## 2 Tests and Methods of Analysis
### Wind Tunnel Tests  
The tests were conducted using the closed-circuit, low-speed, and low-turbulence wind tunnel (San Technologies Co., Ltd., Tochigi, Japan) at the University of Tsukuba (Figure 3). The maximum flow speed of this wind tunnel is 55 m/s, the blower outlet surface is 1.5 m \(\times\) 1.5 m, the flow speed distribution is within \(\pm\)0.5%, and the turbulence is 0.1% or less. Furthermore, the blockage of the measured soccer ball was within 5% of the nozzle size. For example, when the wind speed was set to 25 m/s, the measured mean wind speed was 25.28 m/s, with the measurement position in the range of \(-\)0.46 to 0.45 and a wind speed distribution within \(\pm\)0.5%. Similarly, the degree of turbulence downstream of the nozzle at a wind speed of 25 m/s was 0.05 to 0.06, within approximately 0.1%, so the error from the ball position was believed to have a negligible effect on the wind speed [17]. Furthermore, in this study, the dynamic pressure can be measured automatically with 0.1 Pa intervals using the Pitot-static tube placed above the measuring portion of the soccer ball. In addition, because the position of the ball during the measurement procedure is set almost at the center of the nozzle cross-section to adjust the distance between the nozzle and the ball to zero, the flow generated from the Pitot tube may not have a direct effect on the flow around the ball. Furthermore, the sting used in this study was 0.6 m long and 0.02 m wide. Moreover, because the effect of sting vibrations is reduced to a small value by placing the six-component force detector behind the soccer ball, the magnitude of the sting forces was ignored in this study. The tests were conducted by placing three used and three new balls in the wind tunnel. In this study, aerodynamic measurements for each soccer ball were performed for wind speeds (Reynolds number) from 7 m/s (approximately \(Re\) = 1.0 \(\times\) 10\({}^{5}\)) to 35 m/s (approximately \(Re\) = 5.5 \(\times\) 10\({}^{5}\)) at intervals of 1 m/s [17]. The relationship between the Reynolds number and wind speed is as follows:  
\[Re=UD/v \tag{1}\]  
where Reynolds number (\(Re\)) is a dimensionless number defined as the ratio of inertial force to viscous force, \(U\) is the wind speed (m/s), \(D\) is the ball diameter (m), and \(v\) is the kinematic viscosity (m\({}^{2}\)/s).  
Figure 2: Soccer balls used in the tests. (**A**–**C**) are the Telstar 18, Merlin, and Derbystar used balls, respectively. (**D**–**F**) are the Telstar 18, Merlin, and Derbystar new balls, respectively.  
These forces were measured using a sting-type six-component force detector (LMC-61256, Nissho Electric Works). In addition, through the aerodynamic forces measured in the tests [1; 5; 6; 7; 8], the drag coefficient (\(C_{D}\)), lift coefficient (\(C_{L}\)), and side force coefficient (\(C_{S}\)) were calculated using Equations (2)-(4):  
\[C_{D}=\frac{2D}{\rho\mathrm{I}^{2}A} \tag{2}\]  
\[C_{L}=\frac{2L}{\rho\mathrm{I}^{2}A} \tag{3}\]  
\[C_{S}=\frac{2S}{\rho\mathrm{I}^{2}A} \tag{4}\]  
where \(\rho\) is the air density, expressed as \(\rho\) = 1.2 kg/m\({}^{3}\); \(U\) is the flow velocity, and \(A\) is the projected area of the soccer ball, expressed as \(A\) = \(\pi\)\(\times\)\(0.11^{2}\) = 0.038 m\({}^{2}\).