## 4 Results and Discussions  
### The Influence of the Initial Pressure on Nozzle Startup  
The movement of gas in the nozzle can be regarded as the movement of gas in the shock tube with a variable cross-section. For a given reservoir region pressure, it can be seen from theoretical relation Equation (9) that the Mach number of the moving shock wave can be increased by reducing the initial pressure (Pe). \[p_{5e}=[1+\frac{2\gamma_{e}}{\gamma_{e}-1}(M_{\rm s}^{2}-1)]\cdot[1-\frac{\gamma _{5}-1}{\gamma_{e}+1}a_{5}(M_{\rm s}-\frac{1}{M_{s}})]^{-\frac{2\gamma_{5}}{ \gamma_{5}-1}} \tag{9}\]  
This indicates that the shock wave can be accelerated by reducing the initial pressure, so as to shorten the nozzle startup time. In order to analyze the influence of the initial pressure on the nozzle startup, the duration of the nozzle starting process was calculated under the conditions of \(P_{5}=1.333\) MPa and \(T_{5}=4120\) K with different initial pressure. Through the analysis of the calculation results, it was found that the higher the initial pressure, the lower the velocity of the waves in the nozzle, including the starting shock wave, the second shock wave, and the left-traveling expansion wave. The velocity distribution of the nozzle axis (left) and Ma contour (right) in different initial pressures at the same moment is shown in Figure 7 and reveals that the increase of the initial pressure not only changes the wave velocity but also the shape of the waves, both of which have different levels of influence on the nozzle startup duration. Because the flow field in the center of the nozzle is not directly disturbed by the low-velocity zone, the central startup duration is related to the velocity of the waves, while the complete startup duration is more closely related to the shape of the waves. The calculation results of the duration of the nozzle starting under different initial pressure show that the relationship between the nozzle center startup duration, instability duration and initial pressure is complex. As the initial pressure increases from a small value, the center startup duration does not change, but when the initial pressure exceeds a certain value, the center startup duration begins to increase. At the same time, the central instability duration decreases at first and then increases, and there is a critical pressure to minimize it, as shown in Figure 8.  
Figure 6: Comparison of the CFD (computational fluid dynamics) results and experimental data. (**a**) Heat Flux. (**b**) Pitot pressure. When the initial pressure is less than 100 Pa, the center startup duration remains unchanged at about 1.6 ms. In this stage, the center instability duration decreases gradually with an increase in the initial pressure. However, when the initial pressure is greater than 100 Pa, the center startup duration and instability duration begin to increase. As can be seen in Figure 8, when the initial pressure is increased to 200 Pa, the center startup duration increases from 1.6 to 2.6 ms, and the center instability duration increases from the lowest 0.72 to 1.54 ms. In other words, when the initial pressure is about 100 \(Pa\), the flow field in the center of the nozzle stabilizes quickly, and the starting wave has the shortest action time of the model. The stability of the center flow field directly depends on the velocity of the left-traveling expansion waves. Under different initial pressure, the factors that affect the velocity of the left-traveling expansion waves are different, resulting in the existence of a critical initial pressure in the starting of the nozzle center. When the initial pressure is greater than the critical pressure, the starting shock wave, the secondary shock wave, and the left-traveling expansion waves all decelerate during the nozzle starting process. The deceleration of the left-traveling expansion waves directly increases the center startup duration.