# Application of a Novel High-Order WENO Scheme in LES Simulations
## 3 Application Cases
### Supersonic Compression Ramp  
The SWTBLI represents a critical aerodynamic and thermodynamic process in the supersonic inlet flow. It can lead to significant alterations in heat transfer in the external flow and produce unsteady pressure loads, potentially shortening the lifespan of structures. Internally, it may trigger large-scale, unsteady separated flows, increasing total pressure loss and causing flow distortion that can lead to engine surge. The compression ramp model, despite its simplicity, captures essential SWTBLI phenomena such as boundary layer separation, reattachment, and turbulence enhancement due to inverse pressure gradients [50]. The highly three-dimensional flow field features intricate multi-scale interactions, making it an ideal test case for validating CFD methods. The WENO-ZQ5 scheme is employed for the LES of this case. Figure 15 illustrates the configuration of the computational domain and mesh, where the inlet boundary layer thickness is specified as \(\delta\) = 6.7 mm. The computational domain consists of a flat-plane section (auxiliary domain) and a 24\({}^{\circ}\) compression ramp section (principal domain) [50]. The coordinate system is origin-based at the ramp, with upstream and downstream lengths of 7.738, a spanwise width of 2.158, and a wall-normal height of 5.238. The TBL is generated in the auxiliary domain using the RRM, and the resulting turbulence data are utilized as the inlet boundary conditions for the principal domain. Figure 15 presents a simplified schematic of the grid for the ramp domain. The mesh is uniformly distributed in the spanwise direction and refined in the streamwise direction near the ramp. In the wall-normal direction, the grid spacing near the wall is designed to ensure that the height of the first cell satisfies \(\Delta z^{+}\leq 1\)[54]. The mesh dimensions at the inflow plane of the principal domain are identical to those at the recycling plane, with the flow field data from the auxiliary domain's recycling plane directly employed as the inlet boundary condition for the principal domain. Figure 14: Visualization of the coherent structures in supersonic TBL. Figure 15: Computational domain and mesh for the SWTBLI. To facilitate comparison with the DNS results, Table 5 presents the grid information for the LES of the compression ramp flow, as well as for the DNS conducted by [PERSON] et al. [51] under similar inflow conditions. In this study, three grid resolutions, namely LES-G1, LES-G2, and LES-G3, were employed to assess grid independence. The grid points in the wall-normal direction remain consistent across all three grids, while a uniform distribution is maintained in the spanwise direction. The refinement primarily affects the grid distribution near the corner region. Notably, the finest mesh resolution in the current simulations remains substantially lower than that of the DNS grid. Once the SWBLI flow field reaches a fully developed state, sampling begins. Given the characteristic frequency of the turbulent boundary layer, \({L_{\infty}}/{\delta}\), the time interval \(0.001{L_{\infty}}/{\delta}\) is selected to ensure adequate time resolution. A total of 400,000 time steps are computed, spanning a total time of \(400{L_{\infty}}/{\delta}\). For instance, in the case of LES-G3, approximately 13 days were required to complete 400 time periods using 177 cores. To verify the accuracy of the computed results, the data from these \(400{L_{\infty}}/{\delta}\) are averaged and compared with the DNS results [51] and experimental data [50]. Figure 16 illustrates the variation of the mean wall pressure (\(p_{w}/p_{\infty}\)) and the mean skin friction coefficient (\(C_{j}\)) along the streamwise direction within the principal domain. For the wall pressure distribution, it is evident that all three grid resolutions can accurately predict the pre-separation wall pressure, the pressure plateau within the separation region (a characteristic of separated flow), and the final inviscid pressure solution downstream.