## 3 Results and Discussion
### Unsteady Cd, Cs, and Cl Obtained using CFD  
The unsteady \(Cs\) and \(Cl\) of a nonspinning ball in the case of Nonspin B obtained using CFD (\(\text{Re}=2.80\times 10^{5}\)) exhibited bigger large-scale fluctuations than unsteady \(Cd\) (Figure 9a). The unsteady \(Cs\) and \(Cl\) in the case of Nonspin B yielded negative and positive values in 1 s. However, \(Cs\), \(Cl\), and \(Cd\) for a spinning ball in the case of Spin B (\(Sp=0.2\) (8 rps) and \(\text{Re}=4.1\times 10^{5}\) (28 m/s)) exhibited small large-scale fluctuations (Figure 9b). In particular, the large-scale fluctuation of the unsteady \(Cs\) for a spinning ball was smaller than that for a nonspinning ball. Moreover, the unsteady \(Cs\) in the case of Spin B yielded only positive values in 1 s. The unsteady \(Cs\) of the nonspinning and spinning balls in the other cases exhibited the same trends. The average unsteady \(Cs\) of the nonspinning ball was 0.054 (s.d. = 0.078) in the case of Nonspin A, 0.039 (s.d. = 0.090) in the case of Nonspin B, and 0.034 (s.d. = 0.050) in the case of Nonspin C (Figure 10). In contrast, the average unsteady \(Cs\) of the spinning ball was 0.222 (s.d. = 0.046) in the case of Spin A, 0.282 (s.d. = 0.036) in the case of Spin B, and 0.301 (s.d. = 0.048) in the case of Spin C (\(\text{Re}=4.25\times 10^{5}\) in all the cases). The average unsteady \(Cs\) of the spinning ball was significantly larger than that of the nonspinning ball (p \(<\) 0.01). FFT analysis indicated that the peak amplitudes of \(Cs\) in the cases involving a nonspinning ball were located in the frequency range from \(\sim\)2 Hz to \(\sim\)5 Hz (Figure 11). Figure 9: Drag, side, and lift force coefficients of (**a**) nonspinning soccer ball for \(\text{Re}=2.80\times 10^{5}\) and (**b**) spinning soccer ball for \(\text{Re}=4.25\times 10^{5}\) (spin rate = 8 rps) obtained using CFD. On the other hand, in the cases involving a spinning ball, the location of the peak amplitude of _Cs_ was unclear and the results indicated the presence of a strong direct current component in _Cs_. According to the above results, _Cs_ of a nonspinning soccer ball fluctuates with greater irregularity with a change in its direction (change in sign) (from \(\sim\)2 Hz to \(\sim\)5 Hz), but _Cs_ of a spinning soccer ball fluctuates only slightly in either direction or size and remains relatively stable. The stability of _Cs_ of a spinning soccer ball is presumably caused in part by the suppression of fluctuations in the shifting separation points owing to the rotation of the ball. It has been reported that the power spectral data of a circular cylinder in the subcritical regime exhibit strong peaks associated with the common vortex shedding frequency, both in the near-wake and the surface measurements, which are in good agreement with the Strouhal number data [44]. As the Reynolds number approaches the critical regime, the increasingly energetic boundary layer is known to be less sensitive to flow fluctuations; hence, the associated line of flow separation on the  
Figure 11: Amplitude of _Cs_ obtained from fast Fourier transform (FFT) analysis in the cases involving (**a**) nonspinning and (**b**) spinning balls. Figure 10: Comparison of the average side force coefficients (_Cs_) of nonspinning and spinning soccer balls (p \(<\) 0.01). The Re values in the case of Nonspin A, Nonspin B, and Nonspin C are \(1.25\times 10^{5}\), \(2.80\times 10^{5}\), and \(4.00\times 10^{5}\), respectively, and the Re values in the case of Spin A, Spin B, and Spin C are \(4.25\times 10^{5}\) (spin rate = 4, 8, and 12 rps, respectively). cylinder surface exhibits a weak oscillating pattern [44]. We speculate that a similar phenomenon may occur on an actual free flight soccer ball. The results of this study show that although a spinning soccer ball is deflected by shifting of the boundary layer separation points (including the Magnus effect), the flight trajectory of the ball is regular and stable, and hence, a curled shot or pass can be aimed precisely at the outset of its course.