## 5 Results and Discussion
### Internal and External Boundary Pressure Distribution  
In order to own the evolution process of the detonation wave in tube, the pressure distribution in different curvature pipes is shown in Figure 8 from \(\theta\) = 15\({}^{\circ}\) to \(\theta\) = 60\({}^{\circ}\). In the 6 pictures, \(\theta\) = 15\({}^{\circ}\) and \(\theta\) = 20\({}^{\circ}\) is in a similar situation, the same trend of the pressure is shown on the upper wall and lower wall. The pressure of the outer wall increases and then decreases, while the inner wall increases slowly, which explains why the small curvature bending has hardly any effect on the propagation of detonation. At the same time, the pressure of the inner and outer walls is not changed greatly. The pressure on the inner wall of diagram \(\theta\) = 30\({}^{\circ}\) decreases gradually, and with the increase of curvature, the rarefaction effect on the inner wall is strengthened gradually. However, there is a critical detonation point different to \(\theta\) = 15\({}^{\circ}\) and \(\theta\) = 20\({}^{\circ}\) where the pressure of inner wall suddenly increased in the late stage of detonation propagation and then decreased. All \(\theta\) = 40\({}^{\circ}\), \(\theta\) = 45\({}^{\circ}\) and \(\theta\) = 60\({}^{\circ}\) are similar and have a big jump in the downstream of the inner wall where the pressure change increased more than 7.3 atm. Owing to the rarefaction of the inner wall, the detonation wave is extinguished, while the detonations are overdriven on the outer wall. A larger pressure gradient is generated between inner and outer walls and moves to the upper wall along with the propagation of detonation wave. Then a strong transverse wave is produced and compacts with the inner wall leading to a reflection which reinitiates the unburned gas rapidly.  
Figure 7: The change of quenching position with bend curvature.  
Figure 8: The history pressure curves of inner and outer bend pipe.  
From the pressure in Figure 8\(\theta\) = 40\({}^{\circ}\), \(\theta\) = 45\({}^{\circ}\) and \(\theta\) = 60\({}^{\circ}\), it can be found that a peak pressure on outer wall exists when the detonation wave begins entering the pipe, which indicates that the detonation has a Maher reflection increasing the pressure. As the detonation wave propagates through the downstream of the pipe, the pressure on the outer wall decreases gradually, while a peak pressure appears on the inner wall of the downstream pipe. This peak pressure is much higher than that in the upstream pipe. It shows the beginning of the detonation. At the same time, it is shown that when the detonation wave propagates in the bend with a certain curvature, the range of pressure increase appears both in the upstream and downstream of the bend, but the pressure on the inner wall of downstream is much larger than the outer wall. Therefore, considering the pressure of the material surface in the industrial design, these two positions of the bend pipe should be taken into account and the inner wall of downstream especially needs an extra reinforcement.  
From the maximum pressure curve in Figure 9, it is obvious that the 30\({}^{\circ}\) curvature is critical value. This also verifies that reinitiation will appear when the degree of pipe is more than 30\({}^{\circ}\), which is also shown in Figure 6. The peak of the pressure in the 45\({}^{\circ}\) bend pipe is the largest one and this fits to the result in Figure 8. The results show that the pressure reinitiation at this ratio is larger than that other curvature on the inner wall. Based on this conclusion, the curvature of bend surface considered in actual engineering damage should try to avoid 45\({}^{\circ}\).