Experimental Study on the Coupling Dynamics of Metal Jet, Waves, and Bubble During Underwater Explosion of a Shaped Charge

Yu Tian; A-Man Zhang; Liu-Yi Xu; Fu-Ren Ming

doi:10.1016/j.eng.2025.04.001

Engineering ›› 2025, Vol. 50 ›› Issue (7) :168 -187. DOI: 10.1016/j.eng.2025.04.001

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Experimental Study on the Coupling Dynamics of Metal Jet, Waves, and Bubble During Underwater Explosion of a Shaped Charge

Yu Tian ^a^,^b
, A-Man Zhang ^a^,^b^,^*
, Liu-Yi Xu ^a^,^b
, Fu-Ren Ming ^a^,^b

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Abstract

Unlike conventional spherical charges, a shaped charge generates not only a strong shock wave and a pulsating bubble, but also a high strain rate metal jet and a ballistic wave during the underwater explosion. They show significant characteristic differences and couple each other. This paper designs and conducts experiments with shaped charges to analyze the complicated process. The effects of liner angle and weight of shaped charge on the characteristics of metal jets, waves, and bubbles are discussed. It is found that in underwater explosions, the shaped charge generates the metal jet accompanied by the ballistic wave. Then, the shock wave propagates and superimposes with the ballistic wave, and the generated bubble pulsates periodically. It is revealed that the maximum head velocity of the metal jet versus the liner angle α and length-to-diameter ratio λ of the shaped charge follows the laws of 1/(α/180°)^0.55 and λ^0.16, respectively. The head shape and velocity of the metal jet determine the curvature and propagation speed of the initial ballistic wave, thus impacting the superposition time and region with the shock wave. Our findings also reveal that the metal jet carries away some explosion products, which hinders the bubble development, causing an inward depression of the bubble wall near the metal jet. Therefore, the maximum bubble radius and pulsation period are 5.2% and 3.9% smaller than the spherical charge with the same weight. In addition, the uneven axial energy distribution of the shaped charge leads to an oblique bubble jet formation.

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Keywords

Shaped charge / Underwater explosion / Metal jet / Waves / Bubble / Coupling dynamics

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Yu Tian, A-Man Zhang, Liu-Yi Xu, Fu-Ren Ming. Experimental Study on the Coupling Dynamics of Metal Jet, Waves, and Bubble During Underwater Explosion of a Shaped Charge. Engineering, 2025, 50(7): 168-187 DOI:10.1016/j.eng.2025.04.001

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1. Introduction

A shaped charge comprises a metal liner and high-energy explosive. It has a cavity located at the opposite end of the detonation point, which can be conical, hemispherical, or other in shape. Since Nobel [1] invented explosives, the shaped charge has had a long development history and has attracted several scholars to conduct related research [[2], [3], [4], [5]]. Compared with the conventional charge, the shaped charge produces a shaped-charge effect during explosions. When detonated, the spherical detonation wave propagates from the detonation point along the charge, and the explosion energy will converge along the axial direction at the cavity, which is the shaped-charge effect. A thin layer of metal is attached to the portion of the explosive that faces the cavity, which is the metal liner of the shaped charge. When the detonation wave impacts the top of the liner, the metal liner is squeezed and gathers along the axis. At the same time, it will melt and flow because of the heat and the work generated by the explosion. Finally, a liquid metal fluid, the metal jet, will be formed at high temperatures and speeds. When detonated in water, the shaped charge produces a metal jet, a strong discontinuity shock wave, and a high-pressure bubble. Their interactions are more complicated than the conventional charges in underwater explosions. Although studies of shaped charge explosions in air have been carried out extensively regarding the formation of metal jets [[6], [7], [8]] and the penetration to target plates [[9], [10], [11]], only a few studies have been conducted on explosions in water [12,13], which focus on the formation of metal jets in water and compare the penetration capability when exploding in air and water.

Based on the morphological characteristics, the metal jet is categorized into shaped charge jet (SCJ), explosive formed projectile (EFP), as well as jetting projectile charge (JPC). Different types of metal jets have different characteristics in shape, velocity, and penetration capability. SCJ is a slender metal jet formed when the liner angle of the conical liner is 30°–70°, accounting for only about 1/3 of the liner mass. The jet head has a high velocity but it is prone to fracturing, and the jet tail has a slug with a low velocity, limiting its penetration capability [14]. Current experimental studies on SCJ typically rely on X-ray or high-speed cameras to investigate the formation and movement of SCJ in the air [[15], [16], [17]], the water penetration [[18], [19], [20]], the target penetration [[21], [22], [23], [24]], and the multilayer targets of water and plate penetration [25,26]. The results revealed that the head velocity of SCJ in the underwater explosion was higher, and the penetration capability was stronger than in the air [27]. EFP is a metal jet similar to a “projectile” formed when the half liner angle ranges from 75° to 80°. The velocity is about 1.5–2 km·s⁻¹, and the mass accounts for 80% of the liner [14], so it is stable. The foundation and development of EFP experienced a long history covering the concept [28], formulation [29], and optimization [30,31]. Previous studies mainly focused on flight characteristics [32,33], water penetration [[34], [35], [36]], target plate penetration [[37], [38], [39]], and concrete penetration damage [[40], [41], [42]] when EFP formed in the air. It was found that EFP was short and thick in air while it was like a “crescent moon” in water [43], and the velocity of EFP decreased sharply when it penetrated water because of water resistance [44]. JPC is the third metal jet, with a small slug and a large diameter jet formed when the liner angle ranges from 70° to 120°. The velocity and penetration capability of JPC are between SCJ and EFP. It accounts for over 50% of the liner mass, and the jet’s head velocity is about 2 to 5 km·s⁻¹ [14], affording a strong penetration ability. The literature presents several experimental studies on the penetration of concrete [45,46] and target plates [[47], [48], [49], [50]] when the shaped charge of JPC exploded in the air. However, few studies have been carried out on underwater explosions, and the only research is also based on numerical simulation. Chen and Lu [51] claimed that JPC experienced velocity decrement and mass loss, and velocity storage capacity and erosion resistance could be improved by raising the thickness of the liner or distance from the charge to the target during water penetration.

Additionally to metal jets, shock waves and bubbles are also generated by shaped charges detonated in water. The shock wave produced is governed by strong discontinuity, high peak, and fast attenuation. While, the bubble has rapid expansion, collapse, and high-speed jet characteristics. Although shock waves and bubbles have been extensively studied in underwater explosion of conventional charges, they have rarely been investigated for shaped charges. Shock waves in underwater explosions were theoretically studied by Cole [52], and followed by many modifications [53,54]. Besides, experiments were also conducted to examine the characteristics of the shock wave [[55], [56], [57], [58]]. An alternative approach involves numerical simulation, allowing for different boundaries or initial conditions [[59], [60], [61]]. Despite the existing works offering a full understanding of the conventional charge, shock waves of shaped charges in underwater explosions have rarely been studied. The findings indicated that, for shorter standoff distances, the shock wave preceded the metal jet in reaching the target [62], and there were reflection and transmission of shock waves at the liner [63]. Compared to shock waves, the study of bubbles in underwater explosions is relatively complicated, as the bubble pulsation and bubble jet of conventional charge under different initial and boundary conditions have significant differences. At the beginning of the 20th century, Rayleigh [64] deduced a rigorous and feasible bubble dynamics equation. Later, several works [[65], [66], [67], [68]] considered the effects of fluid compressibility, surface tension, viscosity, migration, bubble interaction, and so on, and derived the more complete bubble dynamics equations. The dynamic behavior of bubbles in a free field [69,70], near a free surface [71,72], near a solid wall [73,74], and close to a complex boundary [[75], [76], [77]] has also been investigated. In addition, bubbles are also used in other fields [[78], [79], [80], [81], [82]]. Regarding the shaped charge detonation in underwater explosions, a metal jet alters the initial and boundary conditions of both shock waves and bubbles. Different loads are coupling, and the relevant research is rarely reported.

Based on the above analysis, understanding metal jets, ballistic waves, shock waves, bubbles, and their coupling characteristics produced when detonating shaped charges underwater remains insufficient. The relevant studies are limited by high strain rates, large density ratios, and multi-scale in space and time, which are challenging for conventional numerical and theoretical studies. Therefore, experimental studies can be more conducive to exploring the shaped charge’s novel physical phenomena and mechanisms during the underwater explosion. Spurred by the current research gap, this paper conducted experimental studies on the coupling dynamics of metal jets, waves, and bubbles of shaped charges in underwater explosions. In addition, the Eulerian finite element method was applied to investigate the formation and initial penetration of the metal jet. The remainder of this paper is organized as follows. 2 Experimental setup, 3 Eulerian finite element method introduce the experimental setup and the numerical methods, respectively. Section 4 analyzes the coupling dynamics of the metal jet, ballistic wave, shock wave, and bubble of a shaped charge in an underwater explosion. Section 5 compares the underwater explosion dynamics behaviors of shaped charges under different liner angles and charge weights. Finally, Section 6 encapsulates significant phenomena and conclusions.

2. Experimental setup

2.1. Shaped charge for underwater explosion

Fig. 1 illustrates the shaped charge employed for the underwater explosion experiments. It comprises a high-energy explosive, a metal liner, a detonator, a detonator holder, a booster charge, and an air tube, with the related assembly diagram depicted in Fig. 1(a). The trinitrotoluene (TNT) equivalent coefficient of the high-energy explosive is 1.21. The metal liner is made of copper and has a truncated cone, reducing the velocity of the slug and providing space for the formation and acceleration of the metal jet. A detonator with an equivalent TNT of 1 g is precisely positioned by the detonator holder to detonate the shaped charge. A booster charge with an equivalent TNT of 2.6 g is placed between the detonator and the high-energy explosive to fully detonate the high-energy explosive. The air tube acts as a cavity at the front end of the shaped charge and provides space for the full development of the metal jet.

The cross-section and dimensional parameters of the shaped charge are presented in Fig. 1(b). The experiments consider fixed values of d₁ = 6, d₂ = 15, d₃ = 32, d₄ = 34, d₅ = 3, d = 1, h₁ = 40, h₂ = 7.5, and h₄ = 32 mm. The parameters α, h₃, and the weight of the shaped charge m change depending on the case examined. Table 1 reports the parameters and the length-to-diameter ratios (λ, λ = h₃/d₃) for the experiment cases in this paper.

2.2. Experiment arrangement

The experiments were conducted in a square water tank presented in Fig. 2, with a length, width, and height of 8.0 m. The shaped charge was placed horizontally in the middle of the water tank at 1.7 m depth. Two PCB138A05 (PCB Piezotronics, USA) pressure sensors were placed at the measuring points P1 and P2 at 2.5 m in the axial and radial directions of the shaped charge, respectively, to record the pressure during underwater explosions. Their sensitivities are 142.8 and 153.7 mV·MPa⁻¹, respectively, and both have ranges of 35 MPa. The pressure sensors were connected to the ICP-482C05 (PCB Piezotronics) sensor signal adjustment and an oscilloscope with a sampling bandwidth of 200 MHz–1.5 GHz. A sampling frequency of 6.25 Giga samples per second (GS·s⁻¹) was used to capture the electrical signal. Upon the shaped charge detonation, the oscilloscope was triggered to record the pressure curve when the electrical signal exceeded the preset threshold.

Considering the significant difference in time scales between the metal jet, waves, and bubble produced by a shaped charge, two high-speed cameras with different frame rates were utilized to record the experiments. The first one recorded the generation and penetration of the metal jet and the propagation of the ballistic and shock waves. The first camera had a frame rate of 76 000 frames·s⁻¹ and a resolution of 896 × 304 pixels. The maximum inaccuracy in time measurement was the frame interval of 13.14 μs, about 0.88% of the penetration time of the metal jet in water (around 1.5 ms, Fig. 3). The duration of the shock wave was in the range of a few to several tens of milliseconds. Therefore, the error in time measurement was smaller. The frame rate of the second high-speed camera used to record bubble pulsation was 9000 frames·s⁻¹, with a resolution of 1024 × 752 pixels. The time interval was 0.111 ms, accounting for approximately 0.12% of the first pulsation period of the bubble (around 90 ms, Fig. 3).

Before the experiments, we hung a ruler vertically at the shaped charge detonation position in the water tank as a size marker to reference length measurements. The images captured by the high-speed cameras were two-dimensional planes, and the maximum spatial size error was the single pixel length. The pixel length of the high-speed camera for capturing the metal jet and waves was about 3 mm. When measuring the penetration length of the metal jet, the error was about 0.49% (around 0.61 m, Fig. 3), and the error was smaller when measuring the propagation distance of the waves. The pixel length of the high-speed camera used to capture the bubble pulsation was about 2.5 mm, so the maximum bubble radius length error (around 0.55 m, Fig. 3) was about 0.45%.

3. Eulerian finite element method

3.1. Description of the equations

Due to the bright light of the explosion and the metal jet wrapped in the cavitation bubble, the formation process of the metal jet and its initial motion characteristics in the water cannot be seen. Therefore, we adopted an in-house program for numerical simulation using the Eulerian finite element method (EFEM) as an auxiliary explanation for the experiment to investigate the formation and the initial penetration process of the metal jet in water. Precisely, the operator splitting algorithm [83] was used to solve the partial differential equation, that is the Euler equation. The interface of several fluids was handled using the volume of fluid (VOF) approach [84], and the volume fraction function was introduced to describe the volume fraction of each fluid at a particular element. Moreover, the monotone upwind schemes conservation laws (MUSCL) [85] was applied to achieve second-order accuracy in the transport of variables. The Eulerian method was employed to solve the material conservation equations of mass, momentum, and energy. The unified form of these equations is presented below:

(1)

\frac{\partial ψ}{\partial t} + \nabla \cdot Ψ + S = 0

where

ψ

is the conservation variable. t is the time in seconds (s).

\nabla

is the gradient operator.

Ψ

and

S

are the flux function and the source term, respectively, defined as follows:

(2)

ψ = [\begin{matrix} ρ \\ ρ v \\ ρ e \end{matrix}], Ψ = ψ v, S = [\begin{matrix} 0 \\ \nabla p - ρ g \\ p \nabla \cdot v \end{matrix}]

where

ρ

v

, and

p

are the fluid’s density, velocity, and pressure. e refers to the internal energy per unit mass and

g

is the gravitational acceleration.

The momentum conservation equation in Eq. (1) is divided into the following two equations using the operator splitting:

(3)

\frac{\partial ρ v}{\partial t} + \nabla p - ρ g = 0

(4)

\frac{\partial ρ v}{\partial t} + \nabla \cdot (ρ v \otimes v) = 0

where

\otimes

is the outer product symbol. Eqs. (3), (4) are the Lagrangian and Eulerian steps, containing the source and the convective terms, respectively. In Eq. (3), the mesh moves with the fluid, and Eq. (4) calculates the volume transfer of the material between neighboring elements after moving the deformed mesh in the Lagrangian step back to its initial mesh position. Eq. (3) can be expressed using the integration by parts and the Gauss–Green formula:

(5)

\iint_{Ω} ρ \frac{d v_{E}}{d t} Φ_{N} Φ_{E} d s + \iint_{Ω} v_{E} \frac{d ρ}{d t} Φ_{N} Φ_{E} d s = \iint_{Ω} (ρ Φ_{N} g + p \nabla Φ_{N}) d s - \int_{\partial Ω} p n Φ_{N} d l

where

Ω

and

\partial Ω

are the integration domain and the integration boundary.

Φ_{N}

and

Φ_{E}

are the shape functions of nodes “N” and “E.”

v_{E}

is the velocity of node “E.”

n

indicates the unit normal vector pointing from the inside to the outside of

Ω

s

and

l

are the differentiations of

Ω

and

\partial Ω

3.2. Equations of state and constitutive models

Due to the four unknown variables in the three equations, it is necessary to introduce the equation of state to make the equation completely closed. This paper uses the Jones–Wilkens–Lee (JWL) equation to describe the explosion products, formulated as follows:

(6)

p = A (1 - \frac{ω κ}{R_{1}}) \exp^{- (\frac{R_{1}}{κ})} + B (1 - \frac{ω κ}{R_{2}}) \exp^{- (\frac{R_{2}}{κ})} + ω ρ_{0} e κ

where

ρ_{0}

is the explosive’s initial density, with

κ = \frac{ρ}{ρ_{0}}

is the ratio of

ρ

ρ_{0}

. A, B,

R_{1}

R_{2}

ρ_{0}

, and

ω

are the parameters of the high-energy explosives by fitting to the experimental data, as listed in Table 2 [86].

For water, the state is described using the Tammann equation:

(7)

p = (μ - 1) ρ e - μ p_{r}

where

μ

= 7.15 and

p_{r} = 3.39 \times 1 0^{8} P a

are the specific heat ratio and the reference pressure for water [87], respectively.

The Mie–Gruneisen equation is used to describe the state of metal material, expressed as follows:

(8)

p = [1 - 0.5 Γ (κ - 1)] p_{H} + ρ Γ e

where Γ denotes the Gruneisen coefficient, and

p_{H}

denotes the pressure point on the Hugoniot curve, expressed as:

(9)

p_{H} = \{\begin{cases} a_{1} (κ - 1) + a_{2} {(κ - 1)}^{2} + a_{3} {(κ - 1)}^{3} κ > 1 \\ a_{3} (κ - 1) κ \leq 1 \end{cases}

where a₁, a₂, and a₃ are coefficients, expressed as:

(10)

\{\begin{array}{l} a_{1} = ρ_{0} b_{s}^{2} \\ a_{2} = a_{1} [1 + 2 (β - 1)] \\ a_{3} = a_{1} [2 (β - 1) + 3 {(β - 1)}^{2}] \end{array}

where

b_{s}

and

β

are the coefficients related to the metal material. The Mie–Gruneisen equation coefficients of copper are reported in Table 3 [88].

The constitutive model must be introduced for metal material to calculate the deviatoric stress tensor. This paper employs the Johnson–Cook model estimate the yield strength

(σ_{y})

, defined as:

(11)

σ_{y} = (σ_{0} + β_{0} ε_{p}^{γ}) [1 + C_{0} \ln \frac{\dot{ε}}{{\dot{ε}}_{0}}] [1 - {(\frac{e - e_{0}}{M H_{s} (T_{m e l t} - T_{r o o m})})}^{m_{0}}]

where

σ_{0}

is initial yield strength.

β_{0}

and

γ

denote stress hardening coefficients.

C_{0}

is the strengthening coefficient of strain rate.

ε_{p}

is plastic strain,

\dot{ε}

is the strain rate, and

{\dot{ε}}_{0} = 1 s^{- 1}

is the reference strain rate.

e_{0}

and

M

are the initial specific internal energy and mass of material.

m_{0}

is thermal softening coefficient.

T_{m e l t}

and

T_{r o o m}

are melting temperature of material and room temperature, and

H_{s}

denotes specific heat. The detailed parameters of copper are listed in Table 4 [62].

3.3. Boundary conditions

This paper imposes the spherical non-reflecting boundary condition to minimize the influence of reflected waves at the boundary on numerical accuracy. The core idea is to apply a suitable surface pressure at the boundary. Thus, the surface pressure is introduced into the integral on the right-hand side of the equal sign of Eq. (5) to participate in the calculation. The total pressure p [89] can be expressed as:

(12)

p = p_{\infty} - 0.5 ρ {|v|}^{2} + p_{D}

where

p_{\infty} = p_{a} + ρ g h

is the ambient pressure at the depth of the element (

h

) in the infinite distance.

p_{a}

is the standard atmospheric pressure.

p_{D}

denotes dynamic pressure, and from the second order early-time approximation (ETA₂) equation in Ref. [90], it can be calculated as:

(13)

p_{D} + L_{p} c \int p_{D} d t = ρ c v \cdot n_{c}

where

L_{p}

denotes the local curvature of the spherical pressure wave. c and v are the sound speed and material speed, and

n_{c}

denotes the unit vector pointing to the direction of the acoustic wave propagation.

4. Coupling dynamics of a shaped charge

Next, we analyze the coupling dynamics of a shaped charge in the underwater explosion of Case 1, presented in Table 1. During underwater explosions, the shaped charges will form metal jets, ballistic waves, shock waves, and bubbles.

In the direction of the metal jet motion, the metal jet penetrates the water accompanied by the ballistic wave, followed by the propagation of the shock wave, and then the bubble pulsates periodically, as illustrated in Fig. 3. Precisely, the metal jet is generated and penetrates the water within microseconds. Due to water resistance, the metal jet changes from slender to coarse, as shown in Fig. 3(a). Meanwhile, the ballistic wave propagates with the movement of the metal jet in the water. Then, the shock wave begins to propagate and is located behind the metal jet head and the ballistic wave. As the velocity of the metal jet decreases, the shock wave exceeds the metal jet head and superimposes with the ballistic wave, propagating outward together, as shown in Fig. 3(b). Fig. 3(c) depicts the bubble pulsation. Because the flow of the metal jet carries away some of the explosion products and the metal jet hinders the development of the bubble, the bubble wall near the metal jet is depressed inward. Subsequently, the metal jet undergoes severe fragmentation (t = 15.3330 ms), reducing the hindrance to bubble development, with the bubble shape becoming approximately spherical. Subsequently, the bubble forms an oblique upward jet (t = 90.6670 ms).

The metal jet, ballistic wave, shock wave, and bubble generated by the shaped charge in underwater explosions manifest different temporal scales. The metal jet requires microseconds to penetrate, while the ballistic and shock waves propagate within millisecond. Similarly, bubble pulsation occurs within milliseconds. In Case 1, the penetration length of the metal jet in water is approximately 0.61 m, which exceeds the maximum radius of the bubble (0.55 m). The propagation distance of the ballistic and shock waves is further. The following subsections analyze the characteristics of the metal jet, waves, and the bubble of the shaped charge during an underwater explosion.

4.1. Penetration of the metal jet

First, the shock characteristics of the metal jet in Case 1 are elucidated in Fig. 4. When the shaped charge is detonated, the metal jet forms and penetrates water with a conic shape with a sharp head (t = 0.0660 ms). The metal jet has the characteristics of high speed, strain rate, and surface temperature [91]. Due to its high-speed motion, the pressure of the surrounding water decreases below the saturated vapor pressure. In addition, when the metal jet penetrates water at a high temperature, it may vaporize the water quickly, and therefore, complex cavitation is formed around the metal jet under high-temperature and low-pressure. At the trailing edge of the cavitation zone, the explosion product carried away by the metal jet is mixed with water to form a disordered fluid (t = 0.0920 ms). Meanwhile, the slamming force of the water causes the metal jet to break into small metal chips (t = 0.1310 ms). Due to the velocity difference between the head and tail of the metal jet, the jet elongates and finally fractures at the necking position, accumulating metal chips near the head, which becomes thicker (t = 0.2630 to 0.3550 ms). Notably, some metal chips rush out of the head and disperse in all directions from t = 0.5910 to 0.8940 ms.

A numerical simulation investigates the shock characteristics involved in the formation and initial penetration of the metal jet. Fig. 5 compares the numerical simulation and experimental results for Case 1. For the simulation, the mesh size is 0.25d. The area inside the black dotted line in Fig. 5 represents the metal jet and the cavitation surrounding it. The morphology of the metal jet wrapped by the cavitation and the penetration length of the metal jet are in good agreement with the experimental results. At around 0.2 ms, the metal jet is stretched to extremely fine and tends to fracture, and subsequent numerical simulations will no longer be meaningful. Hence, the termination time for the numerical simulation is 0.2 ms.

Fig. 6 numerically simulates the detonation wave progression inside the shaped charge and the formation of the metal jet. When the shaped charge points to the detonation, the detonation wave inside the charge propagates as a spherical wave in Fig. 6(a). The initial explosion pressure reaches 20.1 GPa (t = 0.0005 ms). Concurrently, the shock wave propagates through the water near the detonation point. Subsequently, the explosion pressure continues to rise. At t = 0.0029 ms, the detonation wave initiates to impact the liner, and the liner is squeezed to deform. Since the impedance of the liner is greater than that of the explosive, the detonation wave is reflected by the liner (t = 0.0040 ms). Following that, the detonation wave propagates along the generatrix of the liner. The plastic flow of the liner is induced, resulting in the formation of the metal jet head (t = 0.0050 ms). At t = 0.0060 ms, the explosion pressure peak decreases to 13.4 GPa, primarily concentrated in the truncated section at the top of the liner to form the metal jet head with high velocity. Fig. 6(b) illustrates the formation and velocity evolution of the metal jet, where the coordinate origin of the grid line in the figure is the top of the liner. The extrusion and deformation of the truncated section at the top of the liner leads to the formation of the metal jet head (t = 0.0040 ms). At t = 0.0166 ms, the metal jet is basically formed and reaches the maximum head velocity of 4.02 km·s⁻¹, 2.68 times the sound speed (1.50 km·s⁻¹) in water. The velocity of the jet head is 16.75 times that of the slug. Subsequently, the metal jet is continuously elongated under the velocity gradient. The metal jet penetrates the water at t = 0.0175 ms, and the head velocity of the metal jet is 3.40 km·s⁻¹.

In the experiment, the metal jet is wrapped by cavitation, limiting the understanding of its penetration characteristics in water. Therefore, Fig. 7 presents the initial penetration characteristics of the metal jet in water through numerical simulation. When the metal jet penetrates the water, it will bring out the air in the air tube while generating cavitation around it (t = 0.0200 ms). At t = 0.0360 ms, the metal jet with a high strain rate appears necking, that is the phenomenon where the local cross-sectional area of a material decreases due to tensile stress. Unlike static stretching, metal jets with high velocities and inertia undergo necking everywhere due to the metal jet’s velocity gradient and the liner material’s microscopic characteristics [92,93]. Subsequently, the metal jet stretches, and its head flattens under the water resistance (t = 0.0500 ms).

Considering the bottom of the metal liner as the origin (t = 0 ms, Fig. 4) and the rightward direction representing the positive orientation of the x-axis, Fig. 8 illustrates the penetration length l_j and the head velocity v_j of the metal jet in water for the current experiment and the numerical simulation. l_j, v_j, and time t are dimensionless and calculated based on the charge height h₃, the speed of sound in water c, and h₃/v_jm, where v_jm is the maximum head velocity of the metal jet obtained by numerical simulation. The numerical simulations consider a mesh size (dx) of 0.25d, 0.5d, 0.75d, and 1.0d for convergency testing. The results converge as the mesh size decrease, and the calculated l_j and v_j of the metal jet agree with the experimental results for a mesh size of 0.25d, indicating that the numerical simulation results are convergent. In the subsequent numerical simulations, the mesh size is taken as 0.25d. The penetration length of the metal jet is rapid growth in the initial stage, followed by a significant decrease in its penetration capability owing to severe jet fragmentation, as shown in Fig. 8(a). At t/(h₃/v_jm) = 120, l_j reaches 16.1 times the h₃. In Fig. 8(b), v_j first captured by the high-speed camera is about 1.513 compared to c at t/(h₃/v_jm) = 6.289. Overall, the head velocity of the metal jet initially decreases exponentially and then decreases steadily.

The relationship between the penetration depth of the metal jet into the target plate and the jet length can be obtained [94] according to the simple Bernoulli relation:

(14)

ρ_{j} {(v_{j} - v_{p})}^{2} = ρ_{p} v_{p}^{2}

where

v_{p}

is the metal jet penetration velocity.

ρ_{j}

and

ρ_{p}

are the density of the metal jet and the target plate. When the metal jet has a high speed and is continuous, the relationship between the penetration depth

l_{j} (t)

of the target plate and the jet length

L_{j} (t)

[94] is:

(15)

l_{j} (t) = \sqrt{\frac{ρ_{j}}{ρ_{p}}} L_{j} (t)

where

l_{j} (t)

and

L_{j} (t)

change over time. However, the instability of the metal jet necessitates considering a factor

η

. Then, the final relationship between the penetration depth and the jet length [95] (including the jet fracture block and fracture gap) is:

(16)

l_{j} (t) = η \sqrt{\frac{ρ_{j}}{ρ_{p}}} L_{j} (t)

When water penetration by the metal jet can be regarded as target plate penetration with

ρ_{p} = 998

kg·m⁻³ according to the jet length and the penetration length in the water depicted in Fig. 7, we consider

η = 0.43

. Therefore, at t/(h₃/v_jm) = 120, the jet length is

L_{j} = 12.4 h_{3}

4.2. Propagation of the waves

In the underwater explosion of a shaped charge, the expansion of high-pressure explosion products will radiate a shock wave in water, and the shaped charge will generate a ballistic wave. Fig. 9 illustrates the generation and propagation of the ballistic and shock waves of the shaped charge in Case 1 during the underwater explosion.

Fig. 9(a) highlights that the ballistic wave is generated and propagates outwards. The penetration velocity of the metal jet exceeds the speed of sound in water, generating a shock wave known as a ballistic wave. Furthermore, the shock wave generated by the underwater explosion of the shaped charge propagates outward in an approximately spherical shape in Fig. 9(b). The existence of a metal jet and a ballistic wave will inevitably change the flow field’s pressure during the shock wave propagation. However, this has a minor impact on the shape of the shock wave. After that, the shock wave will be superimposed with the ballistic wave and propagate outward in Fig. 9(c). The shape evolution of the ballistic wave at different moments is depicted in Fig. 10. At t = 0.1050 ms, its shape is like a “funnel.” As the ballistic wave propagates, its curvature gradually decreases, and its shape changes from a “funnel” to a “bowl” (t = 0.3150 ms), indicating that the ballistic wave transitions from an oblique shock wave to a curved shock wave.

Fig. 11 compares the positions of the metal jet head, ballistic wave, and shock wave simultaneously. Before t/(R₀/c) = 7.8750, the ballistic wave propagates close to the metal jet head, where R₀ is the radius of the spherical charge, which equals the shape charge’s weight. During this period, the ballistic wave propagates with a speed determined by the metal jet’s head velocity. Due to water resistance, the head velocity of the metal jet gradually decreases, resulting in the ballistic wave exceeding the metal jet head at t/(R₀/c) = 7.8750. Subsequently, the ballistic wave propagates at a relatively constant speed compared with the metal jet. The similar slope of the position curves of the ballistic and shock waves indicates that their propagation speeds are close. From t/(R₀/c) = 5.9150 − 25.6280, the average speed of shock wave propagation is about 1.68 km·s⁻¹. At t/(R₀/c) = 22.6500, the shock wave exceeds the metal jet head because the head velocity of the metal jet continuously decreases.

4.3. Pulsation of the bubble

Although shaped and conventional spherical charges generate bubbles in underwater explosions, these differ. Hence, to elucidate their different dynamics, we first describe the bubble pulsation process of the conventional spherical charge depicted in Fig. 12. The spherical charge with an equivalent TNT of 52 g is located at a depth of 1.7 m, and the pressure measuring point is 2.5 m from the charge. After the charge is detonated, the bubbles expand rapidly in a spherical shape. The bubble reaches the maximum volume with a maximum bubble radius of R_m = 0.58 m at t = 48.4440 ms. Subsequently, the bubble begins to collapse and forms a bubble jet vertically upward from the lower surface of the bubble (t = 94.4440 ms). A vertical upward protrusion is formed during the bubble rebound stage because the bubble jet is amplified (t = 101.4440 − 108.6670 ms). In this experiment, the protrusion is first observed at t = 97.7710 ms with a velocity of 65.6 m·s⁻¹.

The bubble pulsation process of the shaped charge in Case 1 is presented in Fig. 13. The overall pulsation characteristics of the shaped charge bubble are similar to the spherical charge bubble (Fig. 12). However, the initial bubble and the behavior of the bubble jet in the collapse stage differ. When the metal jet penetrates the bubble wall and flows outward, part of the explosion products in the bubble will be carried away. Simultaneously, the metal jet hinders the bubble expansion. These reasons lead to the inward depression of the bubble wall near the metal jet (t = 1.6670 ms). The bubble jet formed during the collapse is obliquely upward and points to the detonation end (t = 90.6670 ms), rushing out from the top of the bubble and forming a protrusion (t = 101.4440 − 108.6670 ms). The oblique bubble jet forms due to the uneven energy distribution within the bubble. Similar to the characteristics of energy released during an underwater explosion of a cylindrical charge [96], the explosion product of a shaped charge interacts more time with the fluid near the detonation end. Consequently, the flow field near the detonation end has more energy. Moreover, the formation and flow of metal jets lead to a more uneven distribution of bubble energy in the axial direction. The high energy at the detonation end will increase the rotational angular velocity of the bubble’s lower left surface, forming an inclined jet towards the detonation end. In this experiment, the protrusion is photographed for the first time at t = 94.0730 ms, and its head velocity is 52.34 m·s⁻¹, which is 4/5 of that of the spherical charge.

Regarding bubble pulsation, Zhang et al. [67] provided a formula as follows:

(17)

(\frac{c - \dot{R}}{R} + \frac{d}{d t}) [\frac{R^{2}}{c} (\frac{1}{2} {\dot{R}}^{2} + \frac{1}{4} v^{2} + H)] = 2 R {\dot{R}}^{2} + R^{2} \ddot{R}

where

R

is the bubble radius.

\frac{\dot{R}}{2}

\frac{v^{2}}{4}

, and

H

represent the equivalent forces resulting from bubble oscillation, migration, and the surrounding flow field, respectively.

\dot{R}

and

\ddot{R}

are the first and second derivatives of

R

with respect to time.

Fig. 14 compares the bubble radius and migration distance (Z) of the shaped charge (Case 1) and the conventional spherical charge with the theoretical result by Eq. (17). Fig. 14 highlights that the bubble radius curve of the conventional spherical charge agrees with the theoretical result. However, the bubble radius of the shaped charge is slightly smaller than that of the conventional charge and the theoretical result, as presented in Fig. 14(a). The maximum bubble radius is 0.55 m, 5.2% smaller than that of the conventional spherical charge. In the underwater explosion of the shaped charge, the energy of the explosion products is partially converted into the kinetic energy of the metal jet, and some of the explosion products are also carried away by the flowing metal jet. The above phenomena decrease the total energy of the bubble and reduce the bubble radius. The bubble migration curve presented in Fig. 14(b) indicates that the bubble migrates upward rapidly when it collapses to the minimum volume. Nevertheless, there is no significant difference in bubble migration between the shaped and the conventional spherical charges, and both agree with the theoretical result.

Fig. 15 illustrates the free-field pressure p of the shaped charge and the conventional spherical charge recorded by the pressure sensor at measuring point P2 during the underwater explosion. The measuring point P2 is located at 2.5 m in the radial direction of the charge (Fig. 2). Moreover, we choose the ambient pressure

p_{\infty} = p_{a} + ρ g h

at the water depth of the charge

h

and the first pulsation period T of the bubble as the scale for the dimensionless pressure p and time t. Fig. 15(a) presents the shock wave pressure of the two charges, which is compared with the theoretical result by the Zhang et al. [67]. The results infer that the shock wave pressure attenuation of the two charges aligns well with the theoretical result, but the pressure peak differs. Given that the length-to-diameter ratio λ of the shaped charge is 1.18, and the total energy in the axial direction exceeds the total energy in the radial direction during the underwater explosion, the peak of the radial shock wave of the shaped charge is slightly lower than that of the conventional charge and the theoretical result. Figs. 15(b) and (c) present the pulsating pressure curves of the bubbles in different types of charges during the first and second rebounds. Compared with the conventional spherical charge, the pulsation period and pressure peak of the shaped charge are smaller due to the bubble’s overall energy reduction. Specifically, the first and second pulsation periods of the shaped charge bubble are 3.9% and 3.0% smaller than those of the conventional spherical charge, respectively. Moreover, the first and second pulsation pressure peaks of the shaped charge bubble are 93.4% and 79.4% of the conventional spherical charge bubble.

5. Dynamics behaviors of an underwater explosion of shaped charges under different parameters

Next, we investigate the effects of different parameters of the shaped charge, including the liner angle and the charge weight, on the characteristics of metal jets, waves, and bubbles.

5.1. Effects of the shaped charge liner angle

This section focuses on the differences in the characteristics of metal jets, waves, and bubbles of shaped charges with liner angles α of 45° (Case 2), 75° (Case 1), and 150° (Case 3) in the underwater explosions. Table 1 reports the shape parameters of the shaped charges discussed.

Fig. 16 compares the penetration of metal jets in water and the propagation of ballistic and shock waves. Previous studies divide metal jets into EFP, JPC, and SCJ owing to their unique morphological characteristics. The experiments conducted under the liner angle α of 45°, 75°, and 150° forms SCJ, JPC, and EFP, respectively. The experiments indicated that in the initial penetration stage of the metal jet in water, SCJ has a long and slender shape, while EFP is shorter and thicker, and the thickness of JPC is between the two (t = 0.0790 − 0.1310 ms). Indeed, at t = 0.5780 ms, EFP is separated from the bubble because the penetration velocity of EFP in water is greater than the bubble’s expansion speed. Moreover, SCJ has a large metal chip moving obliquely upward, which may be part of the slug (t = 1.2490 ms). Notably, there are also significant differences in the propagation of waves in different cases. Unlike the ballistic waves generated by SCJ and JPC during water penetration, the ballistic wave induced by EFP is wider in the radial direction (t = 0.0790 ms). The distance between the shock wave and the ballistic wave is close, and it is almost superimposed with the ballistic wave from t = 0.1310 − 0.1840 ms.

Next, we compare the formation process of metal jets of shaped charges under different α against numerical simulation. Fig. 17 illustrates the result, highlighting that α affects the area and angle of the detonation wave exerting on the liner, resulting in different collapse angles of the liner during the extrusion process. The difference in the collapse angle affects the deformation mode of the liner, forming various types of metal jets and determining their initial head velocity. When the liner has a small angle (α = 45°/75°), the deformation mode of the liner is “closed,” for example, the bottom of the liner forms the jet, whereas the top forms the slug. However, for a large liner angle (α = 150°), the deformation mode of the liner is “flipped,” forming a projectile. This explains why when the velocity of EFP is greater than the bubble’s expansion speed, EFP separates from the bubble (Fig. 16). As the liner angle decreases, the energy density of the detonation wave applied to the unit area of the liner increases, increasing the liner collapse velocity, thereby improving the head velocity of the metal jet. The maximum head velocity of the three metal jets is 5.18, 4.02, and 2.57 km·s⁻¹ at 0.0150, 0.0166, and 0.0145 ms, respectively. Furthermore, the corresponding head-to-tail velocity ratios of the metal jets are 518.0, 16.8, and 1.8. Among them, the velocity gradient of EFP is the smallest, and the stretching rate is relatively low. The right column in Fig. 16 indicates the morphology and times when the metal jet begins to penetrate the water.

Finally, the differences in the initial penetration characteristics of different metal jets in water are displayed in Fig. 18. Due to the velocity difference of the three metal jets, SCJ entered the water earliest. Similar to JPC, SCJ also occurs necking (t = 0.0360 ms). Nevertheless, their difference is that the necking of SCJ occurs at the middle and rear section of the jet, while the one of JPC is near the front. Compared with SCJ, the slug velocity of JPC is higher, resulting in JPC almost moving out of the bubble (t = 0.0500 ms). Unlike SCJ and JPC, EFP has no evident necking while penetrating the water. However, its morphology transforms obviously, with its shape changing from flat to a projectile (t = 0.0220 − 0.0390 ms). Moreover, EFP carries some explosive products away the bubble and forms disordered fluid behind the metal jet (t = 0.0500 ms).

Figs. 19(a) and (b) compare the penetration length l_j and head velocity v_j of the three metal jets. Fig. 19(a) reveals a negative correlation between α and l_j, (i.e., as α increases, l_j decrease). At t/(d₃/c) = 120, the penetration length of SCJ (α = 45°) is approximately 1.36 and 2.14 times that of JPC (α = 75°) and EFP (α = 150°). Fig. 19(b) infers that the dimensionless initial head velocities

\frac{v_{j}}{c}

of the three metal jets first captured by the high-speed camera are 2.01, 1.51, and 1.02, respectively. The head velocity of SCJ is about 1.32 and 1.97 times that of JPC and EFP. Moreover, the head velocity decay rate of the metal jet increases as α increases. The metal jet head formed by the shaped charge with a large α is thicker and is subjected to greater water resistance. To clarify the relationship between v_j and α, we fit the curve of the maximum head velocity (simulated) of the metal jet with the liner angle, as depicted in Fig. 19(c). The plot reveals that the ratio of the maximum head velocity of the metal jet to the sound speed in water

\frac{v_{jm}}{c}

and α follow a 1/(α/180°)^0.55 relation for a charge mass of 40 g and a liner angle between 45° and 150°.

Fig. 20 demonstrates the difference in bubble pulsation characteristics of shaped charges under different liner angles. When the bubble begins to expand, the EFP flows out of the bubble, so there is less obstruction to the development of the bubble wall. Therefore, for α = 150°, the inward depression of the bubble wall is smaller than that in other cases (t = 1.8890 − 2.7780 ms). At t = 44.5560 ms, the bubbles in the three cases all reach the maximum radius R_m of 0.55 m. This indicates that the change in the liner angle does not affect the total energy of the bubble. Subsequently, the bubbles collapse, generating obliquely upward bubble jets (t = 90.6670 ms). In general, the difference in the liner angle mainly affects the degree of obstruction of the metal jet and the development of the initial bubble wall.

Fig. 21 displays the pressure curves of shock waves and bubble pulsations at measuring point P2 (Fig. 2) under different liner angles α. Although the pressure peaks of the shock waves are slightly different, the pressure attenuation of the shock waves is similar, as shown in Fig. 21(a). For α = 150°, the length-to-diameter ratio λ of the shaped charge is the smallest, and the distribution of explosion energy in all directions is more uniform. Therefore, the pressure peak of the radial shock wave is slightly higher. In Fig. 21(b), the pressure peaks of bubble pulsations are almost the same, but the pulsation times are inconsistent. Cui et al. [76] claimed that the bubble energy

E_{b}

is the work done when the bubble reaches its maximum volume, formulated as follows:

(18)

E_{b} = \frac{4}{3} π R_{m}^{3} (p_{\infty} - p_{v})

where

p_{v}

is the saturated vapor pressure. In our experiments, the bubble’s energy is almost the same for the three cases examined. As a result, there are no differences in the first pressure peaks of bubble pulsations. On the other hand, the difference in bubble pulsation times is due to the different axial contraction velocities of the bubbles, which can be attributed to the different length-to-diameter ratio λ of the shaped charge.

5.2. Effects of the shaped charge weight

This section explores the characteristics of metal jets, waves, and bubbles in underwater explosions of shaped charges with different weights, that is 40, 60, and 80 g (Cases 1, 4, and 5, respectively), as listed in Table 1. The liner angle α of the three shaped charges is 75°.

The weight of the shaped charge will change the total energy of the explosion, thus affecting the impact characteristics of metal jets, waves, and bubbles. The dynamic behaviors of metal jets and waves generated by shaped charges under different charge weights are compared in Fig. 22, highlighting that the liner will receive more energy to form the metal jet as the total energy of the explosion increases. Consequently, the metal jet’s head velocity and penetration length will increase. However, there is a minor effect on the morphological characteristics of the metal jet, such as the thickness of the metal jet and the position of the disordered fluid. For ballistic waves, there is no significant difference in the shape of the wave due to the low difference in the thickness of the metal jet head. However, as the metal jet head’s velocity increases (t = 0.1050 ms), the initial propagation velocity of the ballistic wave increases. On the contrary, for the shock wave, increasing the charge weight inevitably increases the intensity of the shock wave. Still, the propagation velocity of the near field is similar (t = 0.1580 ms).

Figs. 23(a) and (b) compare the penetration length l_j and head velocity v_j of metal jets of shaped charges with different weights. Fig. 23(a) suggests that the dimensionless penetration length l_j/h₃ of the metal jet decreases as the shaped charge weight increases. This indicates that although m increases in multiples, it does not significantly improve the penetration capability of the metal jet in water. Fig. 23(b) illustrates the head velocity curves of the metal jets in the water penetration process. The dimensionless initial head velocities v_j/c captured by the high-speed camera for the first time are 1.51, 1.64, and 1.75, respectively. As the shaped charge weight increases, more explosive energy is exerted on the liner, which is subjected to greater thrust, increasing the metal jet’s head velocity. However, the head velocity v_j does not change multiples with the increased shaped charge weight. In addition, as m increases, the attenuation rate of the head velocity v_j also increases.

Furthermore, Fig. 23(c) depicts the variation of the maximum head velocity v_jm of the metal jet with the length-to-diameter ratio of the shaped charge. When the angle of the liner is constant, λ can characterize the shaped charge weight m. The maximum head velocity v_jm/c of the metal jet is related to λ^0.16. The relationship between the length-to-diameter ratio and the maximum head velocity of the metal jet v_jm applies to the shaped charge with a liner angle of 75° and the length-to-diameter ratio between 0.95 and 2.17.

Next, we compare the pulsation characteristics of bubbles of shaped charges with different weights in Fig. 24. The results infer that as the charge weight increases, the bubble requires more time to reach its maximum volume and has a greater radius. The maximum radius R_m of the bubbles are 0.55, 0.62, and 0.67 m at 44.5560, 49.6670, and 53.667 ms, respectively. Subsequently, the bubbles undergo collapse and rebound. The last column of Fig. 24 presents that the three bubbles have the same equivalent radius. And the protrusions amplified by the bubble jets during the rebound stage are shown. Fig. 25 compares the head velocity of the protrusions (v_bj) for different shaped charge weights, where the head velocities of the protrusions demonstrate a consistent attenuation rate. However, the protrusion emerges earlier and has a higher head velocity for the shaped charge with a small weight. This is because during the bubble rebound, the larger the length-to-diameter ratio, the greater the energy difference between the two ends of the shaped charge, decreasing the initial angular velocity when the lower of the bubble wall rotates inward to form the bubble jet.

Fig. 26 displays the free-field pressure curves of shaped charges with different weights at the measuring point P1, located at 2.5 m axial of the shaped charge (Fig. 2). The coordinate origin of the time axis is the detonation time of the shaped charge. The experimental results highlight that the pressure peaks of the shock wave and bubble pulsation increase with the increase of shaped charge weight.

The variation in the weight of the shaped charge will change the total explosion energy. Therefore, it is necessary to analyze the energy proportion of shock waves, bubbles, and metal jets to the total energy. Neglecting the thermal energy in the explosion, the total explosion energy of the shaped charge comprises the kinetic energy of the metal jet, the shock wave energy, and the bubble energy. The kinetic energy of the metal jet (

E_{j}

) is defined as follows:

(19)

E_{j} = \frac{1}{2} m_{j} v_{j m}^{2}

where

m_{j}

denotes the weight of the metal jet. The shock wave energy (

E_{w}

) can be obtained from the shock wave curve recorded by the pressure sensor [52], as the following formula:

(20)

E_{w} = \frac{4 π D^{2}}{ρ c} \int p {(t)}^{2} d t

where D = 2.5 m is the distance between the pressure measuring point and the shaped charge. The energy of the bubble

E_{b}

is obtained from Eq. (18). Therefore, the total explosion energy can be given as:

(21)

E_{t} = E_{j} + E_{w} + E_{b}

Fig. 27 illustrates the proportion of the

E_{j}

E_{w}

, and

E_{b}

in the total explosion energy

E_{t}

of the shaped charges with different weights m. As the total energy of the explosion increases, the proportion of the kinetic energy of the metal jet in the total energy decreases, while the proportion of the shock wave energy increases. Conversely, the proportion of bubble energy in the total energy of the explosion remains relatively stable, accounting for about 55% of the total energy.

6. Conclusions

Unlike the underwater explosion of conventional charges (without metal liner), shaped charges generate not only strong shock waves and pulsating bubbles but also high-speed metal jets and ballistic waves. This paper studies experimentally the coupling dynamics between metal jets, ballistic waves, shock waves, and bubbles produced by shaped charges in underwater explosions. In addition, the Eulerian finite element method is employed to investigate the formation and initial penetration of metal jets in water. Furthermore, the effects of the liner angle and the weight of the shaped charge on metal jet penetration, shock wave propagation, and bubble pulsation are analyzed.

The shaped charge generates a metal jet accompanied by a ballistic wave. Subsequently, the shock wave propagates, and the bubble pulsates periodically. The metal jet elongates due to the velocity gradient, causing necking and fracture at various positions. As the metal jet penetrates through water, cavitation forms around it. The penetration length of the metal jet is up to 16.1 times the height of the shaped charge, and its maximum head velocity reaches 2.68 times the speed of sound in water. In addition, the ballistic wave caused by the metal jet is observed, and its generation is attributed to the penetration velocity of the metal jet exceeding the speed of sound in water. The shape and velocity of the metal jet’s head determine the curvature and propagation velocity of the initial ballistic wave. Unlike a conventional spherical charge, the shock wave from a shaped charge can superimpose with the ballistic wave. Furthermore, an inward depression of the bubble wall is observed because the metal jet carries away some of the explosive products in the bubble, hindering its development. As a result, the maximum radius and pulsation period of the bubble are 5.2% and 3.9% smaller, respectively, than those of the conventional spherical charge of the same weight. Moreover, the uneven distribution of axial energy in the shaped charge leads to the formation of an oblique bubble jet.

Different liner angles affect the area and angle of the detonation wave acting on the liner, resulting in two deformation modes: “closed” and “flipped.” Based on the metal jet’s morphological characteristics, shaped charges with the liner angle of 45°, 75°, and 150° produce the SCJ, JPC, and EFP, respectively. The maximum head velocity of the metal jet is related to the liner angle α through the formula 1/(α/180°)^0.55. Unlike the initial ballistic waves generated by SCJ and JPC, the initial ballistic wave produced by EFP has a smaller curvature and a lower propagation velocity, making it more likely to be superimposed with the shock wave. In addition, varying liner angles result in different length-to-diameter ratios for the shaped charges, leading to differences in the shock wave’s pressure peak and the bubble pulsation period.

The increased weight of the shaped charge increases the total explosion energy. Besides, the maximum head velocity of the metal jet is correlated with the length-to-diameter ratio λ^0.16 of the shaped charge. As the total explosion energy increases, the proportion of the kinetic energy of the metal jet to the total energy decreases while the proportion of the shock wave energy increases. Notably, the bubble energy accounts for approximately 55% of the total energy the charge releases. In addition, the smaller the length-to-diameter ratio of the shaped charge, the higher the initial velocity of the bubble jet. This is because the energy difference between the two ends of a shaped charge with a small length-to-diameter ratio is smaller, allowing the lower bubble wall to obtain a higher initial angular velocity of inward rotation.

CRediT authorship contribution statement

Yu Tian: Writing – original draft, Investigation, Formal analysis. A-Man Zhang: Writing – review & editing, Supervision, Funding acquisition. Liu-Yi Xu: Validation, Investigation. Fu-Ren Ming: Writing – review & editing, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was funded by the National Natural Science Foundation of China (52071109).

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