# Experimental Investigation on Flow-Induced Rotation of Two Mechanically Tandem-Coupled Cylinders
## 4 Results and Discussion
### L/D = 4.0 and 4.5  
#### 4.1.1 Rotation Response  
The rotational oscillation response curves with L/D = 4.0 and 4.5 were areshown in Figure 11. Herein, the rotation arc length ratio \(A^{*}\) (dimensionless parameter) was examined to characterize the oscillation level, and the calculation equation of \(A^{*}\) was as follows:  
\[A^{*}=\frac{A}{D}=\frac{\frac{\pi\theta_{max}}{180}\times\frac{L}{2}}{D}=\frac{ \pi L}{360D}\theta_{max} \tag{27}\]  
where \(A\) is the average value of the peak of arc length in 60s of continuous oscillation. \(\theta_{max}\) is the average value of the peak of \(\theta\) in 60s of continuous oscillation. \(L\) is the center distance of the upstream and downstream cylinders. \(D\) is the diameter of the circular cylinder. Figure 10: Comparison between experimental and calculated values of system natural frequency. Additionally, the frequency ratio \(f^{\star}\) and reduced flow velocity \(U_{r}\) were defined as \(f^{\star}=f_{\rm{esc}}/f_{n,air}\), and \(U_{r}=U/f_{n,air}\)\(D\), respectively. (1) \(L/D=4.0\): in the range of \(2.314<U_{r}<4.531\), the \(A^{\star}\) first increased and, then, decreased sharply with increasing \(U_{r}\), and the \(f^{\star}\) showed a continuous increase with the rise of \(U_{r}\). The results showed that the rotational oscillation conformed to the response law of VIV and the maximum \(A^{\star}\) was 0.32 in this velocity range (in the complete \(A^{\star}\)-\(U_{r}\) curve, the first peak was named \(A^{\star}\)\({}_{1 st}\), and the second peak was named \(A^{\star}\)\({}_{2 nd}\), which meant \(A^{\star}\)\({}_{1 st,4,0}=0.32\)). Nevertheless, as \(4.531<U_{r}<5.798\), the \(A^{\star}\) would present an increasing trend again, which meant that there was a re-growth region in the arc length responses. The reason might have been that with the increase of incoming flow velocity, the vortex shedding rate of the upstream cylinder was accelerated, which further promoted the rotational oscillation of the downstream cylinder, and finally led to the enhancement of the oscillation level of the two tandem-coupled cylinders. Note that the peak of \(A^{\star}\) could only reach 0.26 in this re-growth region (\(A^{\star}\)\({}_{2 nd,4,0}=0.26\)). The results indicated that, although the arc length grew again at \(4.531<U_{r}<5.798\), it was actually still in the lower branch of VIV, which meant that the oscillation was still suppressed. Continuing to increase \(U_{r}\) until the maximum of the experiment (\(U_{r}=9.599\)), resulted in a steady decrease in the \(A^{\star}\). It showed that when \(U_{r}>5.798\), the rotational oscillation level of the FIR system decreased once again due to the suppression effect strengthening. The reason might be that, as the velocity continued to increase, the vortex shedding rate of the upstream cylinder became faster, resulting in the lower efficiency of reattachment to the downstream cylinder, and, finally, the overall oscillation level of the FIR system gradually decreasing. (2) \(L/D=4.5\): for \(2.430<U_{r}<5.093\), the \(A^{\star}\) also showed the response law of first-creasing and, then, decreasing. Meanwhile, the \(f^{\star}\) rose continuously. It was shown that the rotational vibration was also in line with the VIV in this velocity range. Additionally, as can be seen from Figure 11, \(A^{\star}\)\({}_{1 st,4,5}=0.56\), which was significantly higher than that when \(L/D=4.0\). When \(5.093<U_{r}<6.091\), the \(A^{\star}\) also had a tendency to grow again, just like when \(L/D=4\), but the secondary growth of \(L/D=4.5\) was less than \(L/D=4.0\), and it only increased to \(0.3\) (\(A^{\star}\)\({}_{2 nd,4,5}=0.3\)). For \(U_{r}>6.091\), the \(A^{\star}\) started to show a downward trend.