## 4 Calculation Results and Discussion
### Case 3: Flat-Nosed Projectile  
The results sorted with respect to time for Case 3 are shown in Figure 7. In order to distinguish the contact surface and shock waves, the figure is divided along the axis of symmetry. The upper half of the figure shows the pressure contour, and the lower half shows the density contour. The numbers of contour levels are set 50 and 70 for pressure contours and density contours, respectively, in order to make the features of interest more prominent. Figure 5: Comparison of numerical and experimental interferograms in the shock-tube experiment. Figure 6: Comparison of numerical and experimental results in the projectile flight experiment. Figure 7a shows the muzzle flow field characteristics at \(t=0.271\) ms. At this time the projectile is located close to the muzzle, the airstream rushes out of the barrel and forms the first-blast around the muzzle. The high-speed airstream shears the still air outside the bore after the ejection from the muzzle and forms the SL. A low-pressure zone is then formed in the rear region, which causes the airstream to flow in reverse towards the external wall of the barrel and forms the primary vortex ring. After ejection from the muzzle, the precursor shock wave (PSW) develops nearly spherically in the air. The front of the SSW and CS develop spherically along the axis. In the radial direction, under the effect of the primary vortex ring, the edge of the SSW is entrained into the low-pressure vortex  
Figure 7: Isopycnic (the lower half) and isobars (the upper half) for flat-ended projectile, (**a**) \(t=0.271\) ms; (**b**) \(t=0.472\) ms; (**c**) \(t=0.590\) ms; (**d**) \(t=0.671\) ms; (**e**) \(t=0.736\) ms; (**f**) \(t=0.989\) ms; (**g**) \(t=1.091\) ms; (**h**) \(t=1.371\) ms.  
core, and the \"coronal\" front is formed. The CS is then entrained into the vicinity of the muzzle so that the air initially in the bore and in front of the projectile is restrained. Therefore, the area between the muzzle and the SSW forms an under-expanded region. Figure 7b shows the muzzle flow field characteristics at \(t=0.472\) ms when part of the projectile has exited the muzzle and the first blast further develops in the external flow field continuously. The energy of the first-blast attenuates gradually during its propagation and replenishing energy is impeded due to the \"blocking\" effect of the projectile on the gas behind the projectile. Therefore, the flow characteristics begin to weaken. This weakening causes the attenuation of the primary vortex ring, declined propagation speed of the PSW, declined height of the SSW, and reduced the range of the CS. Since the movement of the projectile is faster than that of SSW, the projectile gradually catches up to the SSW and squeezes the under-expanded region, causing reduced volume and a corresponding increased density in this region. As the projectile moves, the small vortex formed around the corner point of the muzzle by the initial jet flow is absorbed by the sidewall of the projectile and moves with it. Simultaneously, the SSW deforms under the squeezing effect of the airstream in front of the projectile. Figure 7c shows the muzzle flow field characteristics at \(t=0.590\) ms when the projectile base exits the muzzle and the gunpowder gas in the bore begins to exit which forms the second-blast. The second blast is composed of the LSW, the SSW and the CS. Among these, the CS of the second blast separates the airstream disturbed by the first-blast from the second-blast flow field. A non-circular wavefront is created at this moment because the second blast moves through the non-uniform flow field caused by the first blast. When the projectile nose is located in the expanded region, because of the high airstream velocity and subsonic movement of the projectile relative to the airstream, the BSW does not appear at the projectile nose. When the projectile nose rushes out of the SSW of the first blast, the movement relative to the air in front of the projectile becomes supersonic.