Hypersonic Boundary Layer Instability Calculations for Complex Geometries Using Curved, High-Order Adaptive Meshes

Predicting boundary layer transition in hypersonic flows is critical for accurate estimation of drag and aerothermal heating. This process is governed by complex instability mechanisms that are sensitive to surface curvature, shock waves, and entropy layer interactions. This research develops and verifies a Hybridizable Discontinuous Galerkin (HDG) framework coupled with Adaptive Mesh Refinement (AMR) to perform Input/Output (I/O) analysis on these complex flows. The method is first validated against Linear Stability Theory (LST) for a hypersonic flow over a cylinder, demonstrating accurate capture of transverse curvature effects and oblique wave growth. A computational efficiency study reveals that strategic $h$-refinement provides a $3\times$ speedup over global $p$-refinement for comparable accuracy, highlighting the effectiveness of AMR for localized wave packets. Finally, the method is applied to a blunt cone with a detached bow shock. While the framework accurately resolves low-frequency disturbances and the associated metric terms near the stagnation point, the analysis identifies a memory scaling bottleneck in the direct solver when resolving high-frequency Mack modes. These results establish HDG-AMR as a mathematically robust tool for hypersonic stability while motivating the transition to massively parallel iterative solvers for full-scale 3D geometries.