Correction Method for Initial Conditions of Underwater Explosion

Abstract

In numerical simulations of underwater explosions, inaccuracies in the parameters of the Jones–Wilkins–Lee (JWL) equation of state often result in significant deviations between predicted shock wave pressure peaks or bubble pulsation periods and experimental or empirical results. To achieve the precise forecasting of underwater explosion loads, a corrected method for adjusting the initial conditions of explosives is proposed. This method regulates explosion loads by correcting the initial density and initial internal energy per unit mass of the explosive, offering a straightforward implementation and easy extension to complex scenarios. In addition, the accuracy and feasibility of the proposed method were validated through comparisons with experimental data and empirical formulas from international studies. The numerical framework employs the Runge–Kutta Discontinuous Galerkin (RKDG) method to solve the one-dimensional Euler equations. The spatial discretization of the Euler domain is achieved using the discontinuous Galerkin (DG) method, while temporal discretization utilizes a third-order Runge–Kutta (RK) method. The results demonstrate that the proposed correction method effectively compensates for load discrepancies caused by inaccuracies in the JWL equation of state parameters. After correction, the maximum error in the shock wave pressure peak is reduced to less than 4.5%, and the maximum error in the bubble pulsation period remains below 1.9%.

1. Introduction

Warships are vulnerable to attacks from underwater weapons, such as torpedoes, mines, and depth charges during maritime operations. These weapons, widely deployed in warfare, pose significant threats to naval vessel survivability by inducing destructive underwater explosions near or directly on ships [1]. Structural damage to ships primarily arises from diverse underwater explosion loadings, including shock waves, bubble pulsations, jet impacts, and cavitation effects [2,3,4]. Full-scale ship trials are the most direct and effective means to evaluate the anti-shock performance and protective measures. However, their high cost, prolonged timelines, and operational constraints often lead to uncertain outcomes [5,6]. Ship structures comprise plates, beams, rods, and shells, and some studies have focused on testing hierarchical structural components [7]. For instance, Chen et al. [8] conducted three underwater explosion experiments with varying TNT charge weights to investigate the dynamic response of fixed square steel plates. Their results showed that the plate damage decreased with increases in the strength, thickness, and detonation distance. However, excessive shock wave pressures destroyed pressure sensors, and bubble pulsation data were unobtainable for larger charges.

Fortunately, numerical simulation methods offer efficient, cost-effective alternatives for studying underwater explosion phenomena and remain a focal point in global research. Barras et al. [9] employed the Arbitrary Lagrangian–Eulerian (ALE) method to simulate underwater explosions, emphasizing bubble pulsation dynamics. Zhang et al. [10] developed an axisymmetric smoothed particle hydrodynamics (SPH) model coupled with the boundary element method (BEM) to capture the full physics of cylindrical charge explosions. Zong et al. [11] applied a “scattered wave” acoustic-structure coupling approach to analyze whole-ship structural damage. Gan et al. [12] investigated the dynamic response of hull girders under bubble loading, while Li et al. [13] introduced an improved eigenvalue method for simulating 1D underwater explosions and secondary shock waves. Gao et al. [14] established a fully coupled thermomechanical model to analyze a 13-layer composite under underwater explosion loading, incorporating thermal deformation effects. Other studies combined small-scale experiments with numerical simulations to explore structural damage, dynamic responses, and loading characteristics [4,15,16,17,18]. For example, Lin et al. [19] validated shock wave superposition effects and clamped circular plate damage using LS-DYNA simulations and equivalent underwater explosion tests. Mao et al. [20] studied the dynamic response of pressurized cylindrical shells subjected to near-field underwater explosions, focusing on bubble pulsation and energy transfer. Huang [21] developed a finite element model to analyze the cavitation during bubble pulsation, though discrepancies in bubble pulsation periods persisted between the simulations and experiments. Inaccuracies in loading inputs, particularly those arising from errors in the equation of state parameters of explosives, hinder reliable predictions of structural responses.

An accurate underwater explosion load prediction is critical for anti-explosion ship designs and explosive power regulation. Extensive research has focused on theoretical models and numerical methods for shock waves and bubble pulsations, the primary underwater explosion load components [22]. Taylor’s 1941 theoretical analysis of the incident and reflected pressures on plates [23] laid the groundwork for shock wave studies. Cole [24] consolidated experimental and theoretical advancements in underwater explosions, formulating empirical equations for shock wave loads. Zamyshlyaev [25] derived empirical formulas for bubble pulsation, while Zhang et al. [26] proposed a unified bubble dynamics theory for gravity-influenced environments. Souers [27] revised the Lawrence Livermore National Laboratory (LLNL) detonation EOS, underscoring the need for precise explosive product modeling. Current numerical approaches often assign initial conditions using actual explosive dimensions/densities or a high-pressure bubble matching the explosive’s radius and energy [16,28,29,30]. Alternatively, pressure time histories from empirical formulas are directly input into finite element software [31]. The numerical accuracy of the shock wave pressure and bubble pulsation period is equally critical. The complexity of underwater detonation processes, including ambiguous gas–liquid interfaces, necessitates robust numerical methods to resolve shock wave peaks and bubble pulsation periods. Finite difference, finite volume, discontinuous Galerkin, and Runge–Kutta discontinuous Galerkin methods are widely adopted for their efficacy in handling fluid discontinuities, non-physical oscillations, and cavitation [32,33,34,35,36]. Multiphase flow challenges are addressed via Level Set techniques, front tracking, or the Ghost Fluid Method (GFM) [37,38,39]. Wu et al. [40] proposed an h-adaptive local discontinuous Galerkin (LDG) method for second-order wave equations, refining elements based on pressure gradients to enhance the efficiency and accuracy.

Despite its advantages, numerical underwater explosion modeling faces limitations in accuracy and computational cost, potentially leading to structural overdesign or underprotection. Particularly, in numerical simulations of underwater explosions, inaccuracies in the parameters of the Jones–Wilkins–Lee (JWL) equation of state often result in significant deviations between predicted shock wave pressure peaks or bubble pulsation periods and experimental or empirical results. This study focuses on underwater explosion loads, by first conducting research in a 1D free-field underwater explosion scenario; we isolate interference from structural deformation-induced complexities, such as bubble deformation, jet formation, and cavitation, which are inherent in near-field explosions or 3D structural response analyses. This approach allows for the precise validation of explosive parameters and load characteristics in an idealized environment. Subsequently, applying these validated loads to near-field or 3D applications eliminates inaccuracies arising from load inputs, enabling researchers to exclusively investigate other factors, such as structural behavior and constitutive models. If the explosive parameters satisfy accuracy requirements in 1D free-field simulations, they are more likely to yield reliable results in complex scenarios. This methodology ensures that load prediction errors are decoupled from structural modeling uncertainties, enhancing the robustness of underwater explosion studies for naval architecture and marine engineering applications.

This paper aims to reduce load prediction errors by proposing a corrected initial condition method for explosives, adjusting the initial density and specific internal energy to align simulated shock wave peaks and bubble periods with experimental or empirical benchmarks. Section 2 details governing equations, the correction methodology, and the numerical validation. Section 3 applies the method to underwater explosion cases, demonstrating its accuracy. Conclusions are summarized in Section 4.

2. Correction Methods

To enhance the research efficiency, this study will establish a one-dimensional spherical symmetric free-field model for numerical investigations of underwater explosions. Within this model, both gaseous and liquid phases will be modeled using the one-dimensional Euler equations in conservation form. These governing equations will be closed by employing appropriate equations of state: the JWL equation of state for explosive gases and the Tait equation of state for water as the liquid medium.

Building upon this theoretical framework, the initial conditions of the explosive will be modified through a dual adjustment approach. The correction method is based on the positive correlation between the shock wave pressure and the specific internal energy per unit mass of the explosive. If the initial parameters yield a shock wave pressure peak lower than experimental or empirical results, the specific internal energy per unit mass is incrementally increased. Conversely, if the simulated peak is higher than benchmarks, the specific internal energy is decreased. Iterative recalculations are performed until the error reduces to within 5%. Subsequently, leveraging the inverse relationship between volume and density under mass conservation (with fixed explosive mass), if the simulated bubble pulsation period is shorter than the experimental or empirical value, the density of the detonation products is reduced to increase their volume. Conversely, if the simulated period is longer than the experimental or empirical value, the density is increased to reduce the volume, thereby optimizing the bubble pulsation period until the error falls within 5%. The adjustment magnitude for each step is determined through iterative trials and empirical evaluation, and the schematic representation of this correction process is illustrated in Figure 1.

2.1. Governing Equations of Fluid Dynamics

2.1.1. One-Dimensional Fluid Equations

The modeling of the gas and liquid phases employs the Euler equations to describe the flow dynamics of compressible fluids. The one-dimensional Euler equations in conservation form can be expressed as

$$U_t+f{(U)}_x=S(U)$$

(1)

where the conservative variables U, flux vector f(U), and geometric source term S(U) can be expressed as

$$\begin{array}{l}U={\left[\rho , \rho v, E\right]}^T\\ f={\left[\rho v, \rho v^2+p, v(E+p)\right]}^T\\ S=-{\displaystyle \frac{\alpha }{r}}{\left[\rho v, \rho v^2, v(E+p)\right]}^T\end{array}$$

(2)

The parameter α = 0 corresponds to one-dimensional problems, α = 1 to planar axisymmetric problems, and α = 2 to spherically symmetric problems. Here, t denotes time, x represents the spatial coordinate, ρ is the fluid density, v is the velocity, p is the fluid pressure, and E is the total energy per unit volume. The total energy E is defined as

$$E=\rho e+{\displaystyle \frac{1}{2}}\rho v^2$$

(3)

where e denotes the internal energy per unit mass of the fluid, defined as a function of the fluid pressure p and density ρ, satisfying

$$e=e(p,\mathsf{\rho })$$

(4)

2.1.2. Equations of State

(1) JWL equation of state

The JWL equation of state, which describes the relationship between pressure, density, and energy in detonation gases, can be analogized to the interplay of pressure, volume, and energy during fuel combustion in an internal combustion engine. In both systems, the rapid energy release drives dynamic changes in thermodynamic states, though the timescales and physical mechanisms differ significantly. The pressure p during underwater detonation is expressed by Equation (5)

$$p=A(1-{\displaystyle \frac{\omega {\rho }_g}{R_1{\rho }_{c0}}})\exp (-R_1{\displaystyle \frac{{\rho }_{c0}}{{\rho }_g}})+B(1-{\displaystyle \frac{\omega {\rho }_g}{R_2{\rho }_{c0}}})\exp (-R_2{\displaystyle \frac{{\rho }_{c0}}{{\rho }_g}})+\omega {\rho }_g{e}_g$$

(5)

where A, B, R₁, R₂, and ω are the coefficients of the JWL equation of state, determined by fitting experimental data. These parameters are material-dependent constants. ρ_c0 denotes the initial density of the explosive, ρ_g represents the density of the detonation gas, and e_g is the internal energy per unit mass of the explosive. The explosive material and associated parameters used in this study are listed in Table 1.

(2) Tait equation of state

The Euler equations can be closed using an equation of state (EOS) to establish the relationship between pressure and other flow variables. Water can be described by the Tait equation of state

$$p=D{\left({\displaystyle \frac{\rho }{{\rho }_r}}\right)}^N-\overline{D}$$

(6)

where the reference density ρ_r = 1000  kg/m³; the constant C = 1.0 × 10⁵  Pa; the constant D = 3.31 × 10⁸  Pa; $\overline{D}$ = D − C = 3.309 × 10⁸  Pa; and the constant N = 7.15.

The Tait equation of state can be used to simulate the behavior of water under relatively moderate pressures [41].

Table 1. Parameters of the JWL equation of state.

	ρc0 (kg/m3)	A (GPa)	B (GPa)	R1	R2	ω	eg0 (MJ/kg)	p0 (GPa)
TNT (Wardlaw, 1998 [28])	1630	548.4	9.375	4.94	1.21	0.28	4.2814	7.8039
TNT (Dobratz, 1972 [42])	1630	373.8	3.747	4.15	0.9	0.35	3.681	8.427

Table 1. Parameters of the JWL equation of state.

	ρc0 (kg/m3)	A (GPa)	B (GPa)	R1	R2	ω	eg0 (MJ/kg)	p0 (GPa)
TNT (Wardlaw, 1998 [28])	1630	548.4	9.375	4.94	1.21	0.28	4.2814	7.8039
TNT (Dobratz, 1972 [42])	1630	373.8	3.747	4.15	0.9	0.35	3.681	8.427

2.2. Correction Principles and Procedures

Assuming the explosive is approximated as a sphere, the initial parameters of the JWL equation of state for the explosive include the initial density of the detonation products and the initial specific internal energy per unit mass of the explosive. A free-field spherically symmetric underwater explosion computational model is established. The Euler domain is spatially discretized using the discontinuous Galerkin method, while the temporal domain is discretized via the Runge–Kutta method, yielding preliminary pressure time-history curves at different spatial locations. The peak shock wave pressure is extracted and compared with experimental or empirical formula results. If the error does not exceed 5%, the initial specific internal energy per unit mass of the explosive is retained. Since the peak shock wave pressure is positively correlated with the initial specific internal energy per unit mass, the initial specific internal energy is adjusted iteratively until the error in the peak shock wave pressure is within 5%. For the correction related to the bubble pulsation period, the initial density of the detonation products can be adjusted based on mass conservation to modify the bubble volume, thereby refining the bubble pulsation period until its error relative to experimental or empirical formula results is within 5%. This completes the correction of the initial explosive conditions for the underwater explosion numerical simulation.

To validate and compare the method, two models were selected: the Zamyshlyayev empirical formulas and the Geers–Hunter model. The Zamyshlyayev underwater explosion empirical formulas are semi-empirical and semi-analytical models derived from extensive experimental data. These formulas are primarily used to rapidly estimate key parameters of underwater explosions, such as shock wave pressure peaks, bubble pulsation periods, and energy decay rates.

The Zamyshlyayev empirical formulas for underwater explosions include the following:

(1) Shock wave phase

$$p\left(t\right)=P_m{e}^{-t/\theta },t<\theta$$

(7)

$$p\left(t\right)=0.368P_m{\displaystyle \frac{\theta }{t}}\left[1-{\left({\displaystyle \frac{t}{t_p}}\right)}^{1.5}\right],\theta \le t\le t_1$$

(8)

$$p\left(t\right)=P^{*}\left[1-{\left({\displaystyle \frac{t}{t_p}}\right)}^{1.5}\right]-\Delta P,t_1\le t\le t_p$$

(9)

$$p\left(t\right)={\displaystyle \frac{{10}^5}{\overline{r}}}\left[{\displaystyle \frac{0.686{\overline{P}}_0^{0.96}}{\xi }}+{\displaystyle \frac{5.978{\overline{P}}_0^{0.62}(1-{\xi }^2)}{{\xi }^{0.92}}}-30.1{\overline{P}}_0^{0.65}{\xi }^{0.36}\right]$$

$$-{\displaystyle \frac{1.73\times {10}^{10}}{{\overline{r}}^4{\overline{P}}_0^{0.43}}}\left(1-{\xi }^2\right){\xi }^{0.1},t_p\le t\le T-t_2$$

(10)

Equation (7) represents the pressure waveform during the exponential decay phase, Equation (8) describes the pressure waveform during the reciprocal decay phase, Equation (9) corresponds to the pressure waveform after the post-reciprocal decay phase, and Equation (10) pertains to the pressure waveform during the bubble expansion–contraction phase, where

$$P_m=\left\{\begin{array}{l}4.41\times {10}^7{\left({\displaystyle \frac{Q^{1/3}}{R}}\right)}^{1.5},6\le \overline{r}<12\hfill \\ 5.24\times {10}^7{\left({\displaystyle \frac{Q^{1/3}}{R}}\right)}^{1.13},12\le \overline{r}<240\hfill \end{array}\right.,\theta =\left\{\begin{array}{l}0.45r_0{\overline{r}}^{0.45}{10}^{-3},\overline{r}\le 30\hfill \\ 3.5{\displaystyle \frac{r_0}{c}}\sqrt{\lg \overline{r}-0.9},\overline{r}>30\hfill \end{array}\right.,$$

$$\overline{r}={\displaystyle \frac{R}{r_0}},\overline{t}={\displaystyle \frac{c}{r_0}}t,{\overline{t}}_1={\displaystyle \frac{c}{r_0}}{t}_1,\Delta P={\displaystyle \frac{10}{{\overline{r}}^4}}\left(5635{\overline{t}}^{0.54}-0.113{\overline{P}}_0{\overline{t}}^2\right),P^{*}={\displaystyle \frac{7.173\times {10}^8}{\overline{r}{\left(\overline{t}+5.2-m\right)}^{0.87}}},$$

$$t_p=\left({\displaystyle \frac{850}{{\overline{P}}_0^{0.81}}}-{\displaystyle \frac{20}{{\overline{P}}_0^{1/3}}}+m\right){\displaystyle \frac{r_0}{c}},{\displaystyle \frac{{\overline{t}}_1}{{\left({\overline{t}}_1+5.2-m\right)}^{0.87}}}=4.9\times {10}^{-10}{P}_m\overline{r}\theta {\displaystyle \frac{c}{r_0}},\xi =\sin {\displaystyle \frac{\pi \overline{t}}{2{\overline{t}}_m}},{\overline{t}}_m={\displaystyle \frac{4350}{{\overline{P}}_0^{0.83}}}-{\displaystyle \frac{30.7}{{\overline{P}}_0^{0.35}}}+m,$$

$$P_0=P_{\mathrm{atm}}+\rho g{H}_0,{\overline{P}}_0={\displaystyle \frac{P_0}{P_{\mathrm{atm}}}},T=2.11{\displaystyle \frac{Q^{1/3}}{{\left(P_0/\rho g\right)}^{5/6}}},t_2=3290{\displaystyle \frac{r_0}{P_0^{0.71}}},m=11.4-{\displaystyle \frac{10.6}{{\overline{r}}^{0.13}}}+{\displaystyle \frac{1.51}{{\overline{r}}^{1.26}}};$$

(2) Bubble pulsation phase

$$p(t)=P_{m1}{e}^{-{\left(t-T\right)}^2/{\theta }_1^2},T-t_2<t<T+t_2$$

(11)

where

$${\theta }_1=20.7{\displaystyle \frac{r_0}{P_0^{0.41}}},P_{m1}={\displaystyle \frac{39\times {10}^6+24P_0}{{\overline{R}}_{bc}}},{\overline{R}}_{bc}={\displaystyle \frac{R_{bc}}{r_0}},R_{bc}=\sqrt{R^2+\Delta H^2-2R\Delta H\sin \phi }$$

In the equations, t denotes time; Q is the weight of the TNT explosive; θ is the shock wave decay coefficient; R is the standoff distance; P_m is the peak pressure of the shock wave; H₀ is the depth of the explosive; r₀ is the initial radius of the spherical explosive; ρ_w is the fluid density; c is the speed of sound in water; P_atm is the atmospheric pressure; g is the gravitational acceleration; P_m1 is the peak pressure of the bubble pulsation load; θ₁ is the decay coefficient of the bubble pulsation load; R_bc is the distance from the measurement point to the bubble center; φ is the angle between the line connecting the explosion center and the observation point and the horizontal line; ΔH is the bubble rise distance; and T is the bubble pulsation period.

Geers and Hunter proposed a computational model that divides the underwater explosion load into two phases: the shock wave phase and the bubble pulsation phase. The core objective of this model is to provide a comprehensive mathematical description of the physical processes governing underwater explosions. Using a second-order doubly asymptotic approximation, they derived a system of equations for the bubble rise motion. The functional expressions are as follows:

During the underwater explosion shock wave load phase (t < 7T_c), the pressure function is given by Equation (12)

$$p\left(t\right)={\displaystyle \frac{1}{R}}{\displaystyle \frac{\rho }{4\pi }}{\left({\displaystyle \frac{r_0}{R}}\right)}^F\ddot{V}\left(t\right)$$

(12)

In Equation (12), the bubble expansion acceleration $\ddot{V}\left(t\right)$ can be calculated by the following equation

$$\ddot{V}\left(t\right)={\displaystyle \frac{4\pi r_0}{\rho }}{P}_c[0.8251\exp (-1.338t/T_c)+0.1749\exp (-0.1805t/T_c)]$$

(13)

where $P_c=L{\left(Q^{1/3}/r_0\right)}^{1+F}$, $T_c=l{Q}^{1/3}{\left(Q^{1/3}/r_0\right)}^G$, L, l, F, and G are material constants.

During the bubble pulsation load phase (t ≥ 7T_c), the bubble pulsation pressure is expressed by Equation (14)

$$p(t)={\displaystyle \frac{1}{R}}\rho (\ddot{a}{a}^2+2a\dot{a})$$

(14)

The parameters related to the bubble radius a in the equation can be determined by solving Equations (15)–(19) simultaneously.

$$\dot{a}=-{\displaystyle \frac{{\phi }_{l0}}{a}}-c^{-1}\left({\dot{\phi }}_{l0}-{\dot{a}}^2-{\displaystyle \frac{1}{3}}{\dot{u}}^2-{\displaystyle \frac{2}{3}}{\displaystyle \frac{\dot{u}}{a}}{\phi }_{l1}\right)$$

(15)

$$\dot{u}=-2{\displaystyle \frac{{\phi }_{l1}}{a}}-c^{-1}\left({\dot{\phi }}_{l1}-2\dot{a}\dot{u}\right)$$

(16)

$${\dot{\phi }}_{l0}={\displaystyle \frac{1}{\left(1+\varsigma \right)}}\left[\begin{array}{l}\left({\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\rho }_g}{\rho }}+\varsigma \right)\left({\dot{a}}^2+{\displaystyle \frac{1}{3}}{\dot{u}}^2\right)-{\displaystyle \frac{{\rho }_g}{\rho }}{c}_g{\displaystyle \frac{{\phi }_{l0}}{a}}\\ +{\displaystyle \frac{2}{3}}\left(1+\varsigma \right)\dot{u}{\displaystyle \frac{{\phi }_{l1}}{a}}-Z\end{array}\right]$$

(17)

$${\dot{\phi }}_{l1}={\displaystyle \frac{1}{\left(1+\varsigma \right)}}\left[\begin{array}{l}\left(1+{\displaystyle \frac{{\rho }_g}{\rho }}+2\zeta \right)\dot{a}\dot{u}-\left(1-{\displaystyle \frac{{\rho }_g}{\rho }}\right)ga\\ -{\displaystyle \frac{{\rho }_g}{\rho }}{c}_g\left(2{\displaystyle \frac{{\phi }_{l1}}{a}}+{\displaystyle \frac{{\phi }_{g1}}{a}}\right)+{\displaystyle \frac{3}{8}}{C}_D{\left|\dot{u}\right|}^{E_D}\end{array}\right]$$

(18)

$${\dot{\phi }}_{g1}={\displaystyle \frac{1}{\left(1+\varsigma \right)}}\left[\begin{array}{l}\left(2+{\displaystyle \frac{c_g}{c}}+\varsigma \right)\dot{a}\dot{u}+{\displaystyle \frac{c_g}{c}}\left(1-{\displaystyle \frac{{\rho }_g}{\rho }}\right)ga\\ -c_g\left(2{\displaystyle \frac{{\phi }_{l1}}{a}}+{\displaystyle \frac{{\phi }_{g1}}{a}}\right)+{\displaystyle \frac{3}{8}}{\displaystyle \frac{c_g}{c}}{C}_D{\left|\dot{u}\right|}^{E_D}\end{array}\right]$$

(19)

In the equations, the expressions for the relevant symbols are as follows

$$\varsigma ={\displaystyle \frac{{\rho }_g{c}_g}{\rho c}},P_g=K_c{\left({\displaystyle \frac{V_c}{V}}\right)}^{\gamma },{\rho }_g={\rho }_c\left({\displaystyle \frac{V_c}{V}}\right),c_g=c_c{\left({\displaystyle \frac{V_c}{V}}\right)}^{{\displaystyle \frac{1}{2}}\left(\gamma -1\right)},c_c=\sqrt{{\displaystyle \frac{\gamma K_c}{{\rho }_c}}},$$

$$Z={\displaystyle \frac{1}{\rho }}\left(P_g-P_0+\rho gu\right)+{\displaystyle \frac{1}{3}}\left[{\left({\displaystyle \frac{{\phi }_{l1}}{a}}\right)}^2-{\displaystyle \frac{{\rho }_g}{\rho }}{\left({\displaystyle \frac{{\phi }_{g1}}{a}}\right)}^2\right]$$

where ρ_c is the explosive density, ζ is the ratio of the acoustic impedance of the gas inside the bubble to that of the external fluid, c_g is the gas speed of sound, c is the speed of sound in the fluid field, C_D is the fluid drag coefficient, V_c represents the initial volume of the spherical explosive, V = 4/3πa³ is the current bubble volume, K_c is the adiabatic constant of the explosive (MPa), γ is the specific heat ratio of the gas, and E_D is the adjustment exponent calibrated to match experimental values.

2.3. Equation Solving

Underwater explosions generate intense shock waves, the propagation of which can be analogized to ripple-like pressure waves formed by a stone dropped into water, albeit with orders of magnitude greater intensity and complexity. These shock waves induce intricate fluid dynamics during underwater transmission, including abrupt pressure gradients and potential cavitation zones. And the bubble pulsation can be described as a “breathing sphere”, expanding and contracting due to the energy exchange with the surrounding fluid. To accurately capture these strong physical discontinuities, numerical methods capable of modeling complex flows and resolving the shock wave propagation must be employed.

The solution procedure is illustrated in Figure 2. A fluid Eulerian grid is constructed, initial conditions are defined, and the gas–water interface is captured using the Level Set method. The RKDG method is employed to solve the Riemann problem under complex equation-of-state conditions.

2.3.1. Solving 1D Euler Equations by RKDG Method

This section details the RKDG method for solving the one-dimensional Euler equations in this work, including spatial discretization, temporal discretization, and nonlinear slope limiting.

(1) The spatial discretization of the 1D Euler equations

Multiply Equation (1) by the test function ϕ(x) and integrate over the cell K

$${\displaystyle {\int }_K\left(U_t+f{(U)}_x\right)\varphi (x)dx}={\displaystyle {\int }_{K}S(U)\varphi (x)dx}$$

(20)

By applying the integration by parts, we obtain

$${\displaystyle \frac{𝜕}{𝜕t}}{\displaystyle {\int }_{K}U\varphi (x)dx}+{\displaystyle \sum _{z\in 𝜕K}{\displaystyle {\int }_{z}f(U)\cdot n_{z,K}\varphi (x)d\Gamma }}-{\displaystyle {\int }_{K}f(U){\varphi }_x(x)dx}={\displaystyle {\int }_{K}S(U)\varphi (x)dx}$$

(21)

where z represents the boundary of element K, n_z,K denotes the outward normal vector on the boundary z of element K, and the test function ϕ(x) satisfies the following: $\varphi \in V_h^k(K)$, Here, $V_h^k(K)$ represent the numerical solution and test function spaces, respectively, satisfying $V_h^k(K)=\left\{P:{\left.P\right|}_K\in P^k(K)\right\}$, while P^k(K) denotes the polynomial space of a degree no greater than k. Let the element K = I_i = [x_i−1/2,x_i+1/2].

Continue to simplify Equation (21)

$${\displaystyle \frac{𝜕}{𝜕t}}{\displaystyle {\int }_{x_{i-1/2}}^{x_{i+1/2}}{U}_i\varphi (x)dx}+{\left.f(U_i(x,t))\varphi (x)\right|}_{x_{i-1/2}}^{x_{i+1/2}}-{\displaystyle {\int }_{x_{i-1/2}}^{x_{i+1/2}}f(U_i)\frac{d\varphi (x)}{dx}dx}={\displaystyle {\int }_{x_{i-1/2}}^{x_{i+1/2}}S(U_i)\varphi (x)dx}$$

(22)

The flux term f(U(x_z, t)) can be replaced by a numerical flux. Considering the computational efficiency and the resolution of discontinuities, this study adopts the FORCE flux, which is the arithmetic average of the LF flux and the Richtmyer flux

$$f(U(x_z,t))\approx {\stackrel{⌢}{f}}^{\mathrm{FORCE}}(U^{-},U^{+})={\displaystyle \frac{1}{2}}\left({\stackrel{⌢}{f}}^{\mathrm{LF}}(U^{-},U^{+})+{\stackrel{⌢}{f}}^{\mathrm{RI}}(U^{-},U^{+})\right)$$

(23)

where

$${\stackrel{⌢}{f}}^{\mathrm{LF}}(U^{-},U^{+})={\displaystyle \frac{1}{2}}\left[f(U^{-})+f(U^{+})-\alpha \left(U^{+}-U^{-}\right)\right]$$

(24)

$${\stackrel{⌢}{f}}^{\mathrm{RI}}(U^{-},U^{+})=f(U^{*})$$

(25)

$$U^{*}={\displaystyle \frac{1}{2}}\left(U^{-}+U^{+}-{\displaystyle \frac{\Delta t}{\Delta x}}\left(f(U^{+})-f(U^{-})\right)\right)$$

(26)

U⁻ and U⁺ denote the values of U(x_z,t) at the adjacent cell interfaces of the current time step, and α represents the maximum eigenvalue of the Jacobian matrix ∂f(U)/∂U.

The trial function ϕ(x) is selected as the orthogonal basis functions (Legendre polynomials)

$${\varphi }_0(x)=1,{\varphi }_1(x)={\displaystyle \frac{x-x_i}{\Delta x_i/2}},\dots$$

(27)

where x_i = (x_i−1/2 + x_i+1/2)/2 and Δx_i = x_i+1/2 − x_i−1/2.

Solving the trial function yields

$${\displaystyle \frac{\mathrm{d}{U}_i^{(k)}(t)}{\mathrm{d}t}}=L(U_i^{(k)}(t)),k=0,1,2,\dots$$

(28)

$$\begin{array}{ll}L(U_i^{(k)}(t))& ={\displaystyle \frac{2}{a_k\Delta x}}\left[{\displaystyle {\int }_{-1}^{1}f(U_i^h)\frac{\mathrm{d}{\varphi }_k(\xi )}{\mathrm{d}\xi }\mathrm{d}\xi }-\left({\stackrel{⌢}{f}}_{i+1/2}(U^{-},U^{+}){\varphi }_k(1)-{\stackrel{⌢}{f}}_{i-1/2}(U^{-},U^{+}){\varphi }_k(-1)\right)\right]\\ & +{\displaystyle \frac{1}{a_k}}{\displaystyle {\int }_{-1}^{1}S(U_i^h){\varphi }_k(x)\mathrm{d}\xi }\end{array}$$

where dϕ₀(ξ)/dξ = 0, dϕ₁(ξ)/dξ = 1, dϕ₂(ξ)/dξ = 2ξ, ϕ₀(–1) = ϕ₀(1) = 1, ϕ₁(–1) = −1, ϕ₁(1) = 1, and ϕ₂(–1) = ϕ₂(1) =2/3.

The integral term in $L(U_i^{(k)}(t))$ is evaluated using the three-point Gauss quadrature method. Consequently, $L(U_i^{(k)}(t))$ in Equation (28) can be expressed as

$$\begin{array}{ll}L(U_i^{(k)}(t))& ={\displaystyle \frac{2}{a_k\Delta x}}[{\displaystyle \sum _{m=1}^M{\overline{\omega }}_{m}f(U_i^h({\xi }_m,t))\frac{\mathrm{d}{\varphi }_k({\xi }_m)}{\mathrm{d}\xi }}\\ & -\left({\stackrel{⌢}{f}}_{i+1/2}(U^{-},U^{+}){\varphi }_k(1)-{\stackrel{⌢}{f}}_{i-1/2}(U^{-},U^{+}){\varphi }_k(-1)\right)]\\ & +{\displaystyle \frac{1}{a_k}}{\displaystyle \sum _{s=1}^S{\overline{\omega }}_{s}S(U_i^h({\xi }_s,t)){\varphi }_k({\xi }_s)}\end{array}$$

(29)

This completes the spatial discretization process.

(2) The temporal discretization of the 1D Euler equations

The temporal term is discretized using the TVD Runge–Kutta (RK) method. For third-order accuracy, the TVD RK discretization of Equation (28) is expressed as

$$\begin{array}{l}{U}_i^{(k)}(t_n^1)=U_i^{(k)}(t_n)+\Delta tL(U_i^{(k)}(t_n))\\ U_i^{(k)}(t_n^2)={\displaystyle \frac{3}{4}}{U}_i^{(k)}(t_n)+{\displaystyle \frac{1}{4}}\left[U_i^{(k)}(t_n^1)+\Delta tL(U_i^{(k)}(t_n^1))\right]\\ U_i^{(k)}(t_{n+1})={\displaystyle \frac{1}{3}}{U}_i^{(k)}(t_n)+{\displaystyle \frac{2}{3}}\left[U_i^{(k)}(t_n^2)+\Delta tL(U_i^{(k)}(t_n^2))\right]\end{array}$$

(30)

where Δt is determined based on spatial accuracy. Let P^k denote the order of the Legendre polynomial; the time step satisfies

$$\Delta t\le {\displaystyle \frac{1}{2k+1}}{\displaystyle \frac{\Delta x}{\max (\left|u\right|+c)}}=\mathrm{CFL}\cdot {\displaystyle \frac{\Delta x}{\max (\left|u\right|+c)}}$$

(31)

where CFL is the Courant–Friedrichs–Lewy number. In this study, CFL = 0.3 is selected for the second-order spatial accuracy and CFL = 0.18 for the third-order spatial accuracy.

(3) The nonlinear slope limiter for the 1D Euler equations

Implementing a nonlinear slope limiter is essential to maintain stability in discontinuous problems when using the DG method. Common limiters include flux limiters, TVD/TVB limiters, and WENO/HWENO/simple WENO limiters. Among these, WENO-type limiters aim to eliminate numerical instability in discontinuous regions without degrading the accuracy in smooth regions. However, HWENO and simple WENO limiters may fail to fully suppress oscillations near discontinuities, while WENO limiters introduce an excessive computational cost and dissipation. Therefore, the von Leer TVD limiter with TVB detection is adopted for underwater explosion simulations to balance accuracy and efficiency.

The limiter can be applied to conservative or characteristic variables. Studies show that applying the limiter to characteristic variables enhances numerical stability and reduces the number of cells requiring limiting. Let R_x denote the matrix of right eigenvectors of the conservative vector U. The characteristic variables W are defined as

$$W=R_x^{-1}\cdot U$$

(32)

The TVD limiter (with TVB detection) is formulated as

$$W_i^{(1)}=\mathrm{TVB}\_\mathrm{MINMOD}\left(W_i^{(1)},W_i^{(0)}-W_{i-1}^{(0)},W_{i+1}^{(0)}-W_i^{(0)},dx,M\right)$$

(33)

where

$$R_x=\left[\begin{array}{ccc}1& 1& 1\\ u-c& u& u+c\\ h-uc& h-1/b_1& h+uc\end{array}\right],R_x^{-1}=\left[\begin{array}{ccc}{\displaystyle \frac{1+b_1(u^2-h)}{2}}+{\displaystyle \frac{u}{2c}}& -({\displaystyle \frac{b_{1}u}{2}}+{\displaystyle \frac{1}{2c}})& {\displaystyle \frac{b_1}{2}}\\ -b_1(u^2-h)& b_{1}u& -b_1\\ {\displaystyle \frac{1+b_1(u^2-h)}{2}}-{\displaystyle \frac{u}{2c}}& -{\displaystyle \frac{b_{1}u}{2}}+{\displaystyle \frac{1}{2c}}& {\displaystyle \frac{b_1}{2}}\end{array}\right]$$

h = (E + p)/ρ, b₁ = Γ/c², Γ = p_e/ρ. For JWL gases: Γ = ω, b₁ =ω/c².

$$W_i^{(0)}=R_x^{-1}\cdot U_i^{(0)},W_{i-1}^{(0)}=R_x^{-1}\cdot U_{i-1}^{(0)}, W_{i+1}^{(0)}=R_x^{-1}\cdot U_{i+1}^{(0)},W_i^{(1)}=R_x^{-1}\cdot U_i^{(1)}$$

$$\mathrm{TVB}\_\mathrm{MINMOD}\left(a,b,c,h,M\right)=\left\{\begin{array}{cc}a=a\hfill & \hfill \mathrm{if} \left|a\right|\le M{h}^2\\ a=\mathrm{MINMOD}(a,b,c)\hfill & \hfill \mathrm{otherwise}\end{array}\right.$$

(34)

$$\mathrm{MINMOD}\left(a,b,c\right)=\left\{\begin{array}{cc}\mathrm{sgn}(a)min(\left|a\right|,\left|b\right|,\left|c\right|)\hfill & \mathrm{if} \mathrm{sgn}(a)=\mathrm{sgn}(b)=\mathrm{sgn}(c)\hfill \\ \hfill 0\hfill & \hfill \mathrm{otherwise}\hfill \end{array}\right.$$

(35)

For the third-order accuracy, W_i⁽²⁾ = R_x⁻¹·U_i⁽²⁾,when the nonlinear slope limiter is applied, W_i⁽²⁾ = 0 is enforced. After completing the slope limiting on the characteristic variables, the results are transformed back to conservative variables as follows

$$U_i^{(1)}=R_x\cdot W_i^{(1)},U_i^{(2)}=R_x\cdot W_i^{(2)}$$

(36)

The limiter is applied at each RK substep to ensure stable, oscillation-free results.

2.3.2. Gas–Water Interface Treatment

For gas–water interactions, the method first solves each fluid domain independently under the assumption of a single fluid, then imposes boundary conditions near the interface.

The Level Set equation is used to track the moving interface, where ϕ(x,t) represents the signed distance from each grid point to the interface

$${\displaystyle \frac{𝜕\varphi (x,t)}{𝜕t}}+u\cdot \nabla \varphi (x,t)=0$$

(37)

where u = (u,v,w) is the fluid velocity in the Eulerian grid.

For a 1D interface located at element i, the Riemann problem is defined as

$$U_I=\Re (U_{i-1},U_{i+1})$$

(38)

Here, U_I represents the interface state, defined as U_I = (U_IL, U_IR); U_IL and U_IR denote the left and right states at the interface, respectively, and $\Re$ denotes the Riemann solver. The extrapolation technique is illustrated in Figure 3.

2.4. Numerical Validation of Solution Method

Underwater free-field explosions, particularly spherical charge explosions, serve as fundamental scenarios for studying underwater blast phenomena. To validate the accuracy and feasibility of the numerical approach, this section replicates the simulation case from Wardlaw [28], who employed the Arbitrary Lagrangian–Eulerian (ALE) method to model the explosion of a 28 kg TNT charge at a water depth of 178 m. In the computational model, the TNT charge is replaced by a high-pressure bubble with an equivalent volume and internal energy, where the bubble radius is derived from the mass. The parameters of the JWL equation of state for TNT are listed in Table 1. Wardlaw’s simulations [28] covered both the early-stage pressure field and pressure time-history calculations (C1) and long-term bubble pulsation radius calculations (C2).

Table 2. Initial conditions for subcases C1 and C2.

Case	r (m)	Fluid Left	ρ0L (kg/m3)	p0L (GPa)	u0L (m/s)	Fluid Right	ρ0R (kg/m3)	p0R (Pa)	u0R (m/s)
C1	0.16	JWL	1630	7.80	0	Water	1000.0	1.0 × 105	0
C2	0.16	JWL	1630	7.80	0	Water	1000.38	1.0 × 105	0

summarizes the initial conditions for subcases C1 and C2.

For subcase C1, the computational domain spans [0, 10 m], with an Eulerian grid resolution of 1 mm. Subcase C2 employs a domain of [0, 100 m], where the gas and water regions are discretized with grid sizes of 0.5 mm and 2.5 mm, respectively. To ensure temporal accuracy over prolonged simulations, a third-order RK method is adopted. The results of C1 and C2 are presented in Figure 4 and Figure 5, respectively.

Through numerical simulations and a comparison of the results obtained by the proposed method with Wardlaw’s experimental data, a high degree of agreement is observed, demonstrating that this method can effectively address the multi-medium problem involving JWL gas and water.

3. Numerical Model and Results

3.1. Numerical Simulation Setup

The explosive is approximated as a sphere, and a free-field spherically symmetric underwater explosion model is established. The computational domain spans [0, 50 m], with a locally refined region for detonation products within [0, 0.5 m]. The initial parameters for the JWL equation of state include the initial density and initial internal energy per unit mass of the explosive. The Euler domain is spatially discretized using the DG method, and the temporal domain is discretized using a third-order RK scheme.

3.1.1. Material Parameters

The TNT explosive is modeled using the Jones–Wilkins–Lee equation of state. The JWL parameters, based on Dobratz’s work [42], are as follows: initial density ρ_c0 = 1630  kg/m³, initial internal energy per unit mass e_g0 = 3.681 MJ/kg, and constants A = 373.8 GPa, B = 3.747 GPa, R₁ = 4.15, R₂ = 0.9, and ω = 0.35. The material parameters are summarized in Table 1 [42].

3.1.2. Case Configurations

The experimental data for validating the numerical method are derived from Swift Jr.’s work [43]. Choi et al. [44] further validated their numerical simulations using this experimental dataset, with the results exhibiting a strong agreement between computational predictions and empirical observations. The validation experiments are described as follows: Two free-field tests were conducted using TNT charges equivalent to 0.227 kg, with charge depths of 182.88 m and 91.44 m. Underwater free-field pressure sensors were placed in the water domain at 0.69 m from the initial bubble center. Based on the measured pressure time-history curves at the 182.88 m charge depth and bubble radius time-history curves at both 182.88 m and 91.44 m charge depths, the proposed method was used to correct the initial explosive conditions in the numerical simulations, ensuring the alignment between numerical and experimental results.

For the validation against the Zamyshlyayev empirical formulas, the cases include a 1 kg TNT equivalent charge at a depth of 100 m, with standoff distances of 1.0 m, 4.0 m, and 7.0 m.

For the validation against the Geers–Hunter model, the cases involve a 1 kg TNT equivalent charge at a depth of 100 m, with standoff distances of 2.0 m, 3.5 m, and 5.0 m.

Details of the validation cases are provided in

Table 3. Validation case information.

Case No.	Charge Quantity (kg)	Charge Depth (m)	Detonation Distance (m)	Contrast Object
1	0.227	182.88	0.69	Test of Swift
2	0.227	91.44	0.69
3	1.0	100	1.0	Empirical formula of Zamyshlyayev
4	1.0	100	4.0
5	1.0	100	7.0
6	1.0	100	2.0	Model of Geers–Hunter
7	1.0	100	3.5
8	1.0	100	5.0

.

3.2. Grid Convergence Analysis

To enhance the computational efficiency while ensuring the accuracy of shock wave pressure peaks and bubble pulsation periods, three grid schemes for the gas domain (denoted as Mg1, Mg2, and Mg3) and water domain (denoted as Mw1, Mw2, and Mw3) were proposed. A comparison of the pressure time-history curves and bubble radius time-history curves for different grid schemes is presented in Figure 6 and Figure 7. Specifically, for the gas domain grid sizes of 0.7 mm, 1.0 mm, and 1.4 mm were tested while maintaining a constant water domain grid size of 2 mm. The results demonstrated that a gas domain grid size of 1.0 mm (paired with a water domain grid size of 2 mm) achieved shock wave pressure peaks and bubble pulsation periods comparable to those obtained with the finer 0.7 mm gas domain grid. Consequently, the optimal grid configuration was determined as follows: a gas domain grid size of 1.0 mm (totaling 500 grids) and a remaining water domain grid size of 2 mm (totaling 24,750 grids).

Detailed grid division schemes are summarized in

Table 4. Grid division scheme.

Grid Division Scheme	Gas Domain Grid Size (mm)	Water Domain Grid Size (mm)
Mg1	0.7	2.0
Mg2	1.0
Mg3	1.4
Mw1	1.0	1.4
Mw2		2.0
Mw3		2.8

.

A grid sensitivity study indicates that further refinement does not significantly improve the computational accuracy. Selecting a gas domain grid size of 1 mm for Mg2 and a water domain grid size of 2 mm for Mw2 effectively simulates underwater explosion loads.

3.3. Results

3.3.1. Comparison with Swift Experiment

For the case with a charge depth of 182.88 m, the initial shock wave pressure peak extracted from the pressure time-history curve was 45.2 MPa, showing a 7.0% deviation compared to the experimental value of 48.6 MPa. The bubble pulsation period calculated from the initial parameters was 14.5 ms, resulting in an 8.8% error relative to the experimental period of 15.9 ms. To address these discrepancies, the e_g0 was increased by a factor of 1.3 to 4.7853 MJ/kg. Recalculations with the adjusted parameters yielded an updated shock wave pressure peak of 48.7 MPa, achieving a 0.2% error compared to the experimental result. The bubble pulsation period, determined from the timing of the secondary shock wave peak in the revised pressure time-history curve, was 16.2 ms, corresponding to a 1.9% error relative to the experimental value of 15.9 ms. This confirmed the validity of the initial condition corrections.

For the 91.44 m charge depth case, experimental pressure data were unavailable in Reference [43], so only the bubble pulsation period was corrected. The experimental bubble pulsation period was 26.7 ms, while the initial simulation yielded a period of 24.1 ms (9.7% error). By reducing the explosive density to 1390 kg/m³ (thereby increasing the explosive volume to extend the bubble dynamics), the recalculated bubble radius time-history curve exhibited a pulsation period of 26.9 ms, achieving a 0.7% error.

The errors in pressure peaks and bubble pulsation periods for Case No.1 and Case No.2, before and after corrections to the JWL equation of state parameters, are summarized in

Table 5. Errors in pressure peaks and bubble pulsation periods for Case No.1 and Case No.2 before and after corrections.

Case No.	eg0 Variation	ρc0 Variation	Pressure Peak Error	Pulsation Period Error
1	eg0→1.3eg0	-	7.0%→0.2%	8.8%→1.9%
2	-	1630→1390	-	9.7%→0.7%

. The pressure and bubble radius time-history curves for Case No.1 and Case No.2, both pre- and post-correction, are presented in Figure 8 and Figure 9.

3.3.2. Comparison with Zamyshlyayev Empirical Formulas

The feasibility of the corrected initial conditions for underwater explosion simulations was validated by comparing results with those derived from the Zamyshlyayev empirical formulas (Section 2.2). Validation cases employed a 1 kg TNT equivalent charge at a depth of 100 m with standoff distances of 1.0, 4.0, and 7.0 m (Case No.3–Case No.5). The errors in pressure peaks and bubble pulsation periods for Case No.3–Case No.5, before and after corrections to the JWL equation of state parameters, are summarized in

Table 6. Errors in pressure peaks and bubble pulsation periods for Case No.3–Case No.5 before and after corrections.

Case No.	eg0 Variation	ρc0 Variation	Pressure Peak Error	Pulsation Period Error
3	-	1630→1400	1.9%→1.1%	10.6%→0.2%
4			5.0%→1.7%	9.9%→0.2%
5			0%→4.4%	9.7%→0.2%

. The pressure and bubble radius time-history curves for Case No.3–Case No.5, both pre- and post-correction, are presented in Figure 10, Figure 11, Figure 12 and Figure 13.

For Case No.3–Case No.5, the initial shock wave pressure peaks extracted from the pressure time-history curves were 52.3, 11.3, and 6.8 MPa, respectively. These values deviated by 1.9%, 5.0%, and 0% from the Zamyshlyayev empirical results (53.3, 11.9, and 6.8 MPa, respectively), which incorporated the hydrostatic pressure at the 100 m depth. Consequently, the specific internal energy per unit mass of the explosive e_g0 was retained.

The bubble pulsation periods, determined from the timing of secondary shock wave peaks in the initial simulations, were 37.9, 39.9, and 41.8 ms, corresponding to errors of 10.6%, 9.9%, and 9.7% relative to the empirical formula results (42.4, 44.3, and 46.3 ms). To mitigate these errors, the explosive density was reduced to 1400 kg/m³, thereby increasing the explosive volume to prolong the bubble pulsation period. Post-correction simulations yielded bubble pulsation periods of 42.5, 44.4, and 46.4 ms, achieving errors of 0.2% for all cases. The revised shock wave pressure peaks were 53.9, 11.7, and 7.1 MPa, with errors of 1.1%, 1.7%, and 4.4% compared to the empirical results.

As the standoff distance increases, the finite propagation time of shock waves in water leads to a growing temporal discrepancy between the detected pulsation peak and the actual bubble collapse moment. For Case No.5, the increased pressure peak error after the correction arises from a trade-off: while the initial conditions already provided highly accurate results for near-field pressure predictions, improving the accuracy of the bubble pulsation period necessitated increasing the bubble volume, thereby elevating the total energy of the gas phase.

3.3.3. Comparison with Geers–Hunter Model

To validate the feasibility of the corrected initial conditions for underwater explosion simulations, results were compared with those derived from the Geers–Hunter model (Section 2.2). Validation cases employed a 1 kg TNT equivalent charge at a depth of 100 m with standoff distances of 2.0, 3.5, and 5.0 m (Case No.6–Case No.8). The errors in pressure peaks and bubble pulsation periods for Case No.6–Case No.8, before and after corrections to the JWL equation of state parameters, are summarized in

Table 7. Errors in pressure peaks and bubble pulsation periods for Case No.6–Case No.8 before and after corrections.

Case No.	eg0 Variation	ρc0 Variation	Pressure Peak Error	Pulsation Period Error
6	-	1630→1525	3.3%→2.5%	6.3%→1.5%
7			0%→0.8%	6.6%→1.7%
8			3.4%→4.5%	6.7%→1.6%

. The pre- and post-correction pressure and bubble radius time-history curves for these cases are illustrated in Figure 14, Figure 15, Figure 16 and Figure 17.

For Case No.6–Case No.8, the initial shock wave pressure peaks extracted from the pressure time-history curves were approximately 23.2, 12.9, and 9.1 MPa, respectively. These values deviated by 3.3%, 0%, and 3.4% from the Geers–Hunter model results (24.0, 12.9, and 8.8 MPa, respectively), which incorporated the hydrostatic pressure at a 100 m depth. Consequently, the specific internal energy per unit mass of the explosive e_g0 was retained.

The bubble pulsation periods, determined from the timing of secondary shock wave peaks in the initial simulations, were 38.6, 39.5, and 40.5 ms, corresponding to errors of 6.3%, 6.6%, and 6.7% relative to the Geers–Hunter model results (41.2, 42.3, and 43.2 ms). To mitigate these errors, the explosive density was reduced to 1525 kg/m³, thereby increasing the explosive volume to prolong the bubble pulsation period. Post-correction simulations yielded bubble pulsation periods of 40.6, 41.6, and 42.5 ms, achieving errors of 1.5%, 1.7%, and 1.6% compared to the Geers–Hunter model. The revised shock wave pressure peaks were 23.4, 13.0, and 9.2 MPa, with errors of 2.5%, 0.8%, and 4.5% relative to the Geers–Hunter model results. Our methodology prioritizes simultaneous compliance with the dual criteria (pressure and period errors ≤ 5%), as exceeding this threshold for either parameter would invalidate the explosive’s calibration. When the initial pressure peak is already accurate (e.g., <3.4% error), further adjustments to improve the bubble pulsation period may slightly increase the pressure error (e.g., from 3.4% to 4.5%) but ensure that both parameters remain within the 5% error threshold. This balance reflects the compromise inherent in multi-parameter optimization, where slight deviations in one parameter are tolerated to achieve global accuracy across all metrics.

The small errors in shock wave pressure peaks, both before and after correction, demonstrate the high accuracy of the numerical simulations in predicting shock wave dynamics. The further reduction in errors post-correction indicates that the selected initial explosive parameters closely align with the Geers–Hunter model, with the correction method exerting a minor yet observable refinement effect on pressure peak calculations.

The larger initial errors in bubble pulsation periods highlight inherent deviations in simulating bubble dynamics. However, adjusting the explosive density significantly reduced these errors, underscoring the pronounced impact of the correction method on improving the bubble pulsation period accuracy. This confirms the method’s efficacy in refining gas-phase energy and volume interactions.

4. Conclusions

The corrected initial conditions method for explosives proposed in this study was validated through comparisons with experimental data and empirical formulas across multiple working conditions.

The results demonstrate that this method effectively compensates for load discrepancies caused by inaccuracies in the JWL equation of state parameters. After correction, the maximum error in shock wave pressure peaks was reduced to less than 4.5%, and the maximum error in bubble pulsation periods remained below 1.9% across all tested conditions. Additionally, the numerical methods employed in this study exhibited a strong accuracy and feasibility in addressing complex multiphase problems, such as underwater explosions, providing an effective numerical framework for related research fields.

By adjusting two key parameters—the initial density and specific internal energy per unit mass of the explosive—errors in shock wave pressure peaks and bubble pulsation periods were significantly reduced, enabling the accurate prediction of underwater explosion loads. Furthermore, validating explosive parameters in 1D free-field scenarios ensures their reliability for generating precise results in more complex applications, such as near-field underwater explosions or 3D structural response analyses. This methodology decouples load prediction errors from structural modeling uncertainties, thereby mitigating challenges arising from multi-factor interactions.

The research presented in this study alleviates computational burdens in advanced underwater explosion simulations by eliminating interference factors, enhancing efficiency. It presents critical implications for naval ship anti-explosion design, explosive power regulation, and safety assessments of underwater structures.