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a b s t r a c tThe flight trajectory of a non-spinning or slow-spinning soccer ball might fluctuate in unpredictable ways, as for example, in the many free kicks of C. Ronaldo. Such anomalous horizontal shaking or rapid falling is termed the 'knuckling effect'. However, the aerodynamic properties and boundary-layer dynamics affecting a ball during the knuckling effect are not well understood. In this study, we analyse the characteristics of the vortex structure of a soccer ball subject to the knuckling effect (knuckleball), using high-speed video images and smoke-generating agents. Two high-speed video cameras were set at one side and in front of the ball trajectory between the ball position and the goal; further, photographs were taken at 1000 fps and a resolution of 1024 Â 512 pixels. Although in a previous study (Taneda, 1978), shedding of horseshoe vortices was observed for smooth spheres in the Reynolds number (Re) range of 3.8 Â 10 5 o Re o 10 6 , in the case of the soccer ball, the vortex structure, which consisted of distorted loop vortices, appeared in the wake behind the ball in the supercritical Re number region. Moreover, after the knuckleballs were airborne, large-scale undulations were observed in the vortex trail visualised with a smoke technique. On the other hand, aerodynamic forces acting on the ball were estimated from the data of the ball's flight trajectory, and a statistically high correlation (r¼ 0.94, p o 0.01) between the fluctuation frequency of the lift and side forces and the undulation frequency of the vortex trail was shown to exist. This fact suggests that the phenomenon of large-scale undulations of the vortex trail is closely related to the cause of the unsteady aerodynamic forces acting on the knuckle ball.

The purpose of this paper is to explain the attractive applicability of the advanced vortex element methods and their contribution to the beginning of the new generation of CFD, with introduction of epoch-making application of the methods to simulation of unsteady flows around bluff bodies and virtual operation of fluid machinery. The vortex methods have been developed and applied for analysis of complex, unsteady and vortical flows in relation to problems in a wide range of industries, because they consist of simple algorithm based on physics of flow. Nowadays, applicability of the vortex element methods to various engineering problems has been developed and improved dramatically and it has become encouragingly clear that the vortex methods have so much interesting features that they provide easy-to-handle and completely grid-free Lagrangian calculation of unsteady and vortical flows without use of any RANS type turbulent models. In this paper, the mathematical background and numerical procedure of a vortex method developed by the group of the present author are briefly explained, and topics of calculated flows around bluff bodies, an oscillating airfoil, a swimming fish, virtual operation of fluid machinery (pumps and water turbines) are introduced

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