# Uncertainty Quantification Approach on Numerical Simulation for Supersonic Jets Performance
## 3 UQ Analysis on Nozzle Discharge Pressure
### Uncertainty Quantification Analysis  
A UQ analysis was carried out considering the discharge pressure as the variable parameter. The Quantities of Interest (QoI) considered as output performance parameters are the Mach number at the nozzle exit (Mexit) and the total pressure drop (\(\Delta\text{$\text{Pt}$}\)). With a fully automated procedure implemented in the Dakota environment, a DoE for the two QoI based on LHS method was performed for the pressure range 80-120 [kPa]. The outlet static pressure values provided by Dakota, according to LHS scheme, were used as the input variable in Fluent CFD software, which allowed us to obtain the corresponding response functions values (see Figure 5). In order to approximate the 'physical' (numerical) model, 121 samples were chosen inside the design space. Equations (4)-(6) show the response functions in the normalized form:  
\[p_{\text{s}}^{*}=\frac{p_{\text{s}}-\text{$\text{Ps}$}_{\text{s min}}}{\text{$ \text{Ps}$}_{\text{max}}-\text{$\text{Ps}$}_{\text{min}}} \tag{4}\]  
\[M_{\text{exit}}^{*}=\frac{M_{\text{exit}}-\text{M}_{\text{exit min}}}{\text{ M}_{\text{exit max}}-\text{M}_{\text{exit min}}} \tag{5}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**CASE** & **Discharge Pressure (p\({}_{\text{s}}\)) [kPa]** & \(\Delta\text{$\text{Pt}$}\) [Pa]** \\ \hline
1 & 80 & 19,270 \\ \hline
2 & 90 & 20,145 \\ \hline
3 & 100 & 31,926 \\ \hline
4 & 110 & 32,675 \\ \hline
5 & 120 & 34,972 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Total pressure drop values with respect to the operative condition. Figure 4: (a) Mach number and (b) PR along nozzle axis. Kriging or 'Gaussian Process' Response Surface Methodology is considered and the response surfaces for the above functions were generated from the training points given by the DoE. The UQ analysis was then performed through Latin Hypercube sampling method, considering 1000 samples taken on the generated response surfaces. In order to ensure the meta-model reliability, a cross-validation analysis was performed. In Dakota, the type of cross-validation which can be exploited is the k-fold cross-validation: at first, the DoE dataset is divided into k partitions and k meta-models are generated, each excluding the \(k\)-th partition of training data. Each surrogate is tested at the points that were excluded in its generation and the user-specified diagnostic metrics are computed with respect to the held out data [14]. In this work, a particular type of k-fold cross-validation was performed, i.e., the Leave-one-out cross-validation or Prediction Error Sum of Squares (PRESS). In this special case, the number of partitions is equal to the number of data points. The results obtained from this analysis, reported in Table 3, include the Root Mean Squared, the Mean Absolute Value and the Maximum Absolute Value of the prediction error (calculated between the observed value and the surrogate model prediction for the training data points). Moreover, three random operating points were selected from the UQ database of 1000 samples, and then, simulated with the CFD solver. The corresponding fluid dynamics results for the QoI were given in input to Dakota as a challenge set of data. The metrics calculated with respect to this dataset are reported in Table 4. \begin{table}
\begin{tabular}{c c c} \hline
**Metrics** & \(\mathbf{M_{exit}}\) & \(\mathbf{\Delta p_{t}\left[Pa\right]}\) \\ \hline Root Mean Squared (RMS) & \(5.06\times 10^{-4}\) & 189.7 \\ \hline R squared & 0.998411 & 0.97978 \\ \hline Mean Absolute Value & \(4.57\times 10^{-4}\) & 119.2 \\ \hline Maximum Value & \(7.00\times 10^{-4}\) & 327.9 \\ \hline \end{tabular}
\end{table} 
Table 4: Metrics for challenge data. Figure 5: Response functions: (**a**) Mach number at nozzle exit, (**b**) Nozzle total pressure drop.