# Numerical Study of Aircraft Wake Vortex Evolution under the Influence of Vertical Winds
## 2 Materials and Methods
### Meshing and Mesh Independence Demonstration  
This chapter performs structured hexahedral meshing of the fluid domain by ICEM-CFD, emphasizing the speed and quality of meshing, and fine-tuning the O-block meshing of the critical airfoil zone domain. Detailed meshing and meshing parameters are given later.. Simulations were solved using the second-order upwind scheme and the SST k-\(\omega\) turbulence model, and consistent variables were tested for all four mesh sizes to assess the mesh dependence. Grid independence was verified by comparative analysis of velocity profiles at 170 m behind the wing. Detailed simulation experiments are shown in Figures 4 and 5. A grid of 8.91 M provided the best efficiency and accuracy trade-off and was therefore selected for the simulation. Figure 4: The vertical velocity and the error along the vorticities line (x/c = 17). Figure 5: Vertical velocity contour at z \(\leq-20\) position (x/c = 17). To select an appropriate grid resolution for our numerical simulations, we conducted tests using four different grid sizes: 4.88 million (abbreviated as 4.88 M), 6.63 million (abbreviated as 6.63 M), 8.91 million (abbreviated as 8.91 M), and 10.91 million (abbreviated as 10.91 M) cells. All conditions except for the grid resolution remained consistent. Table 2 presents the numerical values of the environmental parameters used in the numerical simulations. To enhance the accuracy of simulating vortex dynamics, a second-order upwind dissipation scheme and the SST k-\(\omega\) turbulence model were applied. Vertical velocity along the vortex core line at a position 170 m behind the wing's trailing edge was selected for comparative analysis. Figure 4 illustrates the results for these four grid sizes under the same numerical simulation conditions. The black short line represents the benchmark results obtained with the 10.91 million grid. The error values for the other three grid sizes are also shown. Figure 5 displays the vertical velocity cloud map at \(z\)-axis coordinates \(\leq-20\) m.  
The vertical velocity profiles for all four grid sizes exhibit similar trends and numerical values. One notable difference is that the 4.88 M grid shows more significant velocity oscillations and higher errors, occasionally manifesting spurious vortices outside the vortex core region. As the grid resolution increases, the velocity curves become smoother, and the errors decrease. Notably, the 8.91 M grid results in vertical velocity errors, mainly within the range of 0.1 m/s. The vortex core positions and vorticity values are consistent for all four grid sizes. However, the vorticity decays faster in the 4.88 M grid due to its relatively low resolution. In conclusion, the 4.88 M grid needs to be more coarse, and its simulated flow field around the vortex core needs to be consistent with the actual conditions. The 6.63 M grid shows some improvement but still exhibits higher errors, posing a risk of distortion in long-distance simulations. Both the 8.91 M and 10.91 M grids produce very similar velocity distributions and accurately capture the vortex structure, meeting the grid requirements for experimental simulations. However, the 10.91 M grid demands more computational resources for only a marginal improvement in accuracy. Therefore, we have chosen the 8.91 M grid for further simulation studies. The structured hexahedral meshing of the fluid domain model is performed using ICEM-CFD. Structured grids offer faster grid generation, higher quality, and better suitability for fluid flow and wing surface tension calculations than unstructured grids. The high curvature at the leading and trailing edges of the wing necessitates the generation of finer grids to ensure computational accuracy. However, simply increasing the number of grid nodes significantly reduces the efficiency of numerical calculations. Therefore, the O-block meshing method is adopted to refine the wing's boundary layer mesh.