Numerical Simulation of Water Entry of Wedges in Waves Using A CIP-Based Model

In this study, the water entry of wedges in regular waves is numerically investigated by a two-dimensional in-house numerical code. The numerical model based on the viscous Navier-Stokes (N-S) equations employs a high-order different method—the constrained interpolation profile (CIP) method to discretize the convection term. A Volume of Fluid (VOF)-type method, the tangent of hyperbola for interface capturing/slope weighting (THINC/SW) is employed to capture the free surface/interface, and an immersed boundary method is adopted to treat the motion of wedges. The momentum source function derived from the Boussinesq equation is applied as an internal wavemaker to generate regular waves. The accuracy of the numerical model is validated in comparison with experimental results in the literature. The results of water entry in waves are provided in terms of the impact force of wedge, velocity and pressure distributions of fluid. Considerable attention is paid to the effects of wave parameters and the position of wedge impacting the water surface. It is found that the existence of waves significantly influences the velocity and pressure field of fluid and impact force on the wedges.


Introduction
Wave is one of the most important dynamic factors in ocean and offshore engineering. The effects of waves on the hull and floating structures entering the water surface cannot be ignored, especially in the case of severe sea conditions. When a vessel sails in rough seas or an airplane lands on the sea, the existence of the waves can lead to fully different hydrodynamic behaviors between the water entry in waves and in the calm water. The quasi-V-shaped cross-sections are commonly used near the bottom of vessels. For the sake of simplicity, it is convenient to use a wedge to investigate the water entry in waves. Thus the present work will focus on the effects of waves on the hydrodynamics of wedges during water entry.
Systematic research on water entry could be dated back from the late 19th century when Worthington and Cole (1900) used high-speed flash cameras to observe droplets, water entry of "smooth" and "rough" balls. Since then, this phenomenon has received considerable attention in the past decades. While most of them put emphasis on water entry in calm water (Abrate, 2011;Truscott et al., 2014), only a few studies focus on water entry in waves. Many numerical methods have been developed to solve water entry in waves based on the potential flow theory. On the basis of the boundary element method (BEM), Faltinsen (1993) studied the water entry in waves by the additional mass method. Although this method takes into account the effects of waves on the hydrodynamics during water entry, it neither considers the interaction between waves and objects, nor can it predict the effects of waves on the pressure distribution over the surface of objects. Sun et al. (2015) investigated a twodimensional wedge entering waves obliquely using BEM based on the incompressible velocity potential theory. They found that the gravity effect can change the pressure distribution, the position and the peak value of pressure on wedge during penetration. A time-domain higher-order boundary element method (HOBEM) is developed by Cheng et al. (2018) to investigate the hydrodynamic of a two-dimensional wedge entering waves obliquely with gravity effect in the presence of a uniform current. They found that the current velocity has obvious effects on the horizontal velocity of flow and the wave steepness.
Although great progress has been made in solving water entry in waves with methods on the basis of the potential flow theory, it is difficult for these methods to deal with the violent free-surface flow or breaking waves. The computational fluid dynamics (CFD) methods which solve the N−S equations have become an alternative method to solve the problem. Based on the finite volume method, Bunnik and Buchner (2004) developed a VOF method that can treat three-phase interfaces to investigate the hydrodynamics of the marine structure entering waves. The marine structure is formed by multiple cylinders. Based on the finite volume method, Wang and Wang (2010) treated the free liquid surface as a contact discontinuity in the density domain, and the motion of a suspended horizontal cylinder in waves was simulated by the cut-cell method. The effects of the wave parameters on the vertical impact force were investigated as well. Taking the turbulence model into account, Hu et al. (2017) used the FLUENT software to study the water entry of the flat bottom structure in waves.
From the above discussion, it appears that most of the studies on the water entry of a wedge into waves focused on the wedges entering the wave obliquely. The flow field generated by a wedge entering into waves with the vertical velocity is distinguished from that with oblique velocity. The lack of a systematic study of the flow field for a wedge entering waves with the vertical velocity can be clearly seen, especially, for the velocity field and pressure distribution. The motion of a wedge in waves will affect the propagation of the wave resulting in the deformation of the wave, and the wave force acting on the wedge will affect the motion of the object. Therefore, it is a challenge to simulate the evolution of waves and the location of a moving wedge simultaneously. In this work, an improved numerical model is applied to study the effects of waves on the hydrodynamics of a two-dimensional wedge entering waves. A CIP-based method is adopted to solve the flow solver in the model. The momentum source function derived from the Boussinesq equation is applied to the internal wavemaker. The numerical sponge layer is applied to the numerical wave tank. The tangent of hyperbola for capturing/slope weighting (THINC/SW) scheme is adopted to predict free surface, and an immersed boundary method is employed to treat moving bodies. Wedges with different deadrise angles impacting the water wave surface with constant vertical velocity are simulated by the proposed model, and the impact force of wedges, the velocity and pressure fields of fluid are analyzed.
The rest of the paper is organized as follows. In Section 2, the governing equations, the wave-maker and numerical methods will be presented. The numerical model is validated in Section 3; the water entry of wedges in waves with different deadrise angles and different vertical velocities are discussed in Section 4. Finally, some conclusions are drawn.

Governing equations
The governing equations for two-dimensional incompressible viscous fluid can be described by Navier−Stokes equations ∂u (1) where and t are the velocity components and time, is the fluid density, and p is the pressure. (i=1, 2) presents the coordinates in a Cartesian coordinate system, is the momentum source including the wavemaker and the damping layers, is the gravitational acceleration, and is the viscous term given by The volume function is used to capture the interface among the three phases. The evolution of the volume function is governed by ∂ϕ ϕ m λ where (m = 1, 2, 3) is the volume function of liquid, gas and solid, which is used to update the physical property , such as density and viscosity in every cell by the following expression 2.2 Internal wave-maker and damping layers According to Wei et al. (1999) and Liu et al. (2015), the source function is added to the momentum equation. The following momentum source function is added in the horizontal direction.
where x is the horizontal coordinate, t is the time, is the circular frequency of the wave, and is the parameter related to the wavemaker region. In this paper, , L is the wave length, , and W is the width of the wavemaker region. The surface momentum source term is the largest in the middle of the wavemaker region and decreases as the distance to the middle line increases. When , tends to 0, that is, when , the momentum source function is 0. The momentum source function can be calculated by where . For regular waves, is the wave height and is the calm water depth. and are the parameters of the Boussinesq equation, the relation can be described by where is the position where the velocity is the depth-averaged velocity, and usually takes . At the end of the numerical wave tank, there is a sponge layer to absorb waves, and the damping coefficient of the damping layers is as follows: where is the coefficient, and are the starting position and length of the absorbing region. and are the empirical damping coefficients which can be determined by the numerical test. In this paper, and , where is the time step.

Numerical method
The momentum source function is treated as follows  Ye et al. (2016), a staggered grid system is employed to ensure strong pressure and velocity coupling. The finite difference method is adopted to discretize Eqs. (1)-(4) on a non-uniform mesh. The discretization process is divided into three steps: advection phase, non-advection phase (i) and non-advection phase (ii). The advection term is firstly solved by a third-order finite difference scheme-the CIP scheme (Yabe and Wang, 1991;Yabe et al., 2001); then a second-order central difference scheme is employed to calculate the non-advection phase (i); the pressure distribution is obtained by solving Poisson equation and finally velocity can be updated by a projection method. After solving the flow field, the volume fractional function is calculated by the THINC/SW scheme. Finally, the boundary of a wedge can be treated by an immersed boundary method. More details about the numerical method can be found in Hu and Kashiwagi (2009) and Zhao et al. (2014). The validation is first carried out in the case of the wedge entering into calm water with constant velocity. The numerical results are compared with the experimental results obtained by Tveitnes et al. (2008). In the experiment, the test sections comprised a box section and symmetric wedge section with different deadrise angles . The breadth B of the test sections is 0.6 m, and the height of the box section 0.3 m. The test sections impacted the calm water surface with a constant vertical velocity V. The dimensions of the computational domain are 5B in length and 2B in depth, and the water depth is 0.5 m. The wedge at the onset of the impact is shown in Fig. 1. The domain is sufficiently large in all directions to ensure that the simulation results are independent of the domain size. The mesh density is large both near the free surface along the y-direction and around the wedge in the x-direction, and decreases gradually away from the impact region and the free surface as shown in Fig   F v /d depends on the time step size dt where t is the time, and t = 0 corresponds to the keel point touches the undisturbed water surface for all the following cases. The present numerical results are compared with the experimental and numerical data obtained by Tveitnes et al. (2008). The test sections with a deadrise angle of 30° impacted the calm water surface at a velocity of 0.48 m/s. It can be observed that the numerical results agree well with the experimental results. As the time step decreases, the numerical results are closer to the experimental results and the oscillation gradually decreases in Fig. 3a. Thus dt = T 0 /2000 will be adopted in the following simulations to save CPU time.
To show the mesh resolution convergence, three kinds of mesh sizes used are listed in Table 1. The time step is dt = T 0 /1000. Fig. 3b displays the variations of vertical impact force against the dimensionless penetration depth F v /d during penetration. It is found that the numerical results obtained by three kinds of mesh sizes agree well with previous experimental results, and the mesh convergence is achieved when a small time step is used. To ensure the accuracy of simulations of water entry in water waves, mesh 1 will be adopted in the following simulations.
To further verify the validation of the numerical model, the numerical results are compared with the available experimental data in Tveitnes et al. (2008). The variations of vertical impact force against the dimensionless penetration depth F v /d during penetration is shown in Fig. 4. The comparison confirms that the numerical results obtained by the present numerical model have a good agreement with the experimental data. However, it is found that the experimental results have dynamic noise. The noise may be induced by the vibration at the wedge/fluid interface during the experiments. The dynamics of the wedge surface, the drive system and the fluid system all play a role in generating highfrequency vibration resulting in the noise in the experimental measurements.
In order to analyze the pressure of wedges, the pressure coefficient is defined as: is the density of the water, P is the pressure and the water entry velocity is V. As shown in Fig. 5, the predicted pressure distributions on the wetted surface of present work have been compared with the numerical values from Wagner (1932) and Zhao and Faltinsen (1993). The time corresponds to the moment at which the simulated pressure comes up to the peak values. y is the height of wedge. y/F v = 0 represents the intersection of the undis- Fig. 3. Vertical force of water entry in calm water: , V = 0.48 m/s.  Zhao and Faltinsen (1993) than those conducted by Wagner (1932). The reason is that Wagner neglected the air resistance, air cushion effect and water compressibility, so the peak values are overestimated. The simulation method in Zhao and Faltinsen (1993) is under the frame of potential theory and the gravity is not taken into account, but the present model is a     Fig. 7. Evolution of wave profile: , is water level, A is wave amplitude.
H = 0.08 m η Fig. 8. Evolution of wave profile: , is water level, A is wave amplitude. ∆x = 0.05L ∆y = 0.01H mesh density is large near the free surface. and are adopted as the minimum space step, and the time step is dt = T/2000. Here H is the wave height, L is the wave length, and the wave period is T. It can be seen from Figs. 2 and 3 that the numerical results agree well with previous experimental results. The numerical wave tank established in this paper is accurate and reliable. The minimum grid size and time step used in this paper are larger than the piston wave-making method used by Liu et al. (2019), which indicates that the wave-making method in this paper is more efficient. Then the wave tank will be used to study the water entry of wedge in waves.

Numerical results and discussion
The computation domain and the initial position of the wedge are shown in Fig. 9. The mesh density is large both near the free surface along the y-direction and around the wedge in the x-direction, and decreases gradually away from the impact region and the free surface. According to the study in Section 3, and are adopted as the minimum space step, and the time step is dt = T 0 /2000. All computation configurations are summarized in Table 3. Similar to the experiment in Tveitnes et al. (2008), high velocity is chosen here to maximize the hydrodynamic component of the total force for high deadrise angle sections. The initial position of the keel point of the wedge is x = 12 m, y = 0.141 5 m. According to the evolution of the wave profile at x = 12 m, the appropriate instant is selected to start the movement of the wedge to ensure that it touches the water surface at the wave crest or trough. From Section 3.2, the time t 1 when the wave crest or trough reaches x = 12 m can be obtained, so .  The vertical impact force on the wedges during water entry in waves is compared with those in the calm water surface in Figs. 10−12. The non-dimensional penetration depth is calculated as F v /d. Unless otherwise specified, t=0 means the keel point touching the water surface for all the following cases. Because the wave propagates from the waveward side of the wedge to the leeward side during the wedge impacting water surface in waves, the vertical impact force of the water entry in waves is different from that in the calm water. As the entry velocity increases, the discrepancy becomes larger. When the wedges with different deadrise angles enter the water at the trough, the vertical impact force is greater than that of the peak. In the initial stage of water entry in waves, it is found that the vertical impact force on the wedge with a deadrise angle is smaller than that in the calm water, and it becomes larger after the wedge has entered the water, as shown in Fig. 10. The wedge enters the water at a velocity of 0.48 m/s from the wave trough. When F v /d < 2.017, the vertical impact force for the wave height H = 0.08 m is larger than that for the wave height H = 0.05 m. While F v /d > 2.017, the vertical impact force for the wave height H = 0.08 m is smaller than that for the wave height H = 0.05 m. The wedge enters the water at a velocity of 0.72 m/s from the wave trough. When F v /d < 2.257, the vertical impact force for the wave height H = 0.08 m is larger than that for the wave height H = 0.05 m. While F v /d >2.257, the vertical impact force for the wave height H = 0.08 m is smaller than that for the wave height H = 0.05 m. As shown in Figs. 11 and 12, when the deadrise angles of the wedges are and , the vertical impact force of the wedges impacting the water surface at the wave crest or trough is smaller than that in the calm water. The vertical impact force of the wedge entering the water surface at the wave crest is closer to that in the calm water. The wedge with impact water at a velocity of 0.48 m/s from the wave trough. When F v /d < 2.99, the vertical impact force for the wave height H = 0.08 m is larger than that for the wave height H = 0.05 m. When F v /d > 2.99, the vertical impact force for the wave height H = 0.08 m is smaller than that for the wave height H = 0.05 m. The wedge with impact water at a velocity of 0.72 m/s from the wave trough. When F v /d < 2.669, the vertical impact force for the wave height H = 0.08 m is larger than that for the wave height H = 0.05 m. While F v /d > 2.669, the vertical impact force for the wave height H = 0.08 m is smaller than that for the wave height H = 0.05 m. The wedge with impacts water at a velocity of 1.19 m/s from the wave trough. The vertical impact force for the wave height H = 0.08 m is smaller than that for the wave height H = 0.05 m. While the wedge impacts water from the wave crest, the vertical impact force for the wave height H = 0.08 m is larger than that for the wave height H = 0.05 m. Further explanation can be found in the following section.

Pressure field
Figs. 13−20 display the pressure field for fluid near the wedges. The pressure field in Figs. 13−20 is magnified and the position of the wedges can be determined by the edge of the pressure field. The time corresponds to the instant at which both chines dry, one chine wet, two chines wet and entered the water. The maximum pressure coefficients on the wetted surfaces of wedges are shown in Tables 4 and 5.
It is found that the wedges enter the water with different speeds whether at trough or at crest, the position of the maximum pressure is located near the keel and lower than that in the calm water, and the maximum pressure values decrease as the penetration depth increases. Since the wave propagates forward during water entry, the existence of the wave makes the flow field characteristics on both sides of the wedge different, the pressure distribution on both sides is no longer symmetric. In the initial stage of water entry, when the wedge impacts the water surface at the wave crest, the pressure coefficient on the left side is larger than that on the right side. However, as the wave crest propagates to the right, the water level on the right rises faster, the right side is wetted first, and the pressure coefficient on the right side is larger than that on the left side after the right side has been wetted. As the initial stage of the wedge impacting the water at the crest, the water at the crest has a downward velocity component while propagating forward. Since the wave encounters wedge, the movement of the water particle in the wave is hindered. The water impacting on the left side of the wedge is subject to the leftward and downward resistance and generates acceleration; as a result, the moving direction becomes rightward and downward along the surface of the wedge. Meanwhile the water exerts upward and rightward pressure on the wedge, so in the initial stage of water entry, the left side pressure is larger than that on the right side. As the water passes under the wedge, more and more β 0 = 15 • Fig. 10. Impact force on the wedge during water entry in waves . Fig. 11. Impact force on the wedge during water entry in waves . Fig. 12. Impact force on the wedge during water entry in waves .

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HU Zi-jun et al. China Ocean Eng., 2021, Vol. 35, No. 1, P. 48-60 water moves to the right side of the wedge. As a result, the water level on the right side rises, the right side of the wedge is wetted first. In the later stage of the wedge impacting water surface from the wave crest, waves turn the right motion of the water particle into the left, so the pressure coefficient on the right side increases sharply, and the pressure coefficient on the right side is larger than that on the left side. When the wedge enters the water from the trough, HU Zi-jun et al. China Ocean Eng., 2021, Vol. 35, No. 1, P. 48-60 55 in the initial stage, the pressure coefficient on the left side is smaller than that on the right side. The left side wetted first, and the pressure coefficient on the left side is larger than that on the right side after the left side wetted. In the initial stage of the water entry, the water particle in the trough has an upward velocity during propagating to the left side. The water particle on the right side generates rightward and downward acceleration due to the existence of the wedge. The pressure coefficient on the left side is smaller than that on the right side, and as the wave propagates forward, the water particle on the right side moves to the left side. The water level on the left side rises faster, and more and more water particle gathers on the left side, so the left side is wetted first and the pressure is higher than that on the right side as well. By comparing Fig. 13 with Fig. 14, when the wedge enters the water from the trough, the left and right sides of the wedge are totally wetted earlier than the wedge enters the water from the crest. This is because the water particle in the trough has an upward velocity component during the forward propagation; it is opposite to the movement of the wedge. Moreover, when the wedge is totally wetted, the penetration depth is larger than the wedge enters the water from the crest, and the peak value of the pressure coefficient on the wetted surface is larger as well. The wedges in Figs. 13 and 15 enter the water from the crest, however the wave height in Fig. 15 is larger than that in Fig. 13. As a result, the moment when the left and right sides are totally wet is also later, and the penetration depth is larger. The peak value of the pressure coefficient on the wetted surface is larger after the wedge is totally wet. The wedges in Figs. 14 and 16 enter the water from the trough, however the wave height in Fig. 16 is larger than that in Fig. 14. Therefore the left and right sides of the wedge are totally wetted earlier.
Compared with Figs. 13−16, the velocity of the wedges in Figs. 17−20 is larger. Although the distribution of the pressure field is consistent, there are some differences in the maximum pressure coefficient values on the wetted surface, as shown in Tables 4 and 5. In Fig. 17, the wedge is totally wetted earlier than that in Fig. 13, and the maximum pressure coefficient on the wetted surfaces is also smaller. Compared with Fig. 14, the wedge is totally wetted later in Fig. 18, and the maximum pressure on the wetted surface is larger. However, due to the high water entry velocity, the dimensionless maximum pressure coefficient on the wetted surface is smaller. Compared with Fig. 15, the wedges in Fig. 19 are totally wetted earlier and have a smaller maximum pressure coefficient on the wetted surface. Compared with Fig. 16, the wedge is totally wetted later in Fig. 20, with larger pressure and smaller pressure coefficient on the wetted surface.
In order to compare the pressure field of water entry in waves with those in the calm water surface, Fig. 21 illustrates the pressure field of the wedge with the deadrise angle impacting water surface at the velocity V = 0.72 m/s. Fig. 21b shows the moment when the wedge is totally wetted. The instants in Figs. 21a, 21c and 21d are the same as those in Figs. 18a−18c. In Fig.18c, the left and right sides are totally wetted. Compared with Fig. 20b, it is found that the wedge impacting the calm water surface is wetted earlier. This is because the water rises faster. By comparing Figs. 21a, 21c and 21d with Figs. 18a, 18b and 18c, it is found that the maximum values of the pressure coefficients on wetted surfaces of the water entry in waves are smaller than those in the calm water. The vertical pressure gradient is also smaller than that in the calm water, therefore the vertical impact force is smaller.

Velocity field
The velocity fields during the water entry of the wedge  Only the water velocity is displayed in the velocity field, thus the evolution of the free surface during water entry can also be displayed. It can be seen from the figures that during the penetration, the free surface near the wedge will bulge, forming a head that climbs up along both sides of the wedge, and the head be-comes higher and higher as the penetration depth increases. Because the wave propagates forward during the penetration, the existence of the wave makes the flow field characteristics on both sides of the wedge different. As a result, the velocity field on the left and right sides of the wedge is also asymmetric. The peak values of the velocity field during the β 0 = 30 • Fig. 22. Velocity field for water entry in waves: , H = 0.05 m, wave crest entry.  water entry in the calm water appear in the water rising region, and the water velocity in the vicinity of the keel of the wedge is much smaller than that in the water rising region. While in the initial stage of water entry in waves, before the left and right sides of the wedge get totally wetted, the peak values of the velocity field may appear near the keel of the wedge, as shown in Figs. 22-23 and 25−29. As the penetration depth increases, the peak values of velocity begin to appear in the water rising region. Owing to the existence of an upward or downward velocity component of the water particle during the wave propagation, the velocity of the area between the wave surface and the wedge is much faster than that of the area near the keel. By comparing the wedge impacting wave surface with 0.48 m/s in Figs. 22−25 to those with 0.72 m/s in Figs. 26−29, as the wave height and the impact position are the same, the larger the water impact velocity, the higher the peak values of the velocity field at the moment of wetting on one side of the wedge and wetting on both sides.

Conclusions
In this work, the water entry of wedges in waves has been studied by using a CIP-based method. Firstly, the accuracy of the numerical model is validated. Then the effects of the wave, the deadrise angle and the wedge velocity on the vertical impact force, the free surface, the pressure field and the velocity field during penetration are analyzed. The following conclusions are drawn with wedges with deadrise angles smaller than 30° entering into waves: (1) The vertical impact force of the water entry in waves is different from that in calm water. As the entry velocity increases, the difference increases. The vertical impact force of the wedge impacting the water from the trough with different deadrise angles at different velocities is larger than those from the crest.
(2) The existence of the wave makes the pressure distribution on both sides asymmetrically. When wedges impact the water with different velocity at trough or at crest, the positions of the maximum pressure are located near the keel and lower than that in the calm water, and the maximum pressure values decrease as the penetration depth increases. In the initial stage of water entry in waves, when the wedge impacts the water surface at the wave crest, the pressure coefficient on the left side is larger than that on the right side. However, the right side is wetted first, and the pressure coefficient on the right side is larger than that on the left side after the right side has been wetted. When the wedge enters the water from the trough, the situation should be the opposite.
(3) The existence of the wave makes the velocity field on both sides of the wedge asymmetric. In the initial stage of water entry in waves, before the left and right sides of the wedge are totally wetted, the peak values of the velocity field may appear near the keel of the wedge. As the wave height and the impact position are the same, the larger the water impact velocity, the higher the peak value of the velocity field at the moment of wetting on one side of the wedge and wetting on both sides.