# Uncertainty Quantification Approach on Numerical Simulation for Supersonic Jets Performance
## 4 UQ Analysis on Gas Properties  
### Sensitivity Analysis on Gas Composition  
In the previous part of the paper, an important operating parameter like the nozzle discharge pressure was considered as the input variable to quantitatively appreciate its influence on convergent-divergent performance using UQ approach.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Statistical Moments** & **M** **exit** & **\(\Delta\)Pt** \\ \hline Mean & 1.2 & \(3.1\times 10^{4}\) [Pa] \\ \hline Standard Deviation & 0.1 & \(3.2\times 10^{3}\) [Pa] \\ \hline Skewness & 0.013 & \(-0.98\) \\ \hline Kurtosis & \(-0.8\) & 0.05 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Statistical moments for the response functions distributions.  
Figure 10: Comparison between 1000 and 10,000 samples for the sampling-based UQ analysis. Probability density functions for: (**a**) M exit; (**b**) \(\Delta\)Pt.  
Figure 9: Histograms of the pdfs for the output functions: (**a**) M exit, (**b**) \(\Delta\)Pt; the red curves represent the corresponding Gaussian normal distributions.  
The purpose of this section is to quantify the effect of a change in natural gas composition on the nozzle performance with fixed boundary conditions (total pressure, static pressure and total temperature). The natural gas case is of particular interest for industrial applications; the actual chemical composition of the gas delivered to a site is not constant, but can change significantly on a daily or seasonal base according to the pipeline used and the geographical extraction region.  
Table 7 summarizes some reference data of the most interesting cases for the Italian market: four different real natural gas compositions were selected and their properties were calculated. The variables obtained are the specific heat at constant pressure C\({}_{\text{P}}\) and the molecular weight MW of the mixture. The choice of these variables is related to the option of the fluid specification inside the CFD code that allows an easy automation of the simulation process through scripting.  
Four CFD simulations were performed, keeping constant the nozzle pressure ratio close to the r2 gas-dynamic condition (shock wave at the nozzle exit): Figure 11 shows Mach number contours for the different working conditions.  
Comparing the results to those related to the discharge pressure (p\({}_{\text{s}}\)) sensitivity analysis, it is evident that operating fluid properties (C\({}_{\text{P}}\) and MW) have a lower influence on the flow structure and on the Mach number  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Composition [\% Vol]** & **Italian** & **Libyan** & **North Europe** & **Russian** \\ \hline Methane & 99.61\% & 87.41\% & 91.58\% & 98.08\% \\ \hline Ethane & 0.06\% & 9.81\% & 4.82\% & 0.98\% \\ \hline Carbon Dioxide & 0.02\% & 1.88\% & 1.23\% & 0.10\% \\ \hline Others & 0.31\% & 0.90\% & 2.37\% & 0.84\% \\ \hline
**Properties** & **Italian** & **Libyan** & **North Europe** & **Russian** \\ \hline C\({}_{\text{P}}\) [J/(kg K)] & 2214 & 2031 & 2057 & 2154 \\ \hline MW [kg/kmol] & 16.1 & 19.5 & 18.2 & 16.2 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Natural gas compositions and properties [16].  
Figure 11: Mach number contours: (**a**) Italian, (**b**) Libyan, (**c**) North Europe, (**d**) Russian.  
This is also confirmed by the charts in Figure 12, where the Mach number and PR variation along the nozzle axis are displayed. A slight change of the shock wave position can be observed, caused by the different gas properties.  
On the other hand, due to gas composition change, a significant variation of the total pressure losses can be noticed (see Table 8): more than 10 [kPa] difference within the examined dataset.