# Effect of Lateral End Plates on Flow Crossing a Yawed Circular Cylinder
## 3 Results and Discussion
### Wake Vortex Structure  
The vortex structure is identified with \(Q\)-criterion (the second invariant \(Q\) from the velocity gradient tensor), which is favorable to capture a coherent vortex structure [24]. For a yawed cylinder, the incoming velocity is decomposed into two components, normal, and parallel to the cylinder axis. Considering the vortex production mechanism, the vortices in the wake are likely to result from two channels based on the wall boundary layer separation. One results from the separation of normal flow across a cylinder, hereinafter, named normal vortex, and the other comes out of the separation of axial flow through a cylinder, hereinafter, named axial vortex.  
Under the periodic boundary condition on the two lateral end-plate walls, the vortex structure in the wake is shown in Figure 4a-c respectively at three yaw angles, viz. 15\({}^{\circ}\), 30\({}^{\circ}\), and 45\({}^{\circ}\). It is found that the vortex structure in the wake is independent of the yaw angle. Organized vortices are shed off from the cylinder into the wake in a quasi-two-dimensional manner and the series of rolling axes of the vortex structure are almost parallel to the cylinder axis (dotted line in Figure 4). This scenario of the vortex structure is of no prominent difference from a uniform flow normally crossing a circular cylinder.  
Under the nonslip boundary condition on the two lateral endplate walls, the vortex structure in the wake, as shown in Figure 5a-c respectively at yaw angles of 15\({}^{\circ}\), 30\({}^{\circ}\), and 45\({}^{\circ}\), exhibits a dramatic difference from that under the periodic boundary condition. It is demonstrated that no organized Strouhal vortex is shed in the wake. The vortex structure in the wake is strongly axially varied and exhibits a tapering shape towards the tip end. At a large yaw angle, such as 45\({}^{\circ}\), it even becomes a wedge shape lying on the tip end. Compared with the experiments [25], the numerical simulation is consistent with the experimental flow visualization.  
Furthermore, the inspection of axial flow on the cylinder wall reveals that axial vortex appears in the wake. For clear demonstration, Figure 6 gives the axial vortex structure for a yaw angle of 45\({}^{\circ}\) (with a similar structure at the other two yaw angles of 15\({}^{\circ}\) and 30\({}^{\circ}\)). The top view of \(Q\) contour, shown in Figure 6a, demonstrates that the free axial vortex in the wake is engendered from the separation of the axial boundary layer (herein, the normal vortex is screened). Furthermore, the side view of  
Figure 4: Vortex structure under periodic boundary condition on two end-plate walls.  
contour, shown in Figure 6b, reveals that two axial slender bands with high vorticity are symmetrically attached to the two flanks of the trailing edge.  
In addition, Figure 7 shows the whole vortex structure in the wake viewed in the axial direction, from which the relative geometrical relationship between the normal vortex and the axial vortices is illustrated. Two axial vortices are azimuthally symmetrically separated from the rear wall on the two sides of the trailing edge, whose separation points are azimuthally prior to the shedding point of the  
Figure 5: Vortex structure under nonslip boundary condition of two end-plate walls.  
Figure 6: Axial vortices along the cylinder axis at \(\alpha=45^{\circ}\).  
normal vortex. In terms of the strength and size, the normal vortex is stronger and bigger than the axial vortices.