## 4 Calculation Results and Discussion
### Case 5: Blunt-Nosed Projectile  
The simulation results for the blunt-nosed projectile are shown in Figure 10. Similar to the approach above, the results are compared to the numerical results of Case 3 and Case 4 at corresponding times  
Figure 9: Isopycnic (the lower half) and isobars (the upper half) for cone-ended projectile. (**a**) \(t=0.590\) ms; (**b**) \(t=0.671\) ms; (**c**) \(t=0.736\) ms; (**d**) \(t=0.989\) ms; (**e**) \(t=1.091\) ms; (**f**) \(t=1.371\) ms.  
with the same contour levels. As expected, some phenomena found in the former cases are observed again in this case. However, some differences are observed. For instance, the RMI of the blunt-nosed projectile and the flat-nosed projectile are similar, but the location and the arrival time of the separation shock on the projectile surface for the blunt-nosed projectile are the same as those of the cone-nosed projectile. This is because the BSW intensity of the blunt-nosed projectile is between those of the other two. The Schlieren photos of the last three cases are shown in Table 3. The wake waves can also be clearly identified in the Schlieren photos. Figure 11 shows the acceleration and deceleration history of the projectiles. These results give a good insight into the axial force of the projectile. From the charts, the motion of the projectiles is almost constant between the three projectile models with only a few nuanced differences. Assuming that the projectiles are initially at a uniform motion state, the acceleration of the projectile reaches its maximum in a very short time under the given initial conditions of the flow field. Subsequently, with the decline of the bore gas energy and the increase of the resistance in front of the projectile, the acceleration of the projectile drops continuously. Eventually, the projectile flies steadily, and the acceleration approaches a constant value. It is worth noting that at \(t=0.31\) ms the projectile nose exits the muzzle and enters the expanded low-pressure region of the initial flow field. Next, the air resistance in front of the projectile declines and causes a slight increase in the acceleration of the projectile. In addition, the acceleration of the projectile falls dramatically between \(t=0.5\) ms to \(t=0.7\) ms. This is due to the increase in the resistance in front of the projectile caused by the substantial rise in the pressure and the density from the generation of the BSW as the projectile nose passes through the SSW. The decrease in acceleration is also due to the weakened driving effect of the gas on the projectile caused by the sharp decline in the pressure behind the projectile. This decline in pressure is caused by expansion waves as the gunpowder gas behind the projectile rushes out of the muzzle. This driving effect vanishes after the projectile base exits the SSW. It can be seen in the results that the blunt-nosed and cone-nosed projectiles are almost the same regarding variations in acceleration but are significantly different from the flat-nosed projectile. The initial conditions of the gas behind projectile are constant and, therefore, the observed differences are caused by the different resistance in front of the projectile, which is closely associated with the projectile nose area. The stable values of acceleration of the cone-nosed, blunt-nosed, and flat-nosed projectiles are \(-1361.08\) m/s\({}^{2}\), \(-1849.365234\) m/s\({}^{2}\) and \(-6542.96875\) m/s\({}^{2}\), respectively. In the  
Figure 10: Isopynic (the lower half) and isobars (the upper half) for blunt-ended projectile. (a) \(t=0.736\) ms; (b) \(t=0.989\) ms; (c) \(t=1.091\) ms; (d) \(t=1.371\) ms.  
transitional stage of the stable flight of a projectile (t = 0.7 ms to t = 1.2 ms), the fluctuations of acceleration are related to the wake wave, the DSW, the shock focusing, and interactions between SW and CS. However, the amplitude of the fluctuations is relatively low. During the simulated period, although the acceleration and deceleration are large, the duration time is very short.