A Review of Experimental and Numerical Research on the Slamming Problem of High-Performance Vessels

Abstract

Slamming load is characterized by a high peak and short duration. Severe slamming phenomena are extremely detrimental to the navigation safety of high-speed vessels, thereby constraining the development and application of high-performance ships. Studies on slamming mechanisms, load distribution, prediction, and mitigation methods are particularly essential. This paper provides a comprehensive review of the theoretical, numerical, and experimental research progress on water-entry slamming for high-performance ships. First, the theoretical foundations and numerical simulation methods of slamming are elaborated. Then, existing research findings are summarized from two perspectives: segmented water entry and full-scale wave loads. Finally, unresolved issues and future research directions are identified. The aim is to offer valuable insights for further advancements in high-performance ship slamming studies.

1. Introduction

When navigating in severe sea conditions, ships will generate significant vertical motions, leading to slamming phenomena. During water entry, the hull interacts with the surrounding fluid, generating impact forces, moments, and cavity formation, accompanied by complex hydrodynamic effects such as cavitation, bubble pulsation, spray jets, and intricate fluid–structure interactions [1]. The slamming-induced pressure is characterized by an extremely short duration and high peak magnitude, resulting in highly nonlinear wave loads that cause substantial structural damage. Moreover, the transient slamming force excites elastic vibrations of the hull girder, known as whipping vibration. When the high-frequency whipping-induced bending moments superimpose with the low-frequency wave-induced bending moments, they pose a severe threat to the global longitudinal strength of the hull. According to full-scale measurements and model tests conducted worldwide, the whipping vibration bending moment can reach 30~40% of the total wave-induced bending moment [2]. The slamming pressure is influenced by factors including the hull stiffness, shape, and deadrise angle. In high-curvature regions of the hull, the peak slamming pressure increases significantly while its duration shortens. The slamming pressure intensifies as the deadrise angle decreases. During asymmetric water entry, the side with a smaller angle to the water surface experiences higher pressure [3].

High performance vessels (HPVs) refer to the vessels that feature excellent speed and seakeeping capabilities, along with good economic efficiency and carrying capacity, and have strong adaptability to the environment. In recent years, they have been widely used in both military and civilian fields. However, when encountering severe sea conditions, if slamming occurs more than three times in every 100 encountered waves, the vessel will actively reduce its speed to ensure navigation safety [4]. This demonstrates that slamming phenomena at high speeds have become a bottleneck issue restricting the development of high-speed vessels [5]. The slamming phenomena include bottom slamming, flare slamming, wet-deck slamming, whipping, and so on. In recent years, almost all classification societies have issued the latest structural codes applicable to high-speed vessels. These codes stipulate the design pressure, stress, and component dimensions to adapt to the contemporary development of high-speed ships. Accurate prediction of the slamming load and structural responses has become one of the key prerequisites for the safety of ship design.

Research on ship slamming aims to reveal transient load characteristics, structural response mechanisms, and their coupled effects during high-speed collisions between waves and the hull. There are two computational approaches for analyzing the structural response of a hull under nonlinear slamming loads. One method involves two steps: initially assuming the ship as a rigid body to calculate the total hydrodynamic forces, followed by treating the hull as an elastic body to solve for its vibration response. The other method treats the ship as an elastic body from the outset, employing hydroelasticity theory to solve for the vessel’s global response in waves. This approach simultaneously accounts for slamming loads and feedback effects of structural elasticity on hydrodynamic forces. The recently developed fluid–structure interaction (FSI) methodology [6], which couples computational fluid dynamics with structural mechanics methods, enables the calculation of spatiotemporal distributions of slamming loads, structural deformation, and damage mechanisms. This approach provides critical theoretical support for ship impact-resistant design.

Model testing is also a crucial approach in ship slamming research, primarily including water-entry tests, towing tank tests, and full-scale trials. Water-entry tests involve releasing a ship model freely or at a controlled velocity to impact the water surface using a guide rail or pendulum system. The wave makers are employed to simulate various sea states, combined with a towing system to adjust the model’s speed in towing tank tests. Slamming pressure is measured using piezoelectric sensors or strain gauges, while hull motion is captured via accelerometers. Due to the instantaneous and violent nature of the slamming phenomenon, multiple repeated experiments are required to ensure the reproducibility of the results. This requires the sensors to have a certain ability to resist impact and stable performance. With the application of particle image velocimetry (PIV) [7], researchers could combine high-speed photography to record the internal flow field evolution during water entry, capturing vortex structures. This approach further facilitates the study of slamming pressure distribution and its time-dependent mechanisms. For full-scale trials, pressure sensors or fiber-optic strain sensors are installed at critical position such as the bow and side hull. However, under extreme sea conditions (e.g., typhoons), these sensors must address challenges related to stability and impact resistance.

With the advancement of computer technology, slamming theoretical models have been continuously expanded and refined. The coupling of multiple computational methods has significantly improved the efficiency and reliability of numerical simulations. Meanwhile, the evolution of experimental techniques has enabled researchers to uncover new phenomena and mechanisms, thereby better validating numerical results. Most of the research on ship slamming focuses on planing boats, wedge entry into water, and multi-hull ships [3]. This paper begins with the fundamental theory of slamming, providing an overview and comparison of the distinctions and advantages among several numerical methods. Two key aspects have been reviewed: water entry of high-performance vessel hull segment and whole-ship wave load analysis. Existing research findings are summarized from both numerical simulations and model tests, offering valuable references for future studies in this field. The structure of the paper is shown in Figure 1.

2. Calculation Methods

2.1. Slamming Theory

The study of slamming theory began with von Kármán’s analysis of the slamming loads on seaplanes during landing [8]. The landing process has been simplified as a horizontal cylindrical body with an inclined cross-section entering the water vertically at a certain speed. By introducing the concept of added mass and applying the law of momentum conservation, the calculation formula for the slamming pressure of a wedge-shaped body has been derived. Wagner [9] improved upon von Kármán’s method by introducing the linear free-surface condition, taking into account the effect of wave elevation. He simplified the wedge-shaped body as an equivalent flat plate and solved the pressure distribution on the bottom surface using the velocity potential of flow around a flat plate and Bernoulli’s equation, significantly improving computational accuracy. Although this method is only applicable to wedge-shaped bodies with a deadrise angle of less than 30°, the theoretical framework was further refined and extended in subsequent research [10,11].

An improved approach to Wagner’s theory involves adding additional terms to the velocity potential expression. Logvinovich [12] employed a local asymptotic expansion analysis, taking into account the nonlinear terms in Bernoulli’s equation, and proposed the Logvinovich calculation model (original Logvinovich model). Korobkin [13] further retained the second-order terms related to the deadrise angle of the wedge in the pressure distribution formula of this model, thereby developing the MLM (modified Logvinovich model) method, which can be applied to water-entry problems of wedges with larger deadrise angles. For the flow separation phenomenon after the water entry of a wedge with finite width, Duan [14] introduced auxiliary lines to construct a virtual body surface and incorporated the body surface function into the MLM method. This approach maintains accuracy while improving computational efficiency. Liu [15] derived an explicit solution for the higher-order MLM (modified Logvinovich model) slamming model based on the original MLM, enabling rapid computation of the second-order velocity potential, pressure distribution, and slamming force. This method improves the accuracy of slamming pressure calculations for wedges with larger deadrise angles during water entry. The study provides a new approach for the rapid assessment of slamming problems in structures with high deadrise angle characteristics.

Tassin [16] introduced fictitious body continuation (FBC) into the MLM to study the dynamic evolution of transient pressure on an object under the influence of fluid detachment at the jet root and cavitation behind the body surface, enabling the prediction of long-term transient drag during cavity formation. The new model can analyze water-entry problems involving flow separation and variable velocity. Unlike Fairlie-Clarke and Tveitnes [17], the relationship between object acceleration and force has been calculated by assuming that the added mass remains constant after flow separation. Finally, the FBC method was extended to smooth-body flow separation, including water entry of cylinders and wedges, validating the universality of the model.

For the slamming problem in waves, Wen [18] proposed the modified Wagner model (MWM), which determines the wetted length by solving a free-surface function. Compared to the Logvinovich model employing the curved fictitious body continuation (curved FBC) technique, the MWM model demonstrates superior performance during the slamming phase under linear incident waves, accurately predicting surface loads on the object throughout both the slamming and transition phases.

T.I. Khabakhpasheva [19] investigated the deflection and strain response of conical shells with small deadrise angles during constant-velocity water entry, based on the generalized linear theory for conical shells and the Wagner model for hydrodynamic impact loads. Assuming an infinitely deep axisymmetric potential flow, the singular component of the load was separated, and an improved convergent series was introduced for computation. The study demonstrated that conical shells can be approximated as circular plates only when the deadrise angle is extremely small. This work marked the first successful application of the orthogonal modal method—a simplified structural modal-based approach—to axisymmetric slamming problems in hydroelasticity.

Feng [20] conducted a study on the added mass during the hydroelastic slamming of a wedge. By accounting for variations in the wetted surface curvature, an analytical model for the eigenvalues of the added mass matrix was established (as shown in Figure 2). This enabled the development of a coupled algorithm integrating the modal method with the Wagner analytical solution for added mass computation. Subsequently, the eigenvalues of the added mass were nondimensionalized, and the relationships between the nondimensional coefficients and factors such as boundary conditions, deadrise angle, and added mass matrix dimension were analyzed. The computational results demonstrated that the studied boundary conditions, bottom deadrise angle, and structural stiffness range had no significant influence on the eigenvalues of the added mass. Furthermore, the eigenvalues of the added mass converged at higher dimensions, exhibiting a linear relationship with the wetted surface ratio. Based on these findings, the eigenvalues of the hydrodynamic added mass were used to construct constraint intervals and an optimal theoretical solution for the relaxation factor in an iterative coupling scheme. Through a mathematical model of iterative error, the study revealed the mathematical relationship between iterative error, added mass, and the relaxation coefficient. Based on this relationship, the effective interval for the relaxation factor under iterative stability requirements was determined (as shown in Figure 3).

2.2. Numerical Calculation

Numerical simulation methods can be primarily categorized into boundary element methods (BEM) based on potential flow theory and computational fluid dynamics (CFD) methods based on viscous flow theory, including the finite difference method (FDM), finite element method (FEM), and immersion boundary method [21] (IBM) based on grid description, and mesh-free methods such as smoothed particle hydrodynamics (SPH), moving tarticle semi-implicit (MPS), and lattice Boltzmann method (LBM).

2.2.1. Boundary Element Method

The boundary element method (BEM) is based on potential flow theory. It transforms the governing equations into boundary integral equations and applies fully nonlinear boundary conditions on the domain boundaries. Only the boundaries require mesh discretization, and the boundary integral equations are solved numerically [22]. Compared to methods like the finite element method (FEM), BEM does not require internal domain meshing or mesh updating, significantly reducing computational effort and saving substantial time. Slamming motion is characterized by high frequency and transient behavior, and theoretical predictions are often conducted in the time domain.

Due to the nonlinear boundary conditions during water-entry slamming, Zhao and Faltinsen [23] employed the constant-panel BEM to study the two-dimensional wedge entry problem. They simplified the numerical computation by approximating the jet spray region with multiple straight-line segments. For the wetted surface calculation during slamming, they adopted the direct finite difference method and integrated the second-order Green’s function using a time-stepping scheme. The range of the wedge’s deadrise angle is from 4° to 81°. The results demonstrate that the boundary element method (BEM) achieves good agreement with both the modified self-similar solution [24] and the asymptotic expansion method.

Subsequent studies integrated the BEM with Wagner theory, employing either the mixed Eulerian–Lagrangian (MEL) method or a stretched coordinate system to ensure numerical accuracy and stability during the initial phase of nonlinear free surface simulation. An auxiliary function was introduced to handle the time differentiation of the velocity potential, achieving decoupling between hydrodynamic loads and body acceleration [25]. This approach enabled the computation of three-degrees-of-freedom accelerations for wedge-shaped bodies [26]. Liu [27] investigated the fully nonlinear hydrodynamic problem of a cone vertically and obliquely entering Stokes waves using BEM. The study employed a stretched coordinate system and a modified Eulerian method to track the nonlinear free surface. Computational results showed the relative deadrise angle and relative velocity between the cone and fluid particles in a wave environment varied dynamically over time, while the flow velocity exhibited spatial non-uniformity in waves.

For the hydroelastic problem in slamming phenomena, the boundary element method can be used to solve the pressure integral, and then the pressure load can be transferred to a structural solver (such as the finite element method) to iteratively solve the fluid–structure interaction. An efficient fluid–structure interaction (FSI) strategy has been introduced to calculate the hydroelastic slamming pressure of wedge-shaped sections [28,29]. The strategy coupled the superposition method (for the structural part) and BEM. The mean elastic velocity method has been introduced to account for the influence of elastic response on slamming load distribution and established dynamic coupling equations governing the FSI effects. The introduction of nonlinear pressure terms improved computational accuracy. The study systematically analyzed the effects of plate thickness, water-entry angle, impact velocity (constant speed), and boundary conditions on hydroelastic behavior. This method proves more effective for wedges with larger water-entry angles and weaker hydroelastic characteristics. Furthermore, the approach was extended to numerical simulations of shallow-water elastic wedge impacts [30]. A non-penetration boundary was imposed at the fluid domain bottom to simulate shallow-water effects. The results demonstrated that shallow-water conditions significantly amplify free-surface motion, fluid pressure, and structural dynamic response, with the peak slamming pressure increasing by approximately 18%.

Wen [31] developed a hybrid boundary-finite element (HBF) method to solve the fully nonlinear incompressible potential flow for the entire process of a two-dimensional curved wedge entering water at variable velocity, neglecting the effects of gravity and fluid surface tension. The study proposed analytical expressions for the pressure coefficient and hydrodynamic force during the slamming phase, accounting for acceleration effects. Additionally, the method for modifying the added mass coefficient of a linear wedge was extended to curved wedges, effectively addressing acceleration effects during the transitional phase. A hydrodynamic force calculation formula was established, incorporating corrections for constant-speed slamming coefficients and added mass coefficients.

Overall, the boundary element method offers advantages in efficiency and dimensionality reduction when dealing with water-entry problems. However, the handling of the highly nonlinear characteristics of slamming motion may lead to significant errors in large-deformation regions. For instance, repeatedly applying the MEL method for free-surface tracking can reduce computational efficiency. Additionally, singularities on the body surface boundary require local mesh refinement. When coupled with FEM, BEM may introduce additional errors. Since BEM is based on potential flow theory, it cannot directly simulate viscous flow, potentially underestimating the hydrodynamic damping forces on the body and failing to capture vortex generation and turbulent dissipation.

2.2.2. CFD Method

Mesh-Based Method

Since the 1950s, CFD technology has achieved the visualization and quantitative analysis of complex flow phenomena by discretizing the N-S equation and numerically solving the motion and force conditions of each fluid particle in the flow field. By discretizing the fluid domain using grids, the method often couples with the volume of fluid (VOF) or level set techniques to track free surfaces, making them suitable for viscous fluid simulations.

Reddy [32] calculated the slamming load of a wedge with a 30° deadrise angle during water entry. The finite volume method was adopted to discretize the N-S equation, and the VOF method was used to track the multiphase flow interface. This approach solves the equations for air–water two-phase flow based on the fluid volume fraction. The study analyzed the influence of factors such as entry velocity and grid size on slamming loads during water entry and compared the results with experimental data to validate reliability [33]. Kleefsman [34] applied this method to simulate dam-break flows and wedge water entry, demonstrating excellent agreement with experimental photographs during the entry process.

For the problem of water-entry slamming in waves, Zhang [35] adopted a Cartesian grid framework, utilizing the VOF with piecewise linear interface reconstruction (VOF-PLIC) to capture the free surface and applying the volume-of-solid-based immersed boundary method (VOS-IB) to handle the solid boundary conditions. A relaxation zone was also implemented for wave generation and absorption. First, the model’s reliability was validated by simulating the constant-velocity water entry of a wedge in calm water, with results compared against historical experimental and numerical data. Subsequently, fifth-order Stokes waves were generated, demonstrating excellent agreement between the simulated results and analytical solutions.

Meshless Method

Meshless methods are Lagrangian particle-based approaches that eliminate the need for grids, offering distinct advantages over traditional mesh-based methods when handling free-surface fragmentation, splashing, and multiphase flows, particularly for complex geometries [36].

Smoothed particle hydrodynamics (SPH) is one of the most widely used meshless methods. The method was Initially proposed by Lucy [37], Gingold, and Monaghan [38] for astrophysical problems. This Lagrangian particle-based approach offers significant advantages in simulating multiphase flows. By simply assigning different particle properties to each phase, SPH automatically tracks multiphase interfaces [39] without requiring additional numerical techniques, making it highly effective for multiphase flow simulations. Subsequently, the SPH method was introduced into the field of fluid mechanics and has been widely applied to problems involving large deformations and free-surface flows [40,41,42,43,44,45,46]. Cui [47] adopted the SPH (smoothed particle hydrodynamics) method with open boundary conditions to construct a numerical wave tank, simulating the free water entry of two wedge-shaped bodies with different bottom curvatures under both still water and wave conditions. Based on the Herschel–Bulkley–Papanastasiou (HBP) constitutive model, a two-dimensional numerical model for rigid body impact on non-Newtonian fluids was established. This method effectively captures the motion of non-Newtonian fluids during slamming.

Since the mesh-based method can simulate solid structures and the flow of fluids with small deformations relatively accurately, coupling SPH with FEM can integrate the advantages of both methods. Zhang [48] and Chen [49] employed an adaptive element–particle coupling approach. This method systematically revealed the intrinsic mechanisms of hydroelastic slamming, such as the coupling between flow fields and hydrodynamic loads, the kinematic and dynamic responses of wedge bodies, and the interaction with structural vibrations. Additionally, an empirical formula has been proposed to predict the vibration characteristics of wedge bodies during hydroelastic slamming.

Traditional SPH requires the introduction of artificial viscosity to suppress numerical oscillations when handling shock waves, but this often leads to excessive dissipation. The Riemann SPH method (Riemann-SPH) is an improved smoothed particle hydrodynamics (SPH) algorithm based on Riemann solvers. By incorporating Riemann problem solvers (e.g., HLLC, Roe, etc.) to describe particle interactions, it replaces the artificial viscosity term in conventional SPH, achieving breakthroughs in shock capturing, interface stability, and conservation properties. This method has been widely applied in dam-break flows [50], underwater explosions [51], and water entry of structures [52]. Meng [53] proposed a multiphase Riemann-SPH model for studying water-entry problems. By introducing a truncation threshold for particle density as a special treatment, this model effectively avoids the generation of negative pressure. The study accounts for air effects, and the simulations of vertical displacement and acceleration during the water entry of a ship section agree well with experimental results.

To address the issue of low computational efficiency caused by the high computational cost of the SPH method, Zhang [54] developed a 3D six-degrees-of-freedom (3D 6DOF) SPH model based on GPU parallel programming, successfully simulating complex free-surface flow slamming. This approach significantly improved computational efficiency, but it did not account for gas-phase effects or structural elastic responses.

The MPS method adopts a fully Lagrangian formulation, thus enabling explicit analysis of large deformations at free-surface interfaces [55]. The MPS method has been improved to capture the slamming pressure on two-dimensional marine structures, where slight compressibility was introduced into the modified pressure Poisson equation (PPE) to reduce pressure fluctuations [56]. Shibata [57] and Wan [58] also employed the MPS (moving particle semi-implicit) method to compute the violent fluid–structure interaction problems. Sun [59] investigated the strong hydroelastic interactions between ship hull structures and free surfaces using an enhanced MPS method coupled with a modal superposition approach. In their study, the MPS method was applied to solve the fluid dynamics, while a modified modal superposition method was adopted to analyze the elastic deformation of the ship hull. The study found that the coupled numerical technique employed could accurately capture slamming loads and simulate structural deformation, but the peak strain values it computed were underestimated.

However, the traditional MPS (moving particle semi-implicit) method does not exhibit good convergence in terms of numerical parameters, requiring continuous adjustment of numerical conditions [60]. Additionally, the inconsistency in pressure gradients leads to numerical oscillations in slamming pressure calculations [61]. To address the MPS method’s sensitivity to numerical parameters, Takahito Iida [62] proposed a practical set of numerical conditions for MPS simulations (termed the MPS-slamming conditions) specifically for the 2D wedge slamming problem. By determining the MPS-slamming number, slamming velocity, wedge entry angle, and particle size, the optimal time step can be obtained. Zha [63] proposed an improved high-order MPS method to solve FSI problems during the water entry of elastic wedges. This method has used the conventional serial staggered (CSS) procedure that was a weak two-way coupled scheme, to realize the coupling of the fluid and structure (see in Figure 4). The pressure gradient term has also been modified to ensure first-order consistency and momentum conservation, while adopting a differential particle spacing strategy between the fluid and structural domains to reduce numerical oscillations. Furthermore, a particle convergence metric [64] was introduced to quantify the numerical uncertainty of the enhanced MPS method and evaluate the discretization-induced numerical errors.

The lattice Boltzmann method (LBM) discretizes the Boltzmann equation by transforming the continuous particle distribution function into discrete lattice-based fi(x,t) instead of directly solving the Navier–Stokes equations. Then, it simulates macroscopic fluid behavior through local collision and lattice migration [65]. To investigate the dynamic response of flexible structures under free-surface impact loads in weakly compressible viscous fluids, De Rosis [66] employed the lattice Boltzmann method (LBM) to analyze fluid flow (including free-surface deformation and pressure distribution), used the FEM to solve dynamic stresses and deformations, adopted the time-discontinuous Galerkin (TDG) method for structural dynamics time integration, and implemented an explicit coupling strategy (with synchronized fluid–structure time steps) to achieve fluid–structure interaction. The LBM has also been introduced to calculate the pressure distribution along the curved wedges [67]. The evolution of splash morphology was related to the pressure distribution. The computed pile-up coefficients for slamming wedges obtained by this method show good agreement with experimental measurements from other studies.

In summary, the finite volume method (FVM) grounded in viscous flow theory offers strong mesh adaptability, ensures excellent conservation properties in discretized equations, and can be coupled with the finite element method for solutions. Its primary drawback lies in high computational costs. For particle-based methods, The SPH method, being mesh-free, possesses inherent advantages in handling dynamic large-deformation problems. Its accuracy is unaffected by mesh quality, and it maintains high computational efficiency. Yet, it faces challenges in modeling complex geometries and requires further research for 3D multiphase water-entry problems. The MPS method is concise and efficient, demonstrating robust convergence. However, it suffers from varying degrees of pressure oscillation phenomena.

3. Water Entry of Hull Section

Water entry refers to the process in which an object penetrates the free water surface at a certain velocity, transitioning from air into water. Water impact is a common phenomenon in naval and ocean engineering, with bow-flare slamming being the most widely studied. The impact loads generated during water entry can not only cause localized structural damage to the hull but may also significantly reduce ship speed, thereby affecting voyage time and economic efficiency. Early research on water entry primarily focused on wedge and flat plate impacts, which were later extended by subsequent researchers to multi-hull segments and three-dimensional hull structure entry.

3.1. Wedge Slamming

3.1.1. Experimental Study

Experimental methods serve as a crucial means for investigating the characteristics of slamming processes and validating analytical theoretical models [68]. Among these, wedges are the most extensively studied models. Chuang [69,70] conducted free-fall water-entry experiments on both rigid and elastic bodies, including flat-bottomed and varying deadrise-angle hull sections, as well as rectangular elastic plates. The study revealed the significant influence of air cushioning on flat plates and wedge-shaped bodies with deadrise angles below three degrees. High-speed photography captured the deformation of the water surface as air entrapped underwater. The results showed that the compressible air layer prolongs the duration of slamming pressure. An empirical formula was proposed to relate the peak slamming pressure to the water-entry velocity. Marintek Laboratory [71] conducted free-fall water-entry slamming tests on a wedge with a 30° deadrise angle. The experiments revealed that the maximum slamming force occurred before flow separation, near the root of the spray jet. After flow separation, the peak pressure rapidly decreased. Zhao and Faltinsen conducted free-fall water-entry tests on a typical flared bow section, validating the accuracy of a fully nonlinear hydrodynamic impact theory in numerical simulations. Mojtaba [72] investigated the water entry of wedge-shaped bodies with varying deadrise angles at different entry angles and heights, The study demonstrated that the deadrise angle significantly influences slamming pressure. Traditional asymmetric theories do not apply to all entry angles. Nikfarjam [73] conducted water-entry experiments on wedge-shaped bodies with a deadrise angle of 30° but varying weights, measuring displacement and pressure values at three points on the bottom. By converting the data into non-dimensional depth and pressure coefficients, the study summarized the effects of model weight and deadrise angle on slamming pressure.

Numerous scholars have conducted research on wedges with a 10° deadrise angle. Byoungcheon Seo [74,75] performed drop tests with wedges of varying thicknesses from different heights, systematically investigating the effects of structural flexibility and repeated loading on slamming response. The study established an empirical formula relating impact cycles to cumulative deformation, providing an experimental benchmark for hull-slamming durability design. Furthermore, a comparative study between cylindrical slamming [76] and wedge slamming demonstrated that the impact load on cylindrical models exceeds that of wedges with a 10° deadrise angle, which aligns with theoretical predictions.

Ren [77] conducted hydroelastic slamming experiments on a wedge with a 20° deadrise angle under vertical water entry. By varying impact velocity and bottom plate bending stiffness, the study simultaneously measured wedge motion, jet root expansion, hydrodynamic loads, and structural response, establishing a quantitative relationship between the hydroelastic factor (R) and structural response. This work revealed the modal selection mechanism of energy transfer during slamming. Further research [78] demonstrated that when R exceeds a critical threshold, the trends of pressure and strain undergo abrupt changes. Additionally, structural deformation causes the peak to lag and significantly reduces its amplitude.

Saeed Hosseinzadeh [79] conducted experiments and numerical simulations [80] on the water entry of wedges with varying deadrise angles, including wedges with and without stiffened bottom panels. The study found that maximum strain and deformation occurred during the local wetting phase of the slamming process. The influence of hydroelastic effects on structural response increased with smaller deadrise angles and higher impact velocities. Unstiffened panels exhibited significant fluid–structure interaction (FSI) effects across all impact velocities, whereas stiffened panels only showed pronounced FSI at high impact velocities [81]. Han [82] investigated the distribution and time-record curves of slamming pressure and hydroelastic effects during water entry of stiffened elastic wedges with varying deadrise angles. The results showed that the propagation speed of peak pressure was about 0.8 times of the value predicted by Wagner’s theory. The average propagation speeds of both pressure and stress peaks exhibited a linear relationship with impact velocity [83]. Further numerical analysis [84] revealed that as the deadrise angle increased, the slamming pressure coefficient on the upstream (impact) side significantly increased, while the coefficient on the downstream side decreased. The jet flow region expanded on the upstream side but contracted on the downstream side during asymmetric water entry. The study demonstrated the influence of entry angle on pressure distribution and flow field characteristics.

Thus, the slamming responses are influenced by the deadrise angles. Duan [85] conducted a series of water-entry experiments on rigid and elastic wedges with different entry angles and drop heights. The spatial variation of slamming pressure, influence mechanisms of entry velocity, entry angle, and structural elasticity on pressure propagation characteristics across different deadrise angles have been analyzed. Güzel [86] investigated the effects of hydrophobicity on free surface elevation and slamming loads during the water entry of wedges and cones with varying deadrise angles, with a focus on the dynamic response in the early slamming stage. Jet velocity increases significantly during water entry, and the superhydrophobic effect becomes more pronounced for models with small deadrise angles. The added mass and wetted area do not exhibit a linear positive correlation; instead, their values may decrease as the deadrise angle increases.

Zhang [87] conducted a slamming impact test using 2D partial hull sections with varying curvatures, measuring the time-domain characteristics of slamming pressure at different water-entry heights. The study focused on the influence mechanisms of entry velocity and curvature effects on slamming dynamics. The experiment model and flow chart for the water-entry slamming test are shown in Figure 5 and Figure 6. The experiments revealed that higher-curvature regions (e.g., the sharp bow) exhibited greater peak slamming pressures but shorter pressure rise time. These areas were prone to local buckling, with maximum strain occurring at curvature discontinuities (reaching 60% of the material’s yield limit). But the study did not account for dynamic coupling effects under wave conditions.

For constant-velocity water entry, Meziane [88,89] investigated the influence of panel stiffness on pressure and strain for aluminum wedges with varying deadrise angles under different impact velocities. The study employed a hydraulic impact test rig to achieve quasi-constant-speed entry, examining how the water-entry angle affects structural strength and velocity characteristics. It systematically analyzed the dependence of peak pressure and jet velocity on these parameters. Jain [90] conducted constant-velocity water-entry experiments with wedges, analyzing the formation mechanism of the negative pressure pulse generated at the initial contact between the wedge and the water surface. The study validated the rationality of approximations made when extending the Wagner model to three-dimensional (3D) cases and measured the dynamic response at the air-water interface, caused by pressure buildup in the trapped air layer between the impacting body and the free surface. The results show the pressure coefficient (Cp) depends solely on the solid shape under constant velocity. As stiffness decreases, Cp increases across all deadrise angles. Moreover, the interface pre-deformation induced by the air cushion effect can be fully described by potential flow theory.

The rapid advancement of high-speed imaging technology has led to the widespread application of particle image velocimetry (PIV) [91]. In 2013, Nila et al. [92,93,94] measured the velocity of the flow field during the slamming at various inclination angles and heights, then derived the pressure field by solving the Poisson equation. The experimental set-up and PIV raw image is shown in Figure 7. This allowed them to determine the pressure distribution on the model surface and compare experimental results with numerical simulations to validate the reliability of the method. On this basis, Jalalisendi and Shams [95] analyzed the velocity fields along three vertical planes and five transverse planes of the wedge, reconstructing a 3D velocity field. They then derived the pressure field and compared the reconstructed pressure and hydrodynamic loads with the theoretical values from Wagner’s model. Panciroli [96,97] investigated the hydroelastic phenomena of flexible structures based on PIV technology. The pressure distribution of the model’s bottom was solved by a non-intrusive pressure reconstruction method. The study revealed that wedge flexibility significantly alters hydrodynamic load characteristics, leading to pronounced oscillations in both the pressure distribution and its evolution process on the wetted surface of the impacting body. Zhang [98] employed PIV to accurately capture the influence of bidirectional fluid–structure coupling on the spatial distribution of hydrodynamic loads. The result showed that the pressure field on the wetted surface of the impacting body exhibited pronounced spatial oscillations, while its temporal evolution displayed distinct periodic fluctuations.

Although particle image velocimetry (PIV) can study the transient fluid kinematics of water-entry objects through flow-field visualization, accurately extracting the liquid-phase region under unsteady hydrodynamic loads remains challenging. Guo [99] proposed a deep learning network called MRes-Att-Unet, based on the U-Net architecture, which integrates residual connections and attention mechanisms (previously validated in medical image segmentation). A dedicated dataset was constructed using PIV experiments of two-phase flow during water entry, with further validation in isolated wave-breaking experiments. Additionally, artificial intelligence (AI) has been applied to slamming event identification in ship navigation, where supervised machine learning (ML) methods analyze test signal characteristics to detect slamming events [100]. In summary, applying particle image velocimetry (PIV) to measure acceleration and pressure variations during water-entry slamming tests of ship hulls offers the advantage of capturing pressure distribution across the entire wetted surface, while avoiding data distortion caused by noise filtering of pressure sensors. However, this method is limited to measuring flow-field motion within the laser plane and cannot account for three-dimensional effects during water entry.

3.1.2. Calculation of Fluid–Structure Interaction for Wedge’s Slamming

In recent years, advancements in computational power have enabled the integration of finite element analysis (FEA) with CFD, demonstrating significant potential for engineering applications in ship design and slamming load assessment. In this approach, hydrodynamic problems are solved using CFD software (Ansys Fluent 2024 R1) with overset grid technology, while structural dynamic responses are computed via FEA software (Ansys 2019 R3). Izadi [101] systematically investigated the fluid–structure interaction (FSI) mechanisms of symmetric and asymmetric wedge oblique water entry through bidirectionally coupled FVM–FEM numerical simulations. Jiao [102] employed a CFD–FEM bidirectional coupling numerical tool, integrating the STAR-CCM+ 13.02 with Abaqus, to study the asymmetric oblique water entry of wedges. The relationships among the wedge’s motion, slamming pressure, structural response, and the inclination angle/water-entry velocity have been analyzed. Xiao [103] proposed a bidirectionally coupled FSI method for flexible wedges to investigate the slamming pressures and structural responses of free-falling non-prismatic stiffened steel wedges. OpenFOAM ESI V2306 for Reynolds-averaged Navier–Stokes (RANS) computations has been used to determine hydrodynamic loads and the calculation of structural response has been carried out by Calculix (FEA software). The result showed that elastic deformation can mitigate impact pressure. The stiffness of flexible panels can be enhanced by transverse stiffeners. Pressure duration and peak slamming pressure exhibit a negative correlation. Sun [104] developed a bidirectionally coupled FSI solver for a curved wedge structure, integrating OpenFOAM V1912 (fluid domain), CalculiX Version 2.22 (structural domain), and preCICE 2.4.0 (coupling interface) to achieve high-precision FSI simulations within an open-source framework. The hydroelastic effects in linear wedges are significantly stronger than those in curved wedges and the structural response of the curved wedge exhibits a distinct two-stage characteristic. It can be seen that the stiffness of the wedge has a significant impact on the slamming load, as the large deformation caused by plate vibration alters the evolution pattern of the fluid jet.

This method was further extended to the problem of wedges entering water in waves. Chen [105] employed a CFD–FEM bidirectional coupling numerical approach to investigate the FSI and slamming pressure of a wedge-grid structure during free-fall entry into fifth-order Stokes waves. The slamming loads and structural stresses of the wedge have been calculated during water entry with elastic effects, considering different relative positions between the wedge and waves. Finally, the influence of horizontal and rotational degrees of freedom on the vertical wave-entry results was explored.

Research on approximate analytical expressions for slamming loads has been conducted by several scholars. Sun [106] decomposed the vertical slamming force into velocity-dependent, acceleration-dependent, and gravity-dependent terms. Using CFD methods, they calculated the coefficients for velocity, acceleration, and gravity, enabling the accurate approximation of forces on arbitrary 2D-shaped bodies under various velocity/acceleration combinations except in cases of flow separation followed by rapid body stoppage. Wen [107] employed the FVM combined with the VOF method and the hybrid boundary formulation (HBF) to study slamming and transitional phase characteristics during both constant-speed and variable-speed water entry of wedges. Under the assumptions of an incompressible, inviscid, gravity-free, and surface-tension-negligible fluid with irrotational flow, they derived analytical expressions for slamming forces and hydrodynamic loads during the transitional phase in both constant and variable-speed water-entry process.

3.2. Multi-Hull Section Slamming

3.2.1. Air Cushion Effect

Catamarans or trimarans, as high-performance vessels, are prone to significant slamming phenomena during navigation. Unlike monohulls, multi-hull vessels must account for interactions among hulls, wet-deck slamming, and air-cushion effects. Before water entry, viscous forces in the air induce free-surface deformation, generating flow within the liquid pool. This serves as the initial condition for slamming and exhibits a cushioning effect, manifested as a reduction in the effective contact area, thereby mitigating impact loads [108]. Hasheminasab [109] approximated the catamaran cross-section as two asymmetric wedges connected by a wet deck. The experimental results showed that the spacing between the wedges had a negligible influence on peak pressure, while air cushion effects were proven to significantly alter slamming pressure. Tang [110] employed a semi-enclosed ship model with partially open acrylic plates attached to its surface, adjusting the opening size to control the gas escape efficiency at the model’s front and rear sections. The formation process of the air cushion and the evolution characteristics of the free surface were captured using high-speed photography. In the experiment, by varying the opening size and the initial release height of the model, it was demonstrated that the air cushion effect can delay the occurrence of the pressure peak, reduce the amplitude of the peak pressure, and significantly intensify pressure pulsations.

For high-speed multi-hull vessels, wet-deck slamming is a critical issue when encountering significant vertical motions during navigation. Ma [111] conducted drop tests on a small-waterplane-area twin-hull (SWATH) segment model to research wet-deck slamming. The experiments involved vertical water-entry tests under varying drop heights and velocities, measuring slamming pressures and acceleration data beneath the wet deck. Additionally, particle image velocimetry (PIV) was employed to capture flow-field characteristics. The experimental findings revealed that during wet-deck slamming, air is entrapped into the water, forming an air–water mixture phase due to gas compressibility effects. The results demonstrated that pressure peaks occur during both the hull-bottom slamming and wet-deck slamming stages, while the air-cushion effect effectively mitigates wet-deck slamming pressures. Masoomi [112] proposed integrating “ventilation ducts” into the central bow structure of a catamaran to release accumulated pressure and associated loads. Using a RANS-based numerical method (implemented via the OpenFOAM open-source library) to solve the flow field, the computational results demonstrated that this design could reduce slamming pressure peaks by up to 50%. Swidan [113] investigated the slamming load on the hull bottom and wet deck of a wave-piercing catamaran during water entry at specific speeds, along with an uncertainty analysis of experimental results. This study provided a high-fidelity dynamic load database for the shock-resistant design of wave-piercing hulls and highlighted the limitations of traditional 2D theories in predicting 3D slamming phenomena.

Jiang [114] conducted water-entry impact tests on a segmented trimaran model to further investigate the dynamic interactions among the structure, fluid, and air, as well as the influence of the air cushion on slamming loads. The study employed PIV (particle image velocimetry) technology to analyze the interactions among air, water, and the structure, while pressure sensors were used to measure slamming loads. Additionally, to amplify the air cushion effect, baffles were installed on both sides of the segmented hull to restrict air escape during the experiment. The results demonstrated that the air cushion beneath the connecting bridge significantly reduced slamming pressure. Yu [115] investigated the water-entry slamming characteristics of a trimaran’s segment in regular waves, focusing on the influence of the model’s water entry point on slamming load as shown in Figure 8. The study revealed the cushioning mechanism of the air cushion effect during bubble escape, providing a theoretical basis for optimizing anti-impact structures. The results showed that the wave position at the moment of water entry significantly affects both the peak load and time-domain characteristics, with the most intense fluid motion occurring when the main hull enters the water at the mid-wave position.

3.2.2. Pressure of Wet-Deck Slamming

The calculation of slamming dynamic loads is a critical factor affecting the structural integrity of multi-hull ships [116]. Liu [117] used the CFD method to simulate the slamming process of a trimaran’s typical cross-section, computing the slamming pressure and free-surface evolution. Based on the relationship between bilge curvature and flow separation, a quantitative formula linking bilge curvature to slamming loads was established, and a deck reinforcement zoning criterion was proposed, based on flow-state control. Sun [118,119] adopted a CFD numerical simulation system to calculate the slamming load of a trimaran’s cross-section under various working conditions, including different masses, drop heights, constant-velocity water entry, and so on. The result showed that slamming dynamics exhibit a strong correlation with immersion depth. The impact velocity shows a quadratic relationship with slamming pressure, while acceleration demonstrates a linear dependence on wet-deck slamming pressure. These conclusions align closely with the mechanical patterns implied by Wagner’s theoretical model and the modified Logvinovich model (MLM).

Duan [120] conducted free-drop water-entry experiments on a trimaran cross-section with various drop heights and inclination angles, employing PIV to observe the evolution of the flow field. The study revealed that peak pressure exhibited a quadratic proportionality to impact velocity. A Gaussian distribution function was proposed to fit the pressure time-record curve during the wet-deck slamming phase. Dimensionless pressure coefficients and slamming duration were systematically summarized across different test conditions. Pan [121] investigated the structural response of a trimaran hull girder through vertical water-entry experiments on a trimaran segment, ultimately determining the maximum slamming pressure range and critical zones for this trimaran design. Li [122] applied three flexible coatings of varying thicknesses to a trimaran model. Water-entry tests at different heights demonstrated that the flexible coatings significantly reduced impact loads (by 32–47%). The research also revealed the mechanism of flexible surfaces in dissipating impact energy.

The hull form of a trimaran significantly influences slamming loads. Wang [123] conducted drop tests on two trimaran segment models. One of the models featured a modified hull with an extended main hull to avoid separated bottom jet flow. Comparative analysis of flow-field characteristics and impact pressures revealed that the slamming pressure coefficient on the transverse deck of the original main-hull model was significantly lower than that of the extended main-hull model due to flow-field disturbances caused by main-hull bottom slamming. Similarly, Li [124] modified the main-hull lines to reduce flow-field disturbances during water entry. The experimental results demonstrated that the main-hull form decisively governed the mechanism of wet-deck slamming and profoundly affects both the magnitude and distribution of wet-deck slamming loads.

3.3. Water Entry of 3D Hull Model

Theoretical research on the water entry of two-dimensional (2D) structures has been extensively studied, forming a well-established theoretical framework. However, when applying 2D water-entry theories to predict three-dimensional (3D) structural impacts, significant deviations often arise. In recent years, extensive studies have been conducted on 3D hull water entry, including investigations on bow, stern, and aircraft [125].

Wang [126] conducted water-entry impact tests on a full-scale 3D composite sandwich ship bow with complex geometric features. The bow was fabricated using vinyl ester fiberglass and PVC foam materials. The study analyzed the distribution of slamming loads, dynamic response characteristics, and the effects of varying water-entry velocities. The results indicate that structural optimization of ships should enhance the bending stiffness of the bow curvature transition zone, in order to suppress local buckling. Shan [127] conducted water-entry experiments on a flared bow section using a tank drop test system. The study found that flow separation during slamming induces cavitation in the concave region of the hull, leading to a delay in the slamming occurrence. Additionally, the high-speed jet generated by the falling model traverses the concave area and impacts the flare of the bow, resulting in secondary slamming. The secondary slamming can generate a higher pressure coefficient, with its peak value even exceeding that of the primary slamming. Based on this, the researchers quantified the cumulative effect of secondary slamming, developed a pressure prediction model incorporating flow separation correction, and proposed an engineering evaluation method based on the effective impact angle. Ping [128] conducted a comparative analysis of the hydrodynamic impact performance between bulbous bow and conventional bow rigid hull structures using a combined approach of CFD numerical simulations and vertical water-entry experiments. The study found that the bulbous bow structure induced flow separation 30–40 ms earlier compared to the conventional bow. Additionally, the flow velocity in the separation zone decreased by 25–35%, revealing the potential threat of jet impact on hull safety. Based on these findings, the authors proposed an optimized bow design criterion based on flow pattern control.

Xie [129,130,131] systematically investigated the asymmetric slamming load characteristics of truncated bow and stern structures under combined roll–pitch motions (see Figure 9). The pressure near the stern shaft exhibited multi-peak behavior and high-frequency oscillations, with obvious cavitation effects. Further research revealed a phenomenon was observed at specific bow locations, where pressure decreased with increasing water-entry velocity. Liu [132] conducted water-entry experiments on a stern model, measuring the relationships among immersion depth, acceleration, and slamming loads via sensors. The study demonstrated the influence of cavitation effects and jet flows on pressure distribution dynamics. Mutsuda [133] conducted water-entry experiments on wedge and stern models, combined with numerical analysis. The strongly nonlinear interactions between the hull and free surface during stern slamming have been investigated. These interactions arise from spray jets and flow separation. The study demonstrated that the water-entry angle and relative vertical velocity between the water surface and model are critical factors influencing slamming loads.

In general, research on the slamming phenomena of monohull structures has been extended from two-dimensional models to three-dimensional analysis, particularly focusing on bulbous bow and stern slamming issues. The air cushion effect introduces more significant cavitation phenomena, causing hysteresis in slamming pressure and acting as a buffer. This effect shows more pronounced impacts on multi-hull segmented water entry scenarios. The cross-deck’s forepart undergoes more severe slamming effects. Main-hull entry generates higher slamming pressures near the connecting bridge on side hulls compared to adjacent main-hull regions. Air cushion effects distinctly influence segmented water entry in multi-hull vessels. To suppress the slamming pressure, measures such as installing baffle plates beneath wet decks, applying flexible coatings, and optimizing the body line of the main hull can be adopted. In terms of experimental techniques, PIV technology, with its high frame rate, non-intrusive measurement, and flow field velocity analysis, has become the important experimental method for studying the hydrodynamics of water-entry slamming. It has strong applicability in revealing the flow mechanism and validating numerical models.

4. Wave Load During Navigation

During ship navigation, slamming caused by vertical motions generates significant wave-induced loads. Early research on wave load problems was primarily based on linear theory. However, under high-sea-state conditions, the vertical motions and slamming loads resulting from phenomena such as bottom slamming [134], flare slamming [135], and wet-deck slamming on trimaran connecting bridges exhibit pronounced nonlinear characteristics. The tensile and compressive forces in the hull girder become asymmetric and no longer maintain a linear relationship with wave height. To investigate these nonlinear phenomena, it is necessary to treat slamming and hull motions as a coupled nonlinear system, including the coupling effect of the coupling effects between hull girder bending moments and local slamming impacts [136]. The influence of slamming loads on ship motions should be fully considered.

4.1. Slamming of High-Speed Monohull

4.1.1. Numerical Solution

High-speed crafts often operate in a planing state during navigation. The ‘gliding effect’ leads to highly complex and strongly nonlinear characteristics in the slamming loads, motions, and structural responses, making numerical simulations particularly challenging. Bao [137] integrated the strip theory with the BEM, discretizing the hull into multiple two-dimensional wedge-shaped sections to simulate water entry. The study provided detailed numerical results on pressure distribution, free surface deformation, and force (moment) variations, along with an in-depth analysis of the influence of hull speed parameters. Prini [138] used the strip theory combined with wave-induced load calculations to analyze slamming loads on high-speed crafts, comparing the zero-speed Green’s function method with the Rankine source method incorporating speed correction terms. Validation against experimental data showed that the Green’s function method exhibited better correlation at low speeds, while the Rankine source method was more suitable for high-speed conditions. Parunov [139] investigated a patrol frigate by employing nonlinear strip theory, frequency-domain, and time-domain 3D boundary element methods to compute hull motions. Euler–Bernoulli and Timoshenko beam theories were applied for hull girder stiffness modeling, calculating the wave-induced responses and springing responses of the warship in regular waves. Comparisons with model tests demonstrated that the seakeeping method based on potential flow theory, combined with limited CFD–FEM case corrections, exhibits excellent computational reliability and efficiency.

The 2D + T theory, based on the traditional strip theory and incorporating a three-dimensional free-surface condition with forward speed, demonstrates good accuracy and efficiency in calculating the wave-making and slamming loads of high-speed monohull vessels in motion. Bilandi [140] employed the 2D + T method to calculate the pressure distribution on each cross-section of a planing hull and the normal forces acting on the two-dimensional sections. The reliability of this method was validated through comparison with experimental data. Tavakoli [141] employed the 2D + T method to calculate the motion and slamming loads of a planing craft during navigation, comparing the results with CFD methods and model tests. The study found that for nonlinear responses in vertical motion, the 2D + T theory showed good agreement with experimental data at low to medium speeds. However, as the speed increased, the accuracy of this method declined for long-wavelength waves exceeding twice the ship’s length, with the calculated slamming forces being underestimated. Shao [142] adopted the 2D + T theory to transform the three-dimensional hydrodynamics problem of planing hulls into a two-dimensional slamming problem, specifically, the problem of impact on cross-sections with continuously deformed shapes during water entry. By integrating a modified Logvinovich model (MLM), this approach enables reliable calculation of slamming forces on planing vessels.

Compared with the potential flow theory, CFD technology can more accurately simulate the velocity and pressure distribution of the flow field during slamming under extreme working conditions. Mousaviraad [143] employed the unsteady Reynolds-averaged Navier–Stokes (URANS) method to compute the hydrodynamic performance and slamming loads of a high-speed planing craft. The reliability of the method was validated through comparisons between 2D water-entry simulations and experimental data. The approach was then extended to 3D full-scale ship simulations for predicting wave-induced slamming pressures in seaways. However, these 3D numerical results lacked experimental validation. Subsequent studies mostly combine the finite element fluid–structure interaction (FSI) technology with the computational fluid dynamics (CFD) method. The calculation conditions include those in still water [144] and in waves [145,146]. The influencing factors of slamming loads on the hull bottom and flare [147] are analyzed, and the reliability is verified through comparison with experiments [148]. Lee [149] designed an optimized truss panel based on CFD load assessment and mathematical optimization, then analyzed the motion of a deep-V planing hull in irregular waves. The above research indicates that the maximum slamming load did not occur at the highest wave crest, but rather under conditions with deeper troughs and lower wave heights. The slamming coefficient (defined as the ratio of the hull-girder vertical bending moment to the wave-induced bending moment) increased significantly with wave steepness and speed, peaking near the surge natural frequency.

For the unique configuration of wave-piercing monohulls, the hull is highly susceptible to severe deck wetness and corresponding hydrodynamic loads. Shan [150] simulated the vessel’s motion characteristics in head seas using Star-CCM+ 2020.01, focusing on the bow-flare slamming effects on the surrounding flow field. The study revealed that the collision of two water jets above the bulbous bow generates a butterfly-shaped high-pressure zone, and the slamming jet exerts strong impact loads on the superstructure.

4.1.2. Model Test

Given the nonlinearity of hydrodynamic and FSI problems in planing craft navigation, Begovic [151] integrated a precisely measurable structural model into the hull of a typical planing boat prototype. Through regular wave tests, this method was validated to enable the detailed investigation of the entire slamming FSI process. Camilleri [152] conducted full-scale sea trials on a planing craft, examining slamming impact characteristics and their induced rigid-body motions and structural responses. For the collected data (e.g., pressure, strain, global hull deflection), an automated algorithm was developed to fit statistical models to peak distributions, with goodness-of-fit tests confirming model applicability. Ibrahim [153] conducted hydroelastic model tests on a semi-planing vessel, systematically investigating the mechanism of slamming effects on global hull girder loads under both regular and irregular waves. The study revealed that coupled heave–pitch motions and wave steepness are key parameters governing the amplitude of slamming loads.

The superposition of high-frequency vibration bending moments and wave-induced low-frequency bending moments (induced by whipping) poses a severe threat to the ship’s longitudinal strength. Zou [154] conducted segmented model tests to investigate the slamming loads on flared bows and whipping vibration responses of ships in regular waves. By employing continuous wavelet transform (CWT) analysis, a quantitative relationship model among ship speed, slamming, and whipping responses was established. The experimental results showed that the asymmetric slamming loads were significantly amplified in oblique wave conditions. Whipping vibrations increased the asymmetry of vertical bending moments (VBM) at the midship section, with this effect intensifying as ship speed increased. Mogoga [155] conducted experiments on the hull girder stresses of an aluminum high-speed patrol vessel and found that the coupling effect of slamming and whipping response can increase stress amplitude in fatigue hotspot regions by 40–65%. A dynamic stress equivalent conversion model has been proposed to achieve coupled damage superposition of wave loads and impact loads, quantifying the influence weight of slamming on hull structural fatigue.

4.2. Slamming of Catamaran

During high-speed catamaran operations, the Froude number typically exceeds 0.5, causing heave and pitch motion amplitudes to significantly surpass wave height and wave slope. This may trigger slamming loads and structural resonance [156], leading to structural damage and deck wetness [157]. Slamming loads exhibit strong correlation with vertical motions, with both motion peaks and slamming load peaks occurring within the same encounter frequency range [158].

4.2.1. Influencing Factors of Slamming Load

INCAT has conducted extensive research on wave-piercing catamarans (WPC), including full-scale trials and model basin tests, establishing a comprehensive slamming database. Based on pressure data and the concept of transmissibility, the prediction [159] and identification [160] of slamming loads have been improved. M.R. Davis [161] performed basin tests on a hydroelastic model of a wave-piercing catamaran. The model consisted of seven segments, including a smaller central bow section, connected by elastic joints (shown in Figure 10). The tests revealed that slamming wave impacts predominantly occurred in the aft region of the short central bow. Slamming loads typically reached about 25% of the vessel’s weight, though some peak loads reached 132% of the vessel’s weight under 4 m significant wave height conditions. At the experimental scale, the slamming duration was generally 0.35 s. Shabani [162,163,164] modified the length and height above waterline of the central bow (CB) in this model and conducted head-sea tests in regular waves at high speeds. The results showed that slamming loads increased significantly with longer central bows, while the encounter frequency had a greater influence on the CB than its geometric variations. Raising the bridge clearance amplified hull motions but reduced peak slamming loads in moderate sea states. The model with higher CB exhibited the lowest vertical motion amplitudes. Overall, the short central bow (short CB) design demonstrated the most stable performance in mitigating slamming loads.

SWATH vessels exhibit superior seakeeping performance due to their unique design [165]. However, they remain susceptible to slamming phenomena in high sea states, where structural damage may occur on the wet deck due to local slamming loads, global wave-induced bending moments, and coupled torsional effects. Ma [166] conducted a study on the slamming load characteristics of SWATH wet decks, focusing on the slamming pressure distribution on the cross-deck structure in regular waves. The evolution of flow-field during slamming events has been captured by high-speed imaging. The influence of wavelengths and wave heights on slamming intensity has also been analyzed. After wave impact on the cross-deck, pressure fluctuations exhibit periodic oscillations caused by the compression–expansion effect of the air cushion beneath the structure.

The flow field exhibits highly nonlinear flow characteristics during wave-piercing catamarans’ slamming events due to mutual interference generated by the two demi-hulls and center bow. This complexity renders potential flow theory inadequate for accurate prediction. Consequently, CFD-based numerical simulations have become the primary tool for investigating such phenomena. McVicar [167] employed a FSI numerical simulation approach to predict the global bending response of a WPC under head sea conditions, induced by slamming loads. The hull motions were solved using RANS-based CFD, with validation against experimental data from a hydroelastic segmented model (HSM). A novel methodology was developed to capture the nonlinear time-varying characteristics of added mass during slamming events. Almallah [168,169] investigated a 98 m WPC using RANS-based CFD with the VOF method to capture free-surface effects. The vertical motions, longitudinal bending moments (LBM), and transverse bending moments (TBM) in moderate sea conditions were calculated. The pitch angular acceleration and wave height exhibited an approximately linear relationship with slamming loads and wave height was proportional to LBM slamming loads. Wave height was proportional to LBM slamming loads. The longitudinal distribution of the transverse separation forces (induced during slamming) was identified as the primary cause of transverse racking moments. Numerical simulations in oblique bow sea irregular waves revealed that pitch and roll accelerations exhibit a combined linear influence on the total LBM slamming loads. By establishing a conversion relationship between full-scale strain measurements and global loads using a FEM, the accuracy of slamming load prediction during the ship design phase was optimized [170].

The above research shows that the FSI technology is effective for predicting the motion and slamming loads of high-Froude-number ships, and it has high computational efficiency. This method has a good application prospect in the prediction of the vertical bending moment of actual ships and the structural design of ships during the ship design process.

4.2.2. Suppression Method

There are two measures to mitigate wet-deck slamming, generally, enhancing the structural strength of the connecting bridge and reducing the vessel’s vertical motions. Hajmohammad [171] investigated the dynamic response of wave-piercing catamaran beam elements under fluid impact loads using the higher-order shear deformation beam theory (HSDBT) with uniform thickness and elastic linearity, while accounting for the porosity and structural damping properties of aluminum beams. By employing graphene platelets (GPLs) as reinforcement materials and solving the dynamic deflection of porous GPL-reinforced beams via the differential quadrature method (DQM) combined with the Newmark time-integration scheme, the study revealed that carbon nanotube (CNT)-reinforced panels could reduce structural dynamic deflection by up to 59%.

Building on prior motion response tests of an 86 m high-speed catamaran [172], G. Jacobi et al. conducted full-scale trials [173] and model experiments [174] to develop a ride control system (RCS) for the wave-piercing catamaran, comprising a bow T-foil and stern trim tab. The workflow of the RCS system is shown in Figure 11. By analyzing the relationship between structural stress rate and pitch motion, they achieved slamming prediction based on sea state and model motions, investigating the system’s impact on slamming probability. The results showed that reducing vertical motions significantly alters slamming occurrence. Subsequent tests confirmed a 75% reduction in slamming pressure with this system [175]. Hasheminasab [176] proposed installing a flat spray rail on the central bow to simultaneously reduce slamming pressure and acceleration. Water-entry impact tests demonstrated that the J1-type profile with the spray rail achieved approximately 60% lower peak acceleration and 70% reduced slamming pressure. The study confirms that adding appendages to catamaran hull sections is an effective strategy for mitigating slamming impact loads.

4.3. Slamming of Trimaran

For numerical studies on wave loads of trimarans, in the 1990s, Grassman [177] conducted wave load measurements at various locations on the RV Triton trimaran under two sea states, including the bottom and upper sections of the main hull and side hulls, cross-deck bottom, and junctions between the main hull, side hulls, and cross-deck. The effects of wave encounter angles and significant wave heights on peak slamming forces were analyzed. The study demonstrated the feasibility of trimarans for military applications by validating their structural response to wave-induced loads. Miao [178] employed rigid and flexible 3D FSI models to calculate the motions and wave loads of the RV Triton trimaran, comparing the results with full-scale measurement data. Subsequently, Miao [179,180] applied 3D hydroelasticity theory incorporating transient slamming responses to compute the slamming locations and time-record curves of slamming forces and moments on both the port and starboard sides under the 45° bow-quartering waves.

In the preliminary design of trimaran structures, slamming loads and longitudinal vibrations are two critical factors [181]. To analyze wave-induced loads on trimarans in regular waves, Tang [182,183] proposed a novel nonlinear 3D time-domain Rankine–Green hybrid method, based on the computational principles of ship-wave interaction using Rankine sources and free-surface Green’s functions, for simulating trimaran motions and wave loads under head sea conditions. By incorporating nonlinear factors such as instantaneous wet, steady wave-making, slamming, and so on, the method successfully coupled nonlinear components in time-domain load simulations. The calculations revealed that as ship speed and wave height increase, the slamming phenomena in the side hull regions and cross-deck areas exhibit significant nonlinear growth characteristics. Khoob [184,185] utilized frequency-domain Green’s function potential flow theory with forward-speed correction, to analyze wave-induced loads on trimarans. A global FEM was then constructed to predict motion and load responses. The roll-induced torque and shear forces are more significant in head sea conditions, while transverse bending moments exhibit higher response amplitudes in oblique seas. Additionally, outward-sloping side hulls demonstrated better suppression of wave-induced loads compared to conventional designs.

Ghadimi [186] employed the commercial CFD software Flow-3D V11.2, utilizing the standard Bretschneider spectrum to simulate the seakeeping performance of a wave-piercing trimaran in irregular waves under Sea State 5. The study investigated motion responses under different speeds and wave encounter angles. Xie [187] conducted seakeeping analysis to obtain asymmetric motion responses (heave, sway, and roll) of the vessel, then applied CFD methods to calculate slamming loads on two typical flare bow sections. The study fully accounted for complex flow phenomena such as flow separation and cavitation effects. The results revealed that oblique wave slamming loads exhibit stronger asymmetry and higher impact intensity compared to head sea conditions. Liao [188,189] equivalently distributed the ship’s mass to the hull and analyzed the structural response characteristics of a trimaran under unidirectional regular waves and bidirectional crossing seas by integrating a fluid solver (Star-CCM+) with a structural solver (Abaqus). This approach significantly improved computational efficiency. Chen [190] employed CFD techniques to construct a numerical wave basin generating bidirectional crossing waves as a typical scenario for 3D short-crested waves. By incorporating a time-domain delay function and a proportional-integral-derivative (PID) autopilot model into the numerical simulations, they proposed a 3D nonlinear time-domain hydroelasticity method [191]. This method accounts for nonlinear hydrostatic restoring forces due to instantaneous wetted surfaces and slamming loads, analyzing differences in ship motions under oblique waves and long-crested regular waves. Liu [192] combined model tests with CFD techniques to investigate the slamming characteristics of the cross-deck structure on a trimaran in head seas. The simulations revealed that when the cross-deck emerged from a wave crest, a significant negative pressure was generated in its forward region. Additionally, the water entry of the main hull induced higher slamming pressures on the cross-deck near the side hulls compared to the corresponding regions of the main hull.

In summary, potential flow theory is predominantly employed for slamming pressure prediction on monohull high-speed vessels. For multi-hull ships, however, the highly nonlinear characteristics of free-surface flows necessitate grid-based CFD methods (e.g., FVM, FDM) as the primary numerical approach. These methods are often coupled with structural analysis techniques (e.g., finite element analysis) to further assess localized structural deformation. The slamming near wave troughs tends to induce higher impact loads in irregular waves. In oblique seas, because of the coupling between roll and heave motions, the side hulls of trimaran generate slamming more obviously, with wave splashes generating localized high pressure on the cross-deck structure. Even under relatively moderate sea states, multi-hulls experience significant slamming loads compared to head seas. The appendages (e.g., T-foils, flaps) can simultaneously suppress vertical motions and reduce slamming pressure through synergistic effects.

5. Conclusions

Regarding the slamming problem of high-speed vessels, including studies on segmented water entry and slamming loads during navigation, this paper comprehensively analyzes the progress in theoretical and experimental research. Although current studies have yielded significant findings and summarized the relationships among factors influencing slamming intensity—particularly with the introduction of PIV technology, which has progressively improved experimental measurement techniques—the following aspects warrant further exploration and in-depth investigation.

(1)

Coupling of Numerical Methods. By combining CFD (computational fluid dynamics) with the BEM (boundary element method), the far flow field is modeled using potential flow theory while the near field employs viscous models. This approach balances the high accuracy of CFD in handling nonlinear problems with the computational efficiency of the BEM, reducing dimensionality and more precisely capturing hydroelastic effects. Similarly, hybrid methods coupling particle-based techniques (e.g., SPH, MPS) with grid-based methods can be adopted, applying particle methods at slamming interfaces to accommodate large deformations while retaining grid methods elsewhere. For free-surface tracking, integrating level set with VOF (volume of fluid) enhances air–liquid interface resolution.

(2)

Application of Machine Learning. High-fidelity CFD simulations (e.g., DNS, LES) require an extremely fine mesh resolution and small time steps, making them impractical for rapid multi-condition evaluations in engineering applications. Neural networks can be employed to predict numerical solutions (e.g., pressure peaks, time-history curves) and replace traditional iterative processes, thereby improving computational efficiency for large-scale simulations. However, current slamming tests often rely on repeated experiments to mitigate data fluctuations, resulting in a scarcity of high-quality datasets. Additionally, biases inherent in numerical models may propagate into machine learning models, necessitating further calibration.

(3)

Slamming Flow Field Measurement Techniques. The measurement of flow fields and pressure distribution on ship hull sections is often achieved using particle image velocimetry (PIV). This method enables the real-time measurement of fluid particle velocities and reconstructs the pressure field from the velocity field via pressure reconstruction techniques. It overcomes the limitations of traditional pressure sensors, which are prone to dynamic impact interference, particularly offering advantages in high-speed water-entry scenarios. However, existing pressure reconstruction methods (e.g., multi-path integration schemes) typically assume incompressible flow and neglect viscous effects. In reality, water-entry processes may involve complex phenomena such as cavitation and turbulence, leading to error accumulation. Additionally, PIV is predominantly applied to 2D models, making it challenging to fully capture the flow field characteristics in 3D scenarios (e.g., ship bow water entry).