Dynamic characteristics of unsteady cavity evolution of high-speed projectiles passing through holes in free surface ice flows

Cui, Wenzhi; Zhang, Song; Zhang, Qi; Sun, Tiezhi

doi:10.1063/5.0159776

The free surface condition of water-entry problems will be significantly modified by the presence of an ice sheet on the water surface. In this paper, we employ computational fluid dynamics to simulate the process of a cylinder entering a water body vertically and validate the numerical method by comparing the cavity evolution with experimental data. Four high-speed water-entry cases are considered: an ice-free water surface, an ice sheet with circular holes, and an ice sheet with minor and normal sized petal-shaped holes. The cavity evolution, flow field characteristics, and motion parameters of the projectile after entering the water are analyzed separately, and the effects of the ice sheets with holes on the typical water-entry characteristics are identified. The results show that the cavity closure mechanism shifts from surface to shallow closure, and a funnel-shaped cavity is observed below the surface when ice is present. The size of the cavity is extremely restricted, but the collapse of the cavity is delayed. The pressure and flow characteristics around the projectile are also affected by ice on the water surface, and the stability of the trajectory is enhanced by the existence of the ice sheet. Finally, the positive acceleration of the projectile triggered by high pressure at the tail of the structure is found to be much larger than that in the ice-free case.

I. INTRODUCTION

The water-entry process involves the flow of multiphase fluids and is commonly encountered in the fields of automatic underwater vehicles (AUVs),^1,2 aircraft free surface ditching,^3,4 and ship engineering.^5,6 The process of water entry is a complex problem owing to the strong nonlinearity, and related work on this topic has been conducted for over a century. Worthington and Cole^7,8 took the lead in systematically studying this issue. After that, scholars began to focus on the evolution of the water-entry cavity, impact load, and water-entry trajectory. Interest in exploring the polar environment has been triggered by the thickness of sea ice decreasing by two meters over the past six decades, a reduction of about 66% that is largely due to the warming climate.⁹ AUVs are highly effective for exploring the underwater world. However, due to interference from the ice sheet environment, a series of unique water-entry characteristics may occur during and after the projectile enters the water, necessitating urgent research.

Arctic research has traditionally focused on the development of arctic routes. As a result, the interaction of ice with ship hulls,^10,11 marine structures,^12,13 and ice–paddle contact^14,15 has been the main areas of research. In recent years, attention has been gradually switched to the impact of ice on the process of trans-medium vehicle entry and exit water. Wang et al.^16,17 investigated the influence of floating ice on the dynamics of a cylinder entering the water at low and high speeds, where the floating ice was treated as a rigid body and the cylinder passed through an ice–water mixture without actually collision with the ice. Several scholars have studied the process of collision and erosion between the projectile and sea ice during water entry and exit. Ren and Zhao¹⁸ adopted the boundary data immersion method (BDIM) to simulate the interaction between the fluid and a rigid sphere, with the ice sheet modeled by the bond-based peridynamics method. The damage of the sea ice and the characteristics of the flow field were analyzed after the sphere penetrated the ice and entered the water. Simultaneously, the coupled Eulerian–Lagrangian technique was employed by Cui et al.¹⁹ to realize the process of a high-speed projectile penetrating the sea ice and entering the water. They studied the evolution of the water-entry cavity and the characteristics of loads in the presence of an ice sheet. The damage mechanism of the ice under the interaction with the projectile was also analyzed. Yue et al.²⁰ conducted numerical simulations of a vehicle launched underwater impacting the ice using the arbitrary Lagrangian–Eulerian (ALE) method based on the LS-DYNA code. Hu et al.²¹ introduced a numerical method involving a modified thermodynamic cavitation model to study a projectile entering the water through an ice hole. They found that the drop in temperature accelerates the surface closure and pinch-off of the bubble. In summary, breakthrough results have been achieved in previous studies on projectiles entering water in ice environments. However, although the finite element method is a powerful tool for analyzing the structural damage of sea ice, the flow field evolution is not captured in sufficient detail. The complex water surface environment has an impact on the hydrodynamics at the initial stage of water entry and is likely to affect the longer-term process. Therefore, research on the dynamics of water entry in an ice sheet environment is of great significance.

The research on cavitation and its formation mechanism has been carried out over the past century. In recent years, the emitted acoustic waves and the associated shock formation of oscillating or collapsing bubbles are drawing many researchers' attention.^22,23 Additionally, collapse dynamics of a cavitating bubble between oblique plates has been studied in depth by experiments, which is certainly beneficial for the understanding of the unsteady evolutionary process of cavitation bubble.²⁴ The evolution of the cavity contains cavitation processes and complex dynamic of cavitation bubbles, which is also the focus of attention in traditional water-entry problems. The simple physical phenomena of cavity formation and splashing were the first to arouse interest. May²⁵ conducted water-entry experiments using a low-speed sphere and found that the surface condition of the sphere affects the characteristics of cavity formation. The typical hourglass-shaped air cavity was observed,²⁶ which can be well represented by the canonical vertical water impact of a low-speed sphere. At the same time, cavity shape also depends on the solid–liquid contact angle.²⁷ Along with the cavity formation, another interesting phenomenon is the growth of the splash crown. Related studies have shown that the splash crown does not always close smoothly. When the ambient pressure changes²⁸ or the velocity of the structure²⁹ is relatively high, the closing of the splash crown will be delayed. At the same time, the cavity dynamics are varying as the truncated cone–cylinder body with different angles of attack.³⁰ The liquid composition at free surface also affects the splash and cavity formation.^31,32 As the problem of low-speed water entry has become more popular, research on the associated high-speed process has increased. Different from low-speed water entry, a large amount of vapor is formed by cavitation in the bubble during high-speed water entry. Supercavitation occurs when the vaporous region becomes sufficiently large to encapsulate the projectile.³³ For the complex cavity and bubble flows of high-speed water-entry problem, theoretical analysis and numerical simulations are often employed due to the limitations of experimental conditions. A unified theory for bubble dynamics is first proposed by Zhang et al.,³⁴ the novel theory can calculate complex bubbles problems by considering boundary effects. The effects of boundaries, bubble interaction, ambient flow field, gravity, bubble migration, fluid compressibility, viscosity, and surface tension are considered. A method for modeling the cavity formation and collapse induced by a rigid sphere was introduced by Lee et al.³⁵ They found that the location of deep closure has the weak relationship with impact velocity, and the occurring time of that is even independent with velocity of the sphere. Furthermore, the effect of the compressibility of the fluid on the development of cavitation was investigated by Chen et al.³⁶ during the water-entry process of a projectile at transonic speed. Obviously, considering the compressibility of the fluid, not only the load characteristics of the entering water but also the evolution of the cavitation and the flow field in the bubble all have changed. Although previous research has extensively covered the evolution of water-entry cavitation from low-speed to high-speed cases, the projectiles have always entered the water under the condition of a free liquid surface. In other words, a splash crown is typically formed, and the external air is always able to flow smoothly into the cavity. There are a few studies considering how sea ice on the free surface influences the cavity evolution. Therefore, studying the evolution characteristics of the cavity during the water-entry process under the interference of ice in arctic regions is of great significance.

The head of the structure will be subjected to a large impact load due to the sudden change in medium density during the water entry process. Water entry takes place when ships are falling into the water^37,38 or sailing in an ocean environment.³⁹ Hydrodynamic loads often raise concerns. For the water entry of projectile, Yao et al.⁴⁰ found that the compressibility of the fluid is very important in evaluating the load on a cylinder with a flat bottom at high impact velocities. Additionally, the impact loads and the affected area will decrease as the aeration level grows. The interaction between the fluid and the structure was considered by Sun et al.^41,42 in evaluating the stress on the surface and the noise characteristics of a high-speed projectile. They found that the structure becomes deformed as it impacts the free surface; this deformation is usually ignored during the water-entry process. At the same time, the load on the projectile exhibits obvious fluctuations. Shi et al.^43,44 employed the ALE method to determine the cavity shape and water-entry acceleration of an AUV during high-speed water entry and found that the results were in good agreement with experimental data. The influence of different head shapes, water-entry speeds, water-entry angles, and attack angles on the impact load was then analyzed. Research on the angle of attack in the water allows the attitude of AUVs to be effectively controlled. The frequency-domain characteristics of the impact load have also been analyzed, and the results are useful for designing of the structure of AUVs. Aiming at high impact loads at the head of the projectile, research into protection measures has also been performed.^45–47 The above-mentioned studies mainly focus on the load and deformation of the head of the vehicle at the initial stage of water entry. Zhang et al.⁴⁸ paid attention to the impact of cavitation collapse on the tail of the vehicle after the projectile enters the water. They found that the high pressure generated by the collapse of the cavity had the same magnitude as the high pressure at the stagnation point of the head pressure. Therefore, the huge load effect will not only occur at the moment of the projectile impacting the water surface, but also after entering the water at a certain depth. However, the basis for their study was a free liquid surface. The formation and magnitude of the load when ice is present on the free liquid surface remain unknown.

The stability of the water-entry trajectory is an important component of many water-entry problems. Chen et al.⁴⁹ investigated the effect of various nose shapes, impact velocities, and water-entry attitude angles on the stability of the trajectory. The effects of asymmetrical nose shapes on water-entry ballistic stability were studied experimentally and numerically by Shi et al.⁵⁰ and Wang et al.,⁵¹ respectively, providing meaningful results for the design of AUV nose shapes. Sui et al.⁵² performed experiments to evaluate the stability of the trajectory of truncated cones with different leading-plane areas and cone angles during water entry and systematically characterized the relationship between the trajectory stability and two different parameters. From the above discussion, it is clear that most of the research on ballistic stability has focused on the influence of the head shape and motion parameters. However, unsteady fluid dynamics are an important factor in ballistic instability. Long trajectories are required to accurately determine projectile stability performance. The complex water surface environment has a vague influence on the dynamics of entering the water, and different flow field characteristics will inevitably lead to differences in the motion stability of the projectile. Thus, analyzing the motion characteristics of projectiles for long-distance ballistics under ice environments is of special interest in ensuring the motion control of the vehicle.

In this paper, the process of a water-entry projectile passing through ice with different shaped holes is simulated, and the results are compared with the ice-free case. Therefore, there is a new challenge compared with previous work on the water-entry process. This paper is arranged as follows. Section II introduces the relevant control equations and models involved in the numerical calculations and verifies the accuracy of the numerical scheme for the high-speed water-entry process. Section III analyzes the influence of ice on the evolution of the cavity and the formation of the splash. Additionally, the effect of ice on the hydrodynamic of the water-entry process is studied systematically, and the coupling of the projectile motion and the evolution of the flow field are analyzed. Finally, the analysis results of the numerical simulations reported herein are summarized in Sec. IV.

II. NUMERICAL METHODOLOGY

A. Governing equations

The present work uses a compressible homogeneous multiphase transport equation. The numerical method consists of a continuity equation, momentum equation, turbulence model, and transport equation for phase change, together with a cavitation model and the equations of state for each phase. The multiphase flow obeys the continuity law and momentum conservation equation,

\frac{\partial ρ_{m}}{\partial t} + \frac{\partial (ρ_{m} u_{i} u_{j})}{\partial x_{j}} = 0,

(1)

\frac{\partial (ρ_{m} u_{i})}{\partial t} + \frac{\partial (ρ_{m} u_{i} u_{j})}{\partial x_{j}} = - \frac{\partial p}{\partial x_{i}} + \frac{\partial}{\partial x_{j}} [μ_{m} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}} - \frac{2}{3} δ_{i j} \frac{\partial u_{k}}{\partial x_{k}})] + \frac{\partial}{\partial x_{i}} (- ρ_{m} \bar{u_{i}^{'} u_{j}^{'}}),

(2)

where

ρ_{m}

and

μ_{m}

denote the density and laminar viscosity of the fluid mixture, respectively. The components of the flow are water, air, and vapor, so

ρ_{m}

and

μ_{m}

can be represented as follows:

ρ_{m} = α_{l} ρ_{l} + α_{a} ρ_{a} + α_{v} ρ_{v},

(3)

μ_{m} = α_{l} μ_{l} + α_{a} μ_{a} + α_{v} μ_{v},

(4)

α_{l} + α_{a} + α_{v} = 1,

(5)

where

α_{l}

⁠,

α_{a}

⁠, and

α_{v}

denote the volume fractions of liquid, air, and vapor, respectively;

p

is the flow pressure;

u

is the velocity of the flow in the ith direction in the Cartesian coordinate system; and

i = 1, 2, 3

⁠, respectively. Equations and constitute the Reynolds-averaged Navier–Stokes equations. Furthermore,

\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}

is the shear strain tensor and

- ρ_{m} \bar{u_{i}^{'} u_{j}^{'}}

is the Reynolds stress tensor.

B. Turbulence model

By comparing with the standard

k - ε

turbulence model, the shear-stress transport (SST)

k - ω

introduced by Menter⁵³ has greater precision when predicting the flow near the wall. Additionally, a great deal of practice has shown that the SST

k - ω

model has fairly good generalizability, which is succeed in the applications for numerical simulation of water-entry problem^54,55 and cavitating flows.^56–58 Hence, the SST

k - ω

model is used to capture the transient nonconstant process in the numerical simulations. The model combines the strengths of the

k - ω

approach with the standard

k - ε

turbulence model and can be expressed as follows:

\frac{\partial (ρ_{m} k)}{\partial t} + \frac{\partial (ρ_{m} u_{i} k)}{\partial x_{i}} = P_{k} - β^{*} ρ_{m} ω k + \frac{\partial}{\partial x_{j}} [(μ_{m} + σ_{k 1} μ_{t}) \frac{\partial k}{\partial x_{j}}],

(6)

\frac{\partial (ρ_{m} ω)}{\partial t} + \frac{\partial (ρ_{m} u_{j} ω)}{\partial x_{j}} = γ P_{ω} - β ρ_{m} ω^{2} + 2 ρ_{m} (1 - F_{2}) σ_{ω 2} \frac{1}{ω} \frac{\partial k}{\partial x_{j}} \frac{\partial ω}{\partial x_{j}} + \frac{\partial}{\partial x_{j}} [(μ_{m} + σ_{ω} μ_{t}) \frac{\partial ω}{\partial x_{j}}],

(7)

where

P_{k} = μ_{t} \frac{\partial u_{i}}{\partial x_{j}} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}) - \frac{2}{3} ρ k δ_{i j} \frac{\partial u_{i}}{\partial x_{j}}

⁠, and

P_{ω} = ρ \frac{\partial u_{i}}{\partial x_{j}} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}) - \frac{2}{3} ρ ω δ_{i j} \frac{\partial u_{i}}{\partial x_{j}}

⁠.

v_{t} = \frac{a_{1} k}{max (a_{1} ω; Ω F_{2})}

is the turbulent eddy viscosity under an invariant measure of the strain rate S and the second blending function.

The second blending function

F_{2}

can be defined as follows:

F_{2} = tanh (a r g^{2}),

(8)

arg = max (2 \frac{\sqrt{k}}{0.09 ω y}; \frac{400 v}{y^{2} ω}),

(9)

where

Ω

is the absolute value of the vorticity, y is the distance to the nearest wall, and

v

is the molecular kinematic viscosity.

C. Cavitation model

The water-to-vapor phase transition is considered in this paper. The Schnerr–Sauer cavitation mode is derived from the Rayleigh–Plesset equation by neglecting the effect of cavity growth acceleration, viscosity, and surface tension. The Schnerr–Sauer cavitation model is widely used in the simulation of high-speed water entry problems.^59,60 The mass transfer between the vapor and liquid phases can be expressed as

\frac{\partial (ρ_{v} α_{v})}{\partial t} + \frac{\partial (ρ_{v} α_{v} u_{j})}{\partial x_{j}} = {\dot{m}}^{+} - {\dot{m}}^{-},

(10)

where

{\dot{m}}^{+}

and

{\dot{m}}^{-}

represent the physical processes of evaporation and condensation in the mass transfer, respectively. The equations for

{\dot{m}}^{+}

⁠,

{\dot{m}}^{-}

and the condition of mass transfer between phases are written as

{\dot{m}}^{+} = \frac{ρ_{v} ρ_{l}}{ρ} α_{v} (1 - α_{v}) \frac{3}{R_{B}} \sqrt{\frac{2}{3} \frac{p_{v} - p}{ρ_{l}}}, p \leq p_{v},

(11)

{\dot{m}}^{-} = \frac{ρ_{v} ρ_{l}}{ρ} α_{v} (1 - α_{v}) \frac{3}{R_{B}} \sqrt{\frac{2}{3} \frac{p - p_{v}}{ρ_{l}}}, p > p_{v},

(12)

where

R_{B}

is the radius of the cavity,

p

is the pressure of the liquid surrounding the cavity, and

p_{v}

is the saturation pressure. The phase transfer is triggered when

p \leq p_{v}

⁠.

D. Equation of state

To calculate the coupling of the air density with the variable pressure, the ideal gas equation is introduced. The ideal gas law expresses density as a function of temperature and pressure and can be written as

ρ = \frac{p}{R T},

(13)

where

p

and

T

are the pressure and temperature of air, respectively (⁠

T

is a constant value in our simulations), and

R

is the gas constant.

E. Validation of numerical method

To verify the effectiveness of the numerical method, we conducted a high-speed vertical water entry experiment on a blunt cylindrical body with a diameter of 6 mm, length of 24 mm, and mass of 1.78 g, with a water-entry velocity of 123.3 m/s. The evolution of the cavity during the high-speed water-entry process was recorded using a high-speed camera. The same water-entry conditions as in the experiment were calculated using the commercial software STAR-CCM+, and overlapping mesh technology was used to simulate the free motion of the cylindrical body in the numerical calculations. Spatial discretization is realized by the finite volume method (FVM) to solve numerically the differential equations. The SST $k - ω$ turbulence model was selected, and a 20-layer boundary mesh was established near the wall of the cylinder to ensure that $y^{+} < 1$ ⁠, thus satisfying the requirements of the turbulence model and achieving a uniform transition between the near-wall mesh and the outer mesh. To increase the calculation accuracy and reduce the computational cost, a special mesh refinement was applied to the water-entry trajectory, and a uniform transition was formed with the far-field mesh. In addition, to better capture the formation of drop splashes and spray crowns during the water-entry process, local mesh refinement was performed at the free liquid surface. The size and boundary conditions of the calculation area are shown in Fig. 1(a): the boundaries of the calculation domain are set as symmetric planes to avoid boundary effects; the bottom and top boundaries of the calculation domain are specified as a velocity inlet and pressure outlet, respectively. Furthermore, the mesh division and local mesh refinement are shown in Fig. 1(b), allowing comparisons with the high-speed camera recording of the cavity shape.

FIG. 1.

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Computational domain with (a) boundary conditions and (b) meshing.

Figure 2(a) compares the cavity shapes at four different times during the process of a flat-headed cylinder entering the water. We can observe that there is no a clear splash crown during the high-speed entry of the small projectile into the water, but instead a foggy liquid mass moving in all directions. The open cavity gradually forms before t = 0.55 ms. When t = 1.8 ms, the surface closure occurs, and the cavity pinches off and is pull down below the free surface. The characteristics of the cavity evolution at these key time points are better obtained through numerical simulations. For better comparing the diameter and length of the cavity, where the scale size of the cavity is measured by using a square grid. The size of the sides of each square is equal to the length of projectile. Figure 2(b) compares the displacement–time history curves of the experiment and numerical calculations. The cavity shapes predicted by the numerical simulations at various characteristic times match well with the experimental snapshots, and the maximum displacement error between the numerical scheme and the experiment is less than 5%. This verifies the effectiveness of the numerical method.

FIG. 2.

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Comparisons of experimental data and numerical results. (a) Cavity evolution and (b) displacement of the cylinder after water entry.

Due to the conducted conditions in this paper involving the presence of an ice sheet on the water surface, a similar condition has also been encompassed to validate the accuracy of numerical calculation methods. When there is no cavitation, the accuracy of the numerical method has nothing to do with the speed of water entry. Hence, the case of a cylinder with wedge head entering the water at a speed of 5 m/s has been used. Sketch of the assembly of the projectile and ice sheet is shown in Fig. 3, where D = 16 mm is the diameter of the projectile. At the same time, the comparison of the cavity shape at three different times companied with the velocity of the projectile between the results of the experiment carried out by Hu et al.²¹ and simulation is shown in Fig. 3. It can be found that the numerical results are in good agreement with the experimental results, and the maximum relative error of the velocity is only 3.2%, which shows that the numerical simulation method has high accuracy.

FIG. 3.

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(a) The geometric model assembly and (b) the result of the numerical simulation.

F. Numerical method setup

Figure 4(a) shows the computational domain. The geometrical dimensions are $29.63 D \times 29.63 D \times 81.5 D$ ⁠, where D is the diameter of the projectile. The entire computational domain is divided into two parts: the upper part is defined as the air and the rest is the water domain. The ambient pressure is set to 101 325 Pa, and the pressure of the water is computed as $P = 101 325 + ρ g h$ ⁠. The boundary conditions of this model are the same as for the verification model, so they are not specified in this figure. The xoy section and a magnified mesh of the computational domain are presented in Fig. 4(b). The trimmer mesh model is adopted to generate the mesh. Along the trajectory of the projectile, there is dense enough grid laid out for the calculation of the fluid. Grid encryption goes from dense to sparse, and this is translated evenly to the outside. The mesh at the position of the water surface is also encrypted in the same way to obtain a clear air–water interface. The zoomed overset mesh and prismatic layer mesh, which has a total of 20 layers, are also exhibited. For more effective calculations, the mesh size of the overlapping area is set to be the same as that of the background encryption area.

FIG. 4.

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Sketch of the (a) computational domain and the (b) regional sectional meshing.

To minimize the influence of the grid size on the numerical results, we calculated the water-entry process of a projectile under three different grid resolutions. The time history curve of the vessel's water-entry acceleration was monitored, as shown in Fig. 5. The minimum grid size and the total number of grid cells are given in Table I. The trend of acceleration is almost identical under the three grid resolutions. When the minimum grid size is 56.56 mm, the first peak of the acceleration is slightly different from the other cases. However, the second positive peak of the acceleration is obviously delayed, and its value is higher than in the other two cases. In the local magnified acceleration curve, the results using this grid resolution have significant numerical fluctuations, which is unacceptable. In terms of both the trend and the peaks of the acceleration curve, grid resolutions II and III correspond almost perfectly. Thus, considering the computational cost required for sufficient accuracy, the minimum grid size of 40 mm was selected for the final numerical mesh.

FIG. 5.

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Acceleration of different grid levels.

TABLE I.

Three resolutions used for grid-independence study.

Grid classification	Minimum size of grid (mm)	Number of grids (×10⁶)
Grid level I	56.56	3.04
Grid level II	40	11.57
Grid level III	28	19.09

Grid classification	Minimum size of grid (mm)	Number of grids (×10⁶)
Grid level I	56.56	3.04
Grid level II	40	11.57
Grid level III	28	19.09

To study the influence of surface ice on the entry dynamics of the projectile, ice sheets with circular and petal-shaped holes are placed on the liquid surface. The numerical results of the unconstrained condition of the free liquid surface will be compared with those of the cases with ice sheets. A schematic diagram of the assemblage of the projectile and sea ice is shown in Fig. 6(a), and the circular and petal-shaped holes can be seen in Figs. 6(b) and 6(c), respectively. The pattern on the petal-shaped hole is separated as minor and normal size, which are formed by changing the edge of the circular hole. Namely, points on the circular edge are shifted inward and outward by $δ$ and $2 δ$ and connected by splines. This is repeated every 30° around the circle. The geometric parameters are listed in Table II; the mass of the projectile is 1420.78 kg. The ice-free, circular hole, minor, and normal sized petal-shaped hole cases are named ${Ice}_{fr}$ ⁠, ${Ice}_{ch}$ ⁠, Ice_ph, and ${Ice}_{ph 2}$ ⁠, respectively.

FIG. 6.

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Sketch of (a) water-entry projectile coupling with ice sheet, (b) bird's eye view of the condition of the water surface, and (c) difference between circular and petal-shaped holes on the ice.

TABLE II.

Geometric parameters of the projectile and ice sheet.

Geometric parameter	$L_{s}$	D	$L_{ice}$	$T_{ice}$	$R_{hole}$	$δ$
Size (m)	5.59 m	0.54 m	12	1	0.54	0.04

Geometric parameter	$L_{s}$	D	$L_{ice}$	$T_{ice}$	$R_{hole}$	$δ$
Size (m)	5.59 m	0.54 m	12	1	0.54	0.04

In the present work, the fluid phases consist of water, air, and vapor. The Schnerr–Sauer cavitation model is considered to describe the transition between water and vapor. The volume-of-fluid model is used to capture the interface of individual fluid phases. Herein, the overlapping mesh and the method of dynamic fluid body interaction (DFBI) model are used to simulate the motion of the projectile with six degrees of freedom and to calculate the force and torque of the fluid and the gravity acting on the structure. Based on the finite volume method, the spatial discretization of the fluid domain is realized by meshing. Furthermore, the convection flux is calculated by the hybrid monotone upstream-centered schemes for conservation laws (MUSCL) third-order/central difference scheme, which effectively improves the stability and accuracy of simulations involving high-speed flows. With respect to the discretization of time, the first-order format is used to guarantee the stability of the numerical calculations.

III. RESULTS AND DISCUSSION

A. Effect of free surface ice on cavity evolution

During the process of projectile passing through the water area with or without the restriction of the ice with hole, the cavity undergoing the stage from generation to collapse presents a complex and similar law. The entire process presents significant unsteady characteristics, and ice hole as a special constraint environment affects the cavity evolution during the water entry. Furthermore, the detailed effects of the presence of ice sheet with different shape hole on the evolution of cavity at different stage deserve further investigation. Figure 7 shows the water volume fraction under the various cases during the process of water entry. Figure 7(a) shows the evolution of the cavity under ice-free conditions. During the entire water-entry process, the projectile experiences four stages: impact on the water surface, formation of an open cavity, surface closure, and collapse. At t = 40 ms, the water is displaced by the high-speed impact of the projectile, and splashes form around the impact location. Part of the air entrained by the structure enters the water forms an open cavity. The projectile moves downward and continues to transfer energy to the surrounding liquid, converting its own kinetic energy into the kinetic and potential energy of the surrounding fluid. The cavity further develops and completely wraps around the body of the projectile. Additionally, under the joint action of the inertial force and the external force transferred from the projectile, the liquid splash moves upward at high speed, and some of the splashed droplets become separated from the splash crown. When t = 120 ms, the surface of the splash crown goes inward and closes under the action of the pressure reduction formed by the high-speed air flow behind the projectile (Bernoulli's law). Accompanied by the surface closure of the cavity, at t = 200 ms, upward and downward high-speed jets simultaneously form at the free liquid surface, which is known as the Worthington jet.³³ Above the free liquid surface, a higher-intensity liquid splash is formed, while below the free liquid surface, as the cavity is pulled off, the high-speed jet penetrates the tail of the cavity and continues to move downward. The cavity still completely wraps the projectile at this time and continues to move downward with the projectile. As the depth increases, although the cavity length continues to increase, its radius decreases due to the dissipation of the kinetic energy of the water particles, and the influence of the pressure difference between the inside and outside of the cavity gradually increases.

FIG. 7.

Volume friction of water of the three cases. (a) Ice fr; (b) Ice ch; (c) Ice ph; and (d) Ice ph 2 cases.

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Volume friction of water of the three cases. (a) ${Ice}_{fr}$ ⁠; (b) ${Ice}_{ch}$ ⁠; (c) ${Ice}_{ph}$ ⁠; and (d) ${Ice}_{ph 2}$ cases.

Finally, the cavity enters the stage of collapse at t = 240 ms. The tail of the cavity is truncated by the huge pressure difference, and the collapse of the cavity starts at the tail and propagates downward to the head. This is because the kinetic energy of the water particles at the cross section of the tail cavity has dissipated and the stable cavity at that cross section can no longer be supported. Thus, the tail cavity collapses first and forms a bubble-like cavity at the rear of the main cavity. Subsequently, the cavity begins to enter the collapse stage. The volume of the cavity continuously collapses and decreases, so the cavity trail does not develop any further after 320 ms, and a weaker foamy cavity trail forms. The figures suggest that the cavity at the tail of the projectile moves downward at a very high speed during the collapse stage and catches up with the projectile at t = 380 ms.

Figures 7(b), 7(c), and 7(d) show nephograms of the volume fraction of water in the case of ice with circular and petal-shaped holes on the free liquid surface, respectively. After the projectile hits the water surface, a high-speed water jet flows to the periphery and moves upward. However, under the obstruction of the ice sheet, only a small part of the water jet passes through the opening in the ice and continues to move upward. Most of the fluid is suppressed under the ice, failing to form a perfect splash crown. However, under the different opening shapes, the forms of high-speed splash are different. When the projectile enters the water through an ice sheet with a circular hole, the fluid moves upward along the wall of the smooth circular breach under the high-speed impact of the structure. When the liquid reaches the upper surface of the ice, the fluid energy of the high-speed movement is attenuated. Under the combined effect of the pressure difference and surface tension, the upward-moving water curtain meets at the upper opening of the ice and forms an upward jet flow. Because the distance between the water curtain and the free liquid surface is relatively large (approximately equal to the thickness of the ice), it does not interfere with the cavity shape under the ice. As a result, the cavity remains unclosed at that time. As the projectile moves further downward and transmits large amounts of energy to the surrounding fluid, the liquid acquires continuous kinetic energy and pours out of the hole of the sea ice. The previous process is repeated, forming a continuous and dispersed upward jet. The figure clearly shows that, as the depth of the projectile increases, the intensity of subsequent jets gradually weakens. Furthermore, the closure of the liquid surface under the ice marks the point at which these upward jets no longer flow.

Different from the circular hole, the fluid along the wall of the petal-shaped hole is dispersed by the texture of the wall, thus forming a thinner splash curtain. At the same time, compared with the minor petal-shaped pattern, the splash is clearer and more robust under the case of Ice_ph2. In the presence of ice, no downward high-speed jet is observed after the closure of the cavity. At t = 240 ms, the cavity tail is cut off under the action of the pressure difference. At this time, the volume of the cavity is much smaller than in the ice-free condition. Moreover, instead of a bubble-like cavity wake, a foam-like cavity wake forms in the three cases with ice.

The evolution of the three-dimensional cavity shape is shown in Fig. 8. With the ice-free environment, under the action of the water pressure, the tail of the cavity is clamped soon after the surface pinches off. After t = 320 ms, a series of bubble trails can be seen at the tail of the cavity in all three cases, and the trail is clearest when there is no ice sheet. However, the size of the cavity is relatively small in Figs. 8(b)–8(d), because of the lower gas content inside the cavity. The projectile wrapped by the cavity continues to move downward, and the cavity collapses and falls off over the subsequent 30 ms. The cavity collapse speed is faster than the speed of the projectile and quickly catches up with the tail of the projectile, where upon the tail of the structure is hit by a high-speed jet produced by cavity collapse. At t = 370 ms, most of the surface of the projectile is wetted, and only the shoulder and tail of the projectile are still wrapped by the cavity. When no ice exists, the wetted area of the projectile is larger. Regardless of whether there is ice, the cavity at the shoulder of the projectile expands again at t = 400 ms. In the ice-free environment, the violent oscillations caused by the collapse of the cavity and the impact of the jet flow deflect the attitude of the projectile, with the symmetry of the cavity most seriously damaged at the head.

FIG. 8.

Three-dimensional cavity morphology of the three cases. (a) Ice fr, (b) Ice ch, (c) Ice ph, and (d) Ice ph 2.

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Three-dimensional cavity morphology of the three cases. (a) ${Ice}_{fr}$ ⁠, (b) ${Ice}_{ch}$ ⁠, (c) ${Ice}_{ph}$ ⁠, and (d) ${Ice}_{ph 2}$ ⁠.

Although the above discussion qualitatively shows that there is a difference in the shape of the water-entry cavity produced by the projectile with or without the ice sheet on the surface of the water, it is useful to quantitatively characterize the evolution of the cavity. Therefore, the curve of the cavity length with respect to time is exhibited in Fig. 9. The water-entry depths of the four cases are also shown. The colored curves represent the length of the cavity and the depth of water entry, respectively. The shape of the dot on the line represents different case just as indicated in the legend. The dimensionless numbers $L_{c} / D$ ⁠, $L_{p} / D$ represent the cavity length and water-entry depth. The penetration distance of the projectile in the four different conditions is almost the same, just with a maximum difference of 0.09 m. For the cavity length, the trend of the four curves is almost the same. When the projectile is nearly at the same position (⁠ $L_{p} / D$ almost contain the same value from beginning to the end), there is obvious difference between length of the cavity at three cases, which indicates that the development of cavity is under the great influence of the ice sheet constrained on the free surface. On the other hand, before t = 250 ms, the cavity length continues to increase. Moreover, there is a delay in the time to reach the maximum cavity length when there is no ice. After that, as the projectile speed decreases and the hydrostatic pressure increases, the cavity begins to collapse, and the cavity length decreases until $t \approx$ 375 ms. Thereafter, the cavity length increases again.

FIG. 9.

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The length of the cavity and penetration distance vs time.

There are significant differences in the cavity length. Under the ice-free condition, the cavity length is always greater than that under the ice conditions, but the curves almost coincide when ice exists. Furthermore, the difference in the length of the cavity is much greater than the displacement, and the maximum gap is nearly 3.33 times the projectile diameter (D). As a result, the slight differences in displacement fail to explain the extreme differences in cavity length of the three cases. The process whereby the cavity enters the re-expansion stage in the presence of ice suggests that the process of cavity evolution is anticipated. Furthermore, an obvious law can be found that is negative correlation between the degree of the unsmoothness of ice-hole wall and the length of the cavity.

An interesting phenomenon is that the form of the cavity closure is different. Figure 9 shows the three-dimensional cavity with its contour lines and a local magnified view of the cavity tail formed at the time of cavity closure. From Fig. 10(a), it is clear that although the cavity completely wraps the structure, an obvious difference is that the existence of ice largely inhibits the expansion and development of the cavity. Especially at the tail, the cavity diameter is greatly reduced compared with the ice-free case. Another notable feature is that surface closure occurs after 200 ms in the ice-free condition. During the downward movement of the projectile, the cavity tail separates from the free liquid surface to create the cavity pinch-off phenomenon. However, when ice exists, the downward movement of the projectile pulls the water down to the subsurface of sea ice under the joint action of gravity and the pressure difference, and then a similar phenomenon of surface closure occurs. Apparently, the upward moving water jets are heavily suppressed by the ice sheet. The process of jet hits the closed crown after it fully closed, leading to an explosionlike collision which spreads water particles all over the place. Thus, there is no Worthington jet formed. In addition, due to the closure of the cavity under the ice sheet, the external airflow cannot continue to move into the cavity, so the free liquid surface under the ice is filled with high-speed air and a funnel-shaped cavity forms. This was also observed in the experimental and numerical results of Hu et al.²¹ The phenomenon indicates that the presence of surface ice shifts the form of cavity closure from surface closure to shallow closure. The presence of the petal-shaped pattern on the wall of the hole is accelerating the process of shallow closure, and the larger size of the petal-shaped pattern indicates the earlier closure time. Combined with Fig. 8, it appears that, after the closure of the cavity under the ice, the funnel-shaped cavity continues to expand as high-speed air pours into the free surface beneath the ice. This is the result of the shallow closure truncating the entrance of air flow into the main cavity bubble. At the same time, the size of the funnel-shaped at the case of the Ice_ph2 is the largest at 400 ms.

FIG. 10.

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Three-dimensional cavity of the three cases. (a) Cavity morphology at 200 ms and (b) magnified tail cavity.

Figure 10(a) shows that the cavity has been pinched off after t = 200 ms in the case of the petal-shaped hole. By observing Fig. 10(b), we see that a downward high-speed jet is formed. As the penetration depth of the projectile increases, a Rayleigh–Taylor instability develops on the top of the cavity, and a water jet penetrates the tail of the cavity and enters its interior in the ice-free condition.⁶¹ The wall surface of the cavity is not greatly affected and remains relatively smooth. When ice exists, there is no obvious jet near the pinch-off position, but the tail wall of the cavity is clearly contaminated and even disturbed, which is the result of disturbance of water particle motion caused by the high-speed water jet due to unsteady bubble breaks in cavity throat and inflow of air after the shallow closure of cavity.

B. Structure evolution of unsteady flow field inside the cavity

Section III A mainly discussed the evolution of the cavity during the water-entry process and analyzed the influence of an ice sheet with different hole shapes. The above analysis suggests that ice has little impact on the initial stage of the cavity, but significantly influences the late-stage development of the cavity for long water-entry trajectories. Therefore, this subsection investigates the fluid flow mechanisms of air and water, especially after the surface closure of the cavity. We are trying to explain the complex coupling effect of flow field structure characteristics and cavity evolution process during water entry at the different condition of water surface. Thus, the velocity and pressure field distribution and unsteady vortex structure characteristics under those conditions were deeply compared and analyzed in this section.

The difference in cavity length is obviously small at the initial stage of water entry but has become large by the time of cavity collapse. The duration of cavity collapse under the different surface conditions is presented in Fig. 11. The start point of the column is the moment at which the collapse occurs, and the end point represents the time of wake collapse, where the characteristics of the cavity are shown at the left of the column. The duration of this event is marked at the top of the column. When ice exists, the duration of cavity collapse is longer than in the ice-free case, which suggests that the speed of the tail cavity collapse is greatest when there is no ice. Moreover, the sequence of cavity collapse is consistent with that in Fig. 9. According to the principle of independent expansion, the energy supporting each section of the cavity is provided by the projectile. The duration of the sectional cavity reflects the energy contained in the cavity. Hence, the cavity with the most energy is the most robust in the presence of ice.

FIG. 11.

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Duration of cavity collapse.

During the water-entry process, the internal pressure of the cavity changes with the evolution of the cavity. To investigate the evolution mechanism of the cavity in the different cases, the pressure on the surface of the projectile and its surroundings was monitored during the water-entry process. Figure 12 shows the position of the five monitored flag points, which are mainly concentrated on the shoulder and tail of the projectile.

FIG. 12.

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Position of flag points on the projectile during water entry.

The formation and collapse of the cavity are typical characteristics of cavity evolutions during water entry. The pressure inside the cavity shows distinctive pulsations at different stages. The following discusses the two stages of cavity development and cavity collapse as a means of analyzing the influence of the ice sheet on the pressure inside the cavity. Figure 13 presents the relative positions of the flag points in the cavity after 300 ms, accompanied by the pressure histories at points p1, p2, and p3. The magnified view in the upper-left corner of Fig. 13(b) shows the pressure history at p1 during the initial stage of water entry. The pressure at the head of the projectile decreases rapidly due to the initial formation of the cavity, the interior of which is mainly composed of air entrained by the projectile into the water. The high-speed flow forms a low-pressure area in the cavity. However, in the early stage, regardless of whether ice is present, the pressure at p1 is similar. In other words, the presence of ice does not have a significant effect on the head pressure. However, the pressure changes at p1 and p2 in the ice-free condition exhibit strong nonlinear characteristics, reflecting the instability of the pressure during the development stage of the cavity. After t = 120 ms, the cavity is in the stable development stage, and the pressure histories at p1 and p2 show that the pressure inside the cavity is lower in the presence of ice than when no ice sheet exists. Namely, the pressure drop inside the cavity is greater, which partially explains why the volume of the cavity in the water is smaller in the presence of an ice sheet.

FIG. 13.

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Morphology of the cavity after collapse at the head of the projectile and comparison of the pressure histories in the three cases. (a) Cavity at 370 ms; absolute pressures at point (b) p1, (c) p2, and (d) p3.

After 300 ms, the cavity begins to shrink under the pressure of the water. As a consequence, the pressure inside the cavity generally increases. The pressure in the cavity begins to fluctuate, accompanied by the continuous leakage of gas at the end of the cavity. From then on, the pressure enters the cavity collapse stage. The right side of the red line in Figs. 13(b)–13(d) shows the pressure histories after the collapse of the cavity. Figure 13(a) indicates that the relative positions of the monitoring points and the cavity are different in the different cases. The pressure variations on the surface of the projectile at p1 are different from those at p2 inside the cavity. Comparing Figs. 13(b) and 13(c), it is not difficult to find that the pressure at p1 is much higher than at p2 in the cavity collapse stage, because p1 is outside the shoulder cavity. The cavity at the head of the projectile shrinks as the depth increases and finally collapses. The pressure on the head pulsates and finally becomes stable at a value close to the hydrostatic pressure.

Position p3 clearly exhibits different pressures with and without the ice sheet: p3 is positioned inside the cavity in the ice-free case and outside in the presence of an ice sheet. A significant high pressure occurs after approximately 360 ms when the ice sheet exists; there is only a slight increase in pressure in the ice-free case because p3 is always wrapped by the cavity. When there is an ice sheet on the surface of the water, the cavity shrinks, collapses, and moves backward along the surface of the projectile. The external water particles move from the high-pressure area to the low-pressure region under the action of the pressure difference caused by the collapse of the cavity, so a water jet forms and impacts the structure, which results in the appearance of high pressures at position p3. As can be seen from Fig. 13(a), the cavity at the head of the projectile has not yet developed at this stage. Therefore, the presence of the ice sheet accelerates the collapse of the cavity.

It is crucial to monitor the pressure changes during the evolution of the cavity on the shoulder of the projectile. The relevant data are obtained from the monitoring points on the head of the projectile. Additionally, the monitoring points at the tail of the projectile reflect the pressure pulsations during the collapse stage. The pressure curves at points p4 and p5 are shown in Figs. 14(a) and 14(b), respectively. When there is an ice sheet on the water surface, Fig. 14(a) suggests that the pressure fluctuations at p4 are very strong when the cavity collapses to the tail of the projectile. In the presence of an ice sheet, pressure ripples can be observed at the rear of the projectile from 300 to 325 ms, which is the result of pressure perturbations along the side of the cavity after pinch-off.⁶² After experiencing pressure fluctuations, this position exhibits a large pressure peak. However, p4 does not experience pressure fluctuations in the ice-free case, with the pressure instead rising to a plateau and then experiencing a peak. It can also be seen from the pressure diagram that the tail of the projectile is in a relatively high-pressure environment, although it is still wrapped by the cavity. Additionally, the peak pressure in the presence of ice is more than twice that of the ice-free condition.

FIG. 14.

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Comparison of absolute pressure vs time and the distinct pressure distribution around the projectile. Absolute pressure at point (a) p4 and (b) p5.

From Fig. 14(b), we can observe a significant difference in the pressure change at the tail of the projectile before the formation of the open cavity. When ice is present, the pressure drops earlier and continuously. However, position p5 first experiences high pressure, which then decreases in the ice-free case. This can be explained by the pressure nephograms. There is a region of high pressure initiated by the splash crown, and the tail of the projectile begins to enter the low-pressure area at the upper surface of the ice rather than the free surface. After about t = 300 ms, the cavity begins to collapse and the high-velocity flow collides with the tail. The projectile is subjected to a large impact load, producing a rapid increase in pressure. Additionally, the high pressure occurs earlier and attains a higher value in the presence of an ice sheet compared with the ice-free condition.

The pressure and fluid flow after 70, 120, and 170 ms in the four different cases are presented in Fig. 15. After the structure enters the water, a circular high-pressure area appears at the head of the projectile. This high-pressure area gradually decreases in size with increasing water depth. Additionally, an obvious zone of low pressure appears in the cavity due to the high-speed flow after the projectile enters the water. However, compared with the cases in which there is an ice sheet on the surface of the water, there is a clear pressure gradient inside the cavity in the ice-free case, which evolves as the cavity develops. This is due to the continuous inflow of air from outside before the cavity closure, while the cavity is still in the development stage. Thus, its internal flow field is in the non-stationary stage. However, the presence of the ice restricts the inflow and acts as a rectifier for the flow status above the ice hole. Moreover, a low-pressure zone relative to the outside exists at the opening of the ice. As a result, the outside air flows through the hole continuously, which explains the continuous expansion of the funnel-shaped cavity after the projectile has passed through the ice sheet.

FIG. 15.

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The development of pressure when (a) t = 70 ms, (b) 120 ms, and (c) 170 ms at four different cases.

An important feature in the process of water entry is the existence of large pressure pulsations around the time of cavity collapse, which have strong unsteady characteristics. Figure 16 shows the pressure field at four moments under different surface conditions. As shown in the figure, under the effect of the pressure difference inside and outside the cavity, the tail of the cavity collapses and the water particles around the cavity quickly fill the cavity area. The huge impact of the water jet generates a large pressure of approximately the same order of magnitude as the head pressure. During the shedding of the cavity at the tail, there is a low-pressure region inside the bubbles moving upward. However, the fastest cavity shedding speed and largest cavity volume occur when no ice exists. High-speed upward-moving bubbles affect the pressure distribution of the water, whereupon the surrounding fluid particles all flow toward the tail of the projectile, forming a downward-sloping pressure contour. However, this phenomenon is not observed when there is an ice sheet.

FIG. 16.

Development of pressure after cavity collapse: (a) Ice fr, (b) Ice ch, (c) Ice ph, and (d) Ice ph 2.

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Development of pressure after cavity collapse: (a) ${Ice}_{fr}$ ⁠, (b) ${Ice}_{ch}$ ⁠, (c) ${Ice}_{ph}$ ⁠, and (d) ${Ice}_{ph 2}$ ⁠.

A stable low-pressure environment exists inside the cavity. As the projectile continues to move downward, the pressure inside the cavity gradually increases. This phenomenon is most obvious in the ice-free case. The area of high pressure in the open-water case is largest when the cavity collapses downward to the tail of the projectile. However, the lowest pressure in the open-water case is 5 MPa, nearly half of the 10.1 MPa in the presence of an ice sheet. In addition, the projectile is already wet at t = 350 ms. Two relatively symmetrical high-pressure areas appear on either side of the projectile. As in the previous discussion, when a layer of ice exists, the projectile enters the next stage of water entry at an earlier time.

The pressure field and the velocity field have a coupling relationship. In the process of entering the water, the flow is often disordered. The streamlines and velocity of the air and water flow are shown in Fig. 17. When the projectile enters the water, the water is displaced by the structure, and the water particles move downward after 40 ms. With the formation of an open cavity, a zone of relatively low pressure develops inside the cavity, allowing more air to enter. In the presence of an ice sheet, the air flow rushes along the surface of the ice toward the hole. Due to the inhibition of the ice sheet, the velocity of the air at the opening of the cavity is significantly higher than in the ice-free condition. With the further evolution of the cavity, the velocity field no longer changes significantly until the tail of the cavity shrinks and collapses. When there is no ice sheet, an obvious high-speed jet appears at the tail of the cavity, which is driven by the high pressure. Furthermore, the speed of the jet is about 4–5 times the speed of the projectile at this time. In the presence of ice, the speed inside the cavity is slightly increased, which is consistent with the pressure field after 280 ms shown in Fig. 17.

FIG. 17.

The evolution of velocity field of (a) Ice fr, (b) Ice ch, (c) Ice ph , and (d) Ice ph 2.

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The evolution of velocity field of (a) ${Ice}_{fr}$ ⁠, (b) ${Ice}_{ch}$ ⁠, (c) ${Ice}_{ph},$ and (d) ${Ice}_{ph 2}$ ⁠.

In addition, the air above the free surface is affected by the upward jet of water, and the movement of air becomes whirling in all four cases. The jet has a more concentrated influence on the air in the case of a circular hole in the ice. However, the area of air that is influenced is larger in the case of the petal-shaped hole. The minor petal-shaped hole also induces a more powerful water jet, although the water flow is scattered. Assuming that the same kinetic energy is transferred to the fluid by the projectile during water entry, the water moving upward and through the ice hole is dispersed into droplets by the pattern, and then, the velocity gained by each fluid particle will be much greater, which can explain this phenomenon. Another noteworthy phenomenon is the emergence of high-velocity airflow⁶³ at the opening of the ice hole after the cavity has undergone shallow closure and pinching off, which is consistent with the low pressure at the ice hole.

Water entry is a complex process with strong unsteady characteristics. Vortex structures are a product of large velocity and pressure gradients, which emerge or dissipate with cavitation evolution. They, thus, reflect these unsteady characteristics to some extent. The existence of ice will inevitably affect the velocity and pressure gradients during the process of water entry, and the induced vortex structures provide another representation of the unsteady development of the pressure and velocity fields. Consequently, it is of great significance to study the evolution of vortex structures. Accompanied by the formation of splashes and cavity evolution during water entry, vortices of various scales are generated in many regions. The formation of cavities induces large velocity, pressure, and density gradients, leading to the generation of vortices. Herein, the second invariant of the velocity gradient tensor Q is introduced to describe the typical unsteady characteristics of the vortex structures.⁶⁴ The second invariant Q can be defined as

Q = \frac{1}{2} (ω_{i j} ω_{i j} - S_{i j} S_{i j}),

(14)

where the

ω_{i j} = \frac{\partial u_{i}}{\partial x_{j}} - \frac{\partial u_{j}}{\partial x_{i}}

and

S_{i j} = \frac{1}{2} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}})

are the rotation tensor and symmetric strain rate tensor of the filtered flow, respectively. When

Q > 0

⁠, the rotation effect dominates; otherwise, the strain is dominant. To fully display the development process of the vortex structures during the water-entry process, two- and three-dimensional vorticity scenes based on the iso-surface of

Q = 5000 / s^{2}

are shown in Fig. 18.

FIG. 18.

Evolution of the vorticity distribution of the three different cases, showing the distribution of the Q-value (left) and the morphology of three-dimensional vorticity (right). (a) Ice fr, (b) Ice ch, (c) Ice ph, and (d) Ice ph 2.

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Evolution of the vorticity distribution of the three different cases, showing the distribution of the Q-value (left) and the morphology of three-dimensional vorticity (right). (a) ${Ice}_{fr}$ ⁠, (b) ${Ice}_{ch}$ ⁠, (c) ${Ice}_{ph}$ ⁠, and (d) ${Ice}_{ph 2}$ ⁠.

First, under the influence of the initial upward jetting above the free liquid surface, strong vortices are generated. With the loss of jet energy and the extension of the vortex, the vortex in the air becomes weaker from 120 ms. The air disturbance is more intense when sea ice is present. The area affected by vortices is larger in the case of a circular hole in the ice. However, the region of disturbed air is smaller as the pattern appear on the ice-hole wall, and the energy dissipation of the vortices is fastest when the ice sheet has a petal-shaped hole. The vorticity intensity at the ice hole is high and lasts for a long time due to the continuous flow, which provides energy for the generation of vortices.

Second, with the formation of the cavity, there is always a strong vortex structure at the head of the projectile. The baroclinic torque in the interior of the cavity is affected by the evolution of the cavity and exhibits obvious anomalous characteristics, first increasing and then decreasing. Hence, alternate positive and negative Q-values appear around the projectile at the initial stage under the action of the pressure and velocity gradients. Furthermore, the vorticity inside the cavity continues to increase before the cavity closure. With the further development of the cavity, the vortex strength inside the cavity continues to weaken. However, there are regions at the tail of the cavity in which the eddy strength is stronger. These areas are influenced by the cavity closure effects of spilled air. When cavity closure occurs, the passage of external air into the cavity is blocked. In all four cases, the vorticity in the cavity decreases significantly. However, the vortex intensity of the cavity is the highest in the case of Ice_ph at t = 200 ms.

Third, the cavity continues to move downward with the projectile, and the tail of the cavity continuously falls off and collapses, leading to velocity gradients, pressure gradients, and density gradients. The moment of inertia decreases significantly as the volume shrinks. Angular distance is an important part of vortex generation. Herein, the angular distance increases, and the intensity of vortices grows at the initial time of collapse inside the cavity. More vortex structures appear in the tail of the cavity in the presence of ice, which indicates that the turbulent nature of the fluid is more robust. As the cavity continues to collapse, the energy of the shedding bubbles is continuously dissipated, and so the vortex structures at the wake of the cavity no longer obtain large amounts of energy. Thus, the vortex strength decreases and almost disappears.

C. Motion and load characteristics

The evolution of the cavity during water entry is strongly affected by the water surface conditions. Important characteristics, such as the closure and collapse of the cavity, have a crucial influence on the motion attitude of the projectile, so it is of great significance to analyze the motion parameters of the projectile under different water surface conditions. In this section, we have compared the similarities and differences of each motion parameter of the projectile in the three cases. At the same time, the reasons for this phenomenon are revealed by analyzing the pressure field distribution. In addition, the variation of the pressure at the lower surface of ice sheet with position and time is given, which shows a certain pattern.

The hydrodynamic impact forces are always the focus of attention during the water-entry process. The sudden impact can impair the projectile structure and internal onboard equipment substantially and can also deviate the projectile trajectory.⁶⁵ For reflecting the magnitude of the hydrodynamic impact forces more intuitively, here the impact force is divided by the mass of the projectile to obtain the magnitude of the acceleration to measure the impact load during water entry.^66,67 Figures 19(a) and 19(b) show the axial acceleration and velocity curves of the projectile, respectively. The sudden change in density from air to water subjects the projectile to a large impact load at the moment it contacts the free surface. Therefore, the projectile undergoes a large axial acceleration in the direction opposite to its motion, and the load action lasts for approximately 1.5 ms. The magnified area of the acceleration curve in the axial direction of the vehicle indicates that the existence of the ice sheet has no obvious influence on the peak value of the impact and duration of action by the fluid. The body of the projectile is wrapped by the cavity, and the cavitation closes after about 200 ms in all three cases. Hence, the acceleration curves are largely consistent, although the closing forms of the cavity are different. As the depth of the projectile increases, the cavity undergoes the collapse stage. Under the impact of the high-speed jet, the projectile obtains a positive acceleration. However, when sea ice exists, the projectile reaches the stage of positive acceleration 13 ms earlier than in the ice-free case. At the same time, the magnitude of acceleration is nearly three times that of the no-ice condition. There are also differences in the acceleration between the ${ice}_{ch}$ and ${ice}_{ph}$ conditions. In the case of ${ice}_{ch}$ ⁠, the positive acceleration of the projectile is greater, reaching about 600 $m / s^{2}$ ⁠. However, the pulse width of the peak acceleration under the ice-free conditions is nearly twice that observed when sea ice exists. With the disappearance of the high-pressure area at the tail of the projectile, the positive acceleration decreases and becomes a decelerating motion again. After the pressure stabilizes at the tail, the negative acceleration is greater than that of the ice-free condition, where the projectile is still wrapped by the cavity.

FIG. 19.

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Comparison of (a) axial acceleration and (b) velocity vs time.

As the velocity is obtained by integrating the acceleration curve with respect to time, the velocity curve provides additional information. Although the axial acceleration is broadly equivalent in each case, the velocity of the projectile is slightly lower when the ice sheet exists, as can be observed from the magnified region of Fig. 19(b). Therefore, during the whole water-entry process, the resistance of the body is greater when there is an ice sheet on the surface. In addition, the duration of the positive acceleration is quite short, but the increment in velocity of the projectile is relatively large. Regardless, the presence or absence of a layer of ice has little effect on the speed of the projectile, similar to the discussion in Sec. III A, so there is a negligible difference in displacement.

In addition to the important motion parameter of axial acceleration, the motion stability of the projectile is of widespread concern. The pressure pulsation generated by the cavity collapse and the huge impact of the downward high-speed jet on the tail of the projectile not only causes axial acceleration but also applies a lateral force and rotational moment to the structure. Hence, Fig. 20 shows the offset acceleration and angular acceleration of the projectile after the cavity has collapsed. The distribution of the offset and rotational acceleration is nearly symmetrical on the upper and lower sides of the x-axis. This is because of the continuous pulsation of the pressure field inside the cavity after it collapses and falls off. The offset and rotational acceleration vary significantly in the different water surface environments. First, when a circular ice sheet exists, the peak values of the offset and angular acceleration are greater than those of the ice-free condition, and they occur earlier. However, petal-shaped pattern on the wall of the hole reduces the radial force and moment. In addition, the time of occurrence of these peaks is consistent with the time that the downward acceleration is induced, the result of the high-speed jet action caused by the collapse of the cavity. However, the duration of the peak acceleration is very short when sea ice exists, while the duration of the offset and angular acceleration obtained by the projectile under ice-free conditions is about 2–3 times longer. With the stabilization of the flow field, the fluctuations in acceleration gradually decrease, and their magnitude tends to zero.

FIG. 20.

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Comparison of (a) offset and (b) angular acceleration vs time in the three cases.

The axial force of a projectile can be divided into two parts: viscous and differential pressure force. Although the cavity has collapsed down to the tail of the projectile, the domain of the projectile is traveling downward and is still wrapped by the cavity. Therefore, the main factor in changing the acceleration is the pressure difference between the head and tail of the projectile. The pressure distributions at different times are given in Fig. 15. To analyze the motion mechanism more clearly, the pressure around the projectile, corresponding to the peak moment of motion acceleration, is given in Fig. 21. The high pressure generated by the collapse of the tail cavitation is on the same order of magnitude as that at the head. Under the action of the head–tail pressure difference, the body obtains a large axial downward acceleration. Moreover, when an ice sheet is present, the strength of collapse of the cavity is much greater than that in the ice-free case, which is why there is a clear difference in the tail high pressure with or without ice. The tail high pressure has a larger range and propagates outward in the form of pressure waves when there is an ice sheet, which explains why the acceleration obtained by the projectile is greater.

FIG. 21.

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Pressure at the time of obtaining the positive acceleration in the three different cases.

The strong nonlinearity and pulsation characteristics of the process of cavity collapse destroy the symmetrical form of the cavity, resulting in the asymmetric tail pressure characteristics. This phenomenon appears with and without the ice sheet. Nevertheless, what we can see from Fig. 21 is that petal-shaped pattern on the wall of the ice hole seems to be beneficial to the symmetrical characteristics of cavity. Moreover, the pressure distribution at the tail of the projectile turns to be perfectly symmetrical at the case of Ice_ph2. Due to the pressure difference on either side of the tail of the projectile, the distance between the tail of the projectile and the center of gravity is large. Therefore, a huge rotational moment is generated, which poses a great threat to the ballistic stability. Furthermore, although the pressure distribution on either side of the projectile is very different in the ice-free condition, it cannot cause a large torque due to the small pressure difference. This discussion explains the large offset and angular acceleration of the projectile.

During the motion of the long ballistic trajectory of the projectile, the non-stationary collapse process of the cavity will greatly affect the motion stability of the projectile. To characterize the motion stability of the projectile under different conditions, the figures showing maximum offset distance and rotation angle are shown in Fig. 22. It is obvious that the offset distance and rotation angle are the largest in the ice-free condition. When there is ice sheet on the free surface, it significantly improved ballistic stability of the projectile. At the same time, the petal-shaped pattern on the ice-hole wall reduces motion stability to varying degrees, which is dependent on the size of a petal-shaped pattern.

FIG. 22.

View large Download slide

(a) Offset distance and (b) rotation angle vs time.

As the splash crown formed by the projectile entering the water has a large kinetic and potential energy, evaluating the fluid pressure on the subsurface of the ice is significant in predicting the damage to the ice sheet. To quantitatively analyze the impact of the fluid on the ice sheet during the water-entry process, ten equally spaced (radius length) pressure monitoring points were established along the radial direction on the lower surface of the ice, as shown in Fig. 23(a). The changes in absolute pressure along the radial direction are shown in Figs. 23(b)–23(d). The monitoring points exhibit the same trend: the pressure of the ice on the surface increases under the action of fluid impact at the initial stage of water entry. The pressure then gradually decreases to a lower level and slowly rises to the hydrostatic pressure with increasing water-entry depth. In addition, farther from the hole, the ice is less affected by the fluid. Apparently, the pressure on the lower surface of ice shows a decreasing trend as the distance from the ice hole increases.

FIG. 23.

(a) Location of the flag points on the lower surface; (b)–(d) history of absolute pressure vs time at different locations of three cases.

View large Download slide

(a) Location of the flag points on the lower surface; (b)–(d) history of absolute pressure vs time at different locations of three cases.

IV. CONCLUSIONS

The content of study in this paper is a water-entry process of a slender projectile with a hemispherical nose at the speed of 100 m/s at the constraint environment of polar ice holes, where the constraint conditions we divided into ice with circular and petal-shaped holes, the trajectory of the projectile is relatively long until the cavity collapses to the tail of the structural body and we conducted a series of numerical simulations. By analyzing the cavity evolution alongside the unsteady characteristics of the flow, including the motion characteristics of the projectile after entering the water, we have revealed the influence of an ice sheet with holes on the process of water entry. The main conclusions can be summarized as follows:

The existence of the ice sheet has an enormous influence on the evolution of the cavity. The upward liquid splashing is suppressed, and a different upward jet is formed due to the shape of the ice. Another effect of jet suppression is that the closed form of the cavity changes from surface closure to shallow closure, and a funnel-shaped cavity is formed under the ice sheet. Moreover, a larger funnel-shaped cavity is formed at the case of ice with petal-shaped holes. In addition, the quantity of entrained and incoming air decreases. As a result, both the diameter and length of the cavity are reduced. The roughness at the wall of the ice hole has a significant effect on the length of the cavity. After the cavity is pulled off, no downward high-speed water jet occurs when the ice sheet exists. Thus, in the absence of ice, the cavity collapses lower and more intensely. In conclusion, the presence of ice accelerates the process of cavity evolution.
The ice sheet changes the unsteady characteristics of the flow field during the water-entry process. When ice exists, a stable low pressure quickly appears inside the cavity at the early stage of water entry, resulting in higher-speed airflow inside the cavity. In contrast, there is a pressure gradient in the cavity in the no-ice condition, which causes regional pressure fluctuations around and on the surface of the projectile. At this stage, the vortices around the projectile are stronger when there is no ice, and the entire cavity is filled by vortex structures. After the cavity is fully developed, the vortex intensity inside of it is decreasing at the three cases. Specifically, the air flow is relatively more disturbed inside of the cavity at the case of Ice_ph and Ice_ph2, so a huge amount of the vortex structure still exists at that time. The highest pressure occurs after the happening of collapse at the tail of the cavity. When the cavity collapses at the tail of the projectile, the pressure reaches a minimum, and exhibits obvious symmetry in the presence of an ice sheet. The pressure at the tail of the cavity is largest in the ice-free condition, and the flow state in the tail is disturbed by the falling bubbles, resulting in the formation of vortex structures. When ice exists, the vortex structures in the tail are weaker.
The acceleration trends are largely insensitive to the presence or absence of ice during the water-entry process. However, in the late stage of water entry, the high pressure generated by the collapse of the tail cavity induces a large positive acceleration in the projectile, and this acceleration is intensified by a factor of ∼3 in the presence of an ice sheet compared with the ice-free condition. At the same time, at the cases with ice sheet, the peak values of the deflection and rotational acceleration on the projectile are larger as a result of the higher asymmetric pressure distribution, but the duration of these peaks is shorter than that in the no-ice condition. The presence of ice enhances the stability of the projectile motion to a certain extent. However, in the view of the cases of ice_ch, ice_ph, and ice_ph2, the forces (both axial and normal) and moments exerted by the fluid on the structure are decreasing with the degree of unevenness of the ice-hole wall increasing. Finally, the vicinity of the ice hole is subjected to an enormous splash impact, and the magnitude of the pressure decreases with increasing distance from the hole.

In this paper, we carried out and evaluated the effect of ice sheet with hole on the evolutionary characteristics of the flow field during the projectile with the velocity of 100 m/s water-entry process. However, the current research work is still insufficient to clarify the influence of sea ice on the complex dynamics of water entry. In a further study, a large amount of supporting numerical computational conditions need to be carried out. The effect of some variables should be considered, such as initial velocity and angle, thickness of the ice, and the nose shape of the projectile head. Especially, experiments on this topic is urgent to be carried out, which considers the destructive behavior of sea ice. Based on the research in this paper, we will carry out relevant experiments in the future to enrich the understanding of this problem.

ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (Nos. 52071062 and 52192692).

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Wen Zhi Cui: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Song Zhang: Data curation (equal); Formal analysis (equal); Investigation (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). Qi Zhang: Data curation (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Tiezhi Sun: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Author(s)

Dynamic characteristics of unsteady cavity evolution of high-speed projectiles passing through holes in free surface ice flows

I. INTRODUCTION

II. NUMERICAL METHODOLOGY

A. Governing equations

B. Turbulence model

C. Cavitation model

D. Equation of state

E. Validation of numerical method

F. Numerical method setup

III. RESULTS AND DISCUSSION

A. Effect of free surface ice on cavity evolution

B. Structure evolution of unsteady flow field inside the cavity

C. Motion and load characteristics

IV. CONCLUSIONS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Dynamic characteristics of unsteady cavity evolution of high-speed projectiles passing through holes in free surface ice flows

I. INTRODUCTION

II. NUMERICAL METHODOLOGY

A. Governing equations

B. Turbulence model

C. Cavitation model

D. Equation of state

E. Validation of numerical method

F. Numerical method setup

III. RESULTS AND DISCUSSION

A. Effect of free surface ice on cavity evolution

B. Structure evolution of unsteady flow field inside the cavity

C. Motion and load characteristics

IV. CONCLUSIONS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Citing articles via

Related Content

Resources

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pubs.aip.org

Connect with AIP Publishing

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