## 4 Numerical Results
### Vortex Core Identification and Vortex Modelling  
Although the simulation result using the Reynolds stress model shows the maximum resolved flow field of the tip-vortex, the predicted vortex structure becomes weaker during its convection along the downstream direction due to numerical damping. To deal with this issue, the vortex model is used to reproduce the tip-vortex with high-resolution. First, the \(\lambda_{2}\) criteria are used to identify the vortex core in the flow field. Figure 14 shows the line constructed by tracing the minimum value of \(\lambda_{2}\) on each plane normal to the mean flow. The constructed line matches well with the vortex core line. Figure 15 compares the vortex core trajectories predicted in this manner for the cases with different grids and turbulence models except for the standard k-\(\varepsilon\) models. Hence, there is no noticeable variation in the identified vortex core lines among considered the cases, and the numerical results well matched the experimental result. Figure 13: Comparison of distributions of (**a**) non-dimensional tangential velocity and (**b**) pressure coefficient predicted using different turbulence models on the line passing through the vortex core in a plane 0.5c downstream from wing-tip. Figure 16 shows the pressure coefficient variation along the vortex core trajectory predicted using different grids and turbulence models. As the grids become coarser, the pressure increases. The pressure difference between the different grids also increases as the vortex is convected downstream. These results imply higher vortex dissipation as the grid size increases. It is observed that the pressure predicted using the turbulence models, except for the Reynolds stress model, increases as the vortex is convected in the downstream direction. Furthermore, the pressure increment is maximum and comparable for the cases of the realizable k-\(\varepsilon\) and the k-\(\omega\) SST models. Furthermore, it is relatively lower for the RNG k-\(\varepsilon\) model. Given that the standard k-\(\varepsilon\) model does not accurately simulate the vortex structure and even cannot converge, as shown in Figure 13, it is not considered in this comparison. The pressure distribution predicted using the Reynolds stress model is minimum and appears to stay stable in the downstream direction. It can be inferred from the results in Figures 15 and 16 that the trajectory of predicted vortex core is not sensitive to the grid-resolution and turbulence models. However, the strength of the vortex represented by the minimum pressure is susceptible to grid-resolution and turbulence models. Therefore, in this study, all the following simulations are conducted using the Reynolds stress model with a coarse grid. This ensures the accuracy of numerical solutions at a moderate numerical cost. Figure 17 shows the variation in pressure and turbulent kinetic energy along the vortex core trajectory. Theoretically, the vortex is a rotating flow about the vortex center on which the rotation speed is zero. Therefore, within the vortex core, there is a region where the flow is partially laminar. As shown in Figure 17, this region begins to form between sections (e) to (f) and develops into a complete vortex structure. The origin of tip-vortex begins to develop from the wing-tip. Although the region with the minimum pressure occurs at the wing-tip, the pressure gradually increases as it moves from (a) to (c) and then decreases from (c). The pressure increase between these planes seems to be due to the adverse  
Figure 14: Vortex core trajectory identified using \(\lambda_{2}\) criterion. Figure 15: Vortex core trajectories predicted using different grids and turbulence models and the camera-recorded vortex core curve. Grids types are varied with the Reynolds stress model, and the turbulence models are varied with the coarse grid. pressure gradient along with the chord-direction of wing airfoil sections near the wing-tip.