## 4 Results and Discussion
### Flow Patterns
#### Deflected gap flow
Figure 9d indicates that the length of the recirculation zone of the upstream cylinder is larger than that of anti-phase flow. Such a phenomenon results from the increased space at the gap, which allows the shear layers of the upstream cylinder to fully develop. Co-shedding flowWhen \(L/D>3.5\), the vorticity and time mean streamlines of the three cylinders are shown in Figure 10. Because \(L/D\) is large enough, the interaction between the cylinders is relatively small, and the flow pattern around each cylinder approximates the fashion of flow past a single cylinder. However, in fact, it can be seen from the vorticity contour that the vortex shed from the upstream cylinder still has a tendency of pairing with the corresponding downstream vortex when the flow goes through the downstream cylinder. Figure 12 shows the normalized time-averaged x-directional flow velocity \(U/U_{0}\) in the tail region of the three cylinders, with the x-axis denoting the distance from the center of each cylinder normalized by \(D\). The single cylinder case is also shown for comparison. The wake flow velocity of downstream cylinders is identical to that of a single cylinder. For the upstream cylinder, however, the velocity can be seen to be higher when it passes through the downstream cylinders at about three-quarters of the position, which confirms the vortex-pairing tendency aforementioned. Additionally, the vortex shedding of the downstream cylinders occurs in an in-phase fashion. There is a certain phase lag between the vortex shedding of the upstream cylinder and downstream cylinders, which is determined by \(L/D\)  
Figure 9: Normalized spanwise vorticity \(\omega_{z}D/U\) in the plane \(z=2\) D at (**a**) \(L/D=2.3\); (**b**) \(L/D=2.5\); (**c**) \(L/D=3\), (**d**) time-averaged streamlines at \(L/D=2.5\). With the purpose of understanding the three-dimensional vortex structures, Figure 13 exhibits the instantaneous \(\mathbf{Q}\) criterion [54] of each flow pattern, for the sake of brevity, only one \(L/D\) is shown at each pattern. \(Q\) is defined as  
\[Q=-\frac{1}{2}\Big{(}\|\mathbf{S}\|^{2}-\|\mathbf{\Omega}\|^{2}\Big{)} \tag{19}\]  
where \(\mathbf{S}\) and \(\mathbf{\Omega}\) represent the strain and rotation tensor, respectively. Significant Kelvin-Helmholtz-type vortex structures are observed. Meantime, instantaneous pressure distributions in the middle section at different flow pattern regimes are shown in Figure 14.  
Figure 11: Phase lag of lift force history between two downstream cylinders. Figure 12: Mean velocity (\(u_{x}\)) at the center line of each cylinder at \(L/D=3.5\). The three gray dashed lines represent the front edge, center line, and end edge of the downstream cylinders, respectively. Figure 10: (a) Normalized spanwise vorticity \(\omega_{2}D/U\); (b) time-averaged streamlines at \(L/D=3.5\). The dotted line indicates the central line behind the front cylinder. It is worth noting that, in anti-phase and in-phase flow regimes, the flow pattern behind the two downstream cylinders shows obvious different characteristics. A mirror symmetry of the vortex with the central line is observed in anti-phase flow, while an almost identical vortex shedding form exists in in-phase flow.