## 3 Results and Discussion
### Flow Visualization from CFD  
The boundary layer separation points (line) and pressure distribution in the case of Nonspin B of a nonspinning ball (\(\omega=0\) rad/s, Re \(=2.80\times 10^{5}\)) fluctuated irregularly along the time axis, and the accompanying ball wake streamlines also exhibited a deflecting tendency (Figure 5a). In addition, a large-scale counter-rotating vortex pair was very often formed in the wake of the ball (Figure 5b,c), and the position of the vortex pair changed as the separation points and pressure distribution fluctuated. The other cases--Nonspin A and Nonspin C--showed similar trends. It is considered that the simulations did not accurately model the separation and reattachment phenomena, although the separation point in the case of the subcritical regime (Nonspin A) shifted to the stagnation point of the ball. Figure 5: Example of static pressure and x-vorticity seen on streamlines around a nonspinning soccer ball obtained using CFD from (**a**) side view, (**b**) top view, and (**c**) back view. On the other hand, the boundary layer separation points and pressure distribution in the case of Spin B of a spinning ball (\(\omega=50.2\) rad/s, \(\text{Re}=4.25\times 10^{5}\)) deflected under the influence of rotation, and the accompanying ball wake streamlines also exhibited a deflecting tendency (Figure 6a,b). Similar to the case of a nonspinning ball, a large-scale counter-rotating vortex pair was observed in the wake, but no large fluctuations were seen, and the pair largely remained at the same position (Figure 6c). The other cases--Spin A and Spin C--exhibited similar trends. The asymmetry in the flow direction of the separation points (line) in the abovementioned boundary layers of the ball affected the pressure distribution (Figure 7a,b) and vortex structure, resulting in the side and lift forces acting on the ball [23]. In addition, it is considered that the large-scale counter-rotating vortex pair produced traces of deflection in the separation points and pressure distribution, and the vortex structure may be similar to that of a wing tip [38; 39]. In the near wake structure of the ball obtained from the back view using CFD, a large-scale counter-rotating vortex pair was formed in the wake of the nonspinning ball (Nonspin B; \(\omega=0\) rad/s, \(\text{Re}=2.80\times 10^{5}\)), and unstable movements such as rotation around the axis of travel, breakdown [40; 41], and re-formation [42] could be observed (Figure 8a-c). Similar trends were observed in the other cases involving a nonspinning ball. In the free-flight test, a large-scale counter-rotating vortex pair was observed in the wake of the nonspinning ball (\(\omega=5\) rad/s, \(\text{Re}=4.10\times 10^{5}\)) (Figure 8d-f). The counter-rotating vortex pair exhibited fluctuations with highly unstable movements such as rotation around the axis of travel or breakdown, similar to the cases in the CFD analysis. Figure 6: Example of static pressure and x-vorticity observed on streamlines around a spinning soccer ball using CFD from (**a**) side view, (**b**) top view, and (**c**) back view. Figure 8: Flow visualization of nonspinning soccer balls using (**a–c**) CFD and (**d–f**) free-flight tests, and that of spinning soccer balls using (**g–i**) CFD and (**j–l**) free-flight tests. The ball spins with a positive rotation about the \(Z\)-axis. The Re values for the nonspinning and spinning balls are \(2.80\times 10^{5}\) and \(4.25\times 10^{5}\) (spin rate = 8 rps), respectively. In the CFD results, the color of the surface of the soccer ball varies according to the pressure coefficient, whereas the color of the streamlines varies according to the vorticity. The view is from behind the ball (back view). Figure 7: Example of static pressure observed on (**a**) nonspinning and (**b**) spinning soccer balls using CFD (back view). The spin direction of the spinning soccer ball in (**b**) is anticlockwise from the top view.