1   Introduction

Predicting the hydrodynamic load on a body entering water is important to resolving problems in the field of marine engineering. When a ship moves in water, the hull experiences a high impact by the waves because of the relative motion between the ship and the waves. Thus, water entry adversely affects ship motion and can even damage the ship structure.

The original research on water entry was conducted by Von Karman (1929), who proposed a model based on the momentum theorem, which was a pioneer theory of water entry. Afterward, Wagner (1932) developed a model to predict impact pressure and studied two-dimensional (2D) water impact on a solid body. With regard to the water entry of wedges, Von Karman's method is very similar to that of Wagner's.

Water entry problems have been widely investigated with experimental, theoretical, and numerical methods.

Experiments play an important role in analyzing and resolving problems arising in engineering practice, and many experimental studies on the water entry of different objects, such as wedges and cylinders, have been conducted.

Most studies are related to the water entry of wedges. Zhao et al. (1996) conducted experimental studies on 2D water entry with and without flow separation. Xu et al. (1999) tested wedges freely dropped from different heights to predict the impact effect during water entry; in their experiment, they reported impact characteristics, such as instantaneous velocity and impact acceleration; given that the wedges were free to roll after water entry, the results showed an inertia effect in the roll direction. Judge et al. (2004) performed numerous experiments on the asymmetric water entry of wedges; their experimental results were in good agreement with those obtained using an analytical 2D vortex distribution method. Some experimental investigations on the water entry of wedge-shaped objects with constant velocities have been conducted (Yettou et al., 2006; Tveitnes et al., 2008; Wang et al., 2015). Barjasteh et al. (2016) investigated the asymmetric water entry of wedges, conducting several experiments with different dead-rise angles, inclination angles, and drop heights; the experimental results showed good agreement with different theories on water entry problem and indicated the considerable effect of inclination angle on impact pressure.

Sun and Faltinsen (2009) conducted an experimental investigation on the water entry of a bow flare ship section; their experiment focused on asymmetric entry and reported several characteristics, such as acceleration and impact pressure; the numerical and experimental results showed a very good agreement. Moreover, Wei and Hu (2014, 2015) experimentally investigated the water entry of a horizontal cylinder using a high-speed digital video camera; in their experiment, the hydrodynamic loads acted on the cylinder were reported, and high-quality images were provided for validation by other studies. Li et al. (2020) conducted experimental research on the water entry of various nosed projectiles with high velocities. She et al. (2021) investigated the water entry of a bulbous bow with a time-resolved particle image velocimetry system with good performance in capturing transient flow; they determined the pressure distribution around the bulbous bow during water entry and provided data for other studies.

Compared with experimental research, the theoretical and numerical investigations are time-saving and cost-saving.

Based on fully nonlinear potential flow theory, nonlinear numerical methods (Iafrati and Battistin, 2003; Xu et al., 2008; Riccardi and Iafrai, 2004; Bao et al., 2016; Bao et al., 2017) were developed to determine the free surface profile and hydrodynamic load on symmetric or asymmetric wedges during water entry. Sun et al. (2019) studied the water entry problem of a 2D wedge-shaped tank with liquid on the basis of velocity potential theory; in their work, two separated regions for the internal liquid and open water were created, and the fluid loads in these regions were estimated. Potential theory has also been applied to several water entry problems (Del Buono et al., 2021; Zekri et al., 2021), and Navier-Stokes equations have been used in studying water entry problems (Piro and Maki, 2013; Gu et al., 2014).

Complementary to these numerical studies, different CFD methods have been utilized in investigating water entry.

Wen and Qiu (2018) proposed a constrained interpolation profile to predict a three-dimensional (3D) water entry; in their work, density functions were adopted to improve an interface-capturing scheme, and an MPI-based parallel computing technology was used to accelerate computational speed. Sun and Faltinsen (2006), and Sun (2007) investigated a free surface profile during the water entry of a vertical cylinder; they used the boundary element method (BEM), and the results obtained agreed with experimental results. Wu (2012) suggested a high-order BEM for resolving water entry problems. Moreover, a semi-analytical hydrodynamic impact theory for analyzing the water entry of elastic wedges on the basis of modal superposition method, BEM and Wagner theory was developed (Yu et al., 2019; Zhang et al., 2021; Sun et al., 2021a).

Most CFD applications were implemented through smoothed particle hydrodynamics (SPH), mainly focusing on fluid–structure interaction (FSI) problems with respect to the fully nonlinear fluid flows (Oger et al., 2006; Farsi and Ghadimi, 2015; Luo et al., 2021; Khayyer et al., 2021; Gotoh et al., 2021; Chen et al., 2021). Sun et al. (2015) simulated free surface flow around a rigid body with a weakly compressible SPH method; they proposed a solid boundary technique to improve the stability of numerical solutions. Sun et al. (2018) also proposed a δ+-SPH model for 3D water entry problems and applied adaptive particle refinement to enhance computational efficiency. Meng et al. (2021) suggested a multiphase SPH model for water entry problems and a cutoff value of density for preventing negative pressure. Several GPU-accelerated SPH methods have been developed and used in investigating water entry problems. Zhang et al. (2022a) presented an accelerated SPH framework-based GPU for simulating 3D water entry problems, and Zhao et al. (2023) developed a multiphase and multiresolution SPH method using multi-GPU implementation for FSI problems, simulated the water entry of a projectile, and provided the resolution of up to 120 million particles by using multiple GPUs. All the numerical methods showed good performance in the simulation of different water entry problems.

Furthermore, the moving particle semi-implicit method has also been applied to simulate water entry problems. Zhang et al. (2022b) developed an MPS method with an adaptive variable-size particle model and verified their work by simulating dam break and water entry problems. Zha et al. (2021) suggested an improved higher-order MPS method for FSI problems and applied it to an elastic wedge water entry problem. Compared with the SPH method, the MPS method has not been widely applied to water entry problems.

The present study aims to investigate water entry problems of 2D wedges by using an MPS method and some handling techniques. This study is organized as follows: Section 2 addresses some schemes for improving the MPS method, a symmetric domain technique for reducing computational cost, and a fluid–solid-coupling algorithm using the MPS method. In Section 3, the proposed techniques are verified through several numerical investigations and discussed. Finally, some conclusions are described in Section Ⅳ.

2   Methodology

2.1   MPS method

2.1.1   Original MPS method

The continuity equation and Navier–Stokes equation for incompressible viscous flows are governing equations in hydrodynamics and are expressed as follows:

where ρ, u, p, ν denote the density, velocity, pressure, and kinematic viscosity of the fluid, respectively, and f is the body force.

The original MPS method was proposed by Koshizuka and Oka (1996); in their study, fluid particles are assumed to be interacting with each other, as indicated in the following weight function:

where r denotes the distance between two particles and r_e is the radius of the influential domain. On the basis of this weight function, the fluid density and differential operators are defined.

The particle number density, i. e., the fluid density, is defined as

where r_i, r_j are the positions of particles i and j, respectively.

The gradient and Laplacian operator are formulated as

where ϕ is an arbitrary scalar quantity, d is the number of space dimensions, n⁰ is the particle number density at the initial arrangement, and λ⁰ is the correction parameter of the Laplacian operator.

The computational procedure of the MPS method mainly consists of two stages. First, fluid particles' velocities and positions at the intermediate time step are determined. The intermediate quantities u_i^* and r_i^* are calculated by velocity u_i^k and position r_i^k at the current time step k as follows:

where Δt and g denote time interval and acceleration of gravity, respectively.

Then, the fluid particles' velocities and positions at the next time step are updated. In this stage, the fluid particles' pressure is calculated by solving the pressure Poisson equation (PPE).

where n_i^* is the particle number density at the intermediate time step. The PPE can be solved by different iterative solvers, such as the conjugate gradient (Hestenes and Stiefel, 1952) and generalized minimal residual methods (Saad and Schultz, 1986; Amritkar et al., 2015). After the fluid particles' pressure is calculated, their velocities and positions at the next time step k + 1 are updated as follows:

Given that the original MPS method has many drawbacks, we adopted some modifications to the original MPS method. Checking free surface particles correctly is crucial to ensuring the stability of a solution. To date, substantial effort has been devoted to improving free surface identification (Marrone et al., 2010; Shibata et al., 2015a; Shibata et al., 2015b). Our numerical studies have shown that the identification method proposed by Shibata et al. (2015b) is effective. Thus, we used it to identify free surface particles.

Furthermore, owing to unphysical pressure fluctuation arising from the original PPE source term, many modifications to PPE have been proposed. In this study, we used the PPEs proposed by Sun et al. (2021b) to obtain smooth pressure distribution.

where γ is a relaxation coefficient, which is 0.01 in this study.

In addition, the gradient model from Khayyer and Gotoh (2008) was used in this study.

This gradient model ensures the conservation of momentum and correctly represents the action–reaction physical law.

2.1.2   Some techniques for improving the original MPS method

2.1.2.1   Laplacian compensation in the region close to the wall boundary

One of the shortcomings of the original MPS method is the lack of neighboring particles of fluid particles in the region close to the wall boundary. This problem leads to the incorrect calculation of the Laplacian operator of the fluid particles' pressure and thus decreases the accuracy of the calculated pressure. One way to resolve this problem is to add fluid particles in the region. This approach guarantees the accuracy of pressure calculated by Laplacian compensation.

In the present study, we considered one layer of wall particles and several layers of virtual particles beyond the wall boundary (Figure 1). Virtual particles are generated by the corresponding wall particle, that is,

Figure  1  Generation of virtual particles

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where r_w is the position of a wall particle, r_v is the position of virtual particles generated by this wall particle, l₀ is the initial spacing, n is the normal unit vector to the wall boundary, and s is the distance between the virtual particle and corresponding wall particle.

After several modifications are implemented, the pressure of the virtual particle can be expressed as follows:

where P_v and P_w denote the pressure of virtual and wall particles, respectively; ρ is the fluid density; U and $ \dot{\boldsymbol{U}}$ are the velocity and acceleration of the wall particles, respectively.

The Laplacian operator of pressure at a fluid particle i close to the wall boundary can be calculated as follows:

where Λ_i^r denotes a set of real particles in the support domain of particle i and Λ_i^d denotes a set of virtual particles inside the support domain of particle i. In this equation, the second term within parentheses is the Laplacian compensation of pressure at particle i. Thus, the accuracy of the calculated pressure of fluid particles will be further improved relative to that obtained with the original MPS method.

2.1.2.2   Collision model between fluid and wall particles

Fluid particles tend to penetrate the wall boundary in the original MPS method. This phenomenon is more frequent for fluid particles with high velocities.

To address this problem, a collision model between fluid and wall particles has been suggested. The diagram of the collision model is shown in Figure 2.

Figure  2  Diagram of the collision model between a fluid particle and a wall particle

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If the distance between fluid particle i and wall particle j is smaller than a certain threshold value, fluid particle i collides with wall particle j (Figure 2). The velocity of fluid particle i is corrected as follows:

where Δu denotes the correction value; n is the normal unit vector to the wall boundary; u_i and u′_i are the velocities of a fluid particle before and after the collision, respectively; a and b are the adjusting parameters, which were empirically set at 0.6–0.7 and 0.2–0.5, respectively, in this study.

In addition, we assumed that the collision was an elastic collision and the wall was stationary. In ocean engineering practice, most problems are related to the moving boundary. The collision model can be easily modified for the moving boundary as follows:

where u_i and u_j denote the velocity of fluid and wall particles before the collision, respectively; u″_i is the relative velocity of a fluid particle to a wall particle.

2.1.3   Symmetric domain technique

If a simulation domain flow has a symmetrical characteristic, the cost of the computation can be reduced by considering the "half-domain" rather than the "whole-domain". In other words, only the half of the simulation domain is simulated. In the present study, a handling technique for the symmetric domain is suggested, and a model for symmetric water entry is considered.

Figure 3 shows the schematic of the symmetric domain simulation. The initial arrangement of fluid particles is shown in Figure 3(a). Let a domain of distance 4l₀ away from the Y axis toward the –x direction denote the "image domain", and all the fluid particles inside the image domain denote "image particles" (Figure 3(b)). The image particles are generated by symmetrizing fluid particles within the domain of distance 4l₀ away from the Y axis toward the +x direction. The flow in the region close to the Y axis can be modeled correctly by integrating the physical quantities of image particles into the following equation:

Figure  3  Schematic of symmetric domain simulation

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where the subscripts i′ and i denote the image and fluid particle, respectively; r_x, r_y denote the coordinates of particles in the x, y directions, respectively; P denotes the pressure of particles; v_x, v_y denote the velocity components in the x, y directions, respectively.

These image particles are considered in the calculation of fluid particles close to the symmetric plane. This principle is the same as adding virtual particles in Laplacian compensation. This technique not only reduces computational time but also reduces consumed memory by half, compared with whole-domain simulation. Notably, image particles are dynamically generated in every time step, and handling is only implemented in symmetric flow simulation.

2.2   Fluid–solid-coupling algorithm

2.2.1   Governing equations for fluid–solid-coupling problem

The motion of a solid can be considered the resultant motion of translation and rotation. Therefore, the velocity of any position in a solid can be evaluated by the translational velocity of its centroid and rotational velocity around its centroid. Based on the conservation principle of momentum and moment momentum, the translational and rotational acceleration of a solid's centroid are determined by the following equations (Sun et al., 2018):

where the subscript c denotes the centroid of a solid, U_c is the translational velocity of the centroid, Ω_c is the rotational velocity around the centroid, M is the mass of the solid, I is the moment of inertia, and F_f−sT_f−s are the resultant force and moment acted on the solid from the fluid, respectively.

After the translational velocity U_c and rotational velocity Ω_c are determined with Equations (21) and (22), the velocity at any position over a solid boundary is calculated as follows:

where r_b and r_c are the positions of any point over the boundary and a solid's centroid, respectively.

2.2.2   Fluid–solid interaction

Solving Equations (21) and (22) is crucial for the fluid–solid coupling simulation. The resultant force and moment acted on a solid from a fluid can be calculated if the pressure on the solid boundary is known.

In the present study, the fluid–solid interaction is implemented according to the following process. In every time step, the MPS simulation is first performed, and the pressure of wall particles is determined. Then, the F_f−s and T_f−s values of all the wall particles are calculated by the following formula:

where the subscript i denotes the index of the wall particles, P_i is the pressure of wall particles, ΔS_i is the boundary surface's infinitesimal area, n_i is the normal vector to the boundary, and r_i is the position vector of the wall particles. We can argue that ΔS_i = l₀². The translational velocity U_c and rotational velocity Ω_c of the centroid of the solid are updated by Equations (21) and (22), respectively. Finally, the velocities of wall particles υ_b are calculated using Equation (23), and the positions of the wall particles are updated as follows:

where r_b^new denotes the updated position of wall particles, and Δt is the time step.

The algorithm for fluid-solid coupling is illustrated in Figure 4.

Figure  4  Algorithm for fluid–solid coupling problem

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3   Numerical results and discussions

The main goal of the present study is to investigate the water entry problem by using the MPS method. To validate our suggestions, we numerically studied hydrostatic pressure and dam break problems and verified the proposed MPS method. Then, the water entry problems of 2D wedges were investigated using the MPS method, and the results were compared with experimental and previous numerical results.

3.1   Validation of the present MPS method

3.1.1   Hydrostatic pressure problem

The hydrostatic pressure problem is one of the problems widely used in validating the performance of particle methods. Figure 5 shows the geometry for the hydrostatic pressure problem. The tank is filled with fresh water to 0.4 m. Pressure is captured at a measuring point P and compared with an analytical solution. The simulations are conducted with different particle spacings, i. e. l₀ = 0.006, 0.008, 0.01, 0.012 m. The time step is 1.0 × 10⁻³ s, and the physical simulation time is 5 s.

Figure  5  Geometric model for hydrostatic pressure problem

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Figure 6 shows the comparison between analytic pressure and MPS pressure at measuring point P. In this figure, MPS results correspond to 0.006 m particle spacing. Figure 6(a) depicts the comparison of the analytic pressure and the result of MPS without Laplacian compensation, whereas Figure 6(b) shows the comparison of the analytic pressure and the result of MPS with Laplacian compensation. As can be seen in Figure 6, Laplacian compensation greatly enhances the accuracy of the MPS method, and pressure fluctuation is suppressed when the method is applied.

Figure  6  Comparison between analytic pressure and MPS pressure at measuring point P

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Table 1 shows the averaged relative errors of pressure at measuring point P at different particle spacings. These errors are calculated as the relative deviation of the MPS results from the analytical pressure and time averaged by averaging from 2 s to 5 s. When l₀ = 0.006 m, the relative errors are 0.22 and −8.78 with and without Laplacian compensation, respectively. This result implies the effectiveness of Laplacian compensation and obtained in other particle spacings. Laplacian compensation exerts a good effect on the performance of the MPS method (Table 1), and relative error decreases with particle spacing. Therefore, a small particle spacing provides high accuracy.

Figure 7 shows the particles' arrangement and pressure distribution obtained when t = 5 s and l₀ = 0.006 m. The particles are evenly distributed, and the pressure distribution has smooth and reasonable characteristics.

Figure  7  Particles' arrangement and pressure distribution at t = 5 s

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The above results demonstrate that Laplacian compensation considerably improves the performance of the MPS method.

3.1.2   Dam break problem

The dam break problem is a typical problem for verifying the particle method. For the validation of the present MPS method, the dam break problem is simulated and compared with the experimental data of Lobovsky et al. (2014). The geometry model for the dam break simulation is illustrated in Figure 8. Four measuring points are at the same positions in the experiment (Lobovsky et al., 2014): 0.003, 0.015, 0.03, and 0.080 m along the vertical direction. The particle spacing is 0.005 m, and the number of total particles is 14 964. The time step is set at 5 × 10⁻⁴ s.

Figure  8  Geometry model for the dam break problem

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Figure 9 shows the time histories of pressure at four measuring points obtained with the experiment and MPS method, here H is a height of water in Figure 8, i.e., 0.6 m. As shown in Figure 9, the results of MPS with Laplacian compensation are in good agreement with the experimental data. However, the results of MPS without Laplacian compensation show severe fluctuation, demonstrating that Laplacian compensation enhances the performance of the MPS method.

Figure  9  Time histories of pressure at different measuring points from the experiment and MPS without or with Laplacian compensation

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To validate the collision model between fluid and wall particles, we used the MPS results obtained at t = 0.397 s (Figure 10). When the MPS method was performed without the collision model, two particles penetrated the wall boundary, as shown in Figure 10(a). Conversely, when the collision model was adopted, no fluid particle penetrated the wall boundary. These findings demonstrate that the proposed collision model works efficiently.

Figure  10  Effect of collision model between fluid particles and wall particles (t = 0.397 s)

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Figure 11 depicts the pressure distribution obtained at different times from the present MPS method. The pressure distributions are smooth and reasonable physically, and the proposed MPS method shows good performance in the simulation of the dam break problem.

Figure  11  Pressure distribution obtained at different times from the present MPS

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3.2   Investigation into the 2D wedge water entry problem with the present MPS method

In this subsection, the symmetric and asymmetric water entry of a 2D wedge were investigated using the MPS method and compared with some benchmark experiments and other numerical simulations. In the symmetric water entry problem, the symmetric domain technique was validated.

3.2.1   Symmetric water entry

The experimental model by Zhao et al. (1996) is widely used in investigating water entry problems. It was selected as a criterion in our study. Figure 12 depicts the geometry for the simulation of 2D wedge water entry. The dimensions of the wedge are the same as in the experimental model. The width (b) and dead-rise angle (β) of the wedge are 0.5 m and 30°, respectively, and the mass (M) of the wedge is 72.5 kg. In this simulation, the depth (D) and width (B) of the simulation domain are 1.0 and 2.0 m, respectively, l₀ = 0.01, 0.008, 0.005 m, and the total number of particles are 20 759, 32 201, and 81 517, respectively.

Figure  12  Geometry for the simulation of 2D wedge water entry

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Figure 13 shows the time history of the vertical velocity of a wedge during water entry. The MPS result was compared with the experiment (Zhao et al., 1996) and SPH results (Zhao et al., 2023). As shown in Figure 13, the MPS result agrees well with the experimental and SPH results. Figure 14 illustrates the time history of the vertical force acting on the wedge during water entry. In Figure 14, the vertical force by the MPS simulation is in good agreement with the experiment and below 17.5 ms when l₀ = 0.005 m. However, the MPS result is inferior to the experiment and SPH results in the other particle spacings, possibly because the high-order Laplacian model was not adopted.

Figure  13  Time history of the vertical velocity during water entry

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Figure  14  Time history of the vertical force acting on the wedge during water entry

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For the validation of the results, especially those obtained with the symmetric domain technique, the 2D wedge water entry problem, which was the same as that in another experiment (Yettou et al., 2006), was investigated with the present MPS method. The dimensions of the wedge are the same as those of the wedge used in the experiment (Yettou et al., 2006). Figure 15 shows the geometry of a wedge with two pressure sensors at its edges. The width of the wedge is 1.2 m. We selected one test configuration from the several test configurations of the experiment. The characteristics of the test configuration are as follows: dead-rise angle of the wedge 25°; wedge mass, 94 kg; and drop height, 1.3 m. The width and depth of the simulation domain are 2.0 and 1.0 m, respectively. The MPS simulations were conducted when l₀ = 0.006, 0.008, and 0.010 m.

Figure  15  Geometry of the wedge

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Figure 16 shows the time history of the wedge velocity during water entry from the drop height. In Figure 16, the result of MPS agrees well with that of experiment conducted by Yettou et al., (2006) and SPH (Zhang et al., 2022a) until time reaches approximately 0.55 s. Beyond this time, the MPS result deviates from the experiment and SPH simulation results, but the total trend is similar to that in the experiment.

Figure  16  Time history of the wedge velocity during water entry

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Figure 17 depicts the time history of the wedge velocity predicted by experiments (Zhao et al., 1996; Yettou et al., 2006), Von Karman (1929), Wagner (1932), and the present MPS method. The MPS result is extremely close to that of Wagner's results but does not agree well with the experimental results (Zhao et al., 1996; Yettou et al., 2006). The possible reason is the large particle spacing. However, the trend of the MPS result is similar to that of the experiment results (Figure 18). Figure 18 shows the comparison of the pressure obtained at P₁ and P₂ sensors during the experiment (Yettou et al., 2006), SPH simulation (Zhang et al., 2022a), and the present MPS simulation. Figure 18 shows the suppressed fluctuation in pressure and a trend close to the SPH results at a small particle spacing. However, the numerical results differ from the experimental results.

Figure  17  Time history of the wedge velocity during a certain period of impact after water entry

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Figure  18  Time histories of the pressure at P₁ and P₂ sensors

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Figure 19 shows the pressure distributions obtained at different times during water entry when l₀ = 0.006 m. In Figure 19, the pictures on the left show the results from the whole-domain simulation, and the pictures on the right show the results from the symmetric domain simulation. In these simulations, the total numbers of particles are 113 946 and 57 214, respectively, and the average computational times per time step are approximately 38 and 10 s, respectively. As can be seen in Figure 19, the pressure distributions from the two simulations are extremely similar. This finding shows that the symmetric domain technique not only considerably reduces the computational cost but also has an accuracy comparable to that of the whole-domain simulation. Thus, in the following simulations, the symmetric domain technique was adopted for the flow with symmetric characteristics.

Figure  19  Pressure distributions at different times during water entry (Left: whole-domain simulation; Right: half-domain simulation)

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3.2.2   Asymmetric water entry

The asymmetric water entry of a 2D wedge was investigated with the present MPS method. For comparison, the experiment (Barjasteh et al., 2016) was selected as a criterion, and the simulation was conducted on the same model used in the experiment (Barjasteh et al., 2016). Figure 20 depicts the wedge geometry for asymmetric entry. As to the asymmetric entry, the inclination angle (φ) is not zero. The mass (M) of the wedge is 44 kg, and l₀ = 0.004, 0.006, 0.008 m. The configurations of the simulation are listed in Table 2, where β and φ denote the dead-rise and inclination angles of the wedge, respectively.

Figure  20  Wedge geometry for asymmetric entry

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First, symmetric water entry was investigated for a comparison with the experimental result. Figure 21 shows the time histories of pressure obtained at different sensors in different cases. The numerical results are in good agreement with the experiment at a small particle spacing. Peak pressure varies according to drop height: approximately 30, 47, and 230 kPa at 0.2, 0.5, and 0.75 m, respectively. The times when the peak pressure is observed also vary: approximately 12 ms at a drop height of 0.2 m and below 5 ms at a drop height of 0.5 and 0.75 m. The wedge dropped at a higher height produces a larger peak pressure, and the peak pressure appears at a later time when the wedge was dropped at a lower height.

Figure  21  Time histories of pressure at different sensors in different cases

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Figure 22 shows the pressure distributions obtained at different times with different dead-rise angles. In this figure, the left side corresponds to β = 20°; and the right side, β = 30°. As shown in Figure 22, the wedge with a small dead-rise angle is subjected to a large impact pressure. This finding is physically reasonable.

Figure  22  Pressure distribution at different times when h = 0.5 m (left: β = 20°, right: β = 30°)

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Asymmetric water entry was investigated. Figure 23 shows the time histories of pressure at the P₁, P₂, and P₃ sensors during asymmetric entry. MPS simulation was conducted at a particle spacing of 0.004 m. Pressure measurements at P₁, P₂, and P₃ were plotted from left to right in every figure. Regarding the dead-rise angle, i. e., (a) and (b) or (c) and (d) in Figure 23, not only peak pressure but also the speed of impact occurrence increases with the inclination angle. At the same inclination angle, not only the peak pressure but also the speed of impact occurrence increases with decreasing dead-rise angle. The numerical results somewhat vary among the experimental results particularly when β = 30° but are in good agreement with the experiment results in terms of trend.

Figure  23  Time histories of pressure at P₁, P₂, and P₃ in asymmetric entry. (P₁: green; P₂: blue; P₃: red)

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Figure 24 illustrates the compared results of the symmetric and asymmetric entry of the wedge. The numerical results somewhat vary from the experimental ones, but they have similar trends. When β = 20°, φ = 5° or φ = 0°, the difference between the numerical and experimental results is comparatively large, and the occurrence time of peak pressure slightly differs between symmetric and asymmetric entry. However, when β = 30°, φ = 10° or φ = 0°, the difference is small, and the occurrence time of peak pressure differs between symmetric and asymmetric entry, i.e., peak pressure appears earlier during asymmetric entry. Figure 24 shows that the hydrodynamic characteristics of the wedge during symmetric and asymmetric entry are similar at small dead-rise and inclination angles.

Figure  24  Time histories of pressure at P₁ in symmetric and asymmetric entry

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Figures 25 and 26 show the pressure distributions obtained at different times during the symmetric and asymmetric entry of the wedge at different dead-rise angles. In Figure 25, the pressure distributions have smooth characteristics, except when t = 0.330, 0.336 s. This feature can also be observed in Figure 26. Both figures show nonsmooth pressure distributions around the wedge. These distributions can be attributed to not adopting a high-order Laplacian model. However, in Figure 26, large impact pressure occurs at a small dead-rise angle.

Figure  25  Pressure distributions at different times during symmetric and asymmetric entry (Left: β = 20°, φ = 0°; Right: β = 20°, φ = 5°)

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Figure  26  Pressure distributions at different times in different dead-rise angles (Left: β = 20°, φ = 10°; Right: β = 30°, φ = 10°)

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Overall, the above numerical results show that the present MPS method can provide a reliable solution for the 2D wedge water entry problem.

4   Conclusions

This study examined the water entry of a 2D wedge. The main purpose was to verify the availability of investigations into water entry problems with the MPS method, and some techniques for enhancing the performance of the method were suggested, including the use of Laplacian compensation in the region close to the wall boundary and a collision model between fluid and wall particles. In addition, a symmetric domain technique for reducing computational cost and a fluid–solid-coupling algorithm using the MPS method were proposed.

The proposed techniques were validated through several numerical studies. First, hydrostatic pressure and dam break were simulated using the MPS method, and numerical results were compared with analytical solution and experimental results. Through these simulations, Laplacian compensation was found to greatly improve the performance of the method. Then, the water entry problems of the 2D wedge were investigated using the method. The simulations on water entry problems were compared with some experimental and previous numerical results. In the symmetric entry simulation, not only the accuracy of the MPS method was verified but also the effect of the symmetric domain technique was validated. The symmetric domain simulation resulted in nearly the same accuracy as the whole-domain simulation, whereas using half of the memory and consuming less time than the whole-domain simulation. In the asymmetric entry simulation, similar to the symmetric entry, the results of the MPS method agreed well with the experimental results, and a high impact pressure was obtained at a large inclination angle. Overall, our results suggest that the MPS method with the proposed techniques is suitable for simulating water entry problems.

However, the present study revealed some drawbacks, such as unstable pressure fluctuation and comparatively large errors. The experimental results were obtained from a 3D wedge, whereas our simulation was conducted on a 2D wedge. Moreover, owing to the limited computational capacity, relatively large particle spacings were selected in this study. Nevertheless, these limitations are expected to be overcome by introducing some advanced techniques, such as multi-GPU parallel computing and multiresolution processing. Another important limitation is that a higher-order Laplacian model was not adopted, which can improve the accuracy and stability of the MPS method. A solution to the above troubles will be carried out in future work.