# Study on the Interference Process of Liquid Radial Reflux on the Stability of a Shaped Charge Jet
## 2 Theoretical Analysis  
In this study, jet penetration into the LFCS (see Figure 1) is divided into five stages: jet penetration into the cover plate, jet reaming in the liquid, side-wall reflection interference, bottom plate reflection interference, and liquid spraying and side-wall secondary reflection interference. This study proposes several assumptions to simplify the calculation and formula derivation:  
1. The radial and axial pressures of the initial reaming are equal, and the product of reaming pressure and hole wall area is constant. 2. During penetration, the jet is approximately a cylindrical rod. 3. Considering that the effect of liquid viscosity is small, it is ignored in the analysis of penetration [24]. 4. The effect of shell plastic deformation in the penetration process is ignored. Based on the above assumptions, the five stages of the jet penetration into the LFCS are as follows:  
The first stage is the jet penetration into the cover plate, which can be treated as a steady-stage penetration. The second stage is the jet reaming in the liquid. The residual velocity \(v_{jw}\) and time spent \(t_{jw}\) after \(h_{w}\) penetration of the liquid layer can be obtained as follows by using virtual origin theory [25]:  
\[v_{jw}=v_{jf}\Bigg{(}\frac{z_{o}+H+\delta_{f}}{z_{o}+H+\delta_{f}+h_{w}} \Bigg{)}^{\sqrt{\rho_{w}/\rho_{j}}} \tag{1}\]  
\[t_{jw}=\frac{z_{o}+H+\delta_{f}+h_{w}}{v_{jw}} \tag{2}\]  
Figure 1: Schematic of the interaction between the projectile and the target. where \(v_{jf}\) is the residual velocity after jet penetration into the cover plate, \(\rho_{w}\) is the density value of the liquid, \(\rho_{j}\) is the density value of the SCJ, and \(z_{o}\) is the coordinate value of the virtual origin. Under the unconstrained condition, the radius and time of crater reaming of the SCJ in the liquid under the influence of shock waves can be obtained by using the penetration equation. When the jet penetrates the LFCS, the liquid pressure continuously increases due to the restriction of the shell. The crater radius of the liquid, \(r_{cw}\), is assumed to reach the maximum when the liquid pressure \(P_{w}\) increases to the same value as the reaming pressure. Then,  
\[P_{w}=P_{c}=P_{jw}\frac{r_{j}{}^{2}}{r_{cw}{}^{2}} \tag{3}\]  
where \(P_{jw}\) is the penetration dynamic pressure, \(P_{c}\) is the reaming pressure, and \(r_{j}\) is the jet radius. [PERSON] and [PERSON] found that the relationship between liquid pressure and density [28] is as follows:  
\[P_{w}=\frac{\rho_{w}/\rho_{w0}-1}{2.3\times 10^{-9}-1.7\times 10^{-9}\rho_{w}/ \rho_{w0}} \tag{4}\]  
where \(\rho_{w0}\) is the density of the liquid at atmospheric pressure. Meanwhile, the liquid density at a certain liquid level can be approximated as follows:  
\[\rho_{w}=\rho_{w0}\frac{r^{2}-r_{cw}{}^{2}}{r^{2}} \tag{5}\]  
The crater radius \(r_{cw}\) formed by the jet in the liquid can be obtained through Equations (3)-(5). Given that the penetration velocity \(u\) of the jet is greater than the sound velocity of the liquid \(c_{w}\), a shock wave will be generated in the liquid, and the propagation velocity of the shock wave can be approximated as follows [29]:  
\[v_{s}=\frac{c_{w}}{\cos[\arcsin(c_{w}/u)]} \tag{6}\]  
At equilibrium, \(v_{s}=u\)[25]. The shock wave propagates to the side wall of the cylindrical cavity of the LFCS with the penetration. After being reflected by the side wall, the reflection shock wave propagates in the opposite direction until it meets the crater wall formed by jet penetration (Figure 2). Figure 2: Schematic of the shock wave side-wall reflection in liquid. The third stage is the side-wall reflection interference, in which the reflection shock wave propagates to the crater wall and drives the radial reflux of the liquid to interfere with the SCJ. Figure 3 shows the propagation path of the shock wave in the liquid. In the lateral direction, reflection occurs when the shock wave contacts the side-wall surface, and the shock wave has only one impact action on the shell of the LFCS and the jet.