## 4 Numerical Results
### Application for Real-World Engineering Problems  
In order to assess orthogonal polynomial based multi-fidelity models over engineering problems, two sonic boom uncertainty quantification applications are implemented. Sonic boom analysis is performed using the aeroacoustic propagation method where the general process for sonic boom analysis is depicted in Figure 9. The process consists of two stages. In the first stage, a near-field pressure signature is obtained at several body lengths away from the air vehicle by flow analysis. Then, the pressure signature on the ground is obtained by propagating the near-field pressure signature to the ground with an aeroacoustic propagation code. The noise level is calculated using the pressure signature on the ground. The near-field pressure signature is calculated at 2 to 3 body lengths away from the air vehicle based on the assumptions of the acoustic solution [50]. Figure 8: Comparison of probability density function (\(N_{hf}=2500\), \(N_{lf}=10,000\)). Figure 7: Surrogate model error metrics for Sobol’-Levitan function (\(N_{lf}=10,000\)). In the current applications, the high-order panel method PANAIR [52], which solves the linearised potential flow equations, is used for low-fidelity flow analysis. The Euler solver in SU2 [53], which is developed by the Stanford University ADL Lab, is used for high-fidelity flow analysis. For sonic boom analysis, the sBoom program [50], developed at NASA Langley Research Center, is used. The flow analysis calculations needed in the sonic boom prediction process are time-consuming. Therefore, flow solvers with different fidelity levels are employed for multi-fidelity flow analysis, while the same high-fidelity solver in sBoom is used for sonic boom prediction. Due to the limitations of the PANAIR program used in low-fidelity flow analyses, a wing-body combination is preferred over a fully realistic aircraft model. Therefore, the JAXA Wing-Body (JWB) model designed by JAXA for the Second AIAA Sonic Boom Prediction Workshop (SBPW-II) is used. The JAXA Wing-Body geometry is demonstrated in Figure 10.  
For the Euler analysis performed by SU2, the flow domain is discretised into 12.8 million elements using a hybrid mesh structure. An unstructured mesh is generated in the neighborhood of the air vehicle while the far-field of the solution domain is discretised with Mach-aligned prisms. The Mach number is equal to 1.6 and the Jameson-Schmidt-Turkel (JST) method is used as a convective flux scheme in all Euler analyses. For discretisation  
Figure 10: The JAXA Wing-Body geometry. Figure 9: Aeroacoustic propagation in sonic boom prediction. Reproduced with permission from Ref. [51]. to be used in PANAIR, quadrilateral elements on the surface are created. Approximately 5360 elements are used to model the JWB geometry and wake region. Euler analysis is performed on a workstation with a 2.4 GHz processor with 36 cores and 192 GB of RAM. The solution converged to a \(10^{-7}\) root mean square density at 566 iterations in 65 min. A PANAIR analysis takes approximately 15 min on a 2.4 Ghz processor with one core using supersonic solid surface boundary conditions. The pressure coefficient distributions on the surface obtained as a result of high and low fidelity flow analyses are depicted in Figure 11.  
As an output of the flow analysis, the pressure signatures taken from two body lengths below the aircraft are illustrated in Figure 12a. As observed in the figure, high oscillations are observed in the rear region of the fuselage in the near-field pressure distribution obtained from the PANAIR program. These oscillations are due to the inability of the potential flow method used in the PANAIR program to properly model expansion waves and interactions. The pressure signature on the ground is obtained as the near-field pressure distribution is propagated to the ground using the sBoom code with standard atmospheric conditions; the result of this propagation is shown in Figure 12b.