# Application of a Novel High-Order WENO Scheme in LES Simulations
## 3 Application Cases
### Pak B Low-Pressure Turbine Cascade  
The transition of the boundary layer from laminar to turbulent flow represents a pivotal and intricate aspect within the field of fluid mechanics. Understanding the physical mechanisms behind this transition is vital for studying turbulence formation and has significant practical value in engineering for predicting and controlling transitions. Due to the working environment of low-pressure turbines, the transition significantly influences the boundary layer state on the suction surface. Accurately predicting the separation of laminar flow and the boundary layer transition is crucial for designing high-load, low-loss, low-pressure turbines. The transition is driven by nonlinear instability, necessitating highly accurate methods to capture small-scale vortices. The load factor Zweifield coefficient \(Z_{w}=1.8\) of the Pak B blade profile is selected to test the WENO-ZQ5 scheme, which is a typical high-load turbine blade profile [59]. It comes from the middle section of a Pratt and Whitney crewless air vehicle's low-pressure blade, with the key parameters listed in Table 6. Blade profile data are sourced from reference [60], and the experimental data are from reference [61]. The free stream turbulence intensity (FSTI) is 0.08%, and the Reynolds number, based on the freestream velocity (\(U_{\infty}\)) and the axial chord length (\(C_{\gamma}\)), is 100,000, according to the experiments. Figure 16: Distribution of averaged wall pressure (\(p_{w}/p_{\infty}\)) and skin friction coefficient (\(C_{\gamma}\)) [50]. Figure 17: Wrinkling of the shockwave’s surface (\(Q=1.5(U_{\infty}/\delta)^{2}\). The computational domain and grid distribution for the Pak B cascade are illustrated in Figure 18. The inlet to the leading edge of the blade is approximately one chord length, and the outlet to the trailing edge is about two chord lengths. An O-type mesh topology is employed around the blade, with mesh refinement applied near the blade surface, where the height of the first layer of cells in the wall-normal direction is set to \(\Delta y^{+}=0.3\). To ensure mesh independence, the computational grid was progressively refined along both the axial (x-direction) and spanwise directions (z-direction), to evaluate the impact of mesh resolution on the aerodynamic performance simulation results. The flow-through period is defined as \(T=C_{x}/U_{\infty}\). A time step of \(0.0001T\) is used to ensure that the Courant number remains below 1 throughout the simulation [62; 63]. The LES covered 320 \(T\), with the flow field achieving statistical stability after 100 \(T\). Data collection began after advancing an additional 120 \(T\). For the LES with 4.07 million grid points, utilizing 160 cores, the actual simulation process required approximately one month. One of the challenges in numerical simulations of LPT turbines is the precise prediction of laminar separation bubbles. Figure 19a presents a comparison between the LES results and experimental data [61] for the time-averaged surface static pressure coefficient \(C_{p}=(p_{\infty}-p_{s})/(0.5\rho U_{\infty})\) on the blade with varying grid resolutions. Here, \(p_{\infty}\) is the freestream pressure, and \(p_{s}\) denotes the time-averaged local wall pressure. The LES outcomes across all three grid resolutions exhibit a high degree of concordance with the experimental data, particularly in predicting the location and size of the separation bubbles on the pressure surface (PS). The results obtained from the two finer grids are nearly identical, suggesting that, when balancing computational resources and accuracy, the grid with 4.07 million nodes is the optimal choice for subsequent flow field analyses. The figure further illustrates that the suction side (SS) of the blade experiences a favorable pressure gradient from \(0.06C_{x}\) to \(0.6C_{x}\), followed by a strong adverse pressure gradient. A pressure plateau is observed at the \(0.75C_{x}\)-\(0.85C_{x}\) region of the SS, where separation occurs before reattachment.