Flow Behavior Behind A Freely Suspended Cylinder in the Wake of A Stationary Cylinder

The experiment of flow past a freely suspended circular cylinder in the wake of an upstream stationary cylinder was carried out in a re-circulating water channel using the Particle Image Velocimetry (PIV) technique. The upstream cylinder was fixed, while the downstream cylinder was suspended from a platform and allowed to move freely in the horizontal plane. The centre-to-centre spacing ratio between two tandem cylinders was initially kept at a constant value of 3.0. The instantaneous flow field and the orbital trajectories were analyzed to reveal the effect of the presence of the upstream cylinder and flow velocity on the dynamic response of the downstream suspended cylinder. The results showed that the upstream stationary cylinder has significant effect on modifying the flow patterns behind two tandem cylinders. Different trajectories of the downstream suspended cylinder with variation of flow velocity U were observed, such as: (1) depicting a figure-8 type motion at U = 0.27 m/s; (2) undergoing intermittent oscillations as it travels downstream at 0.3 m/s ⩽ U ⩽ 0.37 m/s; (3) successive moving downstream, no obvious streamwise oscillation observed at U = 0.43 m/s.


Introduction
Flow induced vibrations of circular cylinders are of interest in the area of coastal and offshore engineering, such as risers, pipelines, mooring lines etc. As well known, when the cylinder is flexible or elastically mounted, the interaction of vortex shedding with the structural characteristics of the body can lead to significant vortex induced vibration (VIV) in both the inline and cross-flow direction. This VIV phenomenon has drawn attention due to the increasing risk of fatigue failure in engineering structures.
Most of the early studies on flow-induced vibration of two tandem cylinders were concerned with the flow around elastically mounted cylinders (Zdravkovich, 1985;Mahir and Rockwell, 1996;Brika and Laneville, 1999;Xu et al., 2008). Assi et al. (2006) studied flow-induced oscillations of a circular cylinder mounted on an elastic base to oscillate in the wake of an upstream stationary cylinder using PIV system. They observed the interference galloping phenomenon at the center-to-center spacing ratio of (2−5.6)D (where D is the cylinder diameter). Price et al. (2007) in-vestigated the forced harmonic oscillation of upstream cylinder transverse to the flow direction in a pair of staggered configuration. The flow patterns around the staggered cylinders were significantly modified due to the oscillation of the upstream cylinder, strong periodicities at the frequency of oscillating cylinder and existence of the sub-and superharmonic resonances were observed. Moreover, in the work of Kim et al. (2009), which investigated the flow-induced vibration characteristics of two tandem circular cylinders, they attempted to identify the flow regimes depending on the spacing ratio L/D, fluctuating lift forces and vibration characteristics of the cylinders. Huera-Huarte and Gharib (2011) investigated the dynamic response of two flexible cylinders in tandem arrangement. In their studies, the main excitation mechanism was wake-induced vibration. The rear cylinder showed large amplitudes of response, at reduced velocities over the expected ones at lock-in when a cylinder was undergoing VIV being isolated. Gao et al. (2014) investigated the flow characteristics for a cylinder oscillating freely in the wake of a larger cylinder upstream using the PIV tech-nique. They found that the presence of the upstream larger cylinder had significant effect on the flow patterns, causing a great reduction in both the oscillation amplitude and vortex shedding frequency.
A number of numerical studies were also conducted to investigate the flow-induced vibration of two tandem cylinders at low Reynolds numbers (Papaioannou et al., 2008;Prasanth and Mittal, 2009;Yang and Zheng, 2010). Prasanth and Mittal (2009) studied the flow-induced oscillation of two circular cylinders in tandem arrangement at Re = 100 with a constant centre-to-centre spacing of 5.5D. The results showed that the presence of a downstream cylinder had significant effect on the behaviour of the upstream cylinder. Borazjani and Sotiropoulos (2009) investigated vortex-induced vibrations of two identical elastically mounted cylinders in tandem in the proximity-wake interference regime at Re = 200. The results showed that when the gapflow mechanism was triggered, the 2-DOF system can develop and sustain large VIV amplitudes comparable to those observed in the corresponding (same reduced velocity) 1-DOF system. Bao et al. (2012) investigated the flow-induced vibrations of isolated and tandem elastically mounted cylinders which had two degrees of freedom and a variety of the in-line to the transverse natural frequency ratios. They found that for a downstream cylinder, the in-line dynamic response was more sensitive to the natural frequency ratio than the response in the transverse direction.
Above all, these previous studies have greatly improved our understanding of flow behavior for the elastically mounted cylinder oscillating in the wake of an upstream cylinder. However, less attention has been paid to flow past a freely suspended cylinder in the wake of an upstream stationary cylinder, which also has important applications in the field of oil exploration, such as the riser system and mooring lines connected to the floating system. One question may arise: if the suspended downstream cylinder was allowed to move freely in the horizontal-plane in the wake of an upstream cylinder, what kind of flow behavior will appear? Furthermore, the effect of upstream cylinder on the orbiting response of the downstream freely suspended cylinder has not yet to be understood. Less consideration has been given to the effects of the upstream stationary cylinder on the orbiting response and vortex wake dynamics for flow past a freely suspended downstream cylinder undergoing both streamwise and transverse oscillations in the wake of a stationary cylinder.
Therefore, the purpose of this paper was to investigate the orbital response and wake characteristics of a freely suspended cylinder in the wake of a stationary cylinder at different flow velocities, where the downstream cylinder assembly was constrained to freely oscillate and move in both the transverse and streamwise directions. Flow fields were obtained by using the particle image velocimetry technique (PIV) and the trajectory of downstream cylinder was cap-tured by a video camera. The effect of flow velocities and presence of the upstream stationary cylinder on the orbital response and vortex wake modes of the downstream suspended cylinder were examined and presented herein. Experiments were conducted in a recirculating open water channel with a 6 m long rectangular cross-section of ( ) at Maritime Research Centre, Nanyang Technological University. Fig. 1 shows the schematic sketch of the experimental setup, where the coordinates x and y denote the streamwise and transverse directions, respectively. The two cylinder models were constructed of smooth acrylic circular tubes, with D=30 mm. The top end of the upstream cylinder was fixed on a plate. The top end of the downstream cylinder protruded through a light foam sheet platform and was secured with two small acrylic blocks. In order to allow the platform free to translate in the streamwise and transverse directions, two well-lubricated ballbearing-filled tracks were designed to provide close to frictionless contact between the platform and the tracks (Gao et al., 2015). In details, the tracks were filled with customized stainless steel balls with minimal friction coefficient, and the surfaces of the bearing balls were well lubricated. Special tap with smoothed surface was also applied on the bottom of the platform. To minimize the friction at the interfaces, the cylinder assembly was designed to be as light as possible. In this manner, the downstream cylinder assembly was suspended from below the platform and constrained to freely oscillate and move in both the transverse and streamwise directions. The descriptions of the experimental details can also be found in Gao et al. (2015). The Reynolds number was in the range of (where D was the cylinder diameter, U was the free-stream velocity and was the kinematic viscosity), corresponding to the flow velocity 0.27 m/s ≤ U ≤ 0.43 m/s. The centre-to-centre spacing ratio L/D used in this experiment was kept in a constant value of 3.0.
A PIV system (LaVision model) was used to measure the flow field behind two tandem cylinders. The flow is seeding using the neutrally buoyant hollow glass spheres (Spherocel ® 110P8) with an approximate diameter of 10− 15 μm as the tracer particles, which offered good traceability and scattering efficiency. The particles were illuminated by a light sheet emitted from a Quantel System double cavity Nd: YAG laser (power ~120 mJ per pulse, duration5 ns). The particle images were recorded using a 12-bit charge-coupled device (CCD) camera with a resolution of pixels. The viewing area was chosen to be about . The LaVision Davis software was used for PIV analysis, to process the raw particle images and determine the flow fields. Velocity vectors were determined using the FFT (Fast-Fourier-Transform) method based on cross correlation algorithm with the standard Gaussian sub-pixel fit structured as an iterative multi-grid method. The processing procedure included two passes, starting with a grid size of pixels, stepping down to pixels overlapping by 50%. A set of 7 500 vectors was obtained in the viewing area. Then a median filter was applied to remove possible outliers in the vector map. For the smoothing process, the average filter was chosen to obtain the final vector maps. For each experiment, a set of 1 050 frames of the instantaneous flow fields was acquired at the frequency of 15 Hz. Further details regarding to the PIV post-processing can be found in Gao et al. (2014).
In this experiment, the laser sheet was positioned in the horizontal (x−y) plane at the mid-height of water depth H s = 100 mm. The clearance between the cylinder and the bottom was about 15 mm, resulting into the aspect ratio of 6.17. The effect due to vortex interaction along the span of the cylinder on the shedding behavior at the mid-plane was likely to be minimal (Norberg, 2003;Lau et al., 2004). The aspect ratio was considered to be large enough to ensure the two-dimensional flow at the mid-plane (Slaouti and Gerrard, 1981;West and Apelt, 1993;Lam and Zou, 2010;Wang et al., 2013). The descriptions of the experimental details can be found in Gao et al. (2015).
A Cannon video camera with a sampling rate of 60 Hz was used to capture the trajectory of the downstream freely suspended cylinder, which was mounted above the water channel. A wide angle view, lens (Cannon EF 8-15 mm f/4L fisheye USM Lens) was used to capture a 180-degree view of the horizontal plane including the top of the cylinder. The lens chosen captured distortion free image even at the corners. The dimension of the field of view obtained from the Cannon camera was about 52D×29D. In order to trace the motion trajectory of the cylinder, each experiment was repeated 6 times. The top of the cylinder was painted black and an image processing program using Matlab was used to extract the locus of the moving cylinder. The streamwise and transverse oscillation frequencies of the cylinder were obtained through spectra analysis of the time histories of the displacements in both the streamwise and transverse direc-tions.

Results and discussions
3.1 Orbital response of downstream cylinder Fig. 2 shows the time histories of the non-dimensional transverse displacements of the downstream suspended cylinder in the wake of a stationary cylinder at different flow velocities. The results indicate that the flow velocity U has a significant effect on the orbital response of the downstream cylinder, different trajectories are observed with increasing of U. As shown in Fig. 2a, at flow velocity U = 0.27 m/s, the downstream cylinder undergoes transverse oscillation with very small amplitude of 0.15D periodically, approximately to be sinusoidal, similar to the case of elastically mounted cylinder observed in Assi et al. (2006), Kim et al. (2009) andCarmo et al. (2011). At U = 0.3 m/s, the transverse oscillation response of the downstream cylinder appears to be irregular, with marginal increasing in the maximum amplitude. Further increases in the flow velocity U to 0.33 m/s caused a small increase in transverse oscillation amplitude to be 0.2D. With further increasing of U, the magnitude of the transverse oscillation amplitude is approximately the same for U = 0.33 m/s and 0.37 m/s. It is interesting to note that, different from the cases of elastically mounted cylinder, the downstream suspended cylinder traverses downstream obliquely due to the interaction of vortex shed from the upstream stationary cylinder, which has not been reported in the cases of elastically mounted cylinder. As U increases to 0.43 m/s, the maximum transverse oscillation amplitude increases significantly to about 0.4D. Owing to the large flow velocity, there are only a few transverse oscillation periods observed in the view field. Fig. 3 shows the time history of the streamwise displacements of the downstream suspended cylinder at different flow velocities. Compared with the transverse oscillation response, the streamwise oscillation response behaves much more complex. At U = 0.27 m/s, similar to the cases of elastically mounted cylinder, the downstream suspended cylinder undergoes streamwise oscillation with a regular sinusoidal trace, the streamwise oscillation amplitude is about 0.1D, half of the transverse oscillation amplitude, consistent with the results in Prasanth and Mittal (2009). At 0.3 m/s ≤ U ≤ 0.37 m/s, a rather striking emerging from Figs. 3b−3d is that a "jump" phenomenon of the streamwise displacements is observed for the downstream suspended cylinder. The jump phenomenon is significantly different from that of elastically mounted downstream cylinder, where the suspended downstream cylinder begins to move downstream accompany with streamwise oscillations, far away from the initial position, while the elastically mounted downstream cylinder undergoes streamwise oscillations at the vicinity of the initial position. The gap distance between the two cylinders increases, and becomes suffi-cient for the upstream stationary cylinder to experience vortex shedding behind it. When U increases to 0.43 m/s, the streamwise displacement of the downstream suspended cyl-inder appears to have a linear relationship with t (s). The streamwise oscillation disappeared at large flow velocity, instead of large oscillation amplitudes occurred in the case   GAO Yang-yang et al. China Ocean Eng., 2020, Vol. 34, No. 5, P. 708-717 711 of elastically mounted cylinder. It can be concluded that the streamwise oscillation response of the downstream suspended cylinder is significantly affected by the flow velocity, where both intermittent movement and oscillation phenomenon can be observed when U ≥ 0.3 m/s. The trajectories of the downstream freely suspended cylinder at different flow velocities are shown in Fig. 4. Different trajectory responses of the downstream suspended cylinder with variation of the flow velocity U are observed. At U = 0.27 m/s, the downstream suspended cylinder undergoes an approximate figure-8 type motion like the case of elastically mounted cylinder observed in Papaioannou et al. (2008) and Bao et al. (2012). With increasing of U to the range of 0.3 m/s to 0.37 m/s, the trajectory motion for the downstream cylinder behaves much more complex and significantly different from the case of elastically mounted cylinder, where the intermittent movement and oscillation phenomena are observed, accompanied with an approximate figure-8 type motion. The equilibrium position of the oscillation drifts downstream as the downstream cylinder is subjected to the unsteady wake induced by the upstream stationary cylinder. But when U increases to 0.43 m/s, only a quasi-sinusoidal trace motion of the downstream cylinder is observed since the cylinder moves fast away from the view of field.

Oscillation frequency
The power spectral density function E u of both stream-wise and transverse displacements from the downstream freely suspended cylinder for different flow velocities is presented in Fig.5. For all the examined flow velocities, the streamwise oscillation response (Fig. 5a) appears to behave much more complex than the transverse oscillation response (Fig. 5b), where a single dominant peak is observed for the transverse displacement spectra, while two dominant peaks are observed for the streamwise one except U=0.43 m/s. At flow velocity U = 0.27 m/s, both the transverse and streamwise displacement spectra yield one major peak at , together with another large peak at the frequency of for the streamwise oscillation responses. With further increases in U to 0.3 m/s, the dominant oscillation frequency for the transverse response appears to be a little larger than the first mode oscillation frequency for the streamwise one. Another larger frequency is observed with a more than twice of the transverse oscillation frequency. When U increases to 0.33 m/s, the transverse oscillation frequency is the same as the first mode oscillation frequency of streamwise response , and a little smaller than the half of the second mode oscillation frequency for the streamwise oscillation spectra. The peaks in both transverse and streamise displacement spectra for U=0.33 m/s are considerably larger than those at U = 0.27 m/s and U = 0.3 m/s. When U increases to 0.37 m/s, the streamwise displacement spectrum yields two peaks at and , while a pronounced peak is observed at for the transverse displacement spectrum. When U further increases to 0.43 m/s, different from those at lower U, no peak is observed for the streamwise displacement spectrum, only a dominant peak is observed at for the transverse one.

Instantaneous flow fields
The instantaneous flow fields captured by PIV technique were analyzed to reveal the dynamic response of the suspended cylinder in the wake of an upstream stationary cylinder. Sequential instantaneous streamline patterns and vorticity counters at different flow velocities U are shown in Figs. 6−10. Solid lines and dashed lines represent positive and negative vortices, respectively. The incremental value of the vorticity is 0.5. It can be seen from these figures that with the increase of U, different flow behaviors are observed, characterized by noting the difference in the sequence of instantaneous streamline topologies and vorticity contours.
At U=0.27 m/s, when the downstream cylinder oscillates at the upwards location, as shown in the streamline topology plots in Fig. 6, two eddies are observed behind the upstream cylinder and a single eddy is observed behind the downstream cylinder. When the suspended cylinder oscillates downwards, only a large scale eddy is observed in the gap between two cylinders and a small scale eddy is observed in the far wake of downstream cylinder. In the vorticity contour plots, the oscillation response of the downstream cylinder is relatively weak, and no vortex shedding is observed in the gap between the two cylinders, consistent with the shear layer reattachment flow regime observed for two tandem stationary cylinders at the same spacing ratio in Carmo and Meneghini (2006), Sumner (2010). However, due to the interaction between the shear layers emanating from upstream and the suspended cylinder, the vortices shed from the downstream cylinder appear to be deformed. The asymmetrical vortex shedding deflected downwards is observed behind the downstream cylinder, which is similar to the results of elastically mounted downstream cylinder in the wake of an upstream stationary one in Assi et al. (2006), Kim et al. (2009), Carmo et al. (2011.
When U increases to 0.3 m/s, the flow patterns behave more turbulent than those at U = 0.27 m/s. In the streamline topology plots, as shown in Fig. 7a, multiple eddies are observed behind the upstream cylinder. However, no eddy is observed behind the downstream suspended cylinder. As the suspended cylinder drifts downstream, the influence of upstream stationary cylinder decreases, and a single eddy is observed behind the downstream cylinder in Fig. 7c. In the vorticity contour plots, the vortex roll-up is prevented in the  GAO Yang-yang et al. China Ocean Eng., 2020, Vol. 34, No. 5, P. 708-717 713 gap between two cylinders due to the presence of the downstream suspended cylinder, while the vortex amalgamation phenomenon is observed instead. Multiple smaller vortices are observed behind the downstream cylinder.
As U further increases to 0.33 m/s, in the vorticity plots, as shown in Fig. 8a, the shear layers separated from both the upper and lower side of the upstream stationary cylinder are stretched, an approximate shear-layer-reattachment pattern is also observed. The vortex shed from the downstream cylinder performs irregularly. For increasing spacing within the reattachment regime, the shear layer that reattached on the front side of the downstream cylinder is drawn into the gap from at least one of the two sides. As the suspended cylinder drifts downstream, the separation between the two cylinders increases to be sufficient for the occurrence of vortex shedding from the upstream cylinder, as shown in Fig. 8c.   GAO Yang-yang et al. China Ocean Eng., 2020, Vol. 34, No. 5, P. 708-717 Apart from the difference in the sequence of instantaneous vorticity contours behind two cylinders, there is a qualitative difference in the streamline topologies, where the flow pattern consisting of two coordinate eddies at the initial time changes to no eddy existence when the cylinder moves downstream.
When U further increases to 0.37 m/s, as shown in Fig. 9a, complex vortex pairing in the wake of the down-stream suspended cylinder is observed. The vorticity dynamics are characterized by the merger of vortices of the same sense of rotation and the intense interaction and deformation of vortices of opposite signs. When the suspended cylinder drifts downstream, the separation between two cylinders is sufficiently large to shed vortices, similar to the co-shedding flow regime for two tandem stationary cylinders. In the streamline topology plots, irregular eddies are  Fig. 9. Instantaneous streamline topologies (top row) and vorticity contours (bottom row) behind the freely suspended cylinder at U =0.37 m/s. GAO Yang-yang et al. China Ocean Eng., 2020, Vol. 34, No. 5, P. 708-717 715 observed in the gap between two cylinders, and no eddy is observed behind the downstream cylinder. When U increases to 0.43 m/s, as shown in Fig. 10, the downstream cylinder is constrained to travel downstream fast. In the streamline topology plots, two eddies are observed in the gap between two cylinders at the initial time, while only a small lobe is observed behind the downstream suspended cylinder. As the cylinder moves downward shown in Fig. 10b, the streamline topologies behave differently with two eddies in the upper side of the upstream cylinder. With further movement of the cylinder, as shown in Fig. 10c, only one eddy is observed close to the surface of upstream cylinder.
In the vorticity contour plots, multiple smaller vortices are observed behind the downstream cylinder. As the suspended cylinder drifts downstream, the vortex roll-up is observed in the gap, and multiple small negative vortices are also observed in front of the downstream cylinder. As mentioned above, when U increases from 0.27 m/s to 0.43 m/s, the center-to-center spacing ratio between the upstream and downstream cylinders increases from 3.0 to larger than 10, resulting in the transitions of flow regimes from the shear layer reattachment regime to the co-shedding flow regime. The mean velocities of the downstream cylinder are calculated by the time histories of the displacements of the downstream cylinder, as shown in Fig. 11. At small flow velocity U = 0.27 m/s, the downstream cylinder exhibits the streamwise and transverse oscillation at the initial position, relatively small instantenous velocity is observed. At 0.3 m/s≤U≤ 0.37 m/s, the intermittent movement and oscillation phenomena are observed, accompanied with an approximate figure-8 type motion, the streamwise mean velocity of the downstream cylinder is kept below 0.002 m/s. At  large flow velocity U=0.43 m/s, the streamwise mean velocity of the cylinder is about 0.078 m/s.

Conclusions
Flow past a freely suspended cylinder in the wake of an upstream stationary cylinder has been investigated at different flow velocities. The Particle Image Velocimetry (PIV) technique was employed to capture the flow field and a camera to capture the trajectory of the downstream suspended cylinder. The results showed that the presence of the upstream stationary cylinder had significant effect on the orbital responses and flow patterns of the downstream suspended cylinder. Three different orbital responses for the downstream cylinder were observed, namely (1) depicting a figure-8 type motion at U = 0.27 m/s; (2) undergoing intermittent oscillations as it travels downstream at 0.3 m/s ≤ U ≤ 0.37 m/s; (3) successive moving downstream, no obvious streamwise oscillation observed at U = 0.43 m/s. The vortex wake modes behave much more complex than those behind the stationary and elastically mounted cylinders, where different flow characteristics are observed in terms of different orbital responses of the downstream suspended cylinder.