## 3 Distribution and Evolution of Vortex Structures
### Evolution of Vortex Structures in the Whole Process of VIV  
Figure 3 depicts the displacement response of the box beam's vertical VIV. The growth process of VIV is split into three stages, namely, rapid development stage, gradual convergence stage, and stable vibration stage, in order to investigate the evolution law of vortex during the development of VIV. Then, four moments of 0, \(\pi/2\), \(\pi\), and \(3\pi/2\) are selected for the average state of each stage, corresponding to the equilibrium position with upward motion, the positive maximum amplitude, the equilibrium position with downward motion, and the negative maximum amplitude, respectively. The size and distribution of vortex structures at the above-mentioned four moments in different stages of VIV are shown in Figures 4-8. Figure 3: Vertical displacement response and stage division of VIV. As shown in Figure 4, when this streamlined box girder remains stationary, a huge vortex locates at the upper surface, and it consists of four sub-vortex structures which locate at the leading edge of the roof, the upstream roof, the downstream roof, and the upside of downstream web separately. As the influence of upstream corner and railings, a very small sub-vortex forms at the leading edge of the roof. The two large sub-vortex structures located at the roof are not linked together because of the central barriers. Additionally, as an effect of the downstream railings and corner, the sub-vortex separates from the main vortex structure. The horizontal dimensions of four sub-vortex structures at the upper surface, from left to right, are 0.04 m, 0.24 m, 0.61 m, and 0.14 m, respectively. Different from the  
Figure 4: Contours of time-averaged vortex structures at stationary case. Figure 5: Contours of time-averaged vortex structures at 0 moment (Unit: s\({}^{-1}\)). (**a**) Rapid development stage of VIV. (**b**) Gradual convergence stage of VIV. (**c**) Stable vibration stage of VIV. upper surface, the vortex structures of the lower surface are more dispersed. Several small vortex structures form behind the track maintenances. It is noticeable that a large vortex consisted of two sub-vortex structures locates at the underside of the downstream web, and the horizontal dimension of this vortex is 0.57 m.  
At 0 moment, the vortex distribution at different stages of VIV is shown in Figure 5. As shown in Figure 5a, the vortex distribution during the rapid development stage is similar to that of the static box girder, because the vertical displacement of the girder is small, and the influence of the small motion on the surrounding flow field is not obvious. As shown in Figure 5b, compared with the rapid development stage, there is still a huge vortex on the upper surface of the box girder, but the number of sub-vortex structures in the vortex is reduced to three with the increase in the VIV amplitude. The sub-vortex on the windward side of the upper surface of the box girder is closer to the central guardrail, and its horizontal dimension is 0.12 m. There is only one sub-vortex on the leeward side with the horizontal dimension of 0.73 m, which may be formed by the fusion of two sub-vortex structures on the leeward side at the rapid development stage. Different from the situation in which the tail sub-vortex will fall off at the first stage, the position of the sub-vortex is closer to the upper inclined web of the leeward side. On the bottom surface of the box girder, there is only one vortex on the leeward inclined web, while for the static girder, this sub-vortex at the back of the position has fallen off. At the stable period, as shown in Figure 5c, there are several separated vortex structures on the upper surface of the box girder, the largest vortex is located at the middle and rear of the upper surface with a horizontal dimension of 0.42 m, and this vortex has not yet completely passed through the central guardrail.