Investigation on the cavitation effect of underwater shock near different boundaries

When the shock wave of underwater explosion propagates to the surfaces of different boundaries, it gets reflected. Then, a negative pressure area is formed by the superposition of the incident wave and reflected wave. Cavitation occurs when the value of the negative pressure falls below the vapor pressure of water. An improved numerical model based on the spectral element method is applied to investigate the cavitation effect of underwater shock near different boundaries, mainly including the feature of cavitation effect near different boundaries and the influence of different parameters on cavitation effect. In the implementation of the improved numerical model, the bilinear equation of state is used to deal with the fluid field subjected to cavitation, and the field separation technique is employed to avoid the distortion of incident wave propagating through the mesh and the second-order doubly asymptotic approximation is applied to simulate the non-reflecting boundary. The main results are as follows. As the peak pressure and decay constant of shock wave increases, the range of cavitation domain increases, and the duration of cavitation increases. As the depth of water increases, the influence of cavitation on the dynamic response of spherical shell decreases.


Introduction
A large number of researchers have studied the feature of underwater explosion load and the dynamic response of ship structure subjected to an underwater explosion (Zhang et al., 2011Liang and Tai, 2006;Brett and Yiannakopolous, 2008;Zhang and Liu, 2015;Hung et al., 2009;Ming et al., 2016). But, the research considering the cavitation effect of underwater shock is less and the influences of cavitation on the feature of underwater explosion load and dynamic response of the structure are not clear yet. When the shock wave of underwater explosion propagates to the surface of free surface or the structure, it gets reflected. If the reflected wave is a rarefaction wave, the reflected wave will make the pressure in water fall. When the pressure in water falls below its vapor pressure, cavitation occurs. Lots of experiments of underwater explosion show that the generation and collapse of cavitation have great influence on the dynamic response of ship structure (Cole, 1948;Brett and Yiannakopolous, 2008;Hung et al., 2009). So it is essential to investigate the cavitation effect of underwater shock near different boundaries.
For the cavitation effect of underwater shock, Bleich and Sandler (1970) adopted the bilinear equation of state to simulate the fluid field subjected to cavitation, and gave the analytical solution of the velocity of a plane plate subjected to a shock wave with cavitation. Felippa and Deruntz (1984) established the cavitating acoustic finite elements method to investigate the cavitation effect. The displacement potential was introduced in this method to reduce the variable number of fluid field, and the first order doubly asymptotic approximation equation (Geers and Felippa, 1983) was used to simulate the non-reflecting boundary. Based on the cavitating acoustic finite elements method, Shin and Santiago (1998) investigated the cavitation effect of 2D surface ship subjected to a shock wave of underwater explosion; Wood (1998) investigated the cavitation effect of ship-like structure subjected to a shock wave of underwater explosion; Zong et al. (2014) investigated the cavitation effect near free surface and ship structure. On the basis of the cavitating acoustic finite elements method, Sprague and Geers (2003) developed the cavitating acoustic spectral elements method to simulate the cavitation effect. In the method, the fluid domain was discretized by the basis function of Legendre polynomial (Patera, 1984), the field separation technique (Mäkinen, 1998) was employed to avoid the distortion of the incident wave propagating through the mesh and the curved wave approximation was applied to simulate the non-reflecting boundary. Sprague and Geers (2006) simulated the dynamic response of ship-like structure subjected to a shock wave of underwater explosion with cavitation based on the cavitating acoustic spectral elements method. Besides, based on the modified ghost fluid method, Liu et al. (2006) investigated the cavitation effect near the structure; Xie et al. (2007) investigated the cavitation effect near free surface. Wang et al. (2014) investigated the shock wave propagation characteristics and cavitation effects of underwater shock near boundaries based on the coupled Lagrangian-Eulerian method. The modified ghost fluid method and the coupled Lagrangian-Eulerian method can capture the propagation of shock wave and reflected wave well, but it is complicated to capture the interface between different mediums and it is difficult to expand the simulation to threedimensional problems. So the cavitating acoustic finite elements method and the cavitating acoustic spectral elements method are widely used to simulate the cavitation effect.
In the process of numerical simulation, an infinite fluid domain is simulated as a finite fluid domain with non-reflecting boundary. The curved wave approximation is the most widely used to simulate the non-reflecting boundary condition (Zhang et al., 2012;Jen, 2009), including the cavitating acoustic spectral elements method developed by Sprague and Geers (2003). But the curved wave approximation does not consider the added mass effect, and the added mass effect is only simulated by the fluid and near fluid structure interaction surface. Felippa and Deruntz (1984) once adopted the first-order doubly asymptotic approximation as the non-reflecting boundary in the cavitating acoustic finite elements method, but the accuracy of the first-order doubly asymptotic approximation is not satisfactory, due to overestimating the fluid damping (Geers and Felippa, 1983). So, in the first author's previous work (Xiao et al., 2014), an improved cavitating acoustic spectral elements method was obtained by replacing the curved wave approximation with the second-order doubly asymptotic approximation. On the other hand, structure was discretized with the nonlinear finite element method. An improved numerical model of fluid structure interaction was established by developing the user subroutine interface to combine the fluid spectral element method with the structure finite element method.
The cavitation effect of underwater explosion involves complex physical phenomena, such as the generating of cavitation, collapsing of cavitation and cavitation reloading. Owing to the complexity of the problem itself, the feature of cavitation effect near different boundaries and the influence of different parameters on cavitation effect are unclear yet. So based on the improved numerical model of fluid structure interaction established in the authors' previous work, the feature of cavitation effect near different boundaries and the influence of different parameters on cavitation effect are investigated in this paper, aims to provide reference to the evolution of shock resistance of surface ship. The main contents of this paper include: (1) the improved numerical model of fluid structure interaction; (2) the cavitation effect near free surface; (3) the cavitation effect near a plate; (4) the cavitation effect near a spherical shell.

Governing equations of fluid
When the cavitation effect is considered, the linear equation of state is no longer suit for the nonlinear characteristics of fluid field. In this paper, the bilinear equation of state is used to deal with the fluid field subjected to cavitation, and it can be written as (Bleich and Sandler, 1970): where p is the total pressure of fluid, , c is the sound speed in fluid, K is the bulk modulus of fluid, ρ is the density of fluid, s is the densified total condensation, , and ρ 0 is the density of saturated fluid.
For small perturbation, inviscid, irrotational and compressible acoustic fluid, the Euler equation of motion can be written as: (2) f = ∇p eq where u is the displacement vector of fluid, f is the unit volume force. And according to the hydrostatic equilibrium equation, we have , p eq is the hydrostatic pressure. The densified total displacement potential ψ is introduced and defined as: Associating Eq. (2) with Eq.
(3), we obtain: Introducing Eq. (1) into Eq. (4) yields: In addition, the continuity equation of fluid can be written as: then, according to the definitions of the densified total condensation and potential and the assumption of small perturbation, we can obtain: Integrating Eq. (7) in time yields: During underwater shock, the total field can be separated into three component fields (Makinen, 1998): equilibrium, incident and scattered field. Then, we have: where the subscripts 'eq', 'inc' and 'sc' represent the variables related to the equilibrium, incident and scattered field, respectively. The equilibrium field can be obtained by the hydrostatic pressure, and the incident field can be obtained by the propagation of incident wave, so it only needs to solve the scattered field. The bilinear equation of state for scattered field can be obtained by introducing Eq. (9) into Eq. (1): Introducing Eq. (9) into Eq. (5) and Eq. (8), we obtain the governing equations of fluid (Sprague and Geers, 2003): Discretizing the first equation of Eq. (11) with the standard Galerkin approach and using Green's first formula, we can obtain the discrete governing equation of fluid (Sprague and Geers, 2003): where s sc and ψ sc represent the column vector of condensation and displacement potential of the scattered field. Q and H are the coefficient matrices, b sc is the column vector of boundary integral, and can be written as: where Ω is the fluid domain, S is the surface of fluid domain, n is the outward normal vector to, φ is the column vector of the basic functions of Legendre polynomial. Q, H and b sc can be solved by Gauss-Lobatto-Legendre integration.

Boundary conditions of fluid
The fluid boundaries include the free surface boundary, fluid structure interaction boundary and non-reflecting boundary. Sometimes the symmetric boundary is used to simplify the model. The characteristics of the free surface boundary is that the pressure on it is equal to the atmospheric pressure, that is, For the symmetric boundary, the fluid displacements normal to it are equal to zero, namely, ∇ψ sc · n = 0.
(15) So, the boundary integral can be rewritten as: where S nrb is the non-reflecting boundary surface, S fsi is the fluid structure interaction boundary surface, u sc is the normal displacement of fluid of the scattered field. The normal displacement on the non-reflecting boundary surface can be given by the second order doubly asymptotic approximation (Geers, 1978): where an asterisk denotes the temporal integration, κ is the curvature of the non-reflecting boundary surface, , and β and γ are the spatial integral operators and defined as: where q 1 and q 2 are the points on the non-reflecting boundary surface, is the distance vector between point q 1 and point q 2 . The normal displacement on the fluid structure interaction boundary surface can be given by the compatibility of the normal displacement at the fluid structure interface (McCoy and Sun, 1997): where G is the transformation matrix relating the structure and fluid nodes, x n is the normal displacement vector of the structure at the fluid structure interface, it can be obtained from the governing equation of the structure Eq. (22). u inc is the column vector of the normal displacement for incident field at the fluid structure interface.
For spherical wave: For plane wave: ξ where l is the distance between non-reflecting boundary point and incident wave center, is the unit vector of the incident wave propagation direction.

Governing equation of structure
The discrete governing equation of the structure can be represented as (Liang and Tai, 2006;Kalavalapally et al., 2009): where M is the structural mass matrix, C is the structural damping matrix, K is the structural stiffness matrix, x is the column vector of structure displacement, and F is the external force vector.
According to the force equilibrium condition at the fluid structure interface, the external force vector can be written as (Shin, 2004): where A is the fluid area matrix, p inc is the column vector of incident wave pressure, p sc is the column vector of scattered pressure and it can be obtained by the bilinear equation of state for the scattered field Eq. (10).

Validation of numerical model
In order to illustrate the validity of the numerical model, the interaction between an infinite elastic cylindrical shell and a plane exponential decay shock wave is simulated by the numerical model. Because of the complexity of the cavitation effect, it is difficult to obtain the analytical solution. In addition, the experimental results are small. So the validity of the numerical model is illustrated by two aspects of considering and not considering cavitation effect. Huang (1970) gave the analytical solution of the problem when the cavitation effect was not considered, and Felippa and Deruntz (1984) gave the numerical solution of the problem when the cavitation effect was considered. Sketch of the interaction between an infinite cylindrical shell and a planar exponential decaying shock wave is shown in Fig. 1.

Without the cavitation effect
The infinite cylindrical shell is treated as a finite cylindrical shell with the symmetrical boundary condition. The radius of the cylindrical shell is 1 m with the thickness of 0.029 m, the cylindrical shell is composed of steel with the density of 7766 kg/m 3 , Young's modulus of 206.4 GPa and Poisson's ratio of 0.3. The density of water is 997 kg/m 3 and the sound speed in water is 1524 m/s. The peak pressure of the plane shock wave is 1 MPa, and the decay constant is 0.137 ms. Fig. 2 shows comparison of the analytical solution and numerical solution for the radial velocity of a cylindrical shell without cavitation. The dotted line represents Huang's analytical solution and the solid line represents the numerical solution of the present study.
As shown in Fig. 2a, the radial velocity of the cylindrical shell at α=0° increases rapidly under the shock wave loading. As the pressure of the shock wave decays, the radial velocity decreases then oscillates near zero. The radial velocity of the cylindrical shell at α=0° agrees well with the analytical solution. As shown in Fig. 2b, before the stress wave reaches, the radial velocity of the cylindrical shell at α=90° is zero. Then, under the stress wave and the shock wave, the radial velocity of the cylindrical shell at α=90° increases. As the pressure of the shock wave decays, the radial velocity decreases and reverse velocity occurs, and then oscillates near zero. The radial velocity of the cylindrical shell at α=90° also agrees well with the analytical solution.

With the cavitation effect
The radius of the cylindrical shell is 5 m with the thickness of 0.025 m. The density of the cylindrical shell material is 7830 kg/m 3 , Young's modulus is 210 GPa, and Poisson's ratio is 0.3. The density of water is 1024 kg/m 3 , and the sound speed in water is 1500 m/s. The peak pressure of shock wave is 8 MPa, and the decay constant is 5 ms. The hydrostatic pressure is 1 MPa. Fig. 3 gives comparison his-  tories of the numerical solutions for the radial velocity of a cylindrical shell with cavitation at α=0° and α=180°. The dotted line represents Felippa's numerical solution, and the solid line represents the numerical solution of the present study.
As shown in Fig. 3a, the radial velocity of a cylindrical shell at α=0° increases rapidly under the shock wave loading, and then, as the pressure of the shock wave decays, the radial velocity decreases. The radial velocity of the cylindrical shell at α=0° agrees well with Felippa's numerical solution. As shown in Fig. 3b, before the stress wave reaches, the radial velocity of the cylindrical shell at α=180°i s zero. Then, under the stress wave and the shock wave, the radial velocity of the cylindrical shell at α=180° increases. As the pressure of the shock wave decays, the radial velocity decreases. The radial velocity of the cylindrical shell at α=180° agrees well with Felippa's numerical solution. It is worthy to illustrate that Felippa's numerical solution is obtained based on the total field model and the first order doubly asymptotic approximation is adopted as the non-reflecting boundary. But the numerical solution of the present study is obtained based on the scattered field model and the non-reflecting boundary is simulated by the second order doubly asymptotic approximation. It leads to the numerical solution of the present study more precise than Felippa's numerical solution.

Results and discussion
In this section, the cavitation effect near free surface, a plane plate and a spherical shell are investigated based on the numerical model, mainly including the feature of cavitation effect near different boundaries and the influence of different parameters on cavitation effect.

Feature of cavitation effect near free surface
In order to illustrate the feature of cavitation effect near free surface, the fluid pressure and cavitation domain are analyzed. Fig. 4 gives the sketch of interaction between a shock wave and a free surface. The distance between the standoff point and free surface d 0 is 3 m, the distance between test Point 1 and free surface d 1 is 1.5 m, and the distance between test Point 2 and free surface d 2 is 0.75 m.
In order to simplify the calculation model, the fluid domain is treated as a rectangular body with free surface boundary on the top, non-reflecting boundary on the bottom, and surrounded by symmetric boundary. Fig. 5 is the sketch of the calculation model. The calculation model with the length of 6 m is discretized as 300 cube solid elements with 0.06 m×0.06 m×0.06 m. The density of fluid is 997 kg/m 3 , and the sound speed in fluid is 1524 m/s. The peak   pressure of shock wave is 1 MPa, and the decay constant is 0.7 ms. Fig. 6 gives the comparison of the fluid pressure of free field and fluid pressure near free surface at the standoff point. Fig. 7 gives the influence of cavitation on the fluid pressure near the free surface at the standoff point.
As shown in Fig. 6, compared with the fluid pressure of free field, the fluid pressure near the free surface with cavitation is complex. When the rarefaction wave being reflected from the free surface propagates to the standoff point, the fluid pressure near the free surface decreases rapidly, that is the cut off effect of free surface. When the total pressure of fluid falls below the vapor pressure of water, cavitation occurs. Then, as the cavitation develops, at the end of cavitation collapse a pressure wave emerges. When the pressure wave propagates to the standoff point, the cavitation reloading effect forms, and the pressure of cavitation reloading is about 0.1 MPa.
As shown in Fig. 7, the fluid pressure near the free surface at the standoff point without cavitation decreases rapidly under the reflected rarefaction wave. Then a negative pressure with the maximum value of -0.79 MPa occurs. As the incident wave and the reflected rarefaction wave decay, the fluid pressure near the free surface at the standoff point without cavitation tends to be zero. Compared with the flu-id pressure near the free surface with cavitation, there is no cavitation reloading effect to form. Because water almost cannot withstand negative pressure, it is unreasonable to ignore the cavitation effect.
In order to further illustrate the cavitation effect near the free surface, Fig. 8 gives the fluid pressure near the free surface at different locations. Fig. 9 gives the cavitation domain near the free surface at different moments. The solid line represents the upper boundary of cavitation domain, the dotted line represents the lower boundary of cavitation domain, and the vertical axis represents the distance below the free surface.
As shown in Fig. 8, the cut off effect of the free surface and the cavitation reloading effect can be seen at the standoff point, test Point 1 and test Point 2. And the cut off effect of the free surface occurs at test Point 2 firstly, then at test Point 1 and at standoff point finally. That is, the closer to the free surface, the earlier the cut off effect of the free surface is generated. But it is worth noting that the cavitation reloading effect forms at test Point 1 firstly, then at test Point 2 and at standoff point finally. It is because that the close of cavitation domain occurs at test Point 1 nearby. At the same time, a pressure wave emerges. The pressure wave propagates to test Point 2 firstly and then to the standoff point. We can clearly see in Fig. 9 that, as time passes by, the upper boundary of cavitation domain moves downward    XIAO Wei et al. China Ocean Eng., 2017, Vol. 31, No. 4, P. 396-407 401 slowly. But the lower boundary of cavitation domain first moves downward rapidly, and then moves back slowly, at last the cavitation domain closes near 1.5 m below the free surface, that is the position of test Point 1.

Parameter analysis
The influence of shock wave parameters on the cavitation effect near the free surface is illustrated from two as-pects: the peak pressure and decay constant of shock wave. In order to illustrate the influence of the peak pressure of shock wave, three cases with different peak pressures of shock wave are analyzed. The decay constants of shock wave θ are taken as 0.7 ms, and the peak pressures of shock wave p m are taken as 1, 3 and 5 MPa, respectively. Fig. 10a gives the influence history of the peak pressure of shock wave on the cavitation domain near the free surface.
As shown in Fig. 10a, for the case with the peak pressure of shock wave of 1 MPa, the range of cavitation domain is 0.08-3.01 m below the free surface, and the duration of cavitation is 1.8-16.0 ms. For the case with the peak pressure of shock wave of 3 MPa, the range of the cavitation domain is 0.08-3.35 m below the free surface, and the duration of cavitation is 1.8-35.0 ms. For the case with the peak pressure of shock wave of 5 MPa, the range of the cavitation domain is 0.08-3.70 m below the free surface, the duration of cavitation is 1.8-49.8 ms. Namely, as the peak pressure of shock wave increases, the range of cavitation domain increases, and the duration of cavitation increases. But the close position of the cavitation domain keeps unchanged, 1.5 m below the free surface.
In order to illustrate the influence of the decay constant of shock wave, three cases with different decay constants of shock wave are analyzed. The peak pressures of shock wave p m are taken as 1 MPa, and the decay constants of shock wave θ are taken as 0.4, 0.7 and 1.0 ms, respectively. Fig.  10b gives the influence histories of decay constant of shock wave on the cavitation domain near the free surface. As shown in Fig. 10b, for the case with the decay constant of shock wave of 0.4 ms, the range of the cavitation domain is 0.10-1.97 m below the free surface, and the duration of cavitation is 2.0-10.9 ms. For the case with the decay constant of shock wave of 0.7 ms, the range of the cavitation domain is 0.10-2.90 m below the free surface, and the duration of cavitation is 2.0-16.0 ms. For the case with the decay constant of shock wave of 1.0 ms, the range of the cavitation domain is 0.10-3.85 m below the free surface, and the duration of cavitation is 2.0-19.6 ms. Namely, with the increase of the decay constant of the shock wave pressure, the range of the cavitation domain increases, and the duration of cavitation increases. And the close position of the cavitation domain moves downward.

Feature of the cavitation effect near a plane plate
In order to illustrate the feature of the cavitation effect near an air-backed plane plate, the normal velocity and hull pressure at the center of the plane plate subjected to a shock wave are analyzed. Fig. 11 gives the sketch of the interaction between a shock wave and an air-backed plane plate. The thickness of plane plate is 0.02 m, and the plane plate is composed of material with density of 7800 kg/m 3 . The planar plate has only the vertical displacement, and the other five degrees of freedom are equal to zero. The density of fluid is 989 kg/m 3 , and the sound speed in fluid is 1451 m/s. The peak pressure of shock wave is 1 MPa, and the decay constant is 1 ms. Fig. 12 gives the influence of cavitation on the normal velocity and hull pressure at the center of the plane plate. Fig. 10. Influence of (a) the peak and (b) decay constant of shock wave on the cavitation domain near the free surface. Fig. 11. Sketch of the interaction between a shock wave and a plane plate.
The solid line represents the results without cavitation, and the dotted line represents the results with cavitation. As shown in Fig. 12a, compared with the result without cavitation, the normal velocity of the plane plate with cavitation has three outstanding features. That is, the decay time of the normal velocity is longer, the reverse velocity occurs and the reverse velocity decreases rapidly to zero. The generation of cavitation makes the plane plate separate from fluid, resulting in the plane plate only being affected by gravity. So the normal velocity of the plane plate decays slowly. When it decays to zero, the plane plate is still separated from fluid, so a reverse velocity occurs. When the cavitation domain closes, the cavitation reloading occurs. The normal velocity of the plane plate decreases rapidly to zero under the pressure of cavitation reloading.
As shown in Fig. 12b, compared with the result without cavitation, the maximum negative pressure at the center of the plane plate with cavitation is smaller. In addition, there is cavitation reloading effect to occur for the hull pressure of a plane plate with cavitation. The plane plate moves under incident wave, and the motion of the plane plate makes a rarefaction wave generate. The rarefaction wave makes the pressure of fluid fall down, and then cavitation occurs when it falls below the vapor pressure of water. The generation of cavitation makes the plane plate separated from fluid, resulting in the hull pressure of the plane plate oscillating near zero. When the cavitation domain closes, the collapse jet of cavitation strikes on the plane plate. The hull pressure of the plane plate increases then decreases rapidly again, that is the cavitation reloading effect. The maximum hull pressure is 1.28 MPa, and the maximum pressure of cavitation reloading is 0.32 MPa.

Parameter analysis
The influence of shock wave parameters on the cavitation effect near the plane plate is also illustrated from the peak pressure and decay constant of shock wave two aspects. Three cases with different peak pressures of shock wave are taken to illustrate the influence of the peak pressure of shock wave. The peak pressures of shock wave p m are taken as 0.7, 1.0 and 1.3 MPa. And the decay constants of shock wave θ are taken as 1.0 ms. Fig. 13 gives the influence histories of the peak pressure of shock wave on the normal velocity and hull pressure of a plane plate.
As shown in Fig. 13a, for the cases with the peak pressures of shock wave of 0.7, 1.0, and 1.3 MPa, the maximum normal velocities of the plane plate are 0.73, 1.05, and 1.36 m/s, respectively; the maximum reverse velocities are -0.18, -0.31, and -0.44 m/s, respectively; and the time of the reverse velocity decaying to zero is 13.5, 18.0, and 22.6 ms, respectively. Namely, as the peak pressure of shock wave increases, the maximum normal velocity of the plane plate  XIAO Wei et al. China Ocean Eng., 2017, Vol. 31, No. 4, P. 396-407 403 increases, the maximum reverse velocity increases, and the time of the reverse velocity decaying to zero postpones. As shown in Fig. 13b, for the cases with the peak pressures of shock wave of 0.7, 1.0, and 1.3 MPa, the maximum hull pressures of the plane plate are 0.9, 1.28, and 1.67 MPa, respectively; the maximum pressures of cavitation reloading are 0.18, 0.32, and 0.37 MPa, respectively; and the time of the cavitation reloading occurring is 13, 17.6, and 22.2 ms, respectively. Namely, as the peak pressure of shock wave increases, the maximum hull pressure increases, the maximum pressure of cavitation reloading increases, and the time of the cavitation reloading occurring postpones.
In order to illustrate the influence of the decay constant of shock wave on the cavitation effect near the plane plate, three cases with different decay constants of shock wave are taken. The peak pressures of shock wave p m are taken as 1.0 MPa. And the decay constants of shock wave θ are taken as 0.2, 0.6 and 1.0 ms. Fig. 14 gives the influence histories of the decay constant of shock wave on the normal velocity and hull pressure of the plane plate.
As shown in Fig. 14a, for the cases with the decay constants of shock wave of 0.2, 0.6, and 1.0 ms, the maximum normal velocities of the plane plate are 0.66, 0.95, and 1.06 m/s, respectively; the maximum reverse velocities are -0.2, -0.27, and -0.31 m/s, respectively; and the time of the reverse velocity decaying to zero is 6, 12.6, and 18 ms, respectively. Namely, as the decay constant of shock wave increases, the maximum normal velocity of the plane plate increases, the maximum reverse velocity increases, and the time of the reverse velocity decaying to zero postpones.
As shown in Fig. 14b, for the cases with the decay constants of shock wave of 0.2, 0.6, and 1.0 ms, the maximum hull pressures of the plane plate are 1.06, 1.24, and 1.28 MPa, respectively; the maximum pressures of cavitation reloading are 0.13, 0.2, and 0.32 MPa, respectively; and the time of the cavitation reloading occurring is 5.65, 12.25, and 17.65 ms, respectively. Namely, as the decay constant of shock wave increases, the maximum hull pressure increases, the maximum pressure of cavitation reloading increases, and the time of the cavitation reloading occurring postpones.

Feature of cavitation effect near a spherical shell
In order to illustrate the feature of the cavitation effect near a spherical shell, the radial velocity and Mises stress of the spherical shell subjected to a shock wave are analyzed. Fig. 15 gives the sketch of the interaction between a spherical shell and a shock wave.
The spherical shell with the radius of 1 m and the thickness of 0.02 m is laid in 50 m below the free surface. The spherical shell is composed of steel with the density of 7800 kg/m 3 , Young's modulus of 210 GPa, Poisson's ratio of 0.3 and the static yield stress of 235 MPa. The spherical shell is free standing. A charge of 8 kg TNT detonates 6 m away from the center of the spherical shell. The density of water is 1000 kg/m 3 and the sound speed in water is 1500 m/s. Since the spherical shell is a symmetrical structure, in order to reduce the calculation and save time, a quarter of the spherical shell with symmetrical boundary condition is considered, and it is composed of 384 quadrilateral shell elements.
Figs. 16 and 17 show the influences of cavitation respectively on the radial velocity and the Mises stress of a spherical shell at the leading point and trailing point. The  solid lines represent the results with cavitation, and the dotted lines represent the results without cavitation. As shown in Figs. 16a and 17a, the influences of cavitation on the radial velocity and the Mises stress of the spherical shell at the leading point are mainly displayed at 1.0-4.5 ms, and the cavitation makes the oscillations of the radial velocity and the Mises stress of the spherical shell at the leading point increase. As shown in Figs. 16b and 17b, the influences of cavitation on the radial velocity and the Mises stress of the spherical shell at the trailing point are mainly displayed after 1.85 ms, and the cavitation makes the oscillations of the radial velocity and the Mises stress of the spherical shell at the trailing point increase. It is illustrated that the cavita-tion first occurs at the leading point of the spherical shell, and then moves to the trailing point. The generation of cavitation makes the spherical shell separated from fluid, and the collapse jet of cavitation striking on the spherical shell leads to the oscillation of the radial velocity of the spherical shell increase.

Influence of water depth on cavitation effect
In order to analyze the influence of the water depth on cavitation effect, three cases with water depths of 25, 50 and 100 m are taken. Fig. 18 gives the influence of the water depth on the radial velocity of a spherical shell at the leading point and trailing point.   XIAO Wei et al. China Ocean Eng., 2017, Vol. 31, No. 4, P. 396-407 As shown in Fig. 18a, as the water depth increases, the difference of the radial velocity of spherical shell at the leading point with cavitation and without cavitation decreases, namely, and the influence of cavitation on the radial velocity of the spherical shell at the leading point decreases. As shown in Fig. 18b, similar to the result of the leading point, as the water depth increases, the difference of the radial velocity of the spherical shell at the trailing point with cavitation and without cavitation decreases, namely, the influence of cavitation on the radial velocity of the spherical shell at the leading trailing point decreases. As the water depth increases, the hydrostatic pressure increases, the cavitation domain decreases, and the duration of cavitation decreases, so the influence of cavitation on the radial velocity of the spherical shell decreases. Namely, the cavitation in shallow water is more obvious than that in deep water.

Conclusions
Based on the improved numerical model of the fluid structure interaction, the cavitation effect of underwater shock near different boundaries is investigated in the present paper, mainly including the feature of cavitation effect near different boundaries and the influence of different parameters on cavitation effect. The main conclusions are summarized as follows.
(1) As the pressure peak of shock wave increases, the range of the cavitation domain increases and the duration of cavitation increases. But the close position of the cavitation domain keeps unchanged. As the decay constant of shock wave increases, the range of the cavitation domain and the duration of cavitation increase. And the close position of the cavitation domain moves downward.
(2) As the peak pressure and decay constant of shock wave increases, the maximum normal velocity of a plane plate increases, the maximum reverse velocity increases, and the time of the reverse velocity decaying to zero postpones. The maximum hull pressure increases, the maximum pressure of cavitation reloading increases, and the time of the cavitation reloading occurring postpones.
(3) The cavitation near a spherical shell first occurs at the leading point of the spherical shell, and then moves to the trailing point. As the water depth increases, the influence of cavitation on the radial velocity of the spherical shell decreases. The cavitation effect in shallow water is more obvious than that in deep water.