## 2 Materials and Methods
### CFD Analysis Using the Lattice Boltzmann Method
The discrete LB equation, using a specific finite-differencing of time (\(\Delta t=1\)), is written as  
\[f_{i}(x+c_{i}\Delta t,\,\,t+\Delta t)-f_{i}(x,\,\,t)=C_{i}(x,\,t) \tag{3}\]  
\[C(x,\,t)=-\frac{1}{\tau}\left[\,f_{i}(x,t)-f_{i}^{eq}(x,\,\,t)\,\,\right] \tag{4}\]  
Figure 1: (**a**) Three-dimensional soccer ball model obtained by laser scanning and (**b**) the Cartesian grid used to generate a spatial grid (W 20 m x H 20 m x L 40 m) for computational fluid dynamics (CFD) analysis. The flow is from the left to the right. A volumetric boundary scheme was chosen as the fluid-structure interaction method. Here, the particle boundary condition was conducted at the surface itself (i.e., on the facets that make up the geometry description). Each of these facets had a set of extruded parallelograms corresponding to the discrete velocity directions. For the cases involving a spinning ball, the boundary layer was simulated using a sliding mesh model [33]. Turbulence was modeled according to the very large Eddy simulation (VLES) principle [32], which directly simulates resolvable flow scales. Unresolved scales were modeled using the renormalization group form of k-\(\epsilon\) equations with proprietary extensions to achieve VLES time-accurate physics. The lattice in this solver was composed of voxels, which are three-dimensional cubic cells. The lattice also included surfels, which are surface elements that occur in areas where the surface of a body intersects with a fluid. For the cases involving a nonspinning ball, direct numerical simulation (DNS) was employed without a turbulent model. The average drag, lift, and side forces acting on the soccer ball model were calculated from the unsteady drag, lift, and side forces over a period of 1.0 s (the calculation ran from 0.2 s to 1.2 s). The following parameters were further calculated from the CFD and experimental data collected over a range of conditions: wind velocity (\(U\)); force acting in the opposite direction of the wind (i.e., drag \(D\)); force acting perpendicular to the wind direction (i.e., lift \(L\)); and force acting sideways (\(S\)) (i.e., Magnus force) with respect to the frontal view. The aerodynamic forces determined from CFD and through the experiments were converted into the drag force coefficient (\(Cd\)), lift force coefficient (\(Cl\)), and side force coefficient (\(Cs\)) as follows:  
\[Cd=\frac{D}{\frac{1}{2}\rho U^{2}A} \tag{5}\]  
\[Cl=\frac{L}{\frac{1}{2}\rho U^{2}A} \tag{6}\]  
\[Cs=\frac{S}{\frac{1}{2}\rho U^{2}A} \tag{7}\]  
Here, \(\rho\) is the density of air (1.2 kg/m\({}^{3}\)), \(U\) is the flow velocity (m/s), and \(A\) is the projected area of the soccer ball (given by \(\pi R^{2}\), where \(R\) is the radius of the soccer ball). \(Cd\), \(Cl\), and \(Cs\) are measured in the \(+X\), \(+Z\), and \(+Y\) directions, respectively. The ratio of the peripheral velocity to the velocity through the air, \(Sp\), was calculated as  
\[Sp=\frac{\omega R}{U} \tag{8}\]  
where \(\omega\) is the angular velocity of the soccer ball (rad/s) and \(R\) is the radius of the soccer ball (0.11 m). One-way analysis of variance was used to statistically test the average of the \(Cs\) values for the spinning and nonspinning balls. Fast Fourier transform (FFT) analysis was employed to compare the frequency characteristics of \(Cs\).