Use of Helical Strakes for FIV Suppression of Two Inclined Flexible Cylinders in A Side-by-Side Arrangement

The experimental studies on flow-induced vibrations (FIV) reduction of two side-by-side flexible cylinders inclined at 45° by using the helical strakes were carried out in a towing tank. The main aim of the experiment is to check whether the helical strakes with a pitch of 17.5D and a height of 0.25D, which is considered as the most effective vibration suppression device for the isolated cylinder undergoing vortex-shedding, still perform very well to reduce FIV of two inclined flexible cylinders in a side-by-side arrangement. The vibration of two identical inclined cylinders with a mass ratio of 1.90 and an aspect ratio of 350 was tested in the experiment. The center-to-center distance between the two cylinders was 3.0D. The uniform flow was simulated by towing the cylinder models along the tank. The towing velocity varied from 0.05 to 1.0 m/s with an interval of 0.05 m/s. The maximum Reynolds number can be up to 1.6×104. Three cases were experimentally studied in this paper, including two side-by-side inclined smooth cylinders, only one smooth cylinder fitted with helical strakes in the two side-by-side inclined cylinders system and both two cylinders attached with helical strakes. The variations of displacement amplitude, dominant frequency, FIV suppression efficiency and dominant mode for the two side-by-side inclined cylinders with reduced velocity were shown and discussed.


Introduction
Long flexible circular cylinders are widely employed in many engineering applications, such as marine risers in offshore engineering, stay cables in bridge engineering, transmission lines in electric engineering, and heat exchanger tubes in nuclear power engineering. When the multiple cylinders, which are placed in a flow, are allowed to oscillate in the cross-flow (CF) and in-line (IL) directions, the wellknown phenomenon of the flow-induced vibrations (FIV) will occur due to the vortex-shedding. FIV of multiple cylinders has been extensively investigated by a large number of researchers (Zdravkovich, 1988;Moretti, 1993;Païdoussis, 2005;Sumner, 2010;Bearman, 2011) because it can result in serious fatigue damage of the cylinder structures.
The two-cylinder system in a side-by-side arrangement is considered as one simple case of multiple cylinders configuration. The FIV behaviors of two side-by-side elastic-ally mounted rigid or flexible cylinders have been the focus of a great deal of research in the last decade (Wang et al., 2008;Chen et al., 2015;Huera-Huarte and Gharib, 2011;Sanaati and Kato, 2014;Xu et al., 2018a). Wang et al. (2008) performed the experimental investigation on the CF FIV of two side-by-side elastically mounted rigid cylinders in a closed loop wind tunnel at Re=5000-41000 with the spacing ratio (S/D, where S is the center-to-center distance between the two cylinders, D is the cylinder diameter) varying from 1.17 to 1.90. It was found that FIV was reduced due to wake interference when S/D<1.50. Chen et al. (2015) carried out a series of numerical simulations on the FIV of two elastically mounted rigid cylinders in a side-by-side arrangement undergoing uniform flow with S/D=2.0-5.0 and Re=100. It was found that the interaction between the two cylinders was not obvious when the spacing was larger than 4.0D. Huera-Huarte and Gharib (2011) experimentally investigated the FIV of two flexible circular cylinders in a side-by-side arrangement with five spacing ratios (S/D=2.0, 2.5, 3.0, 4.0 and 5.0) in a free-surface water channel. It was found that the interaction between the two flexible cylinders through the wake was very weak when S/D≥3.5 and strong proximity interference existed when S/D<3.5. Sanaati and Kato (2014) conducted the experimental studies on the influence of the spacing on two side-by-side identical flexible cylinders subjected to FIV with two spacing ratios (S/D=2.75 and 5.5) at Re=1400-20000 in a towing tank. It was found that the out-of-phase synchronization existed during high amplitude vibration for S/D=5.5. Xu et al. (2018a) experimentally studied the effect of the spacing on the multi-mode FIV of the two side-by-side flexible cylinders with four spacing ratios (S/D= 3.0, 4.0, 6.0 and 8.0) in a towing tank. It was found that the two cylinders still had remarkably strong interaction with each other in the IL direction as the spacing ratio S/D = 8.0.
It is known that the main aim of investigation on the FIV is to reduce the vibration by means of optical methods and devices. Helical strake is a very famous device for the vibration reduction of an isolated single cylinder undergoing vortex-shedding (Zdravkovich, 1981;Rashidi et al., 2016). It has been found that the suppression performance depends on the geometry of the helical strake, such as the screw head, helical strake pitch, helical strake height, coverage density, helical strake cross-section shape (Sui et al., 2016;Gao et al., 2016;Xu et al, 2017b). There has been many research works on the vortex-induced vibration (VIV) suppression of an isolated cylinder either elastically-mounted or flexible by the helical strakes. However, little attention has been paid to the vibration reduction of multiple cylinders with helical strakes subjected to FIV. Some researchers tried to carry out the experimental investigations on the effectiveness of helical strakes for the FIV suppression of two tandem cylinders (Korkischko and Meneghini, 2010;Assi et al., 2010;Baarholm et al., 2005Baarholm et al., , 2007. It was found that the effective strakes for reducing the VIV of an isolated cylinder lose their effectiveness for controlling the FIV of an elastically-mounted rigid cylinder behind a stationary one in a tandem arrangement (Korkischko and Meneghini, 2010;Assi et al., 2010). In addition, the performance of helical strakes for the FIV suppression of the downstream flexible cylinder in the wake of the upstream one might become worse (Baarholm et al., 2005(Baarholm et al., , 2007. To the best knowledge of the authors, little work has been done on the FIV suppression of two cylinders in a side-by-side arrangement. In practical applications, the flexible cylinder structures are commonly inclined with respect to the direction of the oncoming flow. The inclination angle a is defined as the angle between the cylinder axis and the plane perpendicular to the oncoming flow. Hence a=0° is corresponding to the normal flow configuration. In order to improve the understanding of VIV mechanism of inclined cylinders, the Independence Principle (IP), which assumes that the dynamic re-sponses and hydrodynamic features are essentially driven by the inflow normal component and the axial component has a negligible impact, has been proposed by Hanson (1966) and Van Atta (1968). Owing to the influence of the inflow axial flow component, the vibrations of the isolated inclined cylinder or multiple cylinders undergoing vortex-shedding are more complicated than those of vertical cases. Many studies have been published concerning the validity and accuracy of the IP.  numerically investigated the VIV of an inclined flexible cylinder with a=60° and Re=500 by using the DNS method. It was found that the IP was valid in the cases with a high axial tension. In contrast, the behavior of the fluid-structure system clearly deviated from the IP when the axial tension was low.  further numerically studied the free vibrations of a flexible cylinder with a=80°. Comparison of the vibration responses and fluid forces between the inclined and normal cylinder configurations showed that the behavior of the fluid-structure system drastically departs from the IP. Xu et al. (2018b) conducted laboratory tests to characterize the multi-mode VIV of an inclined isolated flexible cylinder in a uniform flow. It was found that the IP was a reasonable hypothesis for examining the multi-mode VIV of an inclined flexible cylinder for a<30°, and some discrepancies gradually appeared as the inclination angle increases from 45° to 60°.
Recently, it has been observed that the performance of helical strakes for the VIV suppression of inclined cylinders is different from that of the vertical cylinders (Zeinoddini et al., 2015;Xu et al., 2017a). Zeinoddini et al. (2015) carried out the experimental tests on the VIV suppression of an inclined elastically-mounted rigid cylinder attached with helical strakes with P=10.0D and H=0.10D in a towing tank. Five inclination angles, a=0°, ±20°, ±45° were selected in their tests. It was found that the VIV response amplitudes of the straked cylinder decreased as the inclination angle increased. Xu et al. (2017a) performed the towing tank experiments to study the effect of the inclination angle on the VIV suppression of an isolated inclined flexible cylinder. The strake with a pitch of 17.5D and a height of 0.25D, which is generally considered as the most effective configuration for reducing the VIV of a vertical flexible cylinder in water was used in the experiment. It was found that the helical strakes perform increasingly worse for the VIV suppression of an inclined flexible cylinder in water as the inclination angle increases.
Based on the above literature review, it can be concluded that the FIV of two side-by-side inclined flexible cylinders with or without helical strakes is still unknown and needs to conduct a further experimental investigation. In this paper, a series of experimental tests of two side-byside inclined flexible cylinders subjected to the FIV with S/D=3.0 and a=45° were carried out in a towing tank. The 17.5D/0.25D strakes were attached with both flexible cylin-ders to reduce the FIV. The FIV behaviors of two side-byside inclined flexible cylinders were investigated and the effectiveness of helical strakes for the FIV suppression was studied to improve the understanding of the FIV mechanism of two cylinders in a side-by-side arrangement.

Description of towing tank experiment
In the experiment, the smooth cylinder model (see Fig.  1a) was made up of an internal copper pipe and an outer silicone tube. The inner copper pipe with a wall thickness of 1.0 mm and an outer diameter of 8.0 mm was covered by a silicone tube with an external diameter D=16 mm to provide a smooth external surface. The cylinder length L was 5.60 m and the aspect ratio (L/D) was 350. The mass per unit length of the smooth cylinder model was 0.3821 kg/m and the mass ratio was 1.90. The straked cylinder model (see Fig.  1b) was made by fitting three-strand 17.5D/0.25D helical strakes with the smooth cylinder. The helical strake coverage was 100%. Main properties of the cylinders are listed in Table 1. Moreover, it should be pointed out that the 17.5D/ 0.25D helical strake is the most effective configuration for the VIV suppression of flexible cylinders in water (Trim et al., 2005;Gao et al., 2015). However, this geometry of strakes might not perform best for reducing the FIV of two side-by-side inclined flexible cylinders due to the interference from each other and the influence of axial flow. Nevertheless, it is not the main aim of this paper to optimize the strakes in respect of the geometry and coverage. We mainly focused on the performance of helical strakes for the FIV suppression of two inclined flexible cylinders in a side-byside arrangement.
The two cylinder models with the same size, property and axial pre-tension were mounted on the experimental device in a side-by-side arrangement with S/D=3.0. It is well known that the spacing ratio plays an important role in the FIV response of two side-by-side flexible cylinders (Huera-Huarte and Gharib, 2011;Xu et al., 2018a). Huera-Huarte and Gharib (2011) stated that the interaction between the two flexible cylinders through the wake was weak when S/D≥3.5 and the strong proximity interference existed when S/D<3.5. Xu et al. (2018a) experimentally proved that the apparent proximity interference between two side-byside cylinders with high aspect ratio existed in the CF direction when the spacing was smaller than 6.0D. Moreover, the side-by-side arrangement for the two flexible slender cylinders could enhance the IL displacement. The two cylinders still had the strong interaction with each other in the IL direction as the spacing ratio S/D=8.0. It should be pointed out that the main purpose of our experiment is to investigate the FIV suppression effectiveness of two side-by-side cylinders by means of helical strakes, and the influence of the spacing ratio on the FIV suppression is not discussed in this paper. Hence, the center-to-center separation distance was kept 3.0D in the current experimental tests. For this spacing case, the FIV of the two flexible cylinders in our tests may have an obvious influence on each other based on the findings of Huera-Huarte and Gharib (2011) and Xu et al. (2018a). In addition, the experimental tests of two side-byside cylinders with or without helical strakes were conducted for comparison convenience.
There are mainly six parts of our experimental device, including the supporting frame, vertical supporting rod, supporting plate, guiding plate, angle plate and cylinder model. The experimental device has been successfully used to carry out our previous tests of the VIV suppression of the isolated inclined flexible cylinder with four inclination angles (a=0°, 15°, 30° and 45°) (Xu et al., 2017a). Fig. 2 shows the sketch of the experimental device. The horizontal supporting frame was made up of several short steel beams. The angle plate was joined with the vertical supporting rod at its top end. The supporting plate was fixed to the bottom end of the vertical supporting rod. In order to eliminate the disturbing flow generated by the supporting plate and supporting rod, the guide plate was used and fixed on the supporting plate by several long bolts. One end of the cylinder model was connected to a universal joint fixed on the supporting plate. The other end of the cylinder model was connected to a steel wire, which was passed through a pulley and connected to the spring, tensioner and load cell by turn. The application of spring was allowed a gradual variation of the axial tension on the cylinder model during tests. The tensioner was used to adjust the axial tension equaled to 450 N for all cases. The load cell was adopted to measure the varying axial tension. More details about the experimental device can refer to Xu et al. (2017a).
The vibration suppression effectiveness of the isolated inclined flexible straked cylinder subjected to VIV is closely related to the inclination angle. Xu et al. (2017a) found both the maximum displacement amplitudes of the inclined straked cylinder in the CF and IL directions increased gradually as the inclination angle increased from 0° to 45°a nd the suppression efficiency was obviously deteriorated at a large yaw angle. In some situations, the VIV responses of the inclined cylinder can be not only suppressed but also enhanced by helical strakes. It can be concluded that the inclination angle may have a significant influence of the FIV reduction of two side-by-side cylinders due to the inflow axial flow component and the interaction with each other. However, it is impossible to carry out the tests for all the inclinations and spacing ratios in our experiment. We aim to experimentally check whether the helical strakes for the vibration suppression of the two side-by-side cylinders perform as poorly as the isolated cylinder at a large yaw angle in this paper. Hence, the inclination angle was kept 45° by adjusting the angle plate. The upper cylinder model which was labeled as "Cylinder #1" was submerged 1.0 m below the free surface to eliminate the free-surface effect. In addition, the lower cylinder was labeled as "Cylinder #2" in this paper. The bending strains were measured by using 28 resistance strain gages attached to the surface of the copper pipe in both the CF and IL directions to collect the information of the bending strains by half-bridge topology. A total of seven measurement points were evenly distributed along the cylinder model. The sampling frequency of the measuring instrument was 100 Hz which is sufficient to avoid aliasing problems. The sampling duration of each test run was 50 s, counting after the carriage reached a stable towing velocity. The waiting time between two consecutive runs was not less than 15 min to calm down the disturbed water. More than 60 runs were performed across all the cases.

Experimental results and discussions
The bending strains were directly measured by using the strain gages at seven measurement points along the cylinder model in our tests. The modal analysis approach proposed by Lie and Kaasen (2006) was utilized to reconstruct the displacement response based on the strain signals. More details of the displacement reconstruction can refer to Lie and Kaasen (2006) and Xu et al. (2017a). In the following section, the results of two side-by-side inclined smooth cylinders undergoing FIV were firstly discussed, and then the case of one smooth cylinder and another straked cylinder in the two side-by-side inclined cylinders system was experimentally investigated, finally, the FIV suppression of two side-by-side inclined cylinders both attached with helical strakes was studied in the towing tank.
3.1 Two side-by-side inclined smooth cylinders Fig. 3 shows the displacement amplitudes of two sideby-side inclined smooth cylinders in the CF and IL directions versus the reduced velocity. Note that the displacement amplitude is the maximum value of the RMS displacement response and the reduced velocity is defined as V r =U/(f 1 D) (where U is the towing velocity, f 1 is the fundamental frequency of the isolated smooth cylinder). Moreover, the experimental results of the isolated smooth cylinder are also plotted in Figs. 3 and 4 for comparison. The maximum displacement amplitude of the two side-byside smooth cylinders is up to 1.50D in the CF direction and close to the isolated smooth cylinder. It can be clearly seen in Fig. 3 that the CF displacement amplitude of the smooth cylinder #1 is in good agreement with that of the smooth cylinder #2. This behavior indicates that the interaction of two side-by-side inclined smooth cylinders with S/D=3.0 and a=45° is not obvious in the CF direction. Hence the CF vibration is similar to that of the isolated cylinder. However, there is a significant difference between the two side-byside inclined smooth cylinders in the IL direction. The smooth cylinder #1 vibrates with much smaller the IL displacement amplitude than the smooth cylinder #2 does. In addition, the vibration of the isolated inclined smooth cylinder undergoing vortex shedding is not consistent with both cylinders in the two side-by-side cylinders system in the IL direction. The experimental results show that the interaction of the two cylinders in a side-by-side arrangement with S/D=3.0 is very obvious in IL direction. These trends might be caused by the inflow axial flow component and the effect of the two side-by-side cylinders. Fig. 4 presents the dominant frequencies of two side-byside inclined smooth cylinders in the CF and IL directions versus the reduced velocity. It should be pointed out that the CF and IL dominant frequencies, f y and f x , were taken as the largest peak in the spectra plot obtained by using the FFT of the time-varying displacement in the CF and IL directions, respectively. The theoretically fundamental natural frequency of the isolated flexible smooth cylinder in still water f 1 , which can be calculated by (where L is the cylinder length, T is the axial tension, EI is the bending stiffness, and m is the total mass, including the structural mass and added mass) was used to non-dimensionalize the dominant frequency in Fig. 4. Two additional lines are also drawn in Fig. 4, and the diagonal dash line represents the Strouhal frequency corresponding to the vortex-shedding frequency for the stationary cylinder. The horizontal solid lines represent the natural frequency for the cases f y /f 1 =1.0 in the CF direction and f x /f 1 =2.0 in the IL direction. It can be seen that the CF dominant frequencies of the smooth cylinder #1 and cylinder #2 are in reasonable agreement with each other and increase linearly with the reduced velocity. A similar trend in the CF direction was observed in the experimental tests of the isolated inclined smooth cylinder (Xu et al., 2017a). Moreover, there is a high level of agreement between the CF dominant frequencies of the two side-by-side inclined smooth cylinders and the isolated smooth cylinder. The difference of the dominant frequencies in the IL direction for the smooth cylinder #1 and cylinder #2 is quite significant due to the influence of the FIV interaction. The IL dominant frequencies of the isolated inclined smooth cylinder are not similar to those of the two side-by-side cylinders system. This means that the VIV feature of the isolated inclined smooth cylinder is obviously different from the two cylinders in a side-by-side arrangement. Table 2 lists the CF and IL dominant modes of two sideby-side inclined smooth cylinders. For convenient comparison, the results of the isolated inclined smooth cylinder are also listed in the table. It is clear that the CF dominant mode of the smooth cylinder #1 is in the range of Mode 1 to Mode 3. A similar variation of the CF dominant mode is observed for Cylinder #2. The dominant mode of the isolated cylinder in CF direction is in the same range as that of the two side-by-side cylinders. These behaviors are consistent with the trends of the CF dominant frequencies, as shown in Fig. 4. In addition, the maximum dominant mode in the IL direction for Cylinder #1 is Mode 5, which is same as the results of Cylinder #2. The variation of the dominant modes for Cylinder #1 is distinct from that for Cylinder #2. A similar trend has been observed in Fig. 4.   3.2 One smooth cylinder and another straked cylinder in the two side-by-side inclined cylinders system Fig. 5 shows the CF and IL displacement amplitudes for one smooth cylinder and another straked cylinder in the two side-by-side inclined cylinders system versus the reduced velocity. For this case, only Cylinder #2 was fitted with helical strakes. The results of the isolated inclined flexible cylinder with or without strakes are also plotted in Figs. 5 and 6 for comparison. There is a high level of agreement between the smooth cylinder #1 and the isolated smooth cylinder. For the straked cylinder #2, the CF displacement amplitude is lower than that of the isolated inclined cylinder with helical strakes. The IL displacement amplitude with the maximum value of 0.36D is close to that of the isolated inclined straked cylinder. We observed a very interesting phenomenon that the use of helical strake cannot suppress the IL FIV of the inclined cylinder in the two side-byside inclined cylinders system. This phenomenon has been observed in the experiment of the isolated cylinder fitted with 17.5D/0.25D helical strakes and inclined with an inclination angle of 45° (Xu et al., 2017a). Fig. 6 presents the CF and IL dominant frequencies for one smooth cylinder and another straked cylinder in the two side-by-side inclined cylinders system versus the reduced velocity. The results of the isolated inclined cylinder with or without helical strakes are also drawn in the figure for the sake of comparison. It can be seen that the variation of the CF dominant frequency versus the reduced velocity for the isolated inclined smooth cylinder is similar to that for the isolated inclined straked cylinder, which means that the use of helical strakes for the isolated cylinder inclined at a=45°c annot effectively reduce the vortex shedding in the CF direction due to the effect of the axial flow (Xu et al., 2017a). In addition, there is a high level of consistency between the isolated cylinder and two side-by-side inclined cylinders system in the CF direction, as shown in Fig. 6. This trend indicates that the performance of helical strakes for the CF FIV suppression of one smooth and another straked in the two side-by-side inclined cylinders system is as poor as that of the isolated inclined cylinder. However, a different trend has been observed in the IL direction. For the isolated inclined cylinder, the helical strakes can obviously suppress the vortex shedding in the IL direction. The IL dominant frequency of Cylinder #2 was significantly restrained by attaching helical strakes.
In order to investigate the performance of helical strakes for the FIV suppression of Cylinder #2 in the two side-byside inclined cylinders system, the FIV suppression efficiencies, η y and η x , in the CF and IL directions are shown in Fig. 7. The FIV suppression efficiency is defined as ×100%, where and are the maximum response amplitudes of the cylinder without and with helical strakes, respectively. It can be seen that the CF FIV suppression efficiency varies approximately in the range of 20%-80% with an average value of 64.82%. This states that the helical strakes can reduce the FIV of the inclined flexible cylinder #2, but the reduction effectiveness is not perfect. This behavior is different from that of the isolated vertical flexible cylinder with helical strakes. The difference might be attributed to the influence of the axial flow component. The IL FIV suppression efficiency ranges from 23.33% to 65.40% with an average value of 46.53%. This means the helical strakes perform badly for the IL FIV reduction of the inclined flexible cylinder #2. Since the strong interaction between Cylinders #1 and #2 and the effect of the axial flow component, the suppression effectiveness of the IL FIV is much worse than that of CF FIV, as shown in Fig. 7. Table 3 lists the CF and IL dominant modes of one smooth cylinder and another straked cylinder in the two  side-by-side inclined cylinders system. It can be observed that the dominant mode of the smooth cylinder #1 ranges from Mode 1 to Mode 3 in the CF direction and Mode 1 to Mode 5 in the IL direction. By attaching helical strakes with only one cylinder in the two side-by-side inclined cylinders system, the dominant mode of the straked cylinder #2 is in the range of Mode 1 to Mode 3 in the CF direction. A similar variation of the dominant mode for the straked cylinder #2 in the IL direction is obtained in the tests. It can be found only one cylinder fitted with helical strakes in the two sideby-side inclined cylinders system cannot effectively reduce the dominant modes in both the CF and IL directions.
3.3 Two side-by-side inclined straked cylinders Fig. 8 shows the displacement amplitudes of two sideby-side inclined straked cylinders in the CF and IL direc-tions versus the reduced velocity. For the sake of comparison, the results of the isolated inclined cylinder fitted with helical strakes are also plotted in Figs. 8 and 9. It is clear that the inclined straked cylinder #2 and the isolated straked cylinder oscillate with nearly similar maximum displacement amplitude of 0.60D in the CF direction. However, the maximum values of the displacement were observed at different reduced velocities. Moreover, the CF displacement amplitude of the straked cylinder #1 is slightly smaller than that of the straked cylinder #2. This trend means that the performance of helical strakes for reducing the FIV of two inclined cylinders in a side-by-side arrangement is the same with that of the isolated cylinder inclined at a=45°. The IL FIV of two side-by-side inclined straked cylinders is obviously different from the CF FIV. The straked cylinder #2 undergoing vortex shedding vibrates with significant lower displacement amplitude than the straked cylinder #1 which has the maximum displacement amplitude of 0.53D. This  . 7. Variations of the CF and IL FIV suppression efficiencies for the straked cylinder in the two side-by-side inclined cylinders system with the reduced velocity. Fig. 8. Variation of the CF and IL displacement amplitudes for two sideby-side inclined straked cylinders with the reduced velocity. Fig. 9. Variation of the CF and IL dominant frequencies for two side-byside inclined straked cylinders with the reduced velocity.
behavior indicates that the IL FIV of two side-by-side inclined cylinders is not effectively passively controlled by using helical strakes. There is a satisfactory consistency between the straked cylinder #2 and the isolated inclined straked cylinder as V r ≥12.52. Compared with the isolated inclined cylinder fitted with helical strakes, much smaller displacement amplitude is obtained as V r <12.52. Fig. 9 presents the dominant frequencies of two side-byside inclined cylinders both attached with helical strakes in the CF and IL directions versus the reduced velocity. It can be found that both CF and IL dominant frequencies of two side-by-side inclined straked cylinders rise linearly with the reduced velocity. But there is not a double times relationship between the IL and CF dominant frequencies. The variation of the dominant frequencies for two side-by-side inclined straked cylinders with the reduced velocity is similar to that of the isolated inclined cylinder fitted with helical strakes. This feature states that the 17.5D/0.25D helical strakes for the two side-by-side inclined cylinders cannot behave quite well for the FIV reduction, which is the same as the isolated inclined straked cylinder, as illustrated in Xu et al. (2017a).
The CF and IL FIV suppression efficiencies for two side-by-side inclined straked cylinders with the reduced velocity are plotted in Fig. 10 to investigate the effectiveness of helical strakes. The average values of the CF FIV suppression efficiency for Cylinders #1 and #2 are 65.27% and 62.91%, respectively. Nearly a similar FIV suppression efficiency for two side-by-side inclined cylinders mounted on helical strakes was observed in the CF direction. Compared with the results in the CF direction, an obvious different IL FIV suppression efficiency between two cylinders was obtained in our tests. The helical strakes cannot reduce the IL vibration of Cylinder #1 subjected to FIV. For this case of the spacing ratio (S/D=3.0) and inclination angle (a=45°), the use of these passive devices for the FIV suppression can even enhance the vibration in the IL direction. A similar behavior has been observed in previous experiments of the VIV suppression of the isolated cylinder inclined at a=45°b y applying helical strakes (Xu et al., 2017a). Table 4 lists the CF and IL dominant modes of two sideby-side inclined straked cylinders. It can be seen that the dominant modes of the straked cylinders #1 and #2 in both the CF and IL directions vary from Mode 1 to Mode 3. A similar trend has been found in the test of the isolated inclined cylinder attached with helical strakes (Xu et al., 2017a). This behavior is consistent with the results of the dominant frequencies in the CF and IL directions, as shown in Fig. 9. These results indicate that the effectiveness of helical strakes for reducing the CF and IL dominant modes in the two side-by-side inclined cylinders system is not remarkable.

Conclusions
A series of experimental tests were carried out to investigate the FIV suppression of two side-by-side cylinders fitted with 17.5D/0.25D helical strakes and inclined at 45°. Based on the experimental results, including the displacement amplitude, dominant frequency, FIV suppression efficiency and dominant mode, the following conclusions can be drawn.
(1) For the two side-by-side inclined smooth cylinders, the interaction between two cylinders is not obvious in the CF direction but much apparent in the IL direction. This Table 4 Dominant modes of two side-by-side inclined straked cylinders in the CF and IL directions phenomenon has also been observed in the experiment of two vertical cylinders in a side-by-side arrangement (Xu et al., 2018a). In addition, the CF vibrations of both inclined smooth cylinders with the maximum response amplitude of 1.50D are consistent with that of the isolated cylinder in Xu et al. (2017a). The IL FIV of two smooth cylinders is quite different from each other in the experiment due to the inflow axial flow component and the interaction of the two side-by-side cylinders.
(2) When only one cylinder was fitted with helical strakes in the two side-by-side inclined cylinders system, there is a high level of consistency between the smooth cylinder in the two side-by-side inclined cylinders system and the isolated inclined smooth cylinder. The FIV behaviors of the straked cylinder in the two side-by-side inclined cylinders system are similar to those of the isolated inclined smooth cylinder fitted with helical strakes in Xu et al. (2017a). These trends mean that the vibration of the two side-by-side inclined cylinders cannot be effectively suppressed by using helical strakes for only one cylinder at the spacing ratio (S/D=3.0) and inclination angle (a=45°) tested in the experiment.
(3) For the two inclined straked cylinders in a side-byside arrangement, the FIV responses of two cylinders are not the same as each other. Moreover, the application of helical strakes for the FIV suppression of two side-by-side inclined cylinders performs as poorly as the isolated inclined cylinder, and sometimes the use of helical strakes even can enhance the FIV. These findings are very meaningful and important for the design of slender cylinder structures in offshore engineering.
Overall, the FIV suppression of two side-by-side flexible cylinders using helical strakes is an extremely complicated problem. Further experimental studies should be performed to investigate the influence of the spacing ratio on the vibration reduction effectiveness. Moreover, the effect of inclination on the passive FIV suppression of two flexible cylinders in a side-by-side arrangement also needs to be investigated thoroughly.