## 1 Introduction  
The behavior (e.g., the trajectory) of sports balls is known to be significantly affected by aerodynamic characteristics such as drag and lift [1; 2], and various studies (including wind tunnel experiments) have been conducted to investigate the effect of these characteristics [3; 4; 5; 6; 7; 8; 9]. However, relatively few studies have examined the air flow around a ball and the vortex structure and dynamics that generate these aerodynamic characteristics [10; 11], although analysis of the vortex structure and dynamics is very important to elucidate the fundamental aerodynamics of sports balls. Many studies have previously been conducted on the vortex structure of a smooth sphere [12; 13; 14], and the results indicate that a horseshoe vortex, an alternating vortex, or a helical vortex is formed depending on the Reynolds number [15; 16; 17; 18]. However, numerous aspects regarding the formation and transition mechanisms of vortex structures are still unknown.  
Various studies have been conducted on the aerodynamics of soccer balls. A drag crisis occurs under certain conditions during the drag and lift measurements of nonspinning balls for which wind tunnel experiments are relatively easy to set up [19]. Wind tunnel experiments have also been conducted on spinning balls to measure their Magnus force and aerodynamic characteristics [20]. Furthermore, attempts have been made to measure certain aerodynamic characteristics through free-flight experiments (without using ball-supporting devices), which provide conditions similar to the actual conditions encountered by a ball [21]. However, most of these studies measured the drag and lift acting on the ball through wind tunnel experiments, and very few studies have visualized and analyzed the air flow around the ball and the dominant vortex structure that generates the drag and lift forces [22]. In particular, the air flow around nonspinning (or low-spinning) and spinning balls in free flight and the associated dominant vortex structure have not yet been examined [23]. Moreover, curled shots and passes involving spinning balls are frequently used in actual soccer ball games, and hence, studies to elucidate the dominant vortex structure are important.  
Computational fluid dynamics (CFD) has been conventionally used to visualize and analyze the flow around bluff bodies; in general, the vortex structure is simulated and steady-state analysis is carried out using turbulence models such as the realizable k-\(\varepsilon\) model [24; 25]. Recently, the lattice Boltzmann method has become one of the main CFD methods for unsteady analysis [26; 27; 28]. The simulation results obtained from this method are closer to the actual phenomenon than those obtained from steady-state analysis and, hence, the method facilitates the realization of high-resolution visualization and high-precision analysis.  
In this study, we investigated the aerodynamic characteristics and large-scale dominant vortex structure of a soccer ball using a combination of the lattice Boltzmann method, wind tunnel tests, and free-flight experiments. This combination enabled us to analyze and visualize the drag force, lift force, and side force (Magnus force) acting on nonspinning and spinning balls in free flight as well as the air flow around the balls. Our results indicate that the dominant vortex structure in the wake of a spinning soccer ball is a large-scale counter-rotating vortex pair [29], which is similar to the vortex formed around a wing tip [30], and that the side force becomes stable if the fluctuation of the separation points of the ball is suppressed by the rotation of the ball.