## 4 Results and Discussions
### Nozzle Axial Profiles Results  
To examine the pressure readings from the experiments and simulations, we have plotted the pressure profiles in Figure 8. The pressure recordings along the nozzle axis are normalized with the inlet total pressure for each simulation. These pressure profile is typical of the behaviour observed where the flow is fully supersonic in the divergent part of the nozzle. We observe that the pressure profile overlaps, irrespective of the amount of impurity in the gas mixture. The pressure profile of the simulations matches one of the experiments perfectly in the converging region of the nozzle. However, a slight difference can be observed after the throat, where a pressure oscillation occurs in this almost sonic part of the nozzle. The variation in CO\({}_{2}\) properties due to local flow acceleration and phase change can also be considered as a possible mechanism to explain this oscillation problem observed near the nozzle's critical point, as discussed in reference [9]. Still, this oscillation issue in the pressure profiles has not been captured experimentally. Overall, the match with the experimental data is satisfactory.  
Figure 7: Condensation as observed in experimental flow visualization adapted from [PERSON] et al. [11] (**top**); snapshots of gas volume fraction from the numerical simulation of case 1 in the fully supersonic regime (**bottom**). The front of the condensation is located at the throat.  
The temperature and Mach number behavior are shown in Figure 9. We can see that impurities do not affect these profiles very much, although a slight temperature rise can be noted in the divergent part of the nozzle when the amount of impurity is reduced. Indeed, changing the amount of N\({}_{2}\) in the mixture obviously leads to thermodynamic and transport property variations. For example, a specific heat ratio (\(\gamma\)) oscillation can be seen in Figure 10a at the same place (close to the condensation cloud interface) as the pressure oscillation. Similar interface fluctuations of the specific heat capacity due to real fluid non-linearity have been discussed in [PERSON] et al. [16; 41]. The importance of considering the real fluid effect on the thermodynamic and transport properties has been also highlighted by [PERSON] et al. [9]. They demonstrated that the various theoretical relationships often employed for convergent-divergent nozzle calculations, and expressed in terms of the isentropic exponent \(\gamma\) in an ideal gas formulation, must be modified in terms of a modified exponent \(n_{\rm s}=\frac{\gamma}{\beta_{\rm T}B}\) when a real fluid formulation is adopted, where \(\beta_{\rm T}\) is the isothermal compressibility. Figure 10b reveals that \(n_{\rm s}\) is less sensitive to the presence of impurities and, indeed, it is appropriate for basic calculations of compressible flow relationships for a real fluid in converging-diverging nozzles. This shows the importance of having a real fluid model over the ideal gas assumptions in this type of study.  
Figure 8: Normalized pressure profile compared to Lettieri experiments for Case 1 with various % impurities. \(P_{0}=P_{\rm t}\). The throat is located at X = 0.  
Figure 9: Effect of various % impurities composition on (**a**) temperature and (**b**) Mach number profiles of the CO\({}_{2}-\)N\({}_{2}\) mixtures.