Hydrodynamic Analysis of Ship with Well Deck in the Linear Numerical Wave Tank

In recent years, the development and construction of islands and reefs has been proposed by the government and commercial company. However, as a large cargo carrier cannot reach islands and reefs if the harbor is not available, such type of carrier which has well deck is designed to meet the requirements of delivering people and equipment. It is a possible way to connect the island and supply cargo ships. -is paper firstly summarizes the domestic and foreign research progress of hydrodynamic analysis of ships with well deck. -en, based on the CFD (Computational Fluid Dynamics) tools, we set up a linear numerical wave tank and study the hydrodynamic performance of original Wigley-III ship and modifiedWigley-III ship with well deck.-e hydrodynamic effect of the floating body in the well deck has been investigated and discussed.


Introduction
Recently, with the development of computer technology, various numerical modeling tools are becoming possibility. Compared with the physical model test, the study period of the numerical method is short, the cost is small, and the resource occupancy is low. Moreover, the traditional design mode has been changed by the numerical method and its process is becoming simple.
Longuet-higgins and Cokelet [1] used the Euler-Lagrange method to simulate plunging breaker based on the idea of direct solution of boundary integral. Baker et al. [2] added the artificial damping viscous term to the boundary of a specific free surface in the same way as the sponge absorbed wave energy and presented the concept of damped wave elimination region in the twodimensional boundary element mode for the first time. Vinje and Brevig [3] proposed the modal decomposition method, which makes the numerical wave tank be capable of dealing with hydrodynamic calculation and motion simulation of floating bodies. Based on completely nonlinear free surface boundary, Jin and Zou [4] established a new vertical two-dimensional wave equation by taking wave surface rise and free surface velocity potential as the variables to be determined. Yan et al. bib5 used the twopoint method to separate the wave morphology of each component wave after the submerged body and established a two-dimensional fully nonlinear numerical model to simulate the wave effect on a submerged horizontal cylinder based on the in-domain wave-making technology. Fang et al. [6] used the mixed finite difference and finite volume numerical schemes to solve the two-dimensional fully nonlinear distributed water wave equation and established the fully nonlinear Boussinesq wave propagation numerical model based on the MUSTA scheme. Zhang et al. [7] established a two-dimensional viscous numerical water trough based on the CIP method, which can be used to simulate linear, weakly nonlinear, and strongly nonlinear waves. Li and Ning [8] simulated and studied the deformation of long wave train propagation and the energy exchange between waves of various orders.
Bai and Taylor [9][10][11] analyzed the phenomenon of fully nonlinear regular wave and focusing wave, respectively, on vertical cylindrical diffraction and the effect of completely nonlinear wave on fixed and floating structures. Li et al. [12] used high-order spectral methods to simulate three-dimensional fifth-order deep-water Stokes waves to verify accuracy and convergence. Chen et al. [13], based on the CIP method for the gas-liquid two-phase flow numerical model, carried out numerical simulation of the propagation and deformation of solitary waves on coastal slopes. Wu et al. [14] studied the effects of wavelength, water level, effective wave height, and friction resistance on the propagation deformation of near-shore waves.
Many effective methods have been derived for the numerical simulation of the interaction between waves and structures. Presently, the hydrodynamic analysis of ships with well deck is based on the viscous flow theory, which can realize the numerical simulation of the coupled response of ship motion and flow field generated in well deck.
Hopman et al. [15] realized the test of two ship models (with good performance and newly designed) through a typical model test program, made the model in different modal time periods, and simulated the ship model test in a series of wave directions and speeds under two irregular spectral conditions. Subsequently, Bass et al. [16] used the commercial CFD integration code with ship motion code (MOTSIM, a 3D nonlinear time domain program) to simulate the wave interaction in the wetted well deck and verified the numerical simulation results with the model test. Cartwight et al. [17] used SPH (Smoothed Particle Hydrodynamics) method to simulate 2D landing craft containing well deck and explore the influence of well-deck design on landing craft movement. Yoon and Seo [18] conducted model tests on landing craft with different well-deck positions and studied their seakeeping and safety, respectively.
In this work, a simplified ship model is used to verify the effectiveness of the numerical tank. Based on the published Wigley-III ship model data, the numerical simulation of ship diffraction in waves is conducted based on the CFD method. Firstly, the heave and pitch motions of the ship and the modified ship are simulated and compared with the test data of DUT (Delft University of Technology) [19,20]. Secondly, the heave and pitch motion of the Wigley-III ship model are simulated. Finally, an attempt is made to simulate the motion of well deck with floating structures, so as to provide an effective technical scheme for relevant study.

Governing Equation and Calculation Algorithm.
Continuous equation and Navier-Stokes equation were applied as the governing equation of an incompressible flow as follows: where U is the velocity, which has three components in x, y, and z direction, respectively, ρ is the density, t is the time, x is the coordinate, p is the pressure, v is the kinetic viscosity, −u i u i is the turbulent stress, and g is the acceleration due to gravity. ICEM CFD and ANSYS FLU-ENT are used for grid generation and for the numerical simulation, respectively. FLUENT is based on the Finite Volume Method (FVM), in which governing equation has been discretized by the FVM. e Computational Fluid Dynamics (CFD) package, FLUENT, has been successfully utilized to solve the fluid and fluid-structure interactions, and it is used as a basic solver with User Defined Functions (UDF) in this work. To ensure accurate results, we adopt the following settings. (1) e pressure staggering option (PRESTO) discretization scheme is used to solve the pressure. (2) e quadratic upwind interpolation of convective kinematics (QUICK) algorithm is used to discretize the momentum. (3) e pressure implicit with splitting of operators (PISO) algorithm is used for the pressure velocity coupling. (4) e volume of fluid (VOF) method is adopted to capture the water surface. (5) e RNG k-ε model (2eqn) is used to simulate the flow of the incompressible and viscous fluid.
To deal with the wave and structure interaction in the Numerical Wave Tank (NWT), the work adopts the C programming language into the basic solver through UDF. e UDF is used to deal with the wave-generating boundary conditions in the inlet boundary, the wave absorbing zone in the end of the Numerical Wave Tank.

Boundary Conditions.
Generally, the top boundary of the computation domain is taken as a pressure inlet, the bottom boundary of the computation domain is the wall, which depends on the depth, and the left-and right-side boundaries of the computation domain are set as symmetry.
e setting of the left inlet wave boundary of the wavegenerating area and the right-end boundary of the waveeliminating area needs to be adjusted according to the calculation, as presented in Figure 1. e other boundaries are also adjusted accordingly to the calculation which can be performed stably.

Numerical Model and Mesh Discretization for NWT.
e calculation domain is designed as a length of 25 m, a width of 2 m, and a depth of 3 m. e distance between calm water surface and pressure inlet boundary is 1 m and the water depth is 2 m. e end of tank is the wave elimination area, which is 10 m, as shown in Figure 1.
In Figure 1, G1, G2, G3, and G4 are the wave elevation probe, which is located at x � 2.5 m, x � 5 m, x � 10 m, and x � 25 m to monitor the corresponding wave height, respectively. e model is meshed by ICEM CFD module of ANSYS. In the longitudinal direction of the main computational domain, the meshes are distributed equally along the wave propagation direction; however, in the wave-eliminating region, the meshes gradually become bigger, which can reduce the total number of grids, the operation time, and can improve the calculating efficiency, meanwhile which can result in numerical dissipation. In the vertical direction, according to the water wave theory, the grid is encrypted near the water surface, in order to capture the accurate position of the free surface and improve the numerical accuracy. e place, which is far away from the free surface, will become sparse gradually. In the transverse direction, the grid is divided equally, as shown in Figure 2.

Convergent Study for NWT.
In the numerical solution, mesh discretization and time interval are the most important factors which affects the simulation accuracy, and the numerical solutions have to be stable with the improvement of mesh encryption skill and time interval. us, the convergence analysis of the numerical wave tank is carried out and the wave parameter for the simulation is described in Table 1.
e convergence study of time interval has been carried out firstly and has been presented in Figure 3. It can be obtained that convergent results of wave height with different incident wavelength can be found while time interval is less than 0.01 s.
We also consider the study of the mesh density, which is represented by the number of cells Nx in the longitudinal direction within each wavelength and Nz in vertical direction per wave height. We can find that good and convergent results can be obtained when Nx is great than 50 and Nz is great than 25, as shown in Figures 4 and 5. Nx � 50 and Nz � 30 will be applied for the following simulations, and the snapshot of free surface can be found in Figure 6.
We can also find the time history results of wave elevation at different probes (x � 2.5, x � 5, and x � 10) and good agreements can be obtained by comparing the present CFD results and theoretical value, as shown in Figures 7-9, respectively. We can conclude that the present tank can be used for long time simulations. From comparison, we can also find the nonlinear phenomenon because of the nonlinear governing equation. It has typical nonlinear effect that the peaks and troughs of waves show asymmetric.
e efficiency of wave elimination can be found in Figures 10 and 11. High efficiency wave-eliminating result can be obtained in the present model as the wave-eliminating technology has been involved in the end of the tank.  [19,20]. Journée evaluated the motion of the four kinds of Wigley through the experimental analysis. is paper will use the Wigley-III model, and the main dimensions are shown in Table 2.

Wave Diffraction of Wigley-III Ship Model
In Cartesian coordinate system oξηζ, the origin of coordinates o is located on the steady water in the middle of the Wigley-III ship and it can be described mathematically by equation (2). Figures 12 and 13 display the exact shape of the ship in 2D and 3D, respectively: (2)

Boundary Conditions.
e inlet boundary is a velocity inlet, which is defined by UDF. When the wave passes through the wave elimination area, the right end should be stationary; thus, the right end boundary can be set as pressure-outlet. e top of the tank is set as pressure-outlet boundary condition. e bottom and the surface of Wigley-III are set as wall boundary conditions. Both sides of the numerical wave tank are set as symmetrical boundary conditions.

Numerical Simulation and Discussion of Wave Diffraction
Case.
e test data can be used to verify the simulation results in the numerical wave tank. Journée [20] provided the data of the Wigely model in regular wave, which will be adopted for the comparison in this work.

Numerical Model Setting.
In this paper, numerical simulation is carried out by using the Wigley-III ship model under the incident wave and with forward speed. e calculation parameters are shown in Table 3. (X w1a ) and the coefficient of the corresponding force (X w1a ″ ). X w1a and X w1a ″ can be calculated by the following equation:

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By comparing the experimental data, it can be found that the CFD simulation results in this work are in good agreement with the experimental data. Figures 16 and 17 show the heave force X w3a and its coefficient X w3a ″ and their comparison with the test data. e numerical results are in good agreement with the experimental results and the heave force of the ship model in the case of heading wave increases with the wavelength, and the relative error is very small. Figures 18 and 19 show the pitch force and its coefficient under the incident heading wave condition. It can be seen that the calculated results in this work are in good agreement with the experimental data.

Numerical Results at Incident Wave Direction β � 150°.
From the numerical results of wave force in the previous section, the CFD results in this paper agree well with the test data. e method adopted in this paper can simulate the ship sailing in heading waves. However, the sea state is random including wave direction and wave period, and we need to consider the typical wave angle such as β � 150°and 3D BEM [21] results will be used for comparison. Figure 20 shows the coefficient of exciting force on the Wigley-III ship model under six degrees of freedom. As we can see that the dimensionless coefficient of force in waves are in good agreement with the results of the potential flow theory. For the long wave simulation, the CFD results in this paper are slightly larger than that of potential flow theory. e reason [22,23] is that it has a serious inaccuracy in the flow field near the wave surface when the potential flow is used to solve the velocity problem. It has been solved by Duan et al. and the technology has been used in China Numerical Tank.

Hydrodynamic Study of Wigley-III with
Floating Body in the Well Deck

Modified Wigley-III Ship
Model. e public data for the ship with well deck cannot be found in the database, so we set a ship with well deck artificially. In order to analyze         Figure 21. Note that the shape of the present ship model is quite different from that of the actual ship with well deck, and further study should be executed by a more appropriate ship model.

e Model with Small Floating Body in the Well Deck.
Based on the modified Wigley-III ship, a small floating structure has been put into the well deck to analyze its effect and multibody interaction. e floating body in the well deck is simplified with a rectangular floating object for the convenient calculation. e model is shown in Figure 22 and the parameters can be found in Table 4.

Numerical Model Setting.
When the well deck is opened and the floating body exits in the well deck, the hull is mostly at zero forward speed; thus, this part focuses on the motion of the hull without forward speed. In the numerical wave tank, the Wigley-III ship model, the modified Wigley-III ship model, and the modified Wigley-III ship model containing the floating body in the well deck has been calculated and compared. e parameters for the simulation are given in Table 5.

Numerical Results with Heading Waves β � 180°.
e hull, encountering different incident heading waves, has been calculated. e obtained results are shown in Figures 23  and 24. It can be seen that the numerical results are consistent with the experimental data.

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In Figure 23, the results of the modified Wigley-III ship model has a similar trend with the original Wigley-III ship model, and it can be clearly seen that the heave coefficient of the modified Wigley-III ship model is reduced when the wavelength is greater than the length of the ship. It means that the well deck is good for the heave motion.
For the pitch, as shown in Figure 24, the trend of the ship model with the well deck changes significantly with the short waves. e amplitude of pitch motion increases significantly when λ/L � 0.5. We can also find that the well deck has disadvantage for the pitch motion. rough comparing the motions, it is found that the floating body in the well deck has small effect on the overall motion of the ship. Figures 25 and 26 show the heave and pitch of the ships at incident wave direction β � 45°. e coefficient of them has been presented in Figures 27 and 28. It can be seen that the motions of ships tend to be consistent and have same trend.        have serious effect on the modified Wigley-III ship. Later, we may focus on the motion of the floating body in the well deck in the high sea state.

Conclusion
Based on CFD (Computational Fluid Dynamics) tools, this paper uses FLUENT to simulate the Wigley-III ship model in three-dimensional linear numerical wave tank and the modified Wigley-III ship with well deck also has been investigated. rough the calculation we have conclusions as below: (1) Based on the Navier-Stoke governing equation, the VOF method was used to capture the water surface, and the RNG turbulence model was used to solve the numerical wave tank. e effects of mesh discretization and time interval have been presented and discussed. We found that the time interval has much greater effect on the accuracy of numerical simulation when the mesh discretization is fixed. We can obtain accurate results when regular wave propagates in the numerical wave tank by using the present model.
(2) As the public data for ships with well deck is not available, we modified the stern of the original Wigley-III ship model. e numerical simulation for the original Wigley-III ship and modified Wigley-III ship with the well deck has been carried out, and the numerical results are in good agreement with the experimental data for wave diffraction problem and the motion of the modified model is similar to that of the original ship. (3) We also found that the floating body did not have significant effect on the modified Wigley-III ship with the well deck. For the further study, we should consider the ship under the high sea state and predict the motion of the floating body in the well deck.

Data Availability
e data used to support the findings of the study are available from the corresponding author upon reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.