## 2 Numerical Methods
### Boundary Conditions and Initial Conditions  
Numerical boundaries in Figure 1 are applied as follows: the barrel and the projectile apply fixed adiabatic and moving adiabatic wall boundary conditions, respectively; the edges of the C zone use a free boundary condition; the axis of the barrel is treated as symmetry axis boundary condition, and thus only the upper half of the domain is calculated due to symmetry; to reduce the effect of reflected waves caused by computational domain boundaries on the flow field, non-reflecting boundary conditions are applied at the inflow and outflow boundaries.  
The variation of the projectile velocity moving in the bore can be derived using the interior ballistic equation. Based on this, combined with Rankine-Hugoniot relations and standard adiabatic shock relations, the characteristics of the initial flow field between the projectile nose and the muzzle, as well as the location of the projectile, can be obtained. From the projectile velocity-time curve derived from classical interior ballistics theory (as shown in Figure 3), the distance from the projectile base to the bore bottom when the initial shock wave arrives at the muzzle is obtained as \(l_{3}=5540\) mm. In the simulations, this point in time is used as the initial time (t = 0). The initial conditions of the region behind the projectile (zone A), such as the distributions of initial pressure and velocity along the bore, were obtained from the interior ballistic calculation. The initial flow field of zone B was obtained from a calculation of the flow field between the projectile front and the initial shock wave. The initial parameter distributions of zone A and zone B are shown in Figure 4. Besides, the gas in zone C is  
\begin{table}
\begin{tabular}{c c c} \hline
**Variable** & **Value** \\ \hline Outer diameter \(d_{1}\) (mm) & 176 \\ Inner diameter \(d_{2}\) (mm) & 125 \\ Barrel length \(l_{4}\) (mm) & 6450 \\ Outer field length \(l_{5}\) (mm) & 5000 \\ Outer field height \(d_{3}\) (mm) & 6000 \\ \hline \end{tabular}
\end{table}
Table 1: Specification of 125 mm gun.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Variable** & **Flat-Ended** & **Blunt-Ended** & **Cone-Ended** \\ \hline Projectile length \(l_{1}\) (mm) & 420 & 420 & 420 \\ Cylindrical length \(l_{2}\) (mm) & 420 & 300 & 300 \\ Inertia moment \(l_{XX}\) (kg\(\cdot\)m\({}^{2}\)) & 0.01445 & 0.01414 & 0.013669 \\ Inertia moment \(l_{YY}\) (kg\(\cdot\)m\({}^{2}\)) & 0.1272 & 0.07828 & 0.085464 \\ Projectile mass \(m\) (kg) & 7.4 & 7.4 & 7.4 \\ \hline \end{tabular}
\end{table}
Table 2: Specification of 125 mm shell.  
Figure 2: Mesh around the projectile.  
treated as an ideal gas with a temperature of T = 293 K, a pressure of Pa = 1.013 \(\times\) 105 Pa, and a molecular weight of 23.5. During numerical simulations, the projectile speed will subsequently vary according to [PERSON]'s laws of motion.  
For the convenience of discussion and analysis of shock-wave phenomena, in the following figures acronyms are used to identify certain phenomena. Namely, SW for shock wave, PSW for precursor shock wave, SSW for secondary shock wave, CS for contact surface, SL for shear layer, BSW for bow shock wave, LSW for leading shock wave, DSW for diffraction shock wave.