# Study on the Interference Process of Liquid Radial Reflux on the Stability of a Shaped Charge Jet
## 2 Theoretical Analysis
According to the propagation characteristics of the shock wave, the two limiting propagation directions of the shock wave during jet penetration into the liquid layer are as follows: (1) the shock wave propagates along the normal direction of the Mach angle \(\vec{n_{1}}\), and (2) the initial shock wave propagates along the radial direction \(\vec{n_{2}}\). The two shock wave propagation paths form corresponding to the maximum velocity and minimum velocity of the jet interfered by the liquid in the side-wall reflection interference stage, respectively. According to the geometric relationship of the propagation path, we can obtain  
\[r_{j}+v_{s}t_{s1}\cos\beta+r_{cw}=2r \tag{7}\]  
\[r_{j}+v_{s}t_{s2}+r_{cw}=2r \tag{8}\]  
where \(t_{s1}\) and \(t_{s2}\) are the propagation times of the shock wave in two different directions. When the reflection shock wave propagates to the penetration crater wall to make the liquid converge radially, the following expressions exist:  
\[t_{jw}=\frac{z_{o}+H+\delta_{f}+h_{w}}{v_{jw}} \tag{2}\]0  
\[t_{jw}=\frac{z_{o}+H+\delta_{f}+h_{w}}{v_{jw}} \tag{2}\]1  
\[t_{jw}=\frac{z_{o}+H+\delta_{f}+h_{w}}{v_{jw}} \tag{2}\]2  
\[t_{jw}=\frac{z_{o}+H+\delta_{f}+h_{w}}{v_{jw}} \tag{2}\]3  
\[t_{jw}=\frac{z_{o}+H+\delta_{f}+h_{w}}{v_{jw}} \tag{2}\]4  
where \(P_{bs}\) is the closing pressure, \(P_{rs}\) is the intensity of the side-wall reflection shock wave, \(K\) is the reflection coefficient of the shock wave, \(u_{bs}\) is the closing velocity of the liquid  
\[t_{jw}=\frac{z_{o}+H+\delta_{f}+h_{w}}{v_{jw}} \tag{2}\]5 Propagation path of the shock wave. particle, \(k\) is a constant, and \(t_{\text{bs}}\) is the time taken for the liquid particle to move from the crater wall to the jet surface. The jet velocity at the position where the liquid converges to the jet surface can be obtained for the two shock wave propagation paths:  
\[t_{jw}=\frac{z_{o}+H+\delta_{f}+h_{w}}{v_{jw}} \tag{2}\]6  
\[t_{jw}=\frac{z_{o}+H+\delta_{f}+h_{w}}{v_{jw}} \tag{2}\]7  
where \(t_{bs1}\) and \(t_{bs2}\) are the times of the liquid's movement to the jet surface corresponding to the two propagation paths. In the side-wall reflection interference stage, the velocity interval of the jet interfered by the liquid is \(\left[v_{jd2}\big{|}_{h=0}\,,\;v_{jd1}\big{|}_{h=h}\right]\). The fourth stage is the bottom reflection interference, which is marked by the impact of the jet on the bottom plate. The shock wave is then reflected at the interface between the liquid and the bottom plate. The bottom plate reflection shock wave propagates to the liquid crater wall and causes secondary reflection at the free interface, which drives the radial reflux of the liquid. At the interface between the liquid and the bottom plate, we can obtain  
\[t_{jw}=\frac{z_{o}+H+\delta_{f}+h_{w}}{v_{jw}} \tag{2}\]8  
\[t_{jw}=\frac{z_{o}+H+\delta_{f}+h_{w}}{v_{jw}} \tag{2}\]9  
Figure 1:0  
Figure 1:1  
where \(P_{rb}\) is the intensity of the bottom reflection shock wave at the crater wall, \(u_{bb}\) is the liquid closing speed, \(t_{rb1}\) is the time for the bottom plate reflection shock wave to propagate to the crater wall, and \(t_{rb2}\) is the liquid closing time. In combination with Equations (16)-(19), the maximum velocity of the interference jet at the bottom plate reflection interference stage can be obtained as follows:  
Figure 1:2  
Considering the overlap and continuity of the bottom plate reflection interference and the side-wall reflection interference (see Figure 4), the total interference jet velocity interval disturbed by the liquid driven by the side-wall reflection shock wave and the bottom plate reflection shock wave was obtained as \(\left[v_{jd2}\big{|}_{h_{w}=0}\,,\;v_{jd3}\right]\), in which the liquid continuously interfered with the jet. The fifth stage is liquid spraying and side-wall secondary reflection interference. (1) Liquid spraying  
When the jet penetrates through the bottom plate of the LFCS, the liquid is rapidly sprayed out along the penetration crater under the action of the pressure difference.