Response of Two Unequal-Diameter Flexible Cylinders in A Side-by-Side Arrangement: Characteristics of FIV

Till now, little information is available on the flow-induced vibration (FIV) of multiple flexible cylinders with unequal diameters. Some FIV characteristics of unequal-diameter cylinders can be predicted based on the knowledge of equal-diameter cylinders, while there are still other features remaining unrevealed. In this paper, the FIV characteristics of two flexible cylinders with unequal diameters arranged side-by-side are experimentally investigated. The diameter ratio of the small cylinder (Small Cyl.) to the large cylinder (Large Cyl.) is nearly 0.5. The aspect ratios and mass ratios of the two flexible cylinders are 350/181 and 1.90/1.47, respectively. The centre-to-centre spacing ratio in the cross-flow (CF) direction is kept constant as 6.0 and the two cylinders can oscillate freely in both the CF and in-line (IL) directions. The towing velocity varies from 0.05 m/s to 1.00 m/s. The dominant modes and frequencies, CF and IL displacement amplitudes and response trajectories are discussed. Compared with the case of two identical cylinders in our previous study, the FIV responses demonstrate some similarities and differences. The similarities are as follows. Both cylinders exhibit multi-mode vibration features and they interact with each other. Meanwhile, the IL FIV shows a more complex behaviour than that in the CF direction. The difference is that as the diameter of one cylinder is increased, the effect on the smaller cylinder becomes more significant. For Large Cyl., the FIV response is similar to its isolated counterpart, which indicates that Small Cyl. has a negligible effect on the FIV of the larger one. Whereas Large Cyl. perplexes the FIV of Small Cyl. during the vibration process. The spacing would change when both cylinders are oscillating. Proximity interference between the two cylinders and wake shielding effect of the Large Cyl. may occur. The dominant frequencies of Small Cyl. are reduced and the wake-induced flutter of Small Cyl. is observed from the response trajectories at different measuring points.


Introduction
Clustered flexible cylindrical structures, such as marine risers and cables, are important structural components for the production and transportation of oil and gas in offshore engineering. These flexible cylindrical structures are exposed to complex ocean currents, which leads to the wellknown fluid-structure interaction (FSI) phenomenon named the flow-induced vibration (FIV). It is widely acknowledged that the FIV is a main cause of the fatigue damages to the structures and the cylindrical structures become more susceptible to the FIV with the increase in their structural lengths and the subsequent lowered natural frequencies. In order to further understand the FIV of multiple circular cylinders, one may refer to Zdravkovich (1987), Zhou et al. (2001), Bearman (2011), Han et al. (2018) and Xu et al. (2018aXu et al. ( , 2018bXu et al. ( , 2018cXu et al. ( , 2019, Wang et al. (2019), Lai (2019), Lai et al. (2019), Ma et al. (2020), and Zhu et al. (2019Zhu et al. ( , 2020. A pair of side-by-side circular cylinders can be regarded as one of the simplest configurations for investigating the FIV of multiple cylinders. A comprehensive review on the flow around two identical circular cylinders fixed side-by-side in space was conducted by Zhou and Alam (2016). Depending on the spacing ratio S/D (S is the centreto-centre spacing in the transverse direction and D is the cylinder diameter) or the behaviours of the gap flow between the cylinders, three flow regimes were identified. At S/D<1.2, the gap flow was so weak that a single vortex street was formed and no vortex was generated in the gap between the cylinders. For 1.2≤S/D<2.0−2.2, a narrow wake was formed behind one cylinder while a wide wake was observed for the other one. This flow regime was often referred to as a bistable regime because the biased gap flow might switch from one side to the other randomly and both states were stable (Wang and Zhou, 2005). With an increase in S/D, i.e., 2.0−2.2<S/D<4.0−5.0, the gap flow possessed adequate momentum and a coupled wake street appeared (Alam et al., 2003).
When the two cylinders are subject to vibrations, the S/D between them changes with time. Moreover, the FIV of flexible cylinders often involves multiple vibration modes and mode competition, which makes the scenario more intricate. Therefore, some attention was paid to the FIV of two equal-diameter flexible cylinders undergoing oscillations. Sanaati and Kato (2014) performed an experimental investigation into the effects of proximity interference on the FIV of two pre-tensioned flexible cylinders in a side-by-side arrangement. The amplitude responses of both cylinders resembled that of an isolated cylinder at S/D = 2.75. At this small spacing ratio, the CF oscillation frequencies of the upper cylinder differed from those of the lower cylinder and an isolated one. It was observed that even for S/D = 5.5, the wakes of the cylinder were strongly synchronised within the region of large vibration amplitudes. Huera-Huarte and Gharib (2011) considered more spacing ratios, i.e., S/D = 2.0, 2.5, 3.0, 3.5, 4.0 and 5.0 in their experiments. The cylinder models with an aspect ratio of 93.75 were placed in a free-surface water channel and free to vibrate in both the CF and IL directions. When S/D was larger than 3.5, the cylinders showed no synchronized motions and they tend to vibrate independently in the VIV manner. Strong proximity interference was found to exist when S/D < 3.5. Later, Xu et al. (2018a) further discussed the FIV characteristics of two flexible cylinders arranged side-by-side. They investigated the multi-mode FIV of two equal-diameter cylinders with an aspect ratio of 350.0. S/D = 3.0, 4.0, 6.0 and 8.0 were selected in the laboratory tests. Their results indicated that the proximity interference existed in the CF direction when S/D was not larger than 6.0. The IL displacement amplitudes of the two cylinders were enhanced by the strong interaction until S/D = 8.0, which contradicted the previous conclusions by Huera-Huarte and Gharib (2011).
Compared with equal-diameter cylinders, a group of cylinders with different diameters are more common in the drilling and production operations. The interaction between a small cylinder (Small Cyl.) and a large cylinder (Large Cyl.) is more complicated than that between a pair of identical cylinders. Owing to the practical significance in engineering practice, research efforts were taken to investigate the flow around two unequal-diameter cylinders in a side-by-side arrangement via numerical simulations (Rahmanian et al., 2014a, 2014b) as well as experimental tests (Lam and To, 2003;Yokoi and Hirao, 2009;Gao et al., 2010;Lee et al., 2012;Song et al., 2015). Lam and To (2003) experimentally investigated the FIV characteristics of a flexibly-mounted circular cylinder in vicinity of a larger cylinder and subject to cross-flow. Complex interference was observed between the flow over the two cylinders with d/D = 0.5 (d denotes the diameter of Small Cyl. and D stands for the diameter of Large Cyl.). The CF and IL spacing ratios were varied from 0 to 2.82 and from 0 to 4.68, respectively. It was observed that the FIV of the elastically mounted Small Cyl. was greatly suppressed in side-by-side arrangement and tandem or near-tandem arrangement. Yokoi and Hirao (2009) carried out experiments on two side-by-side cylinders with d/D = 0.5 and S/d = 3.5, 4.5 and 6.5. Some representative flow patterns relying on S/d were observed, which was similar to the two equal-diameter cylinder case. The large and small cylinders were found to have different lock-in ranges both wider than that of an isolated cylinder. The vortex interaction, the spatial distributions in the flow and the streamwise evolution of the spectral amplitude along the shear layer behind side-by-side cylinders with d/D = 0.5 were studied by Song et al. (2015). They found that the gap flow always stably turned to Small Cyl. For side-by-side cylinders with d/D = 0.5, the effect on the wide wake was more significant but was less pronounced on the narrow wake. The frequency ratios of the narrow and wide wakes depended strongly on the gap ratio and the diameter ratio, however, they were impertinent to the Reynolds number.
In general, studies on the FIV responses of two unequaldiameter cylinders arranged in parallel are still quite limited. Thereinto, the previous research was mainly focussed on fixed cylinders or the FIV of rigid cylinders. However, rigid cylinders cannot fully respond to the structural motion, which is rather different from flexible cylindrical structures with high aspect ratios and low mass ratios. Furthermore, the vibrations of unequal-diameter cylinders in a side-byside arrangement are associated with more complex wake flow and sophisticated FIV responses, which cannot be directly predicted based on the studies for two vibrating equaldiameter cylinders arranged side-by-side.
In this paper, we try to fill the gap in the literature for the FIV of two unequal-diameter flexible cylinders in a side-by-side arrangement by presenting an experimental investigation. The outline for the rest of the paper is as follows. Section 2 describes the experimental setup and the methods used to post-process the data. In Section 3, the displacement amplitudes, response trajectories, dominant frequencies and dominant modes are discussed in detail. Finally, based on the discussions about the results, some conclusions are drawn in Section 4.

Description of the experiment
The experimental tests were carried out in the State Key Laboratory of Hydraulic Engineering Simulation and Safety at Tianjin University. The carriage is installed on a 137.0 m×7.0 m×3.3 m (length×width×depth) towing tank. The experiment apparatus as illustrated in Fig. 1 was mounted on the towing carriage. The main components of the apparatus included the supporting frame, vertical supporting rods, guide plate dials, cylinder models and the axial tension systems. The towing carriage moved along the smooth tracks to simulate the uniform flow. The flow velocity varied from 0.05 m/s to 1.0 m/s with an increment of 0.05 m/s. The initial axial pretensions in the present tests were 450 N. Two flexible cylinders with unequal diameters were assembled on the guide plate dials. The dials could be rotated to realise the different configurations of the two cylinders. In this experiment, they were adjusted to the side-by-side arrangement and then fixed to the guide plates. Universal joints were adopted to achieve the connection purpose and provide the boundary conditions. More details about the experiment apparatus can be found in Xu et al. (2018a). Each cylinder model consisted of an outer silicone tube and an inner copper pipe as shown in Fig. 2a. The inner copper pipes mainly contribute to the bending stiffnesses of the cylinder models. Whereas, the outer silicone tubes provide smooth external surfaces of the cylinder models, offer insultations to the measuring instrumentation and adjust the outer diameters of the cylinders. Compared with the inner copper pipes, the outer silicone tubes have nearly negligible influence on the bending stiffnesses of the flexible cylinders. Seven measuring points (G1−G7) were arranged equidistantly along the span of the cylinder (see Fig. 2b). The CF and IL strain gauges were attached to the outer surface of the copper pipe to measure the strain signals. The data were collected after the flow velocity was stable with a sampling frequency of 100 Hz for a sampling duration of 50 s. The waiting time between two consecutive runs was 15 minutes for the water in the towing tank to settle. In the experimental tests, a filter was applied to the bending strain in the frequency domain to remove the undesired frequency components. The strain signals were high-pass filtered with a cutoff of 1 Hz to eliminate the motion of the carriage and the vibrations of the supporting structures, and low-pass filtered with a cutoff of 40 Hz to remove the inevitable high-frequency noise. The same technique was also adopted in our previous studies (Xu et al., 2018a(Xu et al., , 2018bWang et al., 2019).
The two unequal-diameter cylinders in the previous studies by Lam and To (2003) and Zhao et al. (2007) possessed a diameter ratio d/D (defined by the ratio of the diameter of Small Cyl. to that of Large Cyl.) around 0.5. Therefore, for comparison purposes with the existing research, the diameter ratio in the present study was also selected as d/D ≈ 0.5. In our previous publications on the FIV of multiple identical cylinders (Xu et al., 2018a(Xu et al., , 2018b(Xu et al., , 2018cWang et al., 2019), the diameter of each cylinder was 0.016 m. In this experimental campaign, the diameter of Small Cyl. was inherited from our aforementioned experiments as d = 0.016 m, whereas that of Large Cyl. was chosen as D = 0.031 m. The key parameters of the cylinder models are tabulated in Table 1. The cylinders had identical lengths L = 5.6 m. The corresponding aspect ratios were L/d = 350 and L/D = 181. The masses per unit length of Small Cyl. and Large Cyl. were m s = 0.3821 kg/m and m l = 1.1071 kg/m and the resulting mass ratios were m s * = 1.90 and m l * = 1.47, respectively. The damping ratios of the two cylinders were determined from the free decay tests and calculated by where Y I and Y I+J are the strain response amplitudes of the Ith and the (I+J)-th cycles. In the present experimental tests, the damping ratios of Small Cyl. and Large Cyl. respectively were = 0.034 and = 0.031. According to our previous experiment on the FIV of two identical flexible cylinders (Xu et al., 2018a), proximity interference existed for the spacing ratio up to 6.0. For the two equal-diameter cylinders system, strong interactions  XU Wan-hai et al. China Ocean Eng., 2020, Vol. 34, No. 4, P. 475-487 477 between the vibrations of the cylinders were observed at S/d = 6.0. Therefore, it is expected that the intense interference between the two unequal-diameter cylinders would also be triggered by setting S/d to 6.0. Moreover, the selection of this particular S/d enables an easier comparison with our previous results for the FIV of two flexible cylinders with equal diameters in Xu et al. (2018a). Here, the spacing ratio is nondimensionalised using the diameter of Small Cyl. due to the fact that it was kept the same as those used in our previous experiments. Zdravkovich (1987) classified the wake of two identical stationary cylinders based on the interference between them and divided the entire S/d and T/d (T is the centre-to-centre spacing in the longitudinal direction) domain into four interference regions, i.e., proximity interference, wake interference, proximity and wake interferences and no interference. Fig. 3 presents the combined maps of the interference regimes for both Large Cyl. and Small Cyl. in the stationary configurations. The boundary profiles of the flow regimes for Large Cyl. are represented by the solid blue line while those for Small Cyl. are denoted by the dotted red line. It can be interpreted that for the particular spacing considered in this study, Small Cyl. is located in the proximity interference region of Large Cyl. In contrast, Large Cyl. lies in the no interference regime of Small Cyl. Once the two flexible cylinders are vibrating in both the CF and IL directions, the classification in Fig. 3 might be inaccurate. In that case, the cylinders could also be affected by wake interference. In this experiment, the strain information was obtained from each measuring point. However, what interested us were the displacement responses. Therefore, a modal analysis method was employed to reconstruct the filtered strain signals into the displacement information of the cylinders.
Taking the reconstruction of the CF displacement for each cylinder as an example, the CF displacement (y) is given by (2) where z is the axial coordinate, t is time, is the modal weight, and is the mode shape. The cylinder is simplified into an Euler−Bernoulli beam with simply supported boundary conditions whose mode shape can be expressed as: w n (t) The modal weight is derived based on the relationship between the curvature and strain as follows: Then, the displacement information can be obtained by substituting w n (t) in Eq. (4) into Eq. (2). More details about the modal approach used in the reconstruction of the displacement responses can be found in Trim et al. (2005) and Lie and Kaasen (2006).

Results and discussions
In this section, the experimental results are presented and important FIV response characteristics such as the dominant modes and frequencies, the maximum root mean square (RMS) displacements and the motions trajectories are analysed in detail.
Conventionally, the reduced velocity (V r ) is defined with respect to the freestream velocity, the diameter of the cylinder and the natural frequency of the cylinder. However, the diameters of the two cylinder models in the present study are not the same and their natural frequencies are also different. For expedient comparison, the reduced velocity in this paper is defined in terms of Small Cyl. as: It was found that the theoretical values of the fundamental natural frequencies (natural frequencies of the first mode) of the two cylinders were in reasonable agreement with their counterparts measured in the free decay tests. In order to make a comparison with our previous studies, the theoretical values of the fundamental natural frequencies for  the two cylinders were adopted in this paper. Those theoretical values can be respectively calculated by Hereby, f s,1 and f l,1 were 2.49 Hz and 1.41 Hz, respectively. As the data at low flow velocities involve random vibrations with higher degrees of uncertainties, the results in the present research are presented for U≥0.15 m/s, corresponding to V r ≥ 3.76 when the measured data are relatively stable.

Dominant modes and frequencies
Tables 2 and 3 present the variations of the dominant vibration modes in the CF and IL directions with the reduced velocity. The CF dominant modes of the two cylinders increase from low to high as V r is increased. It is found that in the CF direction, Small Cyl. can be excited up to the fourth mode while the highest attainable mode of Large Cyl. being the third mode. Owing to the larger diameter of Large Cyl., the highest vibration mode it can reach may be lower than that of Small Cyl. for the velocity range considered in the present study. The dominant mode of Large Cyl. at each V r is exactly the same as that of the isolated large cylinder indicating that Large Cyl. is almost unaffected by Small Cyl. As depicted in Fig. 3, when the two cylinders are at rest with S/d = 6.0 in the side-by-side arrangement, Large Cyl. is in the no interference regime of Small Cyl. Even if they are subject to oscillations, Small Cyl. still has nearly negligible influence on the FIV of Large Cyl. The dominant modes of Small Cyl. in the present study are analogous to those of the upper cylinder in the equal-diameter case of Xu et al. (2018a). In the mode transition regions, the CF response of Small Cyl. is easier to reach the higher mode than that of the upper cylinder in Xu et al. (2018a). For example, Small Cyl. transits to the higher mode at V r = 7.52, 16.28 and 22.54 compared with V r = 8.77, 18.79 and 23.80 in the equal-cylinder case.
Similar to the CF dominant modes, the IL dominant modes of Large Cyl. are also insensitive to the presence and vibrations of Small Cyl. It can be concluded that when S/d is constant, the smaller d/D is, the weaker the interference effect on Large Cyl. becomes. The IL response of Small Cyl. behaves consistently with that in the CF direction in the mode transition regions. Small Cyl. switches to higher modes earlier than the isolated small cylinder in both the CF and IL directions. An abrupt jump in the IL dominant mode from the second mode to the fourth mode occurs at V r = 7.52 for Small Cyl. In contrast, that phenomenon is not observed for the isolated small cylinder until V r = 8.77. At the transitional reduced velocities, it is possible that multiple modes coexist and two adjacent modes may compete with each other, which consequently leads to the mode jumps. Small Cyl. is not excited into the sixth mode in the present experiment, which is different from the isolated small cylinder and the upper cylinder in the equal-diameter case (Xu et al., 2018a). Interestingly, the IL dominant mode of Small Cyl. drops to the fourth mode at high V r associated with a fall in the IL dominant frequency as shown in Fig. 4.
Fast Fourier Transform (FFT) method is adopted to compute the dominant frequencies from the time-varying displacements and the variations of the dimensionless dom-  In the unequal-diameter case, the dominant frequencies of Large Cyl. are lower than those of Small Cyl. Owing to the imparity in the diameters of the two cylinders, their wake widths are different. For S/d < 6.0 in the equal-diameter cylinder case of Xu et al. (2018a), the CF and IL dominant frequency responses of the upper cylinder are more fluctuating compared with those of the isolated cylinder. In terms of two equal-diameter stationary cylinders arranged in parallel, when S/d is around 3−4, the two cylinders are at the boundaries of the proximity interference regions and the vortex shedding frequencies are affected (Zdravkovich, 1987). According to Fig. 3, the freely vibrating Small Cyl. is mostly in the proximity interference region of Large Cyl. in this experiment. Therefore, the vortex shedding of Small Cyl. in the present study also has irregular fluctuations. The comparison of Figs. 4 and 5 indicates that this effect is more evident in its IL FIV response. Figs. 6 and 7 illustrate the amplitude spectra of the CF and IL displacements at different measuring points for Small Cyl. and Large Cyl. at V r = 13.78. Two large spikes are identified in the amplitude spectra in Fig. 7. Different oscillation frequencies exist at different measuring points. The readings of the two large spikes in the amplitude spec-    Fig. 7. Amplitude spectra of the IL displacements at different measuring points for Small Cyl. and Large Cyl. at V r = 13.78. tra for Small Cyl. respectively are 5.67 Hz and 11.10 Hz, where 5.67 Hz dominates. This frequency value is lower than those at other reduced velocities in the fourth mode.
In contrast to Fig. 7, the amplitude spectra of the CF displacements for both cylinders in Fig. 6 demonstrate only one peak revealing that the FIV in the IL direction for the two unequal-diameter flexible cylinders is more complex than that in the CF direction. Similar FIV characteristics were reported by Xu et al. (2018a) for two equal-diameter flexible cylinders in a side-by-side arrangement. It can also be seen from Figs. 6 and 7 that there are a lot of small spikes around the large peaks in the amplitude spectra of the CF and IL displacements for Small Cyl., which indicates that the vortex shedding of Small Cyl. is unstable and more complex than that of Large Cyl. resulting in the more complicated FIV behaviours.
Recalling Table 3, two interesting phenomena emerge. First, there are drops in the IL dominant mode of Small Cyl. leading to discontinuously lower IL dominant modes at V r = 10.02, 11.27, 23.80 and 25.05. In the meanwhile, the IL dominant frequency of Small Cyl. at these reduced velocities are also lower which coincides with the corresponding CF dominant frequency. Fig. 8 shows the direct comparison of the CF and IL dominant frequencies of Small Cyl. The IL dominant mode at the corresponding reduced velocities is also labelled in the figure. At the aforementioned reduced velocities, the IL dominant frequency falls and becomes identical to that in the CF direction. Since the CF and IL amplitudes of Large Cyl. are generally larger as shown in Figs. 9 and 10, there may be instants when Small Cyl. moves into the wake of Large Cyl. and the vortex shedding from Large Cyl. has an effect on Small Cyl. The declination of the IL dominant frequency was not observed for the two equal-diameter cylinders arranged side-by-side (Xu et al., 2018a), which reflects that the gap flow deflection is not the reason for the reduction in the IL dominant mode/frequency. In fact, the decrease in the IL dominant mode/frequency of Small Cyl. indicates the occurrence of the wakeinduced flutter. The wake-induced flutter is a mechanism that can excite the downstream cylinder into two-degree-offreedom vibrations, typically in an elliptical orbit. It can be excited in spite of the unsteadiness of the flow, being sustained only by the steady fluid forces present in the wake (Assi, 2009). Clear descriptions and explanations of this mechanism can be found in Price (1975), Price and Abdallah (1990) and Naudascher and Rockwell (1994). Further discussions about the wake-induced flutter of Small Cyl. will be provided in Section 3.3.
On the other hand, the third and fourth modes of Small Cyl. in the range of V r = 10.02−16.28 are associated with the dominant frequencies of the second and third modes, respectively. Yokoi and Hirao (2009) studied the vortex dynamics of two side-by-side cylinders with unequal diameters. Three different wake modes were identified, i.e., two similar vortex streets, the gap flow deflection and a single vortex street. According to the maximum CF displacements of the two cylinders in Fig. 9, both cylinders are subject to large-amplitude vibrations. When the two flexible cylinders are free to vibrate in both the CF and IL directions, they may encounter the scenario in which S/d < 6.0 if the vibration amplitudes are large.
As is shown in Fig. 11, the IL dominant frequency of Small Cyl. at V r = 15.03, 16.28 and 22.54 are distinctly low. For two stationary cylinders with unequal diameters in a side-by-side arrangement, the gap flow deflection causes the wake behind one cylinder to be wider than that behind the other one. Therefore, it is speculated that Small Cyl. might be followed by a wide wake and the gap flow is mainly biased towards Large Cyl. Moreover, it is found that the IL  XU Wan-hai et al. China Ocean Eng., 2020, Vol. 34, No. 4, P. 475-487 481 dominant frequencies of the two cylinders are essentially equal at V r = 12.52 and 13.78 indicating the possibility that S/d is reduced during the vibrations, so the cylinders may get close enough to form one single vortex street.
Since the lack of information about the flow field and the fluid forces, we can only make such a conjecture. Conversely, Gao et al. (2010) and Song et al. (2015) argued that it was easier for the gap flow to turn towards the smaller cylinder. Furthermore, when d/D = 0.5, gap flow deflection still exists at the initial spacing S/d = 6.0. The smaller d/D is, the larger becomes the critical distance for the gap flow deflection to disappear.

Displacement responses
Figs. 9 and 10 demonstrate the variations of the maximum CF and IL RMS dimensionless displacements for Small Cyl. and Large Cyl. with V r . The maximum CF and IL RMS displacements (Max yRMS and Max xRMS ) can be determined from the following equations.
As shown in Fig. 9, the CF amplitude response of Large Cyl. resembles that of the isolated large cylinder in the V r range considered. Max yRMS /D of Lage Cyl. has a peak value approximately equal to 3.39 and the corresponding value for the isolated large cylinder is 3.21. The overall trend and V r for the decreasing amplitudes are also similar. In contrast, the CF amplitude response of Small Cyl. is quite different from that of the isolated small cylinder. Max yRMS /d peaks at V r = 11.27 with the value of 1.68 while the isolated small cylinder reaches its peak value 1.47 when V r = 23.80. A noticeable drop is observed for the CF response amplitude of Small Cyl. in the V r range of 22.54−25.05.
The IL amplitude response of Large Cyl. in Fig. 10 nearly agrees with that of the isolated large cylinder. For the V r ranges of 10.02−12.52 and 23.80−25.05, the IL displacement amplitude of Small Cyl. increases. The increase in the IL amplitude is also one of the characteristics of the wakeinduced flutter (Prasanth and Mittal, 2009). The peak Max xRMS /d = 0.48 is achieved at V r = 11.27 while the peak value of Max xRMS /D for Large Cyl. is 0.48 and appears at V r = 23.80. The reduced velocities associated with the peak IL amplitudes are not the same. One possible reason for that is the different mass ratios of the cylinders (Huera-Huarte and Gharib, 2011). It is observed that both Max xRMS /d for Small Cyl. and Max xRMS /D for Large Cyl. fall over similar ranges of 0.2−0.5. However, as d/D = 0.5, the dimensional IL displacement amplitude of Large Cyl. is approximately twice that of Small Cyl. Fig. 12 presents the CF and IL response amplitude ratios of the side-by-side cylinders to the corresponding isolated cylinders. The amplitude ratio of Large Cyl. is around 1.0 except for a few slightly higher values. Similar to the dominant modes and frequencies, this also supports the conclusion that for the cases considered in the present study, the presence and vibrations of Small Cyl. have a negligible influence on Large Cyl. Nevertheless, the amplitude response of Small Cyl. is rather different from that of the isolated small cylinder. y RMS_small and x RMS_small are approximately twice those of the isolated small cylinder at V r = 8.77 whereas y RMS_small is about half of y RMS_isolated at V r = 7.52, 22.54 and 23.80. It was found by Xu et al. (2018a) that a strong proximity interference exists in the equal-diameter cylinder case when S/d < 6.0. As the lower cylinder in the present study has a larger diameter, it is presumed that the proximity interference would also be quite strong even if the initial S/d is 6.0. The effect of the strong proximity interference leads to the distinct amplitude response of Small Cyl. from its isolated counterpart.
From the above analyses, it is found that the interference effect between the two unequal-diameter cylinders in a side-by-side arrangement is more pronounced on Small Cyl.  For a better demonstration of the effect of Large Cyl. on Small Cyl., the results for the upper cylinder in the equaldiameter cylinder case by Xu et al. (2018a) are compared in Fig. 13. The variation trends and values in the two experiments are relatively close particularly for the IL RMS amplitude. In the CF direction, y RMS_small stays stably above y RMS of the upper cylinder after V r reaches 13.78. This manifests that Large Cyl. has an effect of magnifying the CF amplitude of Small Cyl. at high reduced velocities.
The spatial distributions of the CF RMS displacements for Small Cyl. and Large Cyl. together with their respective isolated cylinders are exhibited in Fig. 14. It can be seen from the figure that the spatial distributions of the CF RMS displacements of the cylinders are basically symmetrical about the middle sections of their spans at V r = 8.77. With the increase in V r to 15.03, Small Cyl. is influenced by the gap flow between the two cylinders and the spatial distribution of y RMS /d becomes significantly different from that of the isolated small cylinder. The spanwise distribution of y RMS /d is obviously asymmetrical, which is a combination of travelling and standing waves. Similarly, the spanwise distributions of x RMS /d and x RMS /D are shown in Fig. 15. At a relatively low V r of 8.77, the spatial distribution of the IL RMS displacements of the isolated small cylinder is characterised by standing waves. In contrast, the spatial distribution of x RMS /d of Small Cyl. has asymmetrical travelling wave characteristics. The variations of x RMS /D along the spans for Large Cyl. and the isolated large cylinder demonstrate combined travelling and standing wave responses when V r = 8.77. Both Small Cyl. and its isolated counterpart show typical standing wave characteristics at V r = 15.03. When V r is further increased to 23.80, Small Cyl. and Large Cyl. are dominated by the fourth mode and characterised by a combination of travelling and standing waves.

Response trajectories
As discussed in Section 3.1, the IL dominant frequency of Small Cyl. experiences drops and at certain reduced velocities, the IL dominant frequency coincides with the CF one. The declination in the IL dominant frequency usually follows the large-amplitude vibration in the CF direction. Furthermore, as d/D = 0.5 in the present study, the IL amplitude of Large Cyl. is twice that of Small Cyl. Based on the above reasons, Small Cyl. might be located behind Large Cyl.
At V r = 10.02, 11.27, 23.80 and 25.05, the IL dimensionless dominant frequency of Small Cyl. is lower and the value becomes identical to that in the CF direction. This phenomenon was not observed for the upper cylinder in the equal-diameter cylinder case with the same spacing ratio in Xu et al. (2018a). The wavelet transform is utilised to postprocess the vibration signals of Small Cyl. at V r = 11.27. As shown in Fig. 16, there are two frequency components (i.e., 4.33 Hz and 8.70 Hz) in the IL displacement signals of  Small Cyl. In the meanwhile, the frequencies in the CF direction at the corresponding measuring points are the same, with the value of 4.33 Hz. Consequently, the CF frequency is about half of the higher IL frequency component and identical to the lower IL frequency component. Fig. 17 shows the response trajectories of Small Cyl. at four different measuring points (G1, G3, G5 and G7) along the span for V r = 11.27. The response trajectories are ob-tained by plotting the nondimensional CF displacement against the dimensionless IL displacement at each measuring point once the cylinder response reaches the steady state. When the higher IL frequency component dominates, the orbital trajectories at measuring points G1 and G7 exhibit an asymmetrical figure-eight shape. The figure-eight shape indicates the occurrence of dual resonance where the IL and CF dominant frequencies have a ratio around 2:1

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XU Wan-hai et al. China Ocean Eng., 2020, Vol. 34, No. 4, P. 475-487 (Dahl et al., 2010. For the cases where the lower IL frequency component is in domination (G3 and G5), the 1:1 IL to CF dominant frequency ratio results in the asymmetrical elliptical shape trajectories, which indicates that Small Cyl. might be undergoing wake-induced flutter at those two measuring points. According to Assi (2014), Small Cyl. can extract energy from the flow in the elliptical orbit. In the meanwhile, there is a slight increase in the IL amplitude of Small Cyl. at this V r as depicted in Fig. 10. More intense IL vibrations and elliptical trajectories of the downstream cylinder is typical signs for the occurrence of wake-induced flutter. The observations above show that Small Cyl. is probably subject to wake-induced flutter and its response is influenced by the vortex shedding from Large Cyl. According to Assi (2014), the wake-induced flutter is excited in the unstable flow and maintained by a stable fluid force in the wake and it usually appears in tandem and staggered arrangements (Xu et al., 2018b. Similar phenomenon was not observed in our previous study on the FIV of two identical flexible cylinders in a side-by-side arrangement (Xu et al., 2018a). Therefore, the difference in the size of the side-by-side cylinders in the present experiment may give rise to the wake-induced flutter of Small Cyl.
The motion trajectories of Large Cyl. at measuring points G1, G3, G5 and G7 for V r = 23.80 are presented in Fig. 18. The trajectories are acquired in a way similar to that used for Fig. 17. Typical eight shape or C shape figures are observed. The trajectories at G3 and G5 are relatively regular and symmetrical. The characteristics of the trajectories for Large Cyl. resemble those of the lower cylinder in the equal-diameter cylinder case by Xu et al. (2018a).

Conclusions
The FIV of two flexible cylinders with different diameters arranged side-by-side is experimentally investigated. The initial spacing ratio is selected as S/d = 6.0 and the cylinders are free to oscillate in both the CF and IL directions. The FIV characteristics of the small and large cylinders (Small Cyl. and Large Cyl.) are summarised by analysing the dominant modes, dominant frequencies, the maximum  RMS displacements and the orbital trajectories. The experimental data are also compared with those for two identical flexible cylinders in parallel. The conclusions of this paper are as follows.
(1) The FIV responses of Small Cyl. and Large Cyl. demonstrate multi-mode vibration features. The in-line (IL) FIV behaviours are more complex than those in the crossflow (CF) direction. The representative eight shape and C shape trajectories are observed for Large Cyl. The response characteristics of Large Cyl. are analogous to those of the isolated large cylinder reflecting that the effect of Small Cyl. on the FIV of Large Cyl. is relatively weak. Similar to the upper cylinder in the identical cylinder tests, it is easier for Small Cyl. to be excited into higher vibration modes in the mode transition regions.
(2) The FIV of Small Cyl. is greatly affected by the presence and vibrations of Large Cyl. For the side-by-side arrangement with initial S/d = 6.0, it is speculated that a single vortex street is formed at high reduced velocities (V r = 15.03, 16.28 and 22.54) and the gap flow deflection appears at comparatively low reduced velocities (V r = 12.52 and 13.78. The IL dominant frequency of Small Cyl. coincides with the CF one at V r = 10.02, 11.27, 23.80 and 25.05. The wake-induced flutter is observed for Small Cyl. with elliptic orbital trajectories at certain measuring points. For the larger CF and IL displacements of Large Cyl., it is reasonable to believe that Small Cyl. located in the wake of Large Cyl. is a possible scenario. (3) The wake modes behind the two unequal-diameter flexible cylinders arranged side-by-side are inferred based on the present experimental data. Quantitative measures of the flow field and the fluid forces are still missing, which requires further investigations. Nonetheless, the present findings are still meaningful to the design of clustered flexible cylindrical structures with different diameters in offshore engineering. In this study, we only consider a specific case of two cylinders in a side-by-side arrangement with S/d = 6.0. More comprehensive and systematic investigations on the FIV of multiple flexible cylinders with different sizes in different configurations could be carried out in the future.