#### 2.1. 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 × 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 × 10

^{5}), 19.0 m/s (Re = 2.8 × 10

^{5}), and 27.0 m/s (Re = 4.0 × 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 × 20 m × 40 m (W × H × 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 × 10

^{-4} mm at Re = 1.25 × 10

^{5}, 7.13 × 10

^{-5} mm at Re = 2.80 × 10

^{5}, and 5.02 × 10

^{-5} mm at Re = 4.00 × 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 × 10

^{-5} s for the cases involving a nonspinning ball and 2.037 × 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

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

where

$\mathsf{\tau}$ is the relaxation time of the fluid, and

${f}^{eq}\left(x,v,t\right)$ 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. The discrete LB equation, using a specific finite-differencing of time (

$\Delta t=1)$, is written as

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–ε 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:

Here, ρ 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 πR^{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

where

ω 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.