A finite difference method of three characteristic lines of two-dimensional non-isentropic steady flow of cylindrical explosive underwater explosion
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摘要: 针对二维定常可压缩超声速非等熵柱状流，提出一种特征线差分解法，通过在沿马赫线的相容方程中添加沿流线的熵变项以描述非等熵效应，得到等熵流和非等熵流均适用的三族特征线方程组。根据水下爆炸近场特点，建立无限长柱状装药的定常模型，将三族特征线方程组用有限差分法离散求解，通过构造合适的网格保证计算格式可以数值上收敛，由此编制程序并计算几种柱状炸药的水下爆炸近场冲击波。对比有限元模拟结果和实验结果发现，特征线差分法可以比较准确地捕捉冲击波形状并计算冲击波后流场，从而验证了所提出的三族特征线差分法的准确性。Abstract: In the present paper, we proposed a two-dimensional finite difference method (FDM) of characteristic lines to address problems of the non-isentropic steady flow of cylindrical explosive underwater explosion. This method describes the non-isentropic effect by adding an entropy-related variable along the flow line to the pressure-related equation along the Mach line, so that both the isentropic flow and the non-isentropic flow can be described in the same equations of the characteristics. Based on the features of the near-field shock wave we firstly modeled the underwater explosion with an infinitely long cylindrical explosive, then discretized those equations using this finite difference method and constructed an appropriate grid to ensure the numerical convergence, and finally calculated the underwater near-field shock wave for several explosives by programming. The numerical examples showed that the results of this method are consistent with those of the commercial finite element software AUTODYN and those of experiments, suggesting that the FDM of characteristics can capture the shock wave front with relatively high accuracy, and confirming that this method is applicable to solving problems in cylindrical explosive underwater explosion.
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图 2 二维特征线差分法的计算格式
Figure 2. Computational scheme of two-dimensional FDM of characteristics
图 5 柱状TNT水下爆炸在R=2R0处的压力时程曲线
Figure 5. Pressure history curve at R=2R0 in underwater explosion of cylindrical TNT
图 3 4种柱状PETN炸药水下爆炸冲击波形状
Figure 3. Underwater shock wave shape of four kinds of cylindrical explosives PETN
图 4 柱状SEP炸药水下爆炸冲击波形状
Figure 4. Underwater shock wave shape of cylindrical explosive SEP
表 1 几种炸药的爆轰产物JWL状态方程参数
Table 1. Parameters of JWL equation of state for several explosives
炸药 CJ参数 JWL状态方程参数 ρ0/(g·cm-3) D/(km·s-1) pCJ/GPa A/GPa B/GPa R1 R2 ω PETN 0.88 0.880 5.170 6.2 348.62 11.29 7.0 2.0 0.24 PETN 1.26 1.260 6.540 14.0 573.10 20.16 6.0 1.8 0.28 PETN 1.50 1.500 7.450 22.0 625.30 23.29 5.25 1.6 0.28 PETN 1.77 1.770 8.300 33.5 617.05 16.93 4.4 1.2 0.25 TNT 1.630 6.930 21.0 373.77 3.75 4.15 0.9 0.35 SEP 1.310 6.970 15.9 372.00 3.48 4.59 1.06 0.29 
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表 2 水的Mie-Grüneison状态方程
Table 2. Mie-Grüneison equation of state of water
ρ0/(g·cm-3) e0/(J·kg-3) pCJ/GPa A1/GPa A2/GPa A3/GPa B0 B1 T1/GPa T2/GPa 1.000 361.9 6.2 2.2 9.54 14.57 0.28 0.28 2.2 0
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