Numerical Simulation of Underwater Explosions Near a Naval Vessel Based on a Compressible Multi-fluid Model

doi:10.12382/bgxb.2024.0956

Abstract

Abstract:

Studying underwater explosions near warships is crucial for hull structure design,explosion impact damage prediction,and personnel safety.To this end,an improved six-equation compressible multiphase flow model based on the diffuse interface method is proposed to resolve thermodynamic state prediction deviation under shock waves and support anti-shock mechanism research and numerical method optimization.The model is improved via a hybrid energy correction equation and a more accurate gas equation of state.A numerical algorithm on an unstructured grid system is constructed,adopting the second-order MUSCL-Hancock scheme (with least-squares reconstruction and Barth-Jespersen limiter) and two-phase HLLC Riemann solver to solve homogeneous hyperbolic equations,and Newton-Raphson iteration for instantaneous pressure relaxation equation.Results show that after total energy equation correction,the model’s simulation of shock wave velocity and interface is highly consistent with the Euler equation’s exact solution,resolving near-interface numerical oscillation.Compared with experimental data,the improved model has a 1.13% relative error and 0.33% higher accuracy; more accurate SG-EOS parameters are obtained by fitting the shock Hugoniot curve.It also clearly shows underwater explosion phenomena:shock wave propagation,bubble expansion-contraction,and bubble collapse water jet.However,it has deficiencies in bubble interface clarity and jet precision,mainly limited by numerical scheme dissipation under extreme gradients.In conclusion,the improved model effectively enhances the accuracy of simulating underwater explosions near naval vessels,supports in-depth warship anti-shock mechanism research,and lays a solid foundation for future numerical method optimization.

Key words: naval vessel, underwater explosions, six-equation model, compressible multiphase flow, Riemann solver

CLC Number:

U661.1

YU Wanli, YANG Ao, TANG Zhaolie, CHENG Han, ZHANG Zhiyang, LIU Weixing. Numerical Simulation of Underwater Explosions Near a Naval Vessel Based on a Compressible Multi-fluid Model[J]. Acta Armamentarii, 2025, 46(9): 240956-.

Figures/Tables 18

Table 1 Different SG-EOS parameters for liquid water

案例	方法	γ	p_∞/MPa
1	Liou等^[23]方法	2.80	600.00
2	Saurel等^[15]方法	4.40	850.00
3	本文方法	5.45	40.45

Fig.1 Comparison of computed shock Hugoniot curves for shock-particle velocity of liquid water

Fig.2 Computational unit and boundary of quadrilateral unstructured grid

Table 2 Initial conditions on the left and right sides of 1D water-air shock tube problem

位置	α₁	ρ₁/ (kg·m^-3)	ρ₂/ (kg·m^-3)	u/ (m·s^-1)	p₁/ MPa	p₂/ MPa
左边	10^-6	1	1000	0	10³	10³
右边	1-10^-6	1	1000	0	0.1	0.1

Fig.3 Distribution diagram of mixture density,normal velocity,and water volume fraction of a 1D water-air shock tube problem at t=0.24ms

Table 3 Initial conditions on the left and right sides of advection of an interface in a uniform pressure and velocity flow problem

位置	α₁	ρ₁/ (kg·m^-3)	ρ₂/ (kg·m^-3)	u/ (m·s^-1)	p₁/ MPa	p₂/ MPa
左边	10^-8	10	10³	100	10⁵	10⁵
右边	1-10^-8	10	10³	100	10⁵	10⁵

Fig.4 Distribution diagram of mixture density,normal velocity,and air volume fraction of an interface in a uniform pressure and velocity flow problem at t=2.79ms

Table 4 Initial conditions of the water,air,and explosive gas bubble for underwater explosions

介质	α₁	ρ₁/ (kg·m^-3)	ρ₂/ (kg·m^-3)	p_m/ MPa
液态水	10^-6	1000	10³	0.1
空气	1-10^-6	1.225	10³	0.1
爆炸气泡	1-10^-6	1250	10³	10³

Fig.5 Shockwave pressure histories of two gauge targets with standoff distances of 0 and 5.0cm along the explosive center adjacent to the rigid surface

Fig.6 Distribution diagram of gas phase volume fraction in the flow field at different times for underwater explosion near the water surface

Fig.7 Distribution diagram of pressure in the flow field at different times for underwater explosion near the water surface

Fig.8 Distribution diagram of computational grid and configuration diagram of mathematical model around the ship

Fig.9 Distribution diagram of pressure of underwater explosion in a free field at 0.5ms and 1.0ms

Fig.10 Grid resolution test

Fig.11 Pressure distribution diagram of flow field near the ship at different times in the first stage of underwater explosion

Fig.12 The velocity vector plot of the weak transmitted shock wave propagating into air at t=10ms

Fig.13 Time-history curve of shock wave pressure at two measurement targets located 2.0m and 2.5m in a straight line from the center of the explosion

Fig.14 Gas phase volume fraction distribution diagram of flow field near the ship at different times in the second stage of underwater explosion

References 27

[1]	张阿漫, 明付仁, 刘云龙, 等. 水下爆炸载荷特性及其作用下的舰船毁伤与防护研究综述[J]. 中国舰船研究, 2023, 18(3):139-154.
	ZHANG A M, MING F R, LIU Y L, et al. Review of research on underwater explosion related to load characteristics and ship damage and protection[J]. Chinese Journal of Ship Research, 2023, 18(3):139-154. (in Chinese)
[2]	RAJENDRAN R, NARASIMHAN K. Linear elastic shock response of plane plates subjected to underwater explosion[J]. International Journal of Impact Engineering, 2023, 24(1):1042-1053.
[3]	LIU J, AN F J, WU C, et al. Experimental investigations on small-and full-scale ship models with polyurea coatings subjected to underwater explosion[J]. Defence Technology, 2022, 18(7):1257-1268. doi: 10.1016/j.dt.2021.05.011
[4]	HIRT C W, NICHOLS B D. Volume of fluid (VOF) method for the dynamics of free boundaries[J]. Journal of Computational Physics, 1981, 39(1):201-225.
[5]	SETHIAN J A, SMEREKA P. Level set methods for fluid interfaces[J]. Annual Review of Fluid Mechanics, 2003, 35(1):341-372.
[6]	PANCHAL A, BRYNGELSON S H, MENON S. A seven-equation diffused interface method for resolved multiphase flows[J]. Journal of Computational Physics, 2023, 475:111870.
[7]	BAER M R, NUNZIATO J W. A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials[J]. International Journal of Multiphase Flow, 1986, 12(6):861-889.
[8]	CHIOCCHETTI S, PESHKOV I, GAVRILYUK S, et al. High order ADER schemes and GLM curl cleaning for a first order hyperbolic formulation of compressible flow with surface tension[J]. Journal of Computational Physics, 2021, 426:109898.
[9]	QIAN J Z, WANG Y J, ZHANG Y, et al. An entropy consistent and symmetric seven-equation model for compressible two-phase flows[J]. Journal of Computational Physics, 2023, 489:112271.
[10]	SAUREL R, ABGRALL R. A multiphase Godunov method for compressible multifluid and multiphase flows[J]. Journal of Computational Physics, 1999, 150(2):425-467.
[11]	HA C T, PARK W G, JUNG C M. Numerical simulations of compressible flows using multi-fluid models[J]. International Journal of Multiphase Flow, 2015, 74:5-18.
[12]	KAPILA A, MENIKOFF R, BDZIL J, et al. Two-phase modeling of deflagration-to-detonation transition in granular materials:Reduced equations[J]. Physics of Fluids, 2001, 13(10):3002-3024.
[13]	ZHANG F, CHENG J. Analysis on physical-constraint-preserving high-order discontinuous Galerkin method for solving Kapila’s five-equation model[J]. Journal of Computational Physics, 2023, 492:112417.
[14]	HE Z W, LIU H P, LI L. Generic five-equation model for compressible multi-material flows and its corresponding high-fidelity numerical algorithms[J]. Journal of Computational Physics, 2023, 487:112154.
[15]	SAUREL R, PETITPAS F, BERRY R A. Simple and efficient relaxation methods for interfaces separating compressible fluids,cavitating flows and shocks in multiphase mixtures[J]. Journal of Computational Physics, 2009, 228(5):1678-1712.
[16]	YU W L, CHOI J I. Numerical study of underwater explosion shock loading near a rigid dam[J]. Journal of Mechanical Science and Technology, 2024, 38(3):1271-1279.
[17]	BARTH T J, JESPERSEN D C. The design and application of upwind schemes on unstructured meshes[C]//Proceedings of 27th Aerospace Sciences Meeting.Reno,NV, US:AIAA, 1989:1989-0366.
[18]	YU W L, SONG S, XU T, et al. Numerical simulations of blast wave propagation after a high-energy explosion[J]. International Journal of Aeronautical and Space Sciences, 2023, 24(1):1042-1053.
[19]	YU W L, SONG S, CHOI J I. Numerical simulations of underwater explosions using a compressible multi-fluid model[J]. Physics of Fluids, 2023, 35(10):106102.
[20]	YEOM G S, CHOI J I. Efficient exact solution procedure for quasi-one-dimensional nozzle flows with stiffened-gas equation of state[J]. International Journal of Heat and Mass Transfer, 2019, 137:523-533.
[21]	NAGAYAMA K, MORI Y, SHIMADA K, et al. Shock Hugoniot compression curve for water up to 1 GPa by using a compressed gas gun[J]. Journal of Applied Physics, 2002, 91(1):476-482.
[22]	COCCHI J, SAUREL R, LORAUD J. Treatment of interface problems with Godunov-type schemes[J]. Shock Waves, 1996, 5:347-357.
[23]	LIOU M S, CHANG C H, NGUYEN L, et al. How to solve compressible multifluid equations:a simple,robust,and accurate method[J]. AIAA Journal, 2008, 46(9):2345-2356.
[24]	PELANTI M, SHYUE K M. A mixture-energy-consistent six-equation two-phase numerical model for fluids with interfaces,cavitation and evaporation waves[J]. Journal of Computational Physics, 2014, 259:331-357.
[25]	HU X Y, ADAMS N A, IACCARINO G. On the HLLC Riemann solver for interface interaction in compressible multi-fluid flow[J]. Journal of Computational Physics, 2009, 228(17):6572-6589.
[26]	HE Z H, DU Z P, ZHANG L, et al. Damage mechanisms of full-scale ship under near-field underwater explosion[J]. Thin-Walled Structures, 2023, 189:110872.
[27]	NGUYEN V T, PHANT H, DUY T N, et al. Numerical modeling for compressible two-phase flows and application to near-field underwater explosions[J]. Journal of Computational Physics, 2021, 215:104805.