Numerical simulation of column charge underwater explosion based on SPH and BEM combination
Highlights
► Axisymmetric SPH model was established combined with BEM. ► Column charge detonation and bubble jet was simulated via axisymmetric SPH method. ► The whole process of underwater explosion detonated by column charge was simulated. ► Numerical results of various phases of underwater explosion agree well with those of experiment.
Introduction
Underwater explosion generates shock wave, high temperature and high pressure pulsing bubble and high-speed jet formed by bubble, with pressure and temperature up to dozens of GPa and 3000 K respectively. Its physical phenomena is extremely complex, which involves various complicated matters such as transient state, high temperature and high pressure, large distortion, and multi-medium flow [1]. Numerical methods including SPH-based two-dimensional underwater detonation, BEM-based bubble pulsing have been formed in the simulation of underwater explosion presently. Nevertheless, underwater attack-type weapons such as torpedo are typically of column charge, making their detonations three-dimensional. Few literatures on numerical simulation of their whole explosion process have been published.
SPH method is usually applied to simulate the phase from explosive initiation to high pressure initial bubble formation. Liu and his cooperator performed numerical simulation of one-dimensional explosive detonation and two-dimensional underwater explosion in the free field based on SPH method [2], [3], [4]. Afterwards, they developed DSPH program to improve precision of calculation [5]. However, the current research on simulation of the detonation phase of underwater explosion is limited to two-dimensional model. It is difficult to solve the problem of three-dimensional underwater explosion for the enormous calculation amount in SPH method. Many three-dimensional underwater explosion problems including column charge explosion are axial symmetrical so that axisymmetric SPH method shall be utilized to convert it to a two-dimensional problem in simulation. Although both A.G. Petschek, D. Molteni and L. Brookshaw developed axisymmetric SPH calculation program [6], [7], [8] based on column coordinate system, precision of calculation results cannot be guaranteed near the symmetrical axis yet. Omang et al. deduced axisymmetric SPH calculation equations [9] from kernel function and its precision is well proved via the Sod shock tube test [10]. But, the current research on axisymmetric SPH method mainly focuses on test of simple problems (such as Sod shock tube) to prove correctness of the program. Application of axisymmetric SPH method in underwater explosion field is still seldom published.
It is usual to simulate the phase from high temperature and high pressure initial bubble formed by underwater explosion to before bubble jet formation via BEM. BEM can fully simulate liquid movement and change of bubble form. However, it is assumed as a spherical charge explosion in the past research of bubble pulsing characteristics [11], [12], i.e. considering initial bubble formed by explosion spherical, while the initial bubble formed by column charge initiation at one end is axisymmetrical ellipsoid. Change of initial conditions may cause change of bubble pulsing characteristics; but research on bubble pulsing characteristics of non-spherical initial form is still scarce.
Due to gravity, bubble generated by underwater explosion will form high-speed jet under the force of static pressure gradient to convert the bubble to multiply connected toroidal bubble. The toroidal bubble will continuously shrink under the action of static pressure, and start to expand under the high internal pressure when it reaches an equilibrium position. It is usual to simulate the phase from bubble jet to toroidal bubble formation via BEM. The vortex ring model put forward by WANG and others [13], [14] is usually utilized to simulate toroidal bubble, in which the singly connected bubble is artificially cut into multiply connected bubble and a votex ring instead of votex sheet is placed within the bubble to simulate its toroidal phase. This method excessively introduces manual intervention and special numerical smoothing technique in the simulation of bubble jet and rebound. The speed of bubble jet is pretty high and bubble surface will yield large distortion in jet, yet SPH method is naturally advantageous to solve this type of problems. For many reasons, papers relevant to SPH method utilized to simulate bubble jet have not been published yet.
Considering the different physical characteristics in different phases of underwater explosion, SPH method and BEM have been combined to perform numerical simulation of the whole process of such a complex physical phenomenon as underwater explosion to fully utilize advantages of these two methods. Axial symmetry based SPH calculation equations are deduced in detail, mirror particle algorithm is proposed to solve the problem of high-speed particle penetrating through the symmetric axis and its detail numerical calculation implementation methods are expounded in this paper. The course from column charge detonation at one end to the formation of non-spherical initial bubble is simulated via axisymmetric SPH method, and bubble pulsing characteristics are simulated via BEM with calculation results of SPH taken as initial condition. Then taking calculation results of BEM prior to bubble jet as the initial condition, axisymmetric SPH method is used to simulate the process from bubble jet to toroidal bubble formation. To verify the feasibility of the numerical method, numerical simulation results and experimental results are compared in this paper.
Section snippets
Kernel function
In SPH method [2], integration expression of function f(r) is defined as follows:where Ω is the integration volume of r, with property of function δ(r − r′) indicated as follows:
If replacing function δ(r − r′) with smooth function W(r − r′, h), the integration function of f(r) can be written as follows:Where 〈〉 indicates the kernel approximation, W(r − r′, h) is smooth function and third-order B-spline function [15] is taken in this
BEM numerical model
Since it is assumed that fluid is irrotational, inviscid and incompressible in the simulation of underwater explosion via BEM [12], Laplace Equation is satisfied within the fluid domain Ω.where ϕ is velocity potential. According to Green Equation, velocity potential of any point in the fluid field Ω can be expressed by velocity potential at boundary S and its normal derivative. With infinite boundary condition [12], boundary integration equation can be expressed as follows:
Charge detonation simulation based on axisymmetric SPH method
Column charge explosion in free field with 0.012 m in diameter and 0.077 m in length is simulated via axisymmetric SPH method in this paper. The calculation model is shown in Fig. 1, where r1 = 0.006 m, r2 = 0.6 m, z1 = 0.077 m, z2 = 1.0 m, d = 0.5 m. In the model, 66,600 particles are distributed uniformly with inter-particle distance of 0.003 m. The TNT Column charge is initiated at its top.
The effectiveness of the proposed mirror particle method to solve the particle penetration problem is discussed firstly.
Conclusion
Axisymmetric SPH numerical model is established in this paper, the whole process of such complicated physical phenomenon as column charge underwater explosion is simulated by combining axisymmetric SPH method and BEM. Column charge underwater explosion experiment is carried out to verify validity of the numerical method. The following conclusions are obtained by research:
- (1)
In the simulation of underwater explosion, the proposed mirror particle algorithm can solve the particle penetration problem
Acknowledgements
This research is supported by the National Science Foundation (50939002), the Excellent Young Scientists Fund (51222904) and Major Basic Research Projects of National Security (613157) of China, to which the authors are most grateful.
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