# Experimental Study on Streamwise Vortex-Induced Vibration of a Flexible, Slender Cylinder
## 4 Results and Discussion  
Figure 5 shows the root-mean-square (RMS) values of the normalized streamwise response amplitude \(x_{RMS}/D\) versus the reduced velocity \(Vr\). Clearly, there are two response branches separated by a low-amplitude region at \(V_{r}\approx 1/(2S_{t})\), where \(S_{t}\) is the Strouhal representing the vortex-shedding frequency of flow past a stationary cylinder. In the present results, the first excitation region is \(1.0<Vr<2.6\), while the second one is \(2.6<Vr<4.0\). The peak value of the normalized RMS amplitude in the first excitation region is roughly 0.07, which is slightly smaller than its counterpart of 0.08 in the second excitation region. The experimental results of [PERSON] et al. [20] and [PERSON] and [PERSON] [23] on the streamwise VIV of a rigid cylinder elastically-mounted on both ends (2D), and a cantilevered flexible cylinder (3D) in a water tunnel, were superimposed in Figure 5. The present results agree well with those of [PERSON] and [PERSON] [23] and the 3-D results of [PERSON] et al. [20], but deviate from the 2D results of [PERSON] et al. [20]. This can be attributed to the different reduced mass-damping parameters adopted in these studies. In the present study, \(Cn=0.167\), which is similar to that of 0.083 in [PERSON] and [PERSON] [23], and 0.24 in the 3D tests of [PERSON] et al. [20], but significantly smaller than 0.98 in the 2D tests of [PERSON] et al. [20]. However, in the second excitation region, the present results show a broader range than those in [PERSON] et al. [20] and [PERSON] and [PERSON] [23]. The peak value is comparable to the 3D results of [PERSON] et al. [20], but obviously larger than those in [PERSON] and [PERSON] [23], and the 2D tests of [PERSON] et al. [20]. A possible reason is the flexible slender cylinder with a very large aspect ratio adopted in this study. It is worth noting that the results presented in Figure 5 are actually a combination of the results with four different axial tensions. As mentioned above, the towing velocity of the carriage in the experiments is from 0.05 m/s to 0.15 m/s, with an increment of 0.05 m/s, restricted by the minimum resolution of the towing tank facility. A total of three reduced velocities were investigated for a given axial tension, which is far from enough to properly depict the variation of response amplitude. However, by carefully selecting the axial tensions and combining the results with different axial tensions, more data points can be reasonably distributed in the range of \(1.0<Vr<4.0\), due to different \(f_{1}\) adopted in the normalization of the reduced velocity. In general, the reasonable agreement between different results provides strong evidence of the reliability of the experimental set-up and the validity of the post-processing approaches for the measurement data. At the transition region, i.e., \(V_{r}\approx 2.5\), between two excitation branches, the streamwise vibration amplitude showed quite a large variation between different experiments. The minimum RMS amplitude was nearly zero in the 2D results of [PERSON] et al. [20], while the maximum value could be as high as \(x_{RMS}/D\approx 0.06\) in the 3D results of [PERSON] et al. [20]. The authors attributed it to the aspect ratio (AR) effects, i.e., the manner in which the depth of the valley between the first and second excitation region decreases with decreasing AR. When the AR <10, only a single excitation region was found in the cantilevered cylinder tests of [PERSON] et al. [20]. [PERSON] et al. [33] concluded from their cantilevered cylinder experiments that the streamwise amplitude did not show a significant reduction in the transition region, when AR < 21 and this trend became even more prevailing with decreasing AR. The results of this study, i.e., \(x_{RMS}/D\approx 0.01\) with AR = 195.5, provide supporting data for the conclusion. Figure 6 presents the dimensionless streamwise and transverse maximum amplitudes versus the reduced velocity, together with the normalized frequencies in both directions.