# Application of a Novel High-Order WENO Scheme in LES Simulations
## 2 WENO-ZQ Scheme and Numerical Tests
### Unsteady Convergence Compared with WENO-JS Scheme  
The convergence properties of high-order schemes are investigated using the three-dimensional viscous vortex (TGV) case, quantitatively addressing the two issues highlighted in the introduction. TGV [48] is a standard benchmark in fluid mechanics, commonly used to assess the capabilities of viscous flow solvers in simulating vortex structure evolution, turbulent transition, and kinetic energy dissipation. The simplicity of its initial and boundary conditions, combined with the availability of analytical solutions, make it an ideal benchmark for assessing the accuracy and convergence of numerical methods. The domain of interest is a cube with a side length of \(2\pi L\), which contains the vorticity of the initial flow field with a smooth distribution. The detailed flow and initial conditions used in this study are described in reference [48]. Simulations are conducted using the initial conditions outlined in Equation (18) to demonstrate the convergence of various high-order schemes in achieving an unsteady solution. \[\left\{ \begin{array}{l}\rho=\rho_{0}\\ u=V_{0}\sin\left(\frac{x}{L}\right)\cos\left(\frac{y}{L}\right)\cos\left(\frac{z }{L}\right)\\ v=-V_{0}\cos\left(\frac{x}{L}\right)\sin\left(\frac{y}{L}\right)\cos\left(\frac{ z}{L}\right)\\ w=0\\ p=p_{0}+\frac{1}{16}\rho_{0}V_{0}^{2}\left(\cos\left(\frac{2x}{L}\right)+2 \right)\left(\cos\left(\frac{2x}{L}\right)+\cos\left(\frac{2y}{L}\right)\right) \end{array} \right. \tag{18}\]  
where \(\rho\) denotes the density, while \(u\), \(v\), and \(w\) represent the velocities in the three spatial directions. The pressure is indicated by \(p\), and \(L\) refers to the length of the computational domain. Subscript '0' indicates reference values. The Reynolds number for the flow is defined as \(\text{Re}=\rho_{0}V_{0}L/\mu=1600\), where \(\mu\) is the viscosity coefficient. The Mach number is \(Ma=V_{0}/\sqrt{\gamma p_{0}/\rho_{0}}=0.1\) and \(\gamma=1.4\). In this field, kinetic energy transfers from the larger scales to the smaller scales until it reaches a sufficiently small scale, where it ultimately dissipates into internal energy due to physical viscosity. Periodic boundary conditions are employed. The evolution of the TGV over time is analyzed using four different mesh sizes: \(32\times 32\times 32\), \(64\times 64\times 64\), \(128\times 128\times 128\), and \(256\times 256\times 256\), to evaluate grid convergence and the accuracy of numerical methods for different mesh densities. For the \(32^{3}\) case, the dimensionless time step is \(\Delta t=0.001L/V_{0}=0.001t_{c}\), ensuring that the Courant number remains below 1. A total of 20,000 time gsteps were simulated, with the calculation extending to \(t=20t_{c}\). The time step was halved with each doubling of the grid resolution. The approximate computational time required for each case utilizing various fifth-order schemes is provided in Table 2.  
Figures 5-8 illustrate the TGV structures computed using the WENO-JS and WENO-ZQ schemes at \(t_{final}=20t_{c}\). Coherent structures in the field are observed employing a \(Q\)-criterion, with iso-surfaces delineated at \(Q(L/V_{0})^{2}=0.005\) and color-coded based on velocity magnitude. The results reveal that the computed flow fields exhibit good symmetry in the vortex structures, indicating that both WENO schemes effectively preserve the flow field's symmetry. The turbulence vortex structures obtained with the WENO-ZQ scheme show superior resolution compared to those obtained with the WENO-JS scheme on the same mesh. For example, Figures 7c and 8b demonstrate that the WENO-ZQ5 scheme with a \(64^{3}\) mesh captures turbulence vortex structures similar to those obtained with the WENO-JS scheme using a \(128^{3}\) mesh.