## 4 Results and Discussions
### Transient Flow Characteristics  
The flow develops transiently in the de Laval nozzle as the outlet pressure relaxes to the estimated (\(P_{exit}\)) value given in Table 1. The flow acceleration from subsonic to fully supersonic regimes in the nozzle diverging part is explained here based on the numerical results of Case 1 with the four following steps:  
1. A subsonic regime in both the converging and diverging part of the de Laval nozzle. In this first step, the flow accelerates progressively in the converging part, but it decelerates along the diverging part. The Mach number is still lower than one in the entire nozzle, and has its maximum value at the throat, as shown Figure 5b. In addition, the pressure profile is also shown in Figure 5a to progressively decrease in the converging part, but it increases along the diverging part towards (\(P_{exit}\)). Figures 5c and 6a show that the condensation is not present in the nozzle in this subsonic regime and, consequently, the nozzle is only filled in the gas phase.
2. A sonic regime where the flow has been accelerated up to a Mach number of one (see Figure 5e) happens simultaneously with the throat pressure decreasing to a critical value (approximately 3.3 MPa), as shown in Figure 5d. Figures 5f and 6b show that the presence of condensation is still not observed just at the moment of meeting critical conditions.
3. A supersonic/subsonic transition regime. In fact, because the pressure at the nozzle outlet (\(P_{exit}\) = 1.973 MPa) is still lower than the pressure value at the throat (3.3 MPa), the flow continues to accelerate and reaches the supersonic regime in the diverging section, as shown in Figure 5h,g. In these figures, a shock wave created just after the throat, and moving towards the nozzle outlet inside the divergent part, is successfully captured when the outlet pressure is relaxed towards (\(P_{exit}\)). The presence of condensation is observed in Figure 6c, as well as the profiles of gas volume fraction in Figure 5i.
4. A fully supersonic regime in the diverging part of the de Laval nozzle is finally obtained when the shock wave leaves the computational domain, as shown in Figure 5j.k. However, the converging part remains in the subsonic regime. The nozzle is filled with condensation from the throat to the exit downstream. Such fully supersonic acceleration of the flow through a converging-diverging nozzle is used in practical applications such as rocket engine lift-off.  
Figure 5: Evolution of Pressure (**a**,**d**,**g**,**j**), Mach (**b**,**e**,**h**,**k**) and GVF (**c**,**f**,**i**,**i**,**i**) profiles as observed during various regimes of flow acceleration in the nozzle from the transient numerical simulation of case 1: (**a**–**c**) subsonic regime, (**d**–**f**) sonic regime, (**g**–**i**) supersonic with shock in the nozzle and (**j**–**l**) fully supersonic with shock outside of nozzle. Note that the flow acceleration period is 0.1 s, during which the different flow regimes displayed in the different lines of the figure follow one another.  
Figure 6: Condensation and shock wave behavior during various regimes of flow in nozzle as observed in simulation of case 1. Condensation is located in the supersonic diverging part of the nozzle behind the shock wave. Shown here is the illustration of the observed gas volume fraction, transitioning from (**a**) subsonic regime, (**b**) sonic regime, (**c**) supersonic with shock in the nozzle and (**d**) fully supersonic with shock outside of nozzle. Note that the flow acceleration period is 0.1 s, during which the different flow regimes displayed in this figure succeed one another.