# Application of a Novel High-Order WENO Scheme in LES Simulations
## 2 WENO-ZQ Scheme and Numerical Tests
### Calibration the Coefficient \(C_{w}\) for the WALE Model  
Although the WENO-ZQ scheme has good spectral characteristics, as a three-dimensional viscous CFD solver, it is necessary to consider the dissipation characteristics of mutually coupled systems, such as the flux method, interface value reconstruction, the viscous flux calculation method, and the turbulence model. The grid-generated turbulence experiment conducted by [PERSON] and [PERSON] [46] (referred to as the CBC experiment) provides a fundamental turbulence idealization known as DHIT. DHIT is a simple yet important test case used to calibrate numerous CFD solvers. [PERSON] and [PERSON] [42] highlighted that the value of the dissipation coefficient \(C_{w}\) affects the magnitude of subgrid turbulent kinetic energy dissipation, necessitating its calibration to ensure the model accurately simulates the turbulence dissipation process. A straightforward method to determine \(C_{w}\) involves assuming that the WALE model provides the same subgrid turbulent kinetic energy dissipation as the classical Smagorinsky model, given by  
\[C_{w}^{2}=C_{s}^{2}\frac{\left\langle\sqrt{2}\big{(}S_{ij}S_{ij}\big{)}^{3/2} \right\rangle}{\big{\langle}S_{ij}S_{ij}\overline{\partial P_{1}}/\overline{ OP_{2}}\big{\rangle}} \tag{17}\]  
where the constant \(C_{s}\) is typically set to 0.18. Therefore, \(C_{w}\) can be numerically assessed using homogeneous isotropic turbulence fields [42; 43]. This section calibrates the dissipation coefficient \(C_{w}\) in the LES-WALE model of the NUAA-Turbo solver using energy spectra from the CBC experiment. The AUSM*-up flux splitting scheme, the fifth-order WENO scheme for reconstructing interface values, and a sixth-order central difference method for calculating viscous fluxes are used.  
Since the measurements from the CBC experiment are dimensionless, a unit cube is employed as the computational domain. The field is initialized at the physical times \(42M/U_{o}\) in the experiment. \(M=0.0508\) m and \(U_{o}=10\) m/s are the size of the turbulence-generating mesh and free stream velocity, respectively. The detailed generation method of the initial flow field is discussed by [PERSON] et al. [47]. The grid is distributed with 64 \(\times\) 64 \(\times\) 64 cells, and the simulations are conducted under periodic boundary conditions. The dimensionless time step is set to \(\Delta t=5\times 10^{-4}\). For the NUAA-Turbo solver, the results match well with experimental results when \(C_{w}=0.45\). Therefore, the dissipation coefficient \(C_{w}\) for the WALE model in this study is set to 0.45. Figure 4 compares the energy spectrum results obtained from different WENO schemes based on the WALE model with \(C_{w}=0.45\) at two subsequent physical times (98 and 171 \(M/U_{o}\)) with the CBC experiment data [46]. The WENO-JS5 scheme exhibits insufficient dissipation at high wavenumbers, as highlighted by the red circle in Figure 4. In contrast, the WENO-ZQ5 scheme shows good consistency with the experimental results within the same wavenumber range.  
Figure 4: Comparison of energy spectra of the experiment by different schemes.