwater" impacted the green water in the current encounter period, as depicted in Figure 20(c). The amount of "backwater" is greater in Case 4 than in Case 2 and Case 3. Therefore, after the formation of the first peak, the second peak of H_GW was stimulated, which is larger than the first peak.

The analysis shows that the wave steepness and "backwater" have a great impact on the value and number of the peak of water height on the deck.

5.3   Influence of the wave height on impact loads

B1–B3 and B5–B7 were selected to evaluate the influence of the wave height on the impact loads at the side and bottom of the bow. The time history of the slamming dynamic pressure is depicted in Figure 19.

When the wave steepness increased from 0.050 to 0.056, the crest value of the slamming pressure significantly increased at B1– B3. B3 had the largest increase at the bow side in Case 3, approximately 20.83%, compared with Case 2.

The time history curves of B3 are smoother compared with those of B1 and B2 at the bow side. This is because B3 is at the low-tumblehome freeboard. Different from the bow-flare slamming, the pressure of B3 did not sharply increase in a very short time, but it smoothly increased in the process of the bow entering the water. Hence, the tumblehome vessel can effectively avoid the sudden increase of pressure at the freeboard.

Figure 19(d), (e), and (f) demonstrate that the slamming pressure sharply increased at the bottom of the bow. In particular, at station 0, the B5 slamming pressure increased by approximately 4 000 Pa in a moment. Large wave steepness caused the water entry and exit from the bow.

Compared to the ship side, the slamming pressure at the bottom of the bow changed more violently in a very short time when the ship entered the water.

5.4   Dynamic process and "wave run-up" of green water

Figures 20–Figures 24 depict the green water of the wave-facing tumblehome hull at four stages under a large wave steepness during an encounter period. The dynamic process and "wave run-up" of the green water were evaluated.

Figure  21  Green water at t = 20.25Te

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Figure  22  Green water at t = 20.50Te

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Figure  23  Green water at t = 20.75Te

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Figure  24  Green water at t = 21.00Te

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During the first 1/4 encounter period (20.00Te – 20.25Te), when the crest of waves reached the ship bow in Case 2, there is no water on the deck behind the wave baffle at t = 20.00Te, as depicted in Figure 20(a). However, in Case 3, Figure 20(b) depicts that there is a large amount of water on the deck left from the previous encounter period behind the wave baffle at t = 20.00Te. When H/λ was 0.067, the deck area covered by water increased, and water even appeared on the deck at the stern, as depicted in Figure 20(c). Meanwhile, the "wave run-up" process is still in progress at t = 20.00Te. At the end of the first 1/4 encounter period (t = 20.25Te), the water surged onto the deck, and the water body was concentrated in the area from the bow to the wave baffle, as depicted in Figures 21(a)–(c).

Hence, when H/λ increases to 0.056, even if there is a wave baffle with a 20° inclination, it will still lead to serious water accumulation on the deck.

In the second 1/4 encounter period (20.25Te–20.50Te), in Case 2, the water on the deck continuously flowed out to the ship side, reducing the total amount of water on the deck, as depicted in Figure 22(a). Compared with Case 2, the total amount of water on the deck increased at t = 20.50Te in Case 3, as depicted in Figure 22(b). In Case 4, the amount of water at the wave baffle significantly increased. Meanwhile, a large amount of water flowed out to the ship side due to the obstruction of wave baffles. The water body tends to climb over the wave baffle at the end of the second 1/4 encounter period, as depicted in Figure 22(c).

In the third 1/4 encounter period (20.50Te–20.75Te), in Case 2, when a water body was obstructed, water parallel to the wave baffle was formed. The water body climbed to the top of the wave baffle, exhibiting the "wave run-up" phenomenon at t = 20.75Te, as depicted in Figure 23(a). However, in Case 3, the large trim amplitude by the stern accelerated the impact of water on the wave baffle. Compared with Case 2, the water body thickness at the wave baffle was significantly increased. The water climbed up and exhibited a plunging trend on the top of the wave baffle at t = 20.75Te, as depicted in Figure 23(b). When H/λ was 0.067, the height of the "wave run-up" was higher than that of Case 3, as depicted in Figure 23(c).

In the last 1/4 encounter period (20.75Te – 21.00Te), in Case 2, the "wave run-up" process ended. As the hull inclined toward the bow, a part of the water attached to the wave baffle returned to the deck and formed water on the deck at t = 21.00Te, as depicted in Figure 24(a). In Case 3, the plunging water body slapped on the deck behind the wave baffle. However, the "wave run-up" process is still in progress in Case 4. Figure 24(c) depicts that the water accumulation on the deck behind the wave baffle was more serious. The water on the deck was utilized in the next green water period.

6   Conclusions

In this study, the green water of a wave-facing tumblehome vessel at Fr = 0.22 was simulated under strong nonlinear regular waves (0.033 ≤ H/λ ≤ 0.067). The motion response, water height on the deck, dynamic process of the green water, and "wave run-up" phenomenon were evaluated under different large wave steepness. The main conclusions are as follows:

1) The predicted time history of the water height on the deck and slamming pressure was verified with the experimental results under H/λ = 0.05. This finding proves the applicability of the CFD method for simulating green water under large wave steepness.

2) When H/λ increased from 0.050 to 0.056 at Fr = 0.22, the amplitude of the heave motion and trim by the bow increased by approximately 30% and 14.57%, respectively. The change in the water height on the deck was most obvious when H/λ increased from 0.033 to 0.050. However, the wave height had little effect on the green water duration. In addition, the wave steepness and "backwater" have a great impact on the value and number of the peak of water height on the deck.

3) The tumblehome vessel can effectively avoid the sudden increase of slamming pressure at the freeboard. Compared to the ship side, the slamming pressure at the bottom of the bow sharply increased. At the bottom of station 0, when H/λ is 0.056, the slamming pressure increased by approximately 4 000 Pa in a very short time.

4) The "wave run-up" in all three cases occurred in the third 1/4 encounter period. As the wave steepness increased to 0.056, the water climbed up, and a plunging-type water body was formed at the top of the wave baffle, causing a large water area on the deck. Meanwhile, an increase in the wave height will lead to the prolongation of the "wave run-up" duration.

This study offers value in terms of the prediction of green water in tumblehome vessels in extreme sea waves. In the future, more evaluations are required on the influence of wavelength, wave baffle shape, and inclination angle on green water.