# Application of a Novel High-Order WENO Scheme in LES Simulations
## 3 Application Cases
### Turbulent Boundary Layer Simulation  
To simulate the evolution of turbulent boundary layer (TBL) flows using LES and DNS, it is essential to provide dynamic inlet boundary conditions that accurately reflect the turbulence characteristics at the inlet. The obtained TBL with specified boundary layer thickness (\(\delta\)) and friction velocity (\(u_{\tau}\)) is pivotal for investigating SWTBLI. This study employs the recycling and rescaling method (RRM) for compressible flows, as refined by [PERSON] et al. [52], to generate turbulent inflow conditions for the TBL. The simulation is conducted using a fifth-order WENO scheme, and the results are compared with those published by [PERSON] et al. [53]. Figure 11 depicts the configuration of the computational domain and mesh for the flat-plate TBL. The dimensions are non-dimensionalized with respect to the boundary layer thickness \(\delta\) in the streamwise (\(x\)), spanwise (\(y\)), and wall-normal (\(z\)) directions. The recycling plane is positioned \(7.31\delta\) downstream from the inflow plane. [PERSON] et al. [54] summarized typical grid sizes for the DNS/LES of wall-turbulent boundary layers using a high-precision computational scheme, indicating that the first cell height near the wall in the wall-normal direction should satisfy \(\Delta z^{+}\leq 1\). This study adheres to these standards, with the grid dimensions at the first layer near the wall at the recycling plane being \(\Delta x^{+}=8.2\), \(\Delta y^{+}=5.3\), and \(\Delta z^{+}=0.46\), with calculations using a local wall friction velocity \(u_{\tau}\) = 32.66 m/s. Table 3 summarizes the grid details and compares them with the grid of the DNS [55] under similar inflow conditions, confirming that the grid meets the requirements for LES. Figure 11: Computational domain and grid configuration. To compare the results with those of different schemes, data for 200 dimensionless time flow fields are analyzed. Figure 12 shows the mean velocity profiles provided by Van Driest transformation [57] at the recycling plane, where \(U_{\rm{vd}}^{+}\) is dimensionless by \(u_{\tau}\). This figure illustrates that in the viscous sublayer, the velocity profiles predicted by the WENO-JS5 and WENO-ZQ5 schemes are consistent with the classical wall law. In the logarithmic layer, the average velocity profile distribution predicted by the WENO-ZQ5 scheme closely aligns with the classical wall law. The distribution of Reynolds stress \(R_{ij}=\sqrt{\frac{\rho}{\rho w}\frac{\sqrt{(u_{\tau}^{\prime}u_{\tau}^{\prime })}}{u_{\tau}}}\) normal to the wall of the recycling plane is compared with [PERSON]'s LES results [53]. As shown in Figure 13, the distributions obtained from both schemes are essentially similar, with peaks predicted to be slightly higher than the reference values. A comparison of the RMSE for the predictions from both schemes reveals that the RMSE of the WENO-ZQ5 scheme is lower, indicating a superior alignment with the reference values in terms of the Reynolds stress distribution. The visualization of coherent structures in Figure 14 is achieved through the \(Q\)-criterion \((Q(\delta/U_{inlet})^{2}=0.1)\), with iso-surfaces colored by the local Mach number. The TBL in this study has a momentum thickness Reynolds number \(\text{Re}_{\theta}=2400\), which classifies it as a moderate-Reynolds-number flow. Both WENO schemes capture the formation of elongated, asymmetric hairpin vortices, consistent with conclusions from the existing literature [58]. The WENO-ZQ5 scheme can capture a richer flow structure, as compared to the WENO-JS5 scheme. In summary, the comparison of velocity profiles, Reynolds stress, and coherent structures demonstrates that both the WENO-ZQ5 and WENO-JS5 schemes effectively generate realistic turbulent boundary layers.