## 2 Materials and Methods
### Numerical Calculation
Grid Resolution: A structured grid with 10 million cells was used, with clustering near the wall to ensure \(y^{+}<1\) in the first grid cell. Boundary Conditions: No-slip boundary conditions were applied at the walls, while periodic boundary conditions were used in the streamwise and spanwise directions. Time Integration: A second-order implicit time-stepping scheme was used with a time step size of \(\Delta t=1\times 10^{-5}\) s, corresponding to a maximum CFL number of 0.5. Parallelization: The solver was parallelized using MPI, with the computational domain decomposed into 512 subdomains. Simulations were performed on a high-performance computing cluster using 512 cores. Validation: The WMLES results were validated against experimental data for turbulent jet flow, showing excellent agreement in mean velocity profiles and Reynolds stresses. The LES solver can capture the size of the vortex. The biggest decisive factor is the size of the mesh. At least four meshes are needed to capture a vortex. Integral length scale:  
\[L_{o}^{RANS}=\left(\frac{k^{3/2}}{\varepsilon}\right)_{RANS}=\left(\frac{k^{1 /2}}{C_{\mu}\omega}\right)_{RANS} \tag{3}\]  
Here, \(k\) is the turbulent kinetic energy, \(\varepsilon\) is the turbulent dissipation rate, and \(\omega\) is the turbulent specific dissipation rate. Mesh size:  
\[L_{grid}=\sqrt[3]{V} \tag{4}\]  
Define a new variable LRV, which is defined as  
\[\text{LRV}=\frac{L_{o}^{RANS}}{L_{grid}} \tag{5}\]  
The value of LRV in the important area of the computational domain in this study is greater than 10; that is, there are at least 10 grids in an integral length scale, which can be considered to meet the requirements. The time analysis and space analysis of the flow of the grid meet the requirements. The entire model is divided using a structured mesh, with a mesh size of 0.17 D. Local refinement is applied to the fragmentation needle region, where the minimum mesh size is 0.005 D and the maximum mesh size is 0.01 D. During the numerical simulation, a velocity inlet is applied below the vertical jet pipe, while the upper part of the vertical jet pipe is set to atmospheric pressure. The outlet boundary is the pressure outlet, and the wall boundary is the no-slip wall. To verify the correctness of the selected mesh size, a mesh independence test was conducted. Polyhedral structured meshes with \(\Delta\)Z/D values of 0.08, 0.17, 0.34, 0.67, and 1.35 were used to analyze the energy dissipation pool. Figure 7 shows the dimensionless jet height for five mesh sizes at a Reynolds number of 32,639, without fragmentation needle arrangement. The analysis shows that the total height of the jet varies with different mesh sizes. As the mesh size increases, the total height of the jet gradually increases. The total height of the jet stabilizes at dimensionless mesh sizes of 0.08 and 0.17, with a relative error of 0.68%. It can be concluded that a further reduction in mesh size had little effect on improving the simulation accuracy. In addition, the computational time increases significantly when the dimensionless mesh size is reduced from 0.17 to 0.08. Therefore, a polyhedral mesh with a mesh size of 0.17 D was used for the simulation. In an LES-type turbulence model,* The kinematic viscosity (\(\
u\)) represents the global, molecular dissipation. * The turbulent viscosity (\(v_{t}\)) represents the local, subgrid-scale dissipation. * The total effective viscosity (\(v_{\rm eff}=\
u+v_{t}\)) ensures that energy is dissipated at all scales, from the largest resolved eddies to the smallest subgrid-scale motions. By combining these concepts, LES accurately captures the local energy dissipation in turbulent flows, ensuring that the energy cascade and dissipation mechanisms are properly represented.