## 4 Results and Discussion
### Flow around Two Stationary Cylinders
#### 4.1.3 Wake Vortex Structure  
Figure 7 is a near-wake vortex pattern of the two tandem cylinders at \(Re=2000\). As can be seen from Figure 7a, in the case of \(d=2D\), the flow behind the cylinder presented a single slender structure,  
Figure 5: The lift and drag coefficients of two tandem cylinders with different spacing. (**a**) root mean square lift coefficient, (**b**) mean drag coefficient.  
Figure 6: The Strouhal number (\(St\)) of two tandem circular cylinders with different spacing ratios.  
and the shear layer on the surface of the upstream cylinder completely enclosed the downstream cylinder. At this time, the two cylinders could be seen as a whole acting on the flow field, and no vortex shedding occurred in the two cylinders. When the spacing ratio increased to \(d=2.5D\), as shown in Figure 7b, the upstream cylinder did not form a vortex between the two cylinders. The shear layer was attached to the downstream cylinder surface, and the vortex shedding mode 2S (S represents a single vortex) was formed at a distance behind the downstream cylinder. At \(d=3D\), as shown in Figure 7c, the upstream cylinder began to shear, creating a vortex between the two cylinders and forming a gap flow. The vortex generated by the upstream cylinder passed through the side wall of the downstream cylinder, and the downstream cylinder wake formed a Karman vortex street. With the increase of the spacing ratio, steady vortices were gradually generated by the upstream cylinder between the two cylinders--that is, the gap flow. The gap flow between the two cylinders was cut off by the downstream cylinder when it moved downstream, which disturbed the flow pattern behind the downstream cylinder and presented the two-vortex shedding flow pattern shown in Figure 7d.  
As shown in Figure 8, the near-wake vortex pattern of the two tandem cylinders at \(Re=6000\)-14,000 is presented. It can be seen from the diagram that the wake patterns of the two cylinders were similar when \(Re=6000\), \(Re=10\),000 and \(Re=14\),000. The vortex pattern was 2S for all the cases, which is the classical Karman street, but the magnitude of vorticity had a slight difference. When \(d=2.5D\), the upstream cylinder could generate vortices between two cylinders, forming a gap flow. The generated vortices bypassed the front wall of the downstream cylinder without being destroyed and could be bypassed along the wall, producing the regular vortex street shown in Figure 8a. When the spacing ratio increased to \(d=3.0D\), as shown in Figure 8b, the wake mode behind the downstream cylinder changed from the single vortex shedding mode to the double vortex shedding mode, resulting in a sharp drop in the lift and drag of the downstream cylinder. With the increase of the spacing ratio, the downstream cylinder was greatly affected by the gap flow between the two cylinders, which can be seen from Figure 8c. When the spacing ratio continued to increase to \(d=7.0D\), the wake state of the upstream cylinder tended to be a single cylinder due to the large gap distance, as can be seen from Figure 8d. The energy of the vortex shedding from the upstream cylinder was small at downstream, which made the wake of the downstream cylinder unable to form an obvious vortex street and weakened the interaction between the two cylinders. The difference is that with the increase of Reynolds number, the vortex shedding frequency of the cylinder increased.  
It can be seen from the above that the spacing ratio affected the flow characteristics of the two tandem cylinders. As the spacing ratio increased, a gap flow occurred between the two cylinders. Under different Reynolds numbers, the spacing ratio of the gap flow was different. When \(Re=2000\), the spacing ratio of the gap flow was \(d_{c}/D=3\); when \(Re=6000\)-14,000, the spacing ratio of the gap flow was \(d_{c}/D=2.5\). At the spacing ratio \(d_{c}\), the lift coefficient, drag coefficient, and \(St\) of the two cylinders had a sudden rise, which indicates that the generation of gap flow had a great influence on them.