1. Introduction
When ships navigate through the sea, they often encounter adverse weather conditions that require sailing through rough waves. The phenomenon of slamming that occurs when the hull collides fiercely with the waves can seriously damage the structural integrity of the ship and pose a threat to the life safety of maritime personnel. In 1994 [
1], the Estonia sank because the wave loads it encountered exceeded the structural limits the vessel could withstand, leading to the damage of the positioning devices and the lock structures of the foredeck shelter-type bow doors. When encountering violent waves, the bow of the ship emerges from the sea and then re-enters the waves at a relatively high speed, colliding with the waves to produce the so-called slamming phenomenon. Based on the different areas affected by the slamming load, slamming can be classified into three types: bow flare slamming; stern slamming; and bottom slamming, with each type of slamming causing different dynamic responses [
2].
For the exploration of slamming issues, scholars have proposed various theoretical and experimental methods for in-depth study. Ochi [
3] proposed a correlation between the length of the hull and the slamming duration, adopting a quasi-static approach to study the dynamic response of the hull beam under slamming loads. Yu Pengyao et al. [
4] used three-dimensional linear potential flow theory and long-term analysis methods of slamming speed to explore the direct calculation method of design loads for bow flare slamming pressure on ships. Hong et al. [
5] conducted research on the dynamic characteristics of bow flare slamming in regular and irregular wave conditions, discussing the spatial distribution and temporal progress of slamming loads on the bow flare. The experimental studies of hull models in wave tanks are of significant guidance and can provide a better understanding of the phenomenon of wave slamming. Hermundstad et al. [
6] proposed a theoretical method that can effectively predict the slamming loads acting on the hull beam and verified the accuracy of this prediction method through model tests. Wang et al. [
7] conducted experimental studies to investigate the bow flare slamming and bottom slamming phenomena of chemical tankers under the action of irregular waves and measured the probability and values of slamming loads, which has important practical value for ship design. Lindemann et al. [
8] conducted research on the dynamic failure characteristics of ship structures under longitudinal and transverse slamming loads through experimental methods, considering the effects of inertia and damping, and improved the existing ideal structural unit method. Wang Xueliang et al. [
9,
10] took a large LNG ship as the research object and compared the wave-induced vibration response of the ship using tank model experimental methods and three-dimensional linear hydroelastic theory, which indicated that the wave slamming loads often cause issues of ultimate strength and fatigue damage in the ship structure. Jiao Jialong et al. [
11] proposed a large-scale segmented-model wave-load testing technique in real sea wave environments and conducted hydroelastic tests on large-scale segmented ship models.
In the area of numerical simulation, Zhu Renqing et al. [
12] conducted numerical simulations on the slamming problem of wedge-shaped bodies entering the water and obtained the effects of wedge stiffness on the water entry process. Yang et al. [
13] studied the dynamic ultimate strength of hull beams under slamming bending moments through numerical simulation methods. The results showed that the duration, amplitude, and impulse of the slamming affect the dynamic response of the hull beam. Mackie [
14] considered the influence of the fluid-free surface in the study of the two-dimensional rigid wedge entering the water and transformed it into a similar flow problem on the complex plane, thereby deriving the shape of the free surface and the slamming pressure. Bilandi et al. [
15,
16] used a combination of the finite volume method and the volume of fluid method to numerically simulate the vertical water entry impact of two-dimensional symmetrical and asymmetrical wedge structures. They also analyzed the hydrodynamic behavior of stepped planing hulls in eight different configurations using CFD methods. The results indicated that specific step heights and placements can significantly reduce the resistance of the vessel, thereby enhancing the performance of high-speed planing boats. Vesselin et al. [
17] used Open FOAM to predict the asymmetric water entry slamming pressure of two-dimensional wedges and the history of liquid surface changes. Zhao Zhongbang [
18] predicted the slamming loads for a typical position at the bow of a VLCC ship using a two-step method. The study established a three-dimensional uniform cross-sectional model and calculated slamming pressure curves and peak values at each point. Zhang Boran [
19] simulated the motion of a three-dimensional hull through numerical waves, compared the pressure data of various points during the slamming process under head-on and oblique waves laterally, and obtained the trend of bow pressure distribution under different navigation conditions. Stavovy [
20], based on the assumption that the slamming speed is equal to the relative speed between the moving body and the wave-slamming surface, derived a method for calculating slamming pressure applicable to all types of hulls. Peng Dandan [
21] studied the impact of different parameters on the dynamic response of two-dimensional and three-dimensional structures, discussing the effects of structural plate thickness and changes in the elastic modulus on slamming pressure. Wang Jiaxia et al. [
22] used STAR CCM+ to study the three-dimensional nonlinear wave slamming loads and conducted numerical predictions and hull optimization for the slamming loads in the bow and stern regions of a large cruise ship navigating in waves. Chillemi et al. [
23] used CFD methodology simulations and parametric optimization algorithms to improve the form design of a racing motorbike. It was shown that the optimized design significantly reduces air resistance and improves downforce, thus enhancing the track performance of the motorbike. Lee et al. [
24] analyzed the structural response of two types of planing hull grillage panels in irregular waves through experiments and numerical simulations. The results showed that the greatest impacts occur in deep troughs with low wave heights. Numerical simulations more accurately predict the vertical acceleration of the center of gravity compared to traditional methods, demonstrating the efficiency of computational approaches and the conservatism of current design practices.
Overall, this thesis employs experimental and numerical simulation methods for its study. The experimental part mainly uses water entry experiments of wedges with different inclination angles to simulate the slamming process of ships in waves, examining how changes in entry speed and wedge inclination affect the slamming impact. The numerical simulation involves the coupling of CFD and FEM, utilizing STAR-CCM+12.02.011-R8 and Abaqus 6.14 software for coordinated two-way fluid–structure interaction analysis, allowing multiple data exchanges per time step until convergence criteria are met, thereby enhancing the calculation’s stability and accuracy. This coupling method can more accurately predict the pressure distribution during the slamming process, the elevation of the free surface, and the dynamic structural response. It provides a scientific basis for ship design and has an important reference value for assessing ship safety under extreme sea conditions. This research not only enhances our understanding of ship slamming response but also advances related numerical simulation technologies.
2. The Numerical Methods
2.1. Governing Equations
The finite volume method (FVM) was imported into STAR-CCM+ software, and the integral form of the governing equation was discretized into a system of algebraic equations in the time and space dimensions. First, the calculation domain was divided into a limited number of adjacent control bodies. These control bodies can be of any polyhedron shape. The discrete governing equations also need to use area integration, volume integration, and time and space derivatives in the calculation process.
It was assumed that the flow is controlled by the RANS equation for viscous three-dimensional flow, where the turbulence effect includes a vortex model and a viscous model. At this time, the continuity equation, momentum equation, and two turbulence characteristics equations needed to be solved. The model selected the Realizable K-Epsilon turbulence model to simulate the effect of turbulence in the fluid, which added some improvements to better handle turbulent flows compared to the Standard K-Epsilon model for high Reynolds number flows, where the governing equations of mass conservation and momentum conservation in integral form can be written as follows:
Conservation of momentum:
where
is the density of water,
is the fluid velocity vector,
is the surface velocity of the control body;
is the unit vector normals perpendicular to the surface of the control body, where
and
are the area and volume of the control body, respectively,
is the stress tensor (representing the velocity gradient and eddy viscosity),
is the pressure,
is the unit tensor.
The volume of fluid model assumes that all immiscible fluids in the control body have the same velocity, pressure, and temperature fields, so only the basic governing equations for the conservation of momentum, mass, and energy of a single-phase flow field in the control body require solving. The conservation equation describing the volume fraction
is
where
is the source or sink of phase
flow field,
/
is the material derivative of phase density
.
2.2. Structural Response Equation
The dynamic response of a three-dimensional ship under a slamming load was investigated through the finite element method, and the effect of the elastic structure response on the slamming load was also taken into account. The three-dimensional model was discretized to obtain a finite element model of the structure. This is also a basic step in finite element calculation. The basic equation of the three-dimensional elastic dynamic equation is based on obtaining the finite element model of the hull structure, and the finite element solution steps for the three-dimensional solid dynamic analysis, which are as follows:
where
is the mass matrix;
is the damping matrix;
is the stiffness matrix;
is the external incentive;
is the node acceleration vector;
is the node speed vector;
is the node displacement vector.
2.3. Fluid–Structure Interactions
The wave-slamming problem of the hull structure belongs to the strong nonlinear, fluid–structure interaction problem. In this paper, the data were automatically transferred at the fluid–solid interface through the cooperative coupling function between software STAR-CCM+ and Abaqus to realize the two-way fluid–structure interaction calculation. STAR-CCM+ solves the fluid equation, and Abaqus solves the structural equation, which belongs to the partitioned fluid–structure interactions. In this article, coupling calculations were coupled using an implicit method between STAR-CCM+ and Abaqus, mainly because the implicit coupling method is suitable for strong coupling calculations between structures and fluids. It allows STAR-CCM+ and Abaqus to exchange data more than once each time and keep iterating until the convergence criteria are met. This coupling scheme is more stable, and the second-order accuracy is higher.
Since the wave-slamming phenomenon is a strong coupling phenomenon, an implicit strong coupling algorithm was selected in the numerical model. As shown in
Figure 1, a single coupling procedure was taken as an example to introduce the process of the implicit coupling time step. Set the motion specification as ‘deformation’ in STAR-CCM+, and set the elastic modulus, plate thickness, initial speed, gravity, time step, and coupling surface of the model in Abaqus. The specific calculation process was as follows: STAR-CCM+ starts the flow field load calculation, then transmits the pressure and shear force to the FEM solver, and then Abaqus performs structural response analysis based on the load of the coupling interface and transfers the displacement and deformation to STAR-CCM+. STAR-CCM+ then moves the mesh by the ‘deformation’ function according to the amount of displacement transmitted. This completes a coordinated coupling for calculation and update and repeats until the maximum physical time is reached.