## 2 Materials and Methods  
### CFD Analysis Using the Lattice Boltzmann Method  
A three-dimensional soccer ball model was constructed (Figure 1a) from data obtained by scanning a real soccer ball (Brazuca, Adidas) using a three-dimensional laser scanner (AICON 3D, Breuckmann GmbH, Germany). For a spinning ball, the flow speed at the velocity inlet was set to 28 m/s (Reynolds number (Re) = \(4.25\times 10^{5}\)), and the spinning rates of the ball were defined as 25.1, 50.2, and 75.4 rad/s (4, 8, and 12 rps); the cases corresponding to these spinning rates were denoted as Spin A, Spin B, and Spin C, respectively in this study. For a nonspinning ball, the flow speeds at the velocity inlet were set to 8.2 m/s (Re = \(1.25\times 10^{5}\)), 19.0 m/s (Re = \(2.8\times 10^{5}\)), and 27.0 m/s (Re = \(4.0\times 10^{5}\)); the cases corresponding to these flow speeds were denoted as Nonspin A, Nonspin B, and Nonspin C, respectively. The flow speeds (ball speeds) in this experiment were based on those observed in actual soccer games [23]. A Cartesian grid was adopted to generate a spatial grid with dimensions of 20 m \(\times\) 20 m \(\times\) 40 m (W \(\times\) H \(\times\) L) comprising nearly 500 million cells (Figure 1b). A sectional grid scale technique was used in this study with a minimum scale of 1 mm and a maximum scale of 4 mm for the cases involving a spinning ball. For the cases involving a nonspinning ball, we defined minimum scales of \(1.63\times 10^{-4}\) mm at Re = \(1.25\times 10^{5}\), \(7.13\times 10^{-5}\) mm at Re = \(2.80\times 10^{5}\), and \(5.02\times 10^{-5}\) mm at Re = \(4.00\times 10^{5}\), and a maximum scale of 4 mm. This grid structure could not represent detailed vortex formations perfectly, but it was used because of computational resource constraints [31]. The outlet pressure was defined as 1013.25 hPa (i.e., atmospheric pressure). The boundary wall of the soccer ball was assumed to obey a no-slip condition, and the outer walls (including the ground surface) were defined as slip walls. The time step for the calculation was \(1.018\times 10^{-5}\) s for the cases involving a nonspinning ball and \(2.037\times 10^{-4}\) s for the cases involving a spinning ball. In this study, aerodynamic simulations were performed using the incompressible flow model of commercial CFD software (PowerFLOW 5.1,Exa Corp., USA) based on the lattice Boltzmann method [26]. The behavior of many-particle kinetic systems can be expressed in terms of the basic mechanical laws governing single-particle motions at the molecular scale. The Boltzmann equation formulates the problem in terms of a distribution function \(f\) (\(x\),\(v\),\(t\)), which is the number density of molecules at position \(x\) and speed \(v\) at time \(t\)[32]. The equation (in the absence of external forces) can be written as  
\[\frac{Df}{Dt}=\frac{\partial f(x,\,v,\,t)}{\partial t}+v\,\
abla f(x,\,v,\,t )=C(x,\,v,\,t) \tag{1}\]  
Here, the total derivative on the left-hand side represents the convective motion of particles, whereas the right-hand side expresses complex intermolecular interactions (collisions). Integration of the distribution function makes it possible to obtain macroscopic variables such as fluid density, speed, and pressure. The collision operator's main purpose is to drive the velocity distribution function towards its equilibrium distribution. The Bhatnagar, Gross and Krook (BGK) collision operator [32] can then be defined as  
\[C(x,\,v,\,t)=-\frac{1}{\tau}\left[\,f(x,v,t)-f^{eq}(x,\,v,\,t)\,\,\right] \tag{2}\]  
where \(\tau\) is the relaxation time of the fluid, and \(f^{eq}(x,\,v,\,t)\) is the equilibrium distribution function. To solve these equations efficiently, we discretized them on a three-dimensional cubic lattice using the D3Q19 model [32]. This model discretized the velocity space into 19 discrete speeds.