# Uncertainty Quantification Approach on Numerical Simulation for Supersonic Jets Performance
## 3 UQ Analysis on Nozzle Discharge Pressure
### Uncertainty Quantification Analysis
\begin{table}
\begin{tabular}{c c c} \hline
**Metrics** & \(\mathbf{M_{exit}}\) & \(\mathbf{\Delta p_{t}\left[Pa\right]}\) \\ \hline Root Mean Squared (RMS) & \(4.15\times 10^{-3}\) & 175.2 \\ \hline Mean Absolute Value & \(2.84\times 10^{-3}\) & 97.9 \\ \hline Maximum Absolute Value & \(1.70\times 10^{-2}\) & 739.7 \\ \hline \end{tabular}
\end{table} 
Table 3: Metrics for cross validation analysis of the response surfaces. The outcomes of the previous analyses were supplemented by the calculation of the relative percentage error (Table 5) between the CFD and the surrogate model response function values. The obtained results certify the reliability of the meta-model. At first, a uniform distribution of the \(\mathrm{p_{s}}\) variable were considered as the input probability density function (pdf): in this case, any pressure values within the established range have the same probability. Results of the uncertainty propagation through the surrogate model are the output functions (\(\mathrm{M_{exit}}\) and \(\mathrm{\Delta p_{t}}\)) discretized pdfs, reported in Figure 6. It is evident that the pdfs of the QoI are very far from the uniform distribution of the input variable (\(\mathrm{p_{s}}\)). The physical explanation of these non-uniform pdfs can be found, considering that the 'r2' state matches with the case \(\mathrm{p_{s}}=100\) [kPa], i.e., the mean of the uncertain input variable distribution. Around this value, two kinds of solutions are highly probable, as evidenced by the exit Mach and \(\mathrm{\Delta p_{t}}\) pdfs (Figure 6):  
* high exit Mach and low losses \(\rightarrow\) overexpanded jet with shock waves outside the nozzle;
* low exit Mach and high losses \(\rightarrow\) shock waves inside the nozzle. This means that even with a large variation of the input variable, the most frequent flow structures can be restricted to two main ranges of \(\mathrm{M_{exit}}\) and \(\mathrm{\Delta p_{t}}\). In order to provide more reliable results for the UQ analysis, a second case with an increased number of samples (10,000, one order of magnitude greater than the previous one) was considered. The comparison among the histograms obtained for the QoI is reported in Figure 7: differences between the two distributions can be detected, but they are so small that the 1000 samples UQ analysis can be considered satisfactory. \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathrm{p_{s}}\) [Pa] & **Percentage Error \(\mathrm{M_{exit}}\)** & **Percentage Error \(\mathrm{\Delta p_{t}}\)** \\ \hline
80,870 & 0.043\% & \(-\)0.053\% \\ \hline
100,451 & 0.015\% & 0.964\% \\ \hline
117,848 & 0.042\% & \(-\)0.056\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Relative percentage error between CFD and response surfaces values for random samples. Figure 6: Histograms of the pdf for the output functions: (**a**) \(\mathrm{M_{exit}}\), (**b**) \(\mathrm{\Delta p_{t}}\). Another case was performed, assigning a normal distribution for the input pdf with a mean value'm' (100 [kPa]) centered on the previously considered variation range and a standard deviation \(\sigma\) chosen in order to have a small perturbation of the pressure: e.g., \(\sigma\approx 1\%\) of the mean value. The resulting pdf for the input variable ps is shown in Figure 8.  
Using again the LHS method for the uncertainty quantification propagation (1000 samples as before), the distributions reported in Figure 9 are obtained for the response functions. It is evident, especially for the losses, that even in this case, the output pdfs are different from the input one (normal distribution). Table 6 summarizes the data of the statistical moments calculated for these distributions. The red curves over the histograms of Figure 9 represent the reference Gaussian continuous distributions associated with the mean and standard deviation values of each discretized distribution obtained from Dakota. Comparing to the normal pdf (in red), the M\({}_{\text{exit}}\) distribution is more flat (negative value of Kurtosis) and slightly asymmetrical.