## 2 Modeling and Computational Approach
### Computational Approach  
Given the axisymmetric nature of the test case, we designed the computational domain as a slender wedge with an angle \(\leq\)5\({}^{\circ}\) and a single cell thickness along the plane of symmetry. This setup process is documented in [45] and illustrated in Figure 1. The domain's length extended to 7.63 D (see Figure 2), while the height of the wedge matched 3.62 D. The spherical bump geometry starts at \(x/c=0\), with the inlet patch located at \(x/c=-1.5\) and the exit boundary at \(x/c=4.4\) in the main geometry. The relevant physical domain extends to \(x/c=2\). To construct a block-structured grid suitable for the bump domain, we employed Salome in conjunction with the snappyHexMesh mesh generator provided by OpenFOAM. OpenFOAM has several solvers for both incompressible and compressible flows (e.g., SonicFOAM and RhoCentralFOAM, etc.). For the current work, we have applied an unstructured finite-volume solver, SonicFOAM, a transient solver for transonic/supersonic, laminar, or compressible flow. Two meshes, \(G_{1}\) and \(G_{2}\), were created with different cell counts in the longitudinal and transverse directions, as shown in Figure 3. Detailed characteristics of these meshes are provided in Table 1. Both meshes used non-uniform grading, with cells stretched in the axial direction and compressed in the transverse direction, particularly near the wall and bump region. The mesh density was enhanced in the expected shock region. The initial cell dimension near the surface was set to 1 mm for both meshes, resulting in a mean value of \(y^{+}\) of 30. Using an axisymmetric grid, we simulate in this way a slice of the 3D axisymmetric bump geometry (a spanwise-periodic configuration with the same cross-sectional profile as given by the three-dimensional bump geometry). The same approach was applied by [PERSON] et al. [41] as well as [PERSON] et al. [46]. Characteristic boundary conditions were imposed at the freestream and outflow boundaries. Efficiently characterizing turbulent inflow conditions remains a persistent challenge for hybrid simulations. A precursor RANS simulation utilizing the \(k\)-\(\omega\) model was employed. This involved extracting mean velocity and other data (see Figure 4) from the mapping plane to provide a proper turbulent inflow at \(x/c=-1\). The set-up involved a composite setting at the inlet boundary. Specifically, velocities within the turbulent boundary layer were directly mapped at a predefined distance downstream, \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Run** & **Grid** & \(N_{cells}\) & \(N_{x}\times N_{y}\times N_{z}\) & \
\begin{tabular}{l} **Min. Wall** \\ **Spacing** \\ \end{tabular} & \(\Delta y^{+}\) \\ \hline \(G_{1}\) & Coarse & 1.08 M & \(300\times 120\times 30\) & 0.00001 & 1.00 \\ \(G_{2}\) & Fine & 5.4 M & \(600\times 150\times 60\) & 0.000001 & 0.85 \\ \hline \multirow{2}{*}{Reynolds numbers} &
\begin{tabular}{l} Mach \\ number \\ \end{tabular} & Char. length & Spatial domain & & \\ \hline \(Re=\) (2.763 M, 5.5 M) & 0.875 & \(c=\) 0.2032 m & 7.63 D \(\times\) 3.62 D \(\times\) 5\({}^{\circ}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The grids applied and specifications of the simulation parameters. Figure 3: Partial view of the grids applied for the bump case: coarse grid \(G_{1}[300\times 120]\), fine grid \(G_{2}[600\times 150]\). while upper-wall velocities were set to a fixed value, as depicted in Figure 4. The computed total mass flow rate in conjunction with the turbulent viscosity profile established a comprehensive inflow profile. The lateral surfaces, on the other hand, employed a wedge boundary condition, which enforces equal and opposite fluxes on both sides. The bump's surface is subjected to a solid wall boundary condition. At the outlet, zero gradients were applied for velocities and turbulent stresses. Since the flow is compressible, an appropriate characteristic-based or pressure-outlet condition was applied at the outlet.