# Uncertainty Quantification Approach on Numerical Simulation for Supersonic Jets Performance
## 4 UQ Analysis on Gas Properties
### Uncertainty Quantification Analysis  
The UQ analysis for the gas composition was performed with the same approach of the previous test. The operating range for the variables C\({}_{\rm P}\) and MW (domain space) was identified from the preliminary sensitivity analysis: the specific heat at constant pressure varies from 2031 [J/(kgK)] to 2214 [J/(kgK)], whereas the MW is in the range 16.1 \(\div\) 19.5 [kg/kmol]. The considered QoI are again the nozzle exit Mach number (M\({}_{\rm exit}\)) and the total pressure drop (\(\Delta\)p\({}_{\rm t}\)--Equation (3)). The DoE was generated with 121 samples according LHS methodology. The response functions scattered distributions are reported in Figure 13 (MW* and Cp* are in normalized form, as in Equations (7) and (8)). Figure 12: (**a**) Mach number and (**b**) PR along the nozzle axis. Figure 13: DoE: (**a**) Mach number at nozzle exit, (**b**) Total pressure losses. \begin{table}
\begin{tabular}{c c} \hline \hline
**Natural Gas** & \(\Delta\)**p\({}_{\rm t}\) [Pa]** \\ \hline Italian & 31,100 \\ \hline Libyan & 21,407 \\ \hline North Europe & 23,933 \\ \hline Russian & 33,579 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Total pressure drop values with respect to gas origin. \[MW^{*}=\frac{MW-MW_{\min}}{MW_{\max}-MW_{\min}} \tag{7}\]  
\[Cp^{*}=\frac{Cp-Cp_{\min}}{Cp_{\max}-Cp_{\min}} \tag{8}\]  
A surrogate model based on the DoE (121 samples) was obtained using Kriging Response Surface Methodology. The resulting response surfaces for the QoI are shown in Figure 14.  
The cross-validation results reported in Table 9 confirm the model reliability. Even in this second application, the validation with three random points of the database was performed. Figure 14: Response surfaces obtained from the DoE for (**a**) \(\mathrm{M_{exit}}\) (**b**) \(\mathrm{\Delta p_{I}}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Metrics** & \(\mathbf{M_{exit}}\) & \(\mathbf{\Delta p_{I}}\) [Pa] \\ \hline Root Mean Squared (RMS) & \(2.87\times 10^{-3}\) & 384.8 \\ \hline Mean Absolute Value & \(1.65\times 10^{-3}\) & 251.2 \\ \hline Maximum Value & \(1.42\times 10^{-2}\) & 1416.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Metrics for cross validation analysis of the response surfaces. The results of the challenge set and the comparison in terms of relative percentage error with respect to the CFD results are reported in Tables 10 and 11 respectively. The low values of the errors certify the reliability of the response surfaces. The UQ analysis was implemented according LHS method (1000 samples) in order to evaluate the component response to input variables statistical distributions. A Gaussian probability distribution function was considered for both variables \(\mathrm{C_{P}}\) and MW. The characteristics of the pdfs are reported in Table 12 and the corresponding trends are shown in Figure 15.  
The mean values \(\mathrm{m_{1}}\) and \(\mathrm{m_{2}}\) of the pdfs were chosen as the arithmetic mean of the selected ranges. The standard deviations \(\sigma_{1}\) and \(\sigma_{2}\) are about 1% of the corresponding mean values in order to simulate the effects of a small perturbation of the involved variables. The uncertainties propagation through the meta-model determines the output distributions for the QoI (see Figure 16); the statistical moments of the resulting pdfs are summarized in Table 13. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Metrics** & \(\mathbf{M_{exit}}\) & \(\mathbf{\Delta p_{t}\left[Pa\right]}\) \\ \hline Root Mean Squared (RMS) & \(1.17\times 10^{-3}\) & 283.7 \\ \hline R squared & \(0.916084\) & \(0.806983\) \\ \hline Mean Absolute Value & \(9.99\times 10^{-4}\) & 192.6 \\ \hline Maximum Value & \(1.81\times 10^{-3}\) & 483.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Metrics for challenge data.