## 3 Results and Discussion  
### Steady Cd and Cs Obtained from CFD  
The steady drag force coefficient (average \(Cd\)) of a nonspinning soccer ball obtained from CFD was \(\sim\)0.39 in the subcritical regime of \(\mathrm{Re}=1.25\times 10^{5}\), \(\sim\)0.19 in the supercritical regime of \(\mathrm{Re}=2.80\times 10^{5}\), and \(\sim\)0.14 at \(\mathrm{Re}=4.00\times 10^{5}\) (Figure 3). The average \(Cd\) of the nonspinning ball obtained in wind tunnel tests was \(\sim\)0.45 in the subcritical regime of \(\mathrm{Re}=1.24\times 10^{5}\) and \(\sim\)0.18 in the supercritical regime of \(\mathrm{Re}=2.82\times 10^{5}\) (the critical Reynolds number was \(2.67\times 10^{5}\) (average \(Cd\) = 0.15)), which were close to the average \(Cd\) values obtained from CFD. We observed a peculiar spike in the subcritical regime owing to the effect of the irregular surface panel shape on the ball, which can affect turbulent reattachment. The maximum measurement error of the average \(Cd\) for verifying the measurement repeatability at the same ball orientation was less than 5% in the subcritical regime, whereas those in the critical and supercritical regimes were lower. Moreover, the maximum measurement error of \(Cd\) at different ball orientations was also less than 5% in the subcritical regime, whereas those in the critical and supercritical regimes were lower. It is considered that the entire surface roughness of the ball in this wind tunnel test was not significantly different from that of a conventional 32-panel ball (Vantaggio; the critical Reynolds number is 2.20-2.50 \(\times\) 10\({}^{5}\)) as the total bond length of the six-panel ball used in this wind tunnel test is similar to that of the conventional ball [35; 36]. Our experimental setup could not measure the average \(Cd\) and \(Cs\) of the spinning ball. Therefore, the average \(Cd\) and \(Cs\) of the spinning ball obtained from CFD were verified using the experimental results reported in a previous study in which a soccer ball with 14 panels was used (Teamgeist (size 5); the critical Reynolds number of the ball was \(2.6\times 10^{5}\)) [19]. The average \(Cd\) of a spinning ball obtained  
Figure 2: Set up of wind tunnel experiment. from CFD was \(\sim\)0.28 at spin parameter (\(Sp\)) of 0.1, \(\sim\)0.34 at \(Sp\) of 0.2, and \(\sim\)0.36 at \(Sp\) of 0.3 (Figure 4a). The average \(Cd\) tended to increase as \(Sp\) increased. These values were slightly larger than the average \(Cd\) of the spinning ball obtained in wind tunnel tests [19]. The average side force coefficient (average \(Cs\)) of the spinning ball obtained from CFD was \(\sim\)0.22 at \(Sp\) of 0.1, \(\sim\)0.28 at \(Sp\) of 0.2, and \(\sim\)0.30 at \(Sp\) of 0.3 (Figure 4b). The average \(Cs\) also tended to increase as \(Sp\) increased, and the values were similar to the average \(Cs\) of the spinning ball obtained in wind tunnel tests [19]. Figure 4: Comparison of (**a**) drag force coefficients and (**b**) side force coefficients of a spinning ball determined from CFD analyses and wind tunnel tests. Figure 3: Comparison of the drag force coefficients of a nonspinning ball determined from CFD analyses and wind tunnel tests (the error bars indicate the standard deviation of \(Cd\)). The average _Cd_ of the nonspinning ball obtained from CFD in this study is similar to that obtained from wind tunnel tests in the present study as well as in previous studies [19; 20] and, hence, it is reasonable to conclude that the CFD calculations are valid. Further, the average _Cs_ of the spinning ball obtained from CFD is slightly larger than the value obtained in wind tunnel tests in this study, but the trends are similar in that the average _Cs_ increases with an increase in _Sp_. Some reports [34; 37] have claimed that the average _Cs_ of the spinning ball increases as Re increases, and considering that the average _Cs_ depends on the ball speed, these calculations may be deemed to be within the permissible range.