## 4 Results and Discussions
### The Influence of the Reservoir Region Condition on Nozzle Startup  
In order to evaluate the influence of the reservoir region temperature and pressure on the nozzle starting parameters (startup time, instability time and critical initial pressure), the following two cases are studied:  
Case 1: \(P_{5}\) = 15 MPa; \(T_{5}\) = 3000 K, 4500 K, 6000 K  
Case 2: \(T_{5}\) = 6000 K; \(P_{5}\) = 5 MPa, 15 MPa, 25 MPa  
A large number of numerical calculations are performed for the starting process under different initial pressure values. Both Case 1 and Case 2 used the above nozzle with a nominal Mach number of 8 and an outlet diameter of 510 mm. The reservoir region component of each case is the equilibrium gas. In Case 1, when the total temperature is 3000, 4500, and 6000 K, the critical initial pressure of the nozzle is 1700, 1000, and 800 Pa, respectively, and the minimum center startup duration is 1.95, 1.5, and 1.25 ms, respectively. At the same time, the center instability duration also decreases slightly. In other words, with an increase in the total temperature, the critical initial pressure of the nozzle decreases, and the minimum center startup duration and instability duration also decrease. The minimum center startup duration is approximately proportional to \(-0.6\) power of the total temperature, as shown in Figure 13--center startup duration. In addition, the curve of the center startup duration with the initial pressure also shows that when the initial pressure is greater than the critical pressure, the center startup and instability duration grow faster under the condition of high total temperature. In other words, the center startup and instability duration of the nozzle is more sensitive to the initial pressure when the nozzle is operated under the high total temperature. The center startup and instability duration of the nozzle depends on the time when the left-traveling expansion waves leave the nozzle completely. Since these waves are left-traveling waves, they move with a velocity of \(u-a\), and travel from the nozzle throat to the outlet in the following time:  
\[t=\int_{0}^{x}\frac{dx}{u-a}=\int_{0}^{x}\frac{dx}{a(M-1)} \tag{10}\]  
Figure 13: The center startup time and instability time of the nozzle variation with the initial pressure (\(P_{5}=15\) MPa). where \(t\) is the time for the left-traveling expansion waves to move from the throat to the nozzle outlet, \(x\) is the distance from the throat to the nozzle outlet along the nozzle axis, \(u\) is the local velocity, a is the local sound velocity, and \(M\) is the local Mach number. The parameters \(u,a\), and \(M\) are all functions of \(x\). When the reservoir region temperature rises, the temperature of the flow field in the nozzle increases, and the local sound velocity increases, and although M depends on the gas expansion ratio, it is only related to the nozzle profile. As can be seen from Equation (10), as the local sound velocity increases, the integral value decreases, and the movement time of the left-traveling expansion wave in the nozzle is reduced. Therefore, the reservoir region temperature is higher, and the nozzle can start more quickly. There is a premise that the initial pressure is less than the critical pressure; if the initial pressure is high, the nozzle center startup duration is also affected by the initial pressure. In Case 2, the reservoir region temperature is 6000 K, and the pressure varies greatly (5\(\sim\)25 MPa), but the difference between the center startup and the instability duration is very small, especially when the initial pressure is less than the critical pressure. The critical initial pressure varies with the different reservoir region pressure, but it is the same as the ratio of the critical initial pressure to the flow pressure at the nozzle outlet when the flow field is stable, which is about 0.4 in this case.