Application of Helical Strakes for Suppressing the Flow-Induced Vibration of Two Side-by-Side Long Flexible Cylinders

Helical strakes have been widely applied for suppressing the vibration of flexible cylinders undergoing vortex-shedding in offshore engineering. However, most research works have concerned on the application of helical strakes for the isolated flexible cylinder subjected to vortex-induced vibration (VIV). The effectiveness of helical strakes attached to side-by-side flexible cylinders in vibration reduction is still unclear. In this paper, the response characteristics of two side-by-side flexible cylinders with and without helical strakes were experimentally investigated in a towing tank. The configuration of the helical strakes used in the experiment had a pitch of 17.5D and a height of 0.25D (where D is the cylinder diameter), which is usually considered the most effective for VIV suppression of isolated marine risers and tendons. The center-to-center distance of the two cylinders was 3.0D. The uniform flow with a velocity ranging from 0.05 m/s to 1.0 m/s was generated by towing the cylinder models along the tank. Experimental results, including the displacement amplitude, the dominant frequency, the dominant mode, and the mean drag force coefficient, were summarized and discussed. For the case where only one cylinder in the two-cylinder system had helical strakes, the experimental results indicated that helical strakes can remarkably reduce the flow-induced vibration (FIV) of the staked cylinder. For the case of two straked cylinders in a side-by-side arrangement, it was found that the performance of helical strakes in suppressing the FIV is as good as that for the isolated cylinder.


Introduction
In practical applications of offshore engineering, when cylindrical bluff structures, such as marine risers, tendons of tension leg platforms and free spanning pipelines, are placed in an oncoming flow, a well-known fluid-structure interaction (FSI) phenomenon called vortex-induced vibration (VIV) may occur. VIV can cause large response amplitudes and high cyclical stresses and finally lead to the affected structure failure in the mode of fatigue. Hence, the oscillation of cylinders undergoing vortex-shedding and the methods for reducing VIV have been thoroughly studied in recent years (Zdravkovich, 1981;Sarpkaya, 2004;Williamson and Govardhan, 2008;Wu et al., 2012;Rashidi et al., 2016;Xu et al., 2019;Zhu et al., 2020).
A multitude of methods for reducing VIV can be found in the literature (Zdravkovich, 1981;Rashidi et al., 2016). As a popular device of VIV suppression for subsea tubulars, helical strake performs quite well. It breaks up the correlation of vortex shedding along the tubular and produces uncorrelated alternating fluid forces along the length of the cylinder. A large number of researchers have paid much attention to the performance of helical strakes for different types of marine cylindrical structures, e.g. short rigid cylinders represented by spar production platforms, and long flexible cylinders represented by risers and tendons. Table 1 lists some important studies on this topic since 2000. According to the flexibility and supporting conditions of the cylinders, these studies can be grouped into four categories, i.e. isolated single rigid cylinder with elastically mounted ends (Constantinides and Oakley, 2006;Korkischko and Meneghini, 2010;Lubbad et al., 2011;Zeinoddini et al., 2015;Senga and Larsen, 2017), isolated single flexible cylinder with fixed ends (Frank et al., 2004;Trim et al., 2005;Vandiver et al., 2006;Quen et al., 2014;Gao et al., 2015Gao et al., , 2016Xu et al., 2017a), multiple rigid cylinders with elastically mounted ends (Korkischko and Meneghini, 2010;Assi et al., 2010), and multiple flexible cylinders with fixed ends (Baarholm et al., 2005(Baarholm et al., , 2007Xu et al., 2018c).
To improve the understanding of the VIV-reduction effect of helical strakes, much research has been performed on the oscillation of an isolated elastically-mounted rigid cylinder fitted with helical strakes. Constantinides and Oakley (2006) employed a second order accurate finite element computational fluid dynamics (CFD) method to study an elastically-mounted straked rigid cylinder subjected to VIV. They found that the helical strakes with P/H=0.25D/15.0D (where P is the strake pitch, H is the strake height and D is the cylinder diameter) can almost eliminate the VIV over the lock-in range of the smooth cylinder. Lubbad et al. (2011) experimentally investigated the efficiency of helical strakes with round-section suppressing the VIV of an elastically-mounted rigid cylinder. It was observed that the triple round-sectioned helical strakes with P/H=0.15D/5.0D, which was considered the best among the tested configurations, can suppress the displacement amplitude by 96.0% in the cross-flow (CF) direction and by 97.0% in the in-line (IL) direction compared with those in the case of smooth cylinder. Zeinoddini et al. (2015) carried out some experimental tests in a towing tank on the CF VIV of smooth and straked cylinders in both vertical and inclined arrangements. Triple strakes with P/H=10.0D/0.10D were used in their tests. It was found that the effectiveness of VIV suppression remained nearly the same for the cases of vertical and inclined cylinders attached with helical strakes. Recently, Senga and Larsen (2017) conducted a series of forced motion tests in a towing tank using cylinders with different hel-ical strake configurations (P/H=17.5D/0.25D and 5.0D/0.14D, respectively) to identify their hydrodynamic coefficients which can be employed in empirical models predicting the response of cylinders with strakes.
Currently, the main concern with VIV suppression in offshore engineering is how to use helical strakes to reduce the oscillation of an isolated long flexible cylinder. Trim et al. (2005) performed experimental tests of VIV suppression of a long marine riser with an aspect ratio of 1405 both in uniform and sheared flows. The helical strakes with P/H=17.5D/0.25D and 5.0D/0.14D were used in their experiments. It was found that both strake configurations were effective in reducing VIV in sheared flow. In uniform flow, the strakes with P/H=17.5D/0.25D performed as well as it did in shear flow, but the strakes with P/H=5.0D/0.14D were not as effective. Quen et al. (2014) pointed out that the effectiveness of VIV reduction of helical strakes on an isolated flexible cylinder is less notable compared with a rigid cylinder. In addition, they found that the pitch of helical strakes contributes less in suppressing the amplitude of vibration while the strake height has a significant influence. Gao et al. (2015) experimentally investigated the effectiveness of VIV suppression of a marine riser attached and fully covered with helical strakes in both uniform and linearly sheared flows. In their experiment, the pitches of the strakes were 5.0D, 17.5D and 20.0D, and corresponding heights were 0.10D, 0.15D and 0.25D. It was found that the riser model with the configuration of P/H=17.5D/0.25D performed best in both flow profiles. Gao et al. (2016) further studied the effects of different helical strake coverage (25%, 50%, 75% and 100%) on the effectiveness of VIV reduction of a flexible riser. They found that as the strake cover- age increased, the suppression efficiency gradually increased, and reached its maximum at 100% coverage. More recently, Xu et al. (2017a) conducted experimental studies on the VIV response of an inclined flexible straked cylinder with different inclination angles to investigate the influence of inclination angle on the effectiveness of VIV suppression. It was found that the suppression efficiencies of the CF and IL displacement amplitudes gradually decreased with an increasing inclination angle. From the above literature review, it can be seen that the performance of helical strakes in reducing the VIV for an isolated flexible cylinder is determined by various parameters, such as the strake pitch, the strake height, the strake coverage, the oncoming flow profiles, the inclination angle between the cylinder axis and the plane orthogonal to the oncoming fluid flow. Moreover, the helical strake with a pitch of 17.5D and a height of 0.25D may be considered as the most effective configuration of isolated single riser and tendon for suppressing the VIV.
Marine risers, which play an important role in deep-sea oil and gas exploitation, are generally found in group arrangements. It is known that the flow-induced vibration (FIV) behaviors of a group of cylinders are much more complex than those of the isolated due to the mutual effects of adjacent cylinders. Some experiments have been conducted to study the FIV response of multiple flexible cylinders in tandem, staggered or side-by-side arrangements in recent years (Allen and Henning, 2003;Huera-Huarte and Bearman, 2011;Gharib, 2011a, 2011b;Sanaati and Kato, 2014;Xu et al., 2018aXu et al., , 2018cXu et al., , 2018dMa et al., 2019aMa et al., , 2019bMa et al., , 2020Wang et al., 2019). Several researchers tried to experimentally investigate the effectiveness of strakes in suppressing the FIV of two tandem cylinders, either elastically-mounted (Korkischko and Meneghini, 2010;Assi et al., 2010) or flexible (Baarholm et al., 2005(Baarholm et al., , 2007Xu et al., 2018c). It was found that the strakes effective in reducing the VIV of an isolated cylinder lost their effectiveness in the case where a rigid cylinder elastically-mounted behind a tandem stationary one (Korkischko and Meneghini, 2010;Assi et al., 2010). In addition, the performance of helical strakes of the downstream flexible cylinder is worse in FIV suppression (Baarholm et al., 2005(Baarholm et al., , 2007Xu et al., 2018c). According to the previous literature review, it can be concluded that fitting strakes with cylinders, which is the typical means of suppressing vibration for an isolated flexible cylinder, may not only be ineffective for the interference-induced oscillations of multiple flexible cylinders but even counter-productive. These findings are consistent with those of Zdravkovich (1981).
As previously mentioned, there is some research work investigating the FIV suppression of tandem cylinders, however, experimental studies on how to reduce the FIV response of multiple cylinders in a side-by-side arrangement by using strakes are still scarce and have been continuously urged by the offshore engineering industry. This study is conducted corresponding to such demands from practical engineering applications. A series of experiments were performed in a towing tank for two side-by-side flexible cylinders. Helical strakes with P/H =17.5D/0.25D were attached to cylinders to experimentally check if the strakes proven to be effective in suppressing VIV of an isolated cylinder are still good choices for suppressing FIV of two flexible cylinders in a side-by-side arrangement.

Experimental setup
As is known, three types of flow regimes can be generated by two side-by-side fixed rigid cylinders depending on the spacing ratio, S/D (where S is the center-to-center separation distance). At very small spacing ratios (S/D<1.1−1.2), one single vortex street is formed and no vortex exists in the gap between two cylinders (Wang and Zhou, 2005); at intermediate spacing ratios (1.1−1.2≤S/D<2.2−2.7), a narrow wake, as well as a wide wake, is observed behind the cylinders and a bistable biased gap flow emerges (Afgan et al., 2011); at large spacing ratios (S/D≥2.2−2.7), a coupled wake street appears (Meneghini et al., 2001;Alam et al., 2003). When the spacing ratio is even greater (S/D≥4.0− 5.0), no obvious interaction between the two wakes of the cylinders occurs (Vakil and Green, 2011). Huera-Huarte and Gharib (2011a) conducted some experiments on the FIV of two side-by-side flexible cylinders in a free-surface water channel. The interaction between the two flexible cylinders through wake in the CF direction was weak when S/D≥3.5 and strong proximity interference existed when S/D<3.5. Xu et al. (2018a) experimentally investigated the effect of spacing ratio on the multi-mode FIV of the two side-by-side flexible cylinders in a towing tank. The remarkably strong interaction between cylinders was observed in the IL direction even with a spacing ratio of up to 8.0. In this study, the center-to-center separation distance was kept 3.0D, so that the cylinders can have an obvious influence on each other in the CF and IL directions based on the findings of Huera-Huarte and Gharib (2011a) and Xu et al. (2018a). Indeed a coupled wake street appeared behind the two side-by-side cylinders in the experimental observation.
As shown in Fig. 1, we performed experimental tests for totally five cases. Firstly, the vibrations of an isolated single smooth cylinder with and without helical strakes were investigated to improve the understanding of the performance of helical strakes for reducing VIV. Then, the characteristics of two side-by-side flexible smooth cylinders undergoing FIV were observed to investigate the interactions of the multiple cylinders. In the end, we carried out the experimental investigation of FIV suppression when only one or both of the side-by-side cylinders were attached with helical strakes, which is the main focus of this paper. The three starts strakes used in the experimental tests have the same pitch (17.5D) and height (0.25D), which is generally con-sidered the most effective configuration for VIV suppression of an isolated flexible cylinder in water (Trim et al., 2005;Gao et al., 2015;Xu et al., 2017aXu et al., , 2017b. The helical strake coverage is 100%.
Our experimental tests were conducted in a 7.0 m wide, 3.3 m deep and 137.0 m long towing tank equipped with a towing carriage. The experimental device, as shown in Fig. 2, was mounted on the carriage. The uniform flow was simulated by towing the cylinder models along the tank. The velocity of towing carriage ranged from 0.05 m/s to 1.0 m/s with an interval of 0.05 m/s. The Reynolds number, which was calculated using the diameter of the smooth cylinder model, was approximately in the range of 800−16000 for the tests. For the convenience of discussion in Section 3, the upper and lower cylinders were labeled as "Cylinder #1" and "Cylinder #2" respectively.
The experimental device mainly consisted of one supporting frame, two vertical supporting rods, two supporting plates, two guiding plates, two pulleys and two steel wires. The vertical supporting rod was used to connect the supporting frame (at the top end) with the supporting plate (at the bottom end). The guide plates and supporting plates were kept parallel to the oncoming flow direction to minimize the disturbing flow induced. The cylinders were pinned at the ends using universal joints. So torsion was forbidden while bending was allowed in both the CF and IL directions. To apply an axial tension, steel wires were adopted to connect the ends of the cylinder models with the guide plate and supporting plate, passing through two small holes. A spring was attached to one end of the cylinder model. The use of these springs was to allow for gradual tension variation during testing. Load cells were used to monitor the variation of the axial tension for the cylinder models. The axial tension, which can be adjusted by the tensioner, was equal to 450 N in this study. Further details of the facilities and apparatus can be found in our previous experimental work (Xu et al., 2018a).
The flexible smooth and straked cylinder models used in this study were the same as the models used in the previous experimental research (Xu et al., 2017a). The smooth cylinder model is a coaxial composite tube, consisting of an inner copper pipe and an outer silicone tube, and has a length of 5.60 m. The copper tube with a diameter of 8.0 mm and a wall thickness of 1.0 mm was covered by the silicone tube with an outer diameter of 16.0 mm. The aspect ratio (length/diameter) and the mass ratio (structural mass/displaced fluid mass) of the cylinder model were 350 and 1.90 respectively. Three strakes with the same pitch of 17.5D and  XU Wan-hai et al. China Ocean Eng., 2020, Vol. 34, No. 2, P. 172-184 the same height of 0.25D were attached to the smooth cylinder to build the straked cylinder model (see Fig. 3). The density of the strake's material is close to the water, meaning that the mass ratio of the cylinder model with helical strakes is nearly the same with the smooth cylinder. Table 2 lists the key parameters of the cylinder model which are the same as those in Xu et al. (2017a). Fourteen pairs of strain gauges were mounted on the outer surface of the copper tube to measure bending strains. Seven of those were oriented in the x-direction (IL) and the others were in the y-direction (CF). As shown in Fig. 3, the measurement points G1−G7 were evenly distributed along the cylinder. The upper cylinder model was submerged 1.0 m below the free surface to eliminate the free surface effect. 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 smaller than 15 minutes so that the disturbed water can calm down. More than 60 runs were conducted in the tests of two side-by-side cylinders.

Results and discussion
In the experimental tests, the bending strains of the cylinder model in the CF and IL directions were directly measured using strain gauges at seven measurement positions. We reconstructed the response displacements by applying a modal approach, which has been widely used in the experi-ments of smooth or straked flexible cylinders undergoing VIV (Trim et al., 2005;Gao et al., 2015Gao et al., , 2016Xu et al., 2018b). The modal approach assumes that the dynamical behavior of the flexible cylinder is approximately linear. Based on the linear decomposition hypothesis, the timevarying VIV displacement response of a cylinder can be taken as the sum of a series of modal components. Taking the reconstruction of CF VIV displacement as an example, the response can be expressed as: where y(z, t) is the time-varying displacement in the CF direction, z is the coordinate along the axis of the cylinder model, t is time, is the modal shape, w n (t) is the modal weight and n is the order of vibration mode.
The pretension cylinder can be simplified as a Bernoulli-Euler beam with pinned-pinned boundary conditions. The modal shape of displacement can be written as: where L is the cylinder length. Based on the geometrical relationship of deformation, the following equation can be obtained: where k is the curvature of the cylinder model. Herein, y′ can be neglected as it is an extremely small value.
ε (z, t) where denotes the strain data which are measured directly from the measurement positions, and R is the outer radius of the copper pipe. The expression of strain and displacement can be written as: Assuming that the strain gauges were attached to a total of M measurement positions (M=7 in our tests), then the vibration of the cylinder can be written as a superposition of N eigenmodes, AW = B. Herein, By using the least square method, the weight function of different mode can be calculated as follows: By applying Eq. (5) and Eq. (10), the displacement of the cylinder can be reconstructed. More details of reconstructing the time-varying displacement based on strain data at measurement points can be found in Lie and Kaasen (2006). The modal approach had been successfully applied to processing the measured data in previous experiments for VIV suppression of an inclined flexible cylinder fitted with helical strakes (Xu et al., 2017a) and two inclined flexible straked cylinders in a side-by-side arrangement (Xu et al., 2018d).
In this section, the experimental results, including the displacement amplitude, the dominant frequency, the dominant mode, and the mean drag force, etc., are presented and discussed to investigate the effectiveness of helical strakes for reducing the FIV of two side-by-side flexible cylinders. The results of the isolated smooth and straked cylinders, which have been published in Xu et al. (2017a), are also briefly introduced for the convenience of comparison.
Before the presentation of experimental results, the definitions of several important parameters are given herein. The reduced velocity is calculated based on the following equation, where U is the towing carriage velocity which is equal to the oncoming flow velocity, f 1 is the fundamental natural frequency of the flexible smooth cylinder in still water. We conducted free decay tests of the flexible smooth cylinder in the tank, and the fundamental natural frequency measured in our tests agrees quite well with the theoretical results obtained by using the following equation, In this study, the theoretical fundamental natural frequency obtained by Eq. (12) was applied to calculate the reduced velocity in Eq. (11). The same way of calculating the reduced velocity has been adopted in our previous experimental work (Xu et al., 2017a).
The spanwise evolutions of the root mean square (RMS) displacements in the CF and IL directions can be calculated according to the following equation, where z i is the position along the axis of the cylinder and s is the number of time samples in the selected time window. In order to obtain accurate results of the RMS displacements, sufficient number of time samples should be used in Eq. (13). The RMS displacements were calculated in the time range of 20−50 s. In addition, the FIV suppression efficiency is defined as: Υ Υ strake where and are the maximum response amplitudes of the smooth and straked cylinders, respectively. Fig. 4 shows the maximum RMS dimensionless CF and IL displacement amplitudes versus the reduced velocity for the case of two smooth cylinders in a side-by-side arrangement. At first glance, the data in Fig. 4 are a little bit scattered in both the CF and IL directions. A further check reveals that both Cylinder #1 and Cylinder #2 exhibit typical upper branch within the first lock-in region in the CF direction, which is the same with the isolated smooth cylinder. These results are inconsistent with the observations by Sanaati and Kato (2014), where two side-by-side flexible smooth cylinders with S/D=2.75 were tested. In their experiment, there is no upper branch in the first CF lock-in region for the upper cylinder. Instead, the upper branch was overtaken by a region in which the amplitude increased gently up to the lower branch region. The peak values of CF displacement amplitudes are nearly 1.71D for Cylinder #1 and 1.42D for Cylinder #2. In addition, it can be found that the smooth Cylinder #1 has a similar maximum value of displacement amplitude to that of the isolated smooth cylinder in the CF direction. The difference of the IL displacement amplitudes between Cylinder #1 and Cylinder #2 is significant with S/D=3.0 in our tests. Compared with Cylinder #2, Fig. 4. Maximum RMS dimensionless displacement amplitudes vs. the reduced velocity for two smooth cylinders.

Displacement amplitude and vibration suppression efficiency
XU Wan-hai et al. China Ocean Eng., 2020, Vol. 34, No. 2, P. 172-184 larger IL displacement amplitude is obtained for Cylinder #1 when V r ≥7.52. These results indicate that there are strong interactions between the two cylinders through wake, which is consistent with the observations by Huera-Huarte and Gharib (2011a) in their experiments of two side-by-side flexible cylinders. Fig. 5 presents the maximum RMS dimensionless CF and IL displacement amplitudes versus the reduced velocity for the case of one smooth and one straked cylinders in a side-by-side arrangement. Note that Cylinder #2 was the cylinder attached with helical strakes. In addition, the results of an isolated cylinder with and without helical strakes are also shown in Fig. 5. It is interesting to note that the CF displacement amplitudes of the smooth cylinder (Cylinder #1) are close to those of the isolated smooth cylinder. The FIV of Cylinder #2 has been effectively suppressed in the CF direction using helical strakes. The CF displacement amplitude of the straked cylinder (Cylinder #2) is approximately the same as that of the isolated straked cylinder. Similar trends are observed in the IL direction. It is thus concluded that helical strakes perform well for suppressing the FIV of the flexible cylinder in a side-by-side arrangement with a smooth cylinder. Fig. 6 gives the results of the maximum RMS dimensionless CF and IL displacement amplitudes versus the reduced velocity for the case of two straked cylinders in a side-by-side arrangement. It can be seen that Cylinder #1 oscillates with a lower displacement amplitude than Cylinder #2 in the CF direction. By comparing with the results of the isolated straked cylinder, it can be found that both Cylinder #1 and Cylinder #2 have very small amplitudes in the presence of helical strakes. The peak value of displacement amplitudes in the IL direction for two straked cylinders is nearly 0.15D, which is much lower than that of the case shown in Fig. 4. It is thus concluded that helical strakes can obviously reduce the oscillation for two flexible cylinders in a side-by-side arrangement.
To further investigate the FIV suppression of helical strakes for two side-by-side cylinders, the vibration suppression efficiencies versus the reduced velocity in both the CF and IL directions are presented and discussed in Figs. 7−9. For the convenience of comparison, the suppression efficiency of the isolated cylinder is given in Fig. 7 as a reference standard. The 17.5D/0.25D strake, which is generally considered as the most effective configuration for vibration suppression of the isolated flexible cylinder undergoing vor-    tex shedding in water (Trim et al., 2005;Gao et al., 2015;Xu et al., 2017aXu et al., , 2017b, performs quite well in our experiment. The averaged suppression efficiencies in the CF and IL directions are 91.33% and 94.84%, respectively. The results indicate that the CF and IL VIV responses are almost completely suppressed.
For the case of one straked and one smooth cylinder in a side-by-side arrangement, it can be observed from Fig. 8 that the FIV reduction effectiveness become slightly worse than the case of the isolated cylinder. The averaged values of suppression efficiencies in the CF and IL directions are 81.78% and 79.38%. These results mean that approximate 80% of the response amplitude for Cylinder #2 can be suppressed by mounting17.5D/0.25D helical strakes.
Moreover, we experimentally investigated on FIV suppression of two straked cylinders in a side-by-side arrangement. The suppression efficiencies versus the reduced velocity in both the CF and IL directions are plotted in Fig. 9. It is clear that the helical strakes can suppress the FIV amplitude significantly in the CF direction for both Cylinder #1 and Cylinder #2. The averaged CF suppression efficiencies could reach up to 98.14 % and 93.19% for Cylinder #1 and Cylinder #2, respectively. The performance of helical strakes for FIV suppression of two side-by-side cylinders is even better than that of the isolated cylinder. This finding is very meaningful for the application of helical stakes for FIV reduction of multiple risers in offshore engineering. It also can be found from Fig. 9 that the suppression effectiveness of helical strakes in the IL direction is as good as that in the CF direction. The IL vibration amplitudes for Cylinder #1 and Cylinder #2 are averagely reduced 82.43% and 95.85%, respectively. It is worth mentioning that the IL suppression efficiency of Cylinder #1 fluctuates between 60% and 90% at the range of V r ≤15.0. As the reduced velocity increases from 15.0, the suppression effectiveness of helical strakes for IL FIV of the Cylinder #1 gradually increases and reaches a plateau.

Dominant frequency and dominant mode
We further investigate the variation of the dominant frequencies and dominant modes for the two side-by-side flexible cylinders without and with helical strakes in both the CF and IL directions with the reduced velocity. It should be pointed out that the CF and IL dominant frequencies, f y and f x , are taken as the largest peak in the spectra plot obtained by applying an FFT to the time series of the displacement response in the CF and IL directions. In this paper, the modal weights can be computed by the modal analysis, and the mode with the largest modal weight is called the dominant mode. The dominant frequency and dominant mode are important for examining the FIV response of long flexible cylinders. Here, the dominant frequency is nondimensionalized using the theoretical fundamental natural frequency of the flexible smooth cylinder in still water f 1 , which can be calculated by Eq. (12). Fig. 10 shows dimensionless dominant frequencies versus the reduced velocity in both the CF and IL directions for the case of two smooth cylinders in a side-by-side arrangement. The dominant frequencies of the isolated smooth cylinder are also plotted in the figure for comparison. An additional diagonal dash line is drawn in these figures, which represents the Strouhal frequency corresponding to the vortex-shedding frequency for the stationary cylinder. It can be clearly observed that the CF dominant frequency of the isolated flexible cylinder increases linearly with the reduced velocity and is in agreement with the vortex shedding frequency. Moreover, the dominant frequency in the CF direction is approximately half of the IL dominant frequency. The CF and IL dominant frequencies of both Cylin-  XU Wan-hai et al. China Ocean Eng., 2020, Vol. 34, No. 2, P. 172-184 der #1 and Cylinder #2 without helical strakes are inconsistency with those of the isolated smooth cylinder. These trends mean that there is no obvious difference of dominant frequencies between two side-by-side smooth cylinders with S/D=3.0. Fig. 11 shows the dominant frequencies versus the reduced velocity in both the CF and IL directions for the case of one smooth and one straked cylinders in a side-by-side arrangement. For the convenience of comparison, the results of the isolated straked cylinder are also drawn in the figure. It can be found that the isolated flexible cylinder with 17.5D/0.25D strakes vibrates at a low value of dominant frequency. In addition, the frequency is nearly independent of the reduced velocity, and there is no clear relationship between CF and IL frequencies. The CF dominant frequency of the smooth Cylinder #1 has a linear upward trend with increasing reduced velocity. Good agreement with the vortex shedding frequency is observed. For Cylinder #1 in the IL direction, the variation of the dominant frequencies with the reduced velocity is similar to that in the CF direction. The two times relationship still exists. The dominant frequency of the straked Cylinder #2 in the two side-by-side cylinders system is lower than that of the smooth Cylinder #1 and larger than that of the isolated straked cylinder, indicating that the use of helical strakes can suppress the dominant frequency of Cylinder #2, but the suppression effectiveness is not as good as the isolated cylinder. Fig. 12 presents the results of dominant frequencies versus the reduced velocity in both the CF and IL directions for the case of two straked cylinders in a side-by-side arrangement. It can be seen that the dominant frequencies of two side-by-side straked cylinders are in a poor agreement with the Strouhal vortex shedding frequency, showing that the used of helical strakes can disturb the normal wake region and result in a change in the dominant frequency. In addition, it can be found that the helical strakes can effectively reduce the CF and IL dominant frequencies of Cylinder #1 and Cylinder #2 subjected to FIV. The performance of helical strakes for suppressing dominant frequencies of two side-by-side cylinders is in reasonable agreement with that of the isolated cylinder (see Fig. 12).
Tables 3 and 4 illustrate the dominant modes of two cylinders with and without helical strakes in a side-by-side arrangement in the CF and IL directions, respectively. The results of the isolated smooth and straked cylinders are also tabulated for comparison. There are a total of 20 cases of reduced velocities calculated by using Eq. (11). For the isolated smooth cylinder, the dominant mode goes up gradually with the increasing reduced velocity, ranging from Mode 1 to Mode 4 in the CF direction and Modes 1−6 in the IL direction. The highest dominant modes for the isolated straked cylinder is Mode 2 in both the CF and IL directions, which are much lower than those of the isolated smooth cylinder. This trend indicates that the CF and IL dominant modes of the isolated cylinder can be suppressed significantly using helical strakes. Overall, the features of the CF and IL dominant modes for two smooth cylinders in a side-by-side arrangement are similar with those of the isolated smooth cylinder, which is consistent with the findings from dominant frequencies in Fig. 10. As for the case of one smooth and one straked cylinders in a side-by-side arrangement, it can be seen that the dominant frequencies of the smooth cylinder (Cylinder #1) in the CF and IL directions are in accordance with the isolated smooth cylinder; the highest dominant modes in the CF and IL directions for the straked cylinder (Cylinder #2) are Mode 3 and Mode 5, respectively, which are slightly lower than those of the isolated smooth cylinder. These observations indicate that the dominant modes for Cylinder #2 have been reduced moderately by helical strakes. For the case of two straked cylinders in a side-by-side arrangement, the dominant modes in the CF Fig. 11. Dimensionless dominant frequencies vs. the reduced velocity for one smooth and one straked cylinders. and IL directions are not higher than Mode 2, implying that the helical strakes can effectively suppress the dominant modes of FIV.

Mean drag force
The mean drag force coefficient C d0 is considered a key parameter to investigate the vibration suppression perform-ance of the helical strakes for flexible cylinders undergoing vortex shedding. It is known that the mean drag force of flexible cylinders is difficult to measure directly by load sensors in the experiments. Huera-Huarte et al. (2006) proposed an indirect finite element model (FEM) of determining mean drag forces by using displacements measured at discrete measurement points along the flexible cylinder.  This methodology has been successfully applied to obtain the mean drag coefficients of the isolated long flexible cylinder subjected to VIV by several researchers (Gu et al., 2013;Han et al., 2017). There is a mean deflection in the IL direction due to the mean drag force acting on the cylinder. According to the beam-bending theory, the governing equation of the flexible cylinder with a pinned-pinned boundary condition for the initial bending deformation can be expressed as: where x 0 (z) is the displacement of the initial bending deformation at point z, which can be obtained by using modal analysis approach based on the bending strain signals at measurement locations. T is the axial pre-tension of the cylinder. f d (z) is the mean drag force at point z. The drag coefficient of the flexible cylinder on each cross-section, C d (z), can be calculated according to the following equation, The mean drag coefficient, C d0 , can be obtained by averaging the coefficient C d (z) over the length of the cylinder, The same method has been employed to calculate the mean drag coefficients of an isolated flexible cylinder fitted with helical strakes in our previous research work (Xu et al., 2017a). Here, the calculation of mean drag coefficients for two side-by-side cylinders with and without helical strakes was conducted using the same FEM technique introduced by Huera-Huarte et al. (2006). More details of this method can be referred to Huera-Huarte et al. (2006). Fig. 13 shows the variation of mean drag force coefficients versus the reduced velocity for the case of two smooth cylinders in a side-by-side arrangement. It can be seen that the mean drag coefficients ranging from 1.35 to nearly 4.0 are much scattered than those of the isolated smooth cylinder, which are also plotted in the figure. In addition, it can be found from Fig. 13 that the mean drag coefficients of the isolated smooth cylinder fluctuate around 2.49. The averaged values of drag coefficients of the smooth Cylinder #1 and Cylinder #2 are 2.27 and 2.38, respectively. There is no obvious difference of drag force coefficients between Cylinder #1 and Cylinder #2. This is to say that the interaction of the two side-by-side cylinders has an insignificant influence on the drag force in our tests. Fig. 14 shows the mean drag force coefficients of one smooth and another straked cylinders in two side-by-side cylinders system versus the reduced velocity. The mean drag coefficients of the smooth Cylinder #1 fluctuate between 1.58 and 2.63 with an averaged value of 2.10. By using the helical strakes for Cylinder #2, the mean drag coefficients are in the range of 1.35 to nearly 4.0 with an averaged value of 2.39. It can be concluded that the application of helical strakes can enhance the mean drag force coefficients for Cylinder #2. Moreover, the mean drag force coefficients of the smooth Cylinder #1 and straked Cylinder #2 are both larger than those of the isolated cylinder fitted with helical strakes, which is also presented in Fig. 14. This phenomenon might be caused by the strong interaction of two side-by-side cylinders in the tests. Fig. 15 shows the results of mean drag coefficients versus the reduced velocity for the case of two straked cylinders in a side-by-side arrangement. For comparison, the mean drag coefficients of the isolated straked cylinder are also presented in the figure. It is interesting that the values   of C d0 for the straked Cylinder #1 and Cylinder #2 are close to each other in all ranges of the reduced velocity with an averaged value of 1.76. This means that the two straked cylinders have nearly the same hydrodynamic characteristics. In addition, it can be found that C d0 of two side-by-side straked cylinders agrees quite well with those of the isolated straked cylinder. This trend indicates that the performance of helical strakes for FIV suppression of two side-byside cylinders in our tests is similar to that of the isolated cylinder.

Conclusions
An experimental investigation was carried out in the towing tank to check the effectiveness of strakes for FIV reduction of two side-by-side flexible cylinders with a spacing of 3.0D. The strakes used in the tests are helical with P/H=0.25D/17.5D. Based on the experimental results, including the displacement amplitude, the FIV suppression efficiency, the dominant frequency, the dominant mode and the mean drag force, the following conclusions can be drawn: (1) For the case of two smooth cylinders in a side-byside arrangement, the maximum displacement amplitudes in the CF direction are nearly 1.71D for Cylinder #1 and 1.42D for Cylinder #2, respectively. The IL displacement amplitudes of Cylinder #1 and Cylinder #2 are significantly different from each other. The dominant modes are in the range of Mode 1-4 in the CF direction and Mode 1-6 in the IL direction. Strong interactions between the two cylinders through wake have been observed in our tests, which is consistent with the experimental result of two side-by-side flexible cylinders in Huera-Huarte and Gharib (2011a).
(2) For the case of one smooth and one straked cylinders in a side-by-side arrangement, the FIV behaviors of the smooth cylinder (Cylinder #1) are in reasonable agreement with those of isolated smooth cylinders. Helical strakes can remarkably suppress the FIV of Cylinder #2. The displacement amplitudes of the straked cylinder (Cylinder #2) are approximately the same as those of the isolated straked cylinders. This means that helical strakes may perform well in suppressing the FIV of the flexible cylinder in a side-byside arrangement with a smooth one.
(3) For the case of two straked cylinders in a side-byside arrangement, the FIV behaviors of Cylinder #1 and Cylinder #2 are similar to those of the isolated straked cylinders. These experimental results indicate that helical strakes can effectively reduce the displacement amplitudes, the dominant frequencies and the dominant modes of Cylinder #1 and Cylinder #2 subjected to FIV.