# Application of a Novel High-Order WENO Scheme in LES Simulations
## 2 WENO-ZQ Scheme and Numerical Tests
### Governing Equations and Numerical Method  
The high-order WENO-ZQ scheme, previously outlined, is incorporated into the in-house computational fluid dynamics (CFD) solver, NUAA-Turbo. NUAA-Turbo is a three-dimensional, viscous, unsteady, compressible, finite-volume RANS/LES-CFD framework solver that utilizes a multi-block structured grid topology. Designed for high-precision numerical simulations of complex flows in turbomachinery, this solver has been widely applied and endorsed by several aerospace engine design organizations in China. The formulation process for the RANS and LES governing equations within the NUAA-Turbo is detailed in reference [38], while references [39; 40] illustrate the application of this solver within the RANS framework. To handle both low- and high-speed flows in aerodynamic analysis, a global preconditioning technique [41] is adopted. The governing equations in a curvilinear coordinate system are expressed in a conservative flux format, as delineated in Equation (13). \[M\Gamma_{q}^{-1}\frac{\partial q}{\partial\tau}+\frac{\partial\big{(}F^{inv}+ F^{vis}\big{)}}{\partial\xi}+\frac{\partial\big{(}G^{inv}+G^{vis}\big{)}}{ \partial\eta}+\frac{\partial\big{(}H^{inv}+H^{vis}\big{)}}{\partial\xi}=0 \tag{13}\]where \(M\equiv\partial Q/\partial q\) represents the matrix responsible for the conversion from conservative variables \(Q=(\rho,\rho u,\rho v,\rho w,\rho e_{t})\) to the primitive variable \(q=(\rho,u,v,w,p)\). The matrix \(\Gamma_{q}^{-1}\) is a constant diagonal matrix, and its elements are determined exclusively by the reference Mach number. Following the application of the Favre filtering technique to Equation (13), the final LES control equations are obtained, incorporating subgrid-scale (SGS) model closure relations [38]. The wall-adapting local eddy-viscosity (WALE) subgrid-scale model [42] is utilized for the LES numerical tests in this study. It can accurately recover the proper \(\gamma^{3}\) near-wall scaling of eddy viscosity without the need for a dynamic procedure. The subgrid eddy viscosity is formulated as follows:  
\[\mu_{sgs}=\text{Re}_{ref}\rho(C_{w}\Delta)^{2}\frac{\left(S_{ij}^{d}S_{ij}^{d }\right)^{3/2}}{\left(S_{ij}S_{ij}\right)^{5/2}+\left(S_{ij}^{d}S_{ij}^{d} \right)^{5/4}} \tag{14}\]  
where \(S_{ij}\) represents the subgrid stress tensor and the grid scale is denoted by \(\Delta\). The dissipation coefficient \(C_{w}\) is a model constant, typically calibrated using decaying homogeneous isotropic turbulence (DHIT) [43], as detailed in Section 2.5. Given that the governing equations in this study are non-dimensionalized, the reference Reynolds number \(\text{Re}_{ref}\) is included in Equation (14). This study employs a structured mesh for the discretization of the computational domain, and the governing equations are discretized using the FVM. For computing the numerical inviscid flux, the advection upstream splitting method (AUSM\({}^{+}\)-up) flux splitting scheme [44] is employed, as expressed below:  
\[\hat{f}_{i+1/2}^{inv}=F_{i+1/2}^{(c)}+P_{i+1/2}=\hat{m}_{i+1/2}\overset{\to}{ \
ot\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! close to those of curve 3. This indicates that choosing any set of positive linear weights summing to one does not affect the optimal fifth-order accuracy of the WENO-ZQ5 scheme. Therefore, for simplicity, the WENO-ZQ scheme will utilize the first set of linear weights (\(\gamma_{1}=0.98\), \(\gamma_{2}=0.01\), and \(\gamma_{3}=0.01\)) unless otherwise specified. Figures 2 and 3 depict the nonlinear analysis of spectral characteristics for different WENO schemes. Figure 2 illustrates that the WENO-ZQ3 scheme exhibits superior spectral properties to the third-order WENO-JS (WENO-JS3) scheme. Additionally, the spectral properties of the third-order MUSCL scheme with a minmod limiter are also included in this figure. The WENO-ZQ3 scheme demonstrates reduced dissipation and dispersion relative to the MUSCL scheme.