Passive VIV Reduction of An Inclined Flexible Cylinder by Means of Helical Strakes with Round-Section

A series of experimental tests of passive VIV suppression of an inclined flexible cylinder with round-sectioned helical strakes were carried out in a towing tank. During the tests, the cylinder models fitted with and without helical strakes were towed along the tank. The towing velocity ranged from 0.05 to 1.0 m/s with an interval of 0.05 m/s. Four different yaw angles (a=0°, 15°, 30° and 45°), defined as the angle between the axis of the cylinder and the plane orthogonal of the oncoming flow, were selected in the experiment. The main purpose of present experimental work is to further investigate the VIV suppression effectiveness of round-sectioned helical strakes on the inclined flexible cylinder. The VIV responses of the smooth cylinder and the cylinder with square-sectioned strakes under the same experimental condition were also presented for comparison. The experimental results indicated that the round-Sectioned strake basically had a similar effect on VIV suppression compared with the square-sectioned one, and both can significantly reduce the VIV of the vertical cylinder which corresponded to the case of a=0°. But with the increase of yaw angle, the VIV suppression effectiveness of both round- and square-section strakes deteriorated dramatically, the staked cylinder even had a much stronger vibration than the smooth one did in the in-line (IL) direction.


Introduction
Slender flexible structures have been widely applied in offshore engineering projects as the oil and gas exploitation goes into the deep sea. Their response features undergoing vortex-induced vibration (VIV) are extremely complex along with multi-modes, traveling wave, multi-frequencies, etc. VIV may cause undesirable vibration amplitude and accelerate structural fatigue damage. Therefore, extensive experimental tests and numerical simulations have been conducted to develop a better understanding of the mechanism of VIV and dynamic response characteristics, such as Zdravkovich (1981), Vandiver (1993), Williamson (1996), Sarpkaya (2004) and Rashidi et al. (2016). These studies further provide a theoretical base for the VIV control by different means.
The helical strakes are successfully employed to passively suppress VIV of flexible cylinders in offshore engineering (e.g., marine risers, cables, and tendons). The mechanism of VIV suppression is to disrupt the process of regular vortex shedding and prevent the shedding from being correlated in the spanwise direction (Gao et al., 2015). According to practical experiences and the previous studies, it has been proved that the helical strake is more advantageous and effective in mitigating VIV as it is omnidirectional. To maximize the VIV suppression effectiveness, much attention has been paid to the optimized configurations of helical strakes, such as the coverage rate, strake's height and pitch. Trim et al. (2005) performed an experimental investigation on the VIV reduction of a long flexible riser. It was found that the helical strakes with a 17.5D pitch and 0.25D height can work effectively for VIV suppression in both uniform and sheared flow. Quen et al. (2014) conducted experimental study on the VIV suppression by different configurations of helical strakes (height=0.05D/0.1D/0.15D; pitch=5.0D/ 10.0D/15D). It was found that a higher height of strake had a more positive effect on the suppression effectiveness. Gao et al. (2016) stated that the VIV suppression efficiency of a vertical flexible cylinder attached with helical strakes increased with the coverage rate, and could peak at 99% in the cross-flow (CF) direction according to their experimental observation. Besides, the cross-section of helical strakes is also of interest and concern by some researchers. Xu et al. (2017b) applied square-and roundcross-sectioned helical strakes for VIV suppression of a vertical flexible cylinder. It was found that the square-and round-sectioned strakes nearly shared the similar VIV reduction effectiveness. Sometimes, the strakes with roundsection represented more excellent effects on the VIV suppression of response frequency than those with square-section.
It is also worth pointing out that the axis of flexible marine structure might not always be perpendicular to the direction of oncoming flow in most engineering situations. In another word, a yaw angle will exist, which is defined as the angle between the axis of slender structure and the plane orthogonal to the oncoming flow. Hence, VIV of the inclined slender cylinders becomes even more complicated due to the influence of the axial secondary flow . Since the existence of yaw angle, the effect of axial secondary flow becomes significant, which might decrease the VIV suppression effect of helical strakes. Helically straked yawed cylinders under different operating conditions have been examined in several experiments. Zeinoddini et al. (2015) experimentally examined an elastically mounted rigid cylinder undergoing vortex-shedding at inclinations of a=0°, ±20°, and ±45°, both with and without helical strakes, the experimental results showed that the peak value of the CF displacement was in inverse proportion to the yaw angle. A similar tendency was also found in the experiments of Franzini et al. (2009Franzini et al. ( , 2013. More recently, Xu et al. (2017a) performed experimental tests in a towing tank, mainly concerning the effect of yaw angle on the VIV reduction of an inclined cylinder with square-sectioned strakes. It was found that the suppressing effect of helical strakes deteriorated with the increase of yaw angle.
In consideration of capital investment and technique level, tiny cylinder structure with round section can be helically fixed on the cylinder to alleviate the VIV in practical engineering. It has been experimentally proved that the square-sectioned strakes nearly share similar VIV reduction effectiveness for the vertical flexible cylinder (a=0°) with the round-sectioned strakes (Xu et al., 2017b). The purpose of this paper is to use the round-sectioned strake for suppressing VIV of an inclined flexible cylinder, and experimentally investigate whether the round-sectioned strake could achieve an equivalent or even better-suppressing effectiveness in comparison with a square cross-sectioned one. It should be pointed out that this work is the deepening and expansion of our previous experimental studies (Xu et al., 2017a(Xu et al., , 2017b(Xu et al., , 2018. To the best knowledge of the authors, it is the first time to carry out experimental tests of the inclined flexible cylinder attached with round-sectioned helical strakes.
The rest of this paper is organized as follows. Section 2 introduces the details of experimental set-up and data processing approaches. Section 3 discusses and analyzes the VIV response behaviors of the inclined cylinders fitted with and without helical strakes, including displacements, dominant frequencies, and dominant modes. Finally, some conclusions are drawn based on the experimental results.

Towing tank experimental description
The experiment was conducted in the towing tank which belongs to the State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University. The size of the tank is 137.0 m×7.0 m×3.3 m (length×width×depth). The uniform oncoming flow in the tests is simulated by towing the cylinder model in the tank. The towing velocity ranged from 0.05 m/s to 1.0 m/s with an interval of 0.05 m/s. The cylinder models fitted with round-and square-sectioned strakes were exposed in uniform flow with four different yaw angles (a=0°, 15°, 30° and 45°) respectively. Also, the smooth cylinder model was tested for comparison. During the test, the cylinder model was connected to a vertical supporting structure with universal joints, which can achieve different yaw angles by adjusting the angle plate. More details of the supporting system and towing devices can be obtained by reference to our previous experimental tests of the inclined smooth cylinder  and straked cylinder with square-sectioned helical strakes (Xu et al., 2017a). The flexible cylinder model was 5.6 m in length and 16 mm in diameter, resulting in an aspect ratio (L/D) of 350. It consisted of inner copper pipe and outer silicon tube as shown in Figs. 1a and 1b. The mass per unit length of the smooth cylinder model, including instrumentation, was 0.3821 kg/m. The axial tension of cylinder model was 450 N. The shapes of strake's cross-section were the geometry variable. The most recognized configuration arrangement of helical strakes with pitch and height of 17.5D/0.25D and three stars was selected in the tests (Trim et al., 2005;Gao et al., 2015). Main parameters of the experimental setup are listed in Table 1. A total of seven measurement points were evenly set along the cylinder (G1-G7) (see Fig. 1c). Each position had two pairs of strain gauges attached to the outside of the copper pipe. During each test, the CF and IL strain responses were measured with a sampling frequency of 100 Hz and the data were sampled during 50 s.
The original measured strain signals cannot be used directly to investigate the vibration of the flexible cylinder undergoing vortex-shedding and need to be pre-processed. Based on the findings by Song et al. (2011) and Gao et al. (2015), the IL strain contains three parts: a tensile strain caused by axial pretension, a bending strain caused by mean drag force and a VIV-induced bending strain. According to the assumptions that the time-averaged bending strain caused by VIV is zero and the time averaged tensile strain caused by axial pretension and time-averaged bending strain caused by mean drag force are temporally invariant, the IL VIV bending strain can be calculated by subtracting the time-averaged values from the strain data. The CF VIV bending strain can be computed using a similar method except that the CF strain signal includes only two parts, i.e., a tensile strain caused by axial pretension and a VIV-induced bending strain.
Then, the bending strain was high-pass filtered with a cut-off of 1.0 Hz to eliminate the low-frequency noise caused by the carriage motion and supporting structure vibration. Moreover, the bending strain was further low-pass filtered with a cutoff of 40.0 Hz to remove the high-frequency electromagnetic noise.
After the pre-processed strain data at the measurement points were obtained, a modal decomposition method was employed to reconstruct the displacement of the cylinder. Based on this method, the displacement can be regarded as the superposition of responses under different modes of vibration. For example, the displacement in the CF direction can be denoted as: where w n (t) is the weight function and ψ n (z) is the eigenmode function, t, z and L denote instantaneous time, coordinate along the axis and the length of cylinder respect-ively. Owing to the pined-pined boundary of the tested cylinder model, the eigenmode function can be written as: 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.
where ε(z, t) is the strain data which is measured directly from the experiments, and R presents the radius of the copper pipe. The expression of strain and displacement can be connected as: Assuming that the strain gauges were attached to a total of M measurement positions, the vibration of the cylinder can be written as a superposition of N eigenmodes, (6) Herein, (9) By using the least square method, the weight function of  XU Wan-hai et al. China Ocean Eng., 2018, Vol. 32, No. 4, P. 413-421 415 different modes can be calculated as follows: By applying Eq. (5) and Eq. (10), the displacement of the cylinder can be reconstructed. More details of the modal decomposition method used in this paper can be found in the research work of Lie and Kaasen (2006).

Experimental results and discussion
To investigate the effect of helical strakes with different cross-sections on the VIV reduction of the inclined cylinder, Fig. 2 shows the max CF RMS displacement amplitude of the inclined smooth and straked cylinders versus the reduced velocity V r . The reduced velocity is obtained by . Herein, U is the towing velocity, f 1 is the theoretical value of fundamental natural frequency, which can be calculated by , T being the axial tension and m being the total mass per unit length, including structural mass and added mass. In addition, it should be pointed out that the highest towing velocity was 1.0 m/s for all the tests. In Fig. 2, V r is calculated using the velocity component that is vertical to the axis of the cylinder, resulting in the absence of experimental results of the inclined cylinders in the high range of reduced velocity. Similar situations will appear in Fig. 2 and all the following figures in this paper. As shown in Fig. 2, the round-sectioned strake significantly suppresses the CF VIV amplitudes of the cylinder when a≤30° and the suppression effectiveness obviously deteriorates as the yaw angle increases from 0° to 45°. To be specific, in vertical case, the peak value of CF VIV amplitude of the smooth cylinder is nearly 1.6D, while the maximum amplitude of the round-sectioned straked cylinder re-mains around 0.125D when the reduced velocity is lower than 20. A similar phenomenon also occurs at the inclination of 15° and 30°, where the vibrations of the smooth cylinder all reach the peak value in the mode resonance region and show a low valley in the mode transition region, and the counterparts of the round-sectioned straked cylinder maintain low values with less mean deviation. When the yaw angle goes up to 45°, a surge of the CF VIV response of round-sectioned straked cylinder appears as V r >15, and the amplitude of displacement even reaches the same level as the smooth cylinder. This phenomenon is understandable since the existence of secondary flow could negatively affect the suppression effectiveness of helical strakes. In addition, the experimental results of the cylinders with squareand round-sectioned strakes are in good agreement with each other. These trends indicate that the performance of round-sectioned strake in suppressing the CF displacement on the inclined flexible cylinder subjected to VIV is similar to that of the square-sectioned strake. Fig. 3 shows the max RMS VIV amplitude in the IL direction versus the reduced velocity. Similarly, the roundsectioned strake successfully reduces the IL vibration of the vertical flexible cylinder which corresponds to a=0°. For the smooth cylinder with a=0°, the peak of the max RMS IL displacement is roughly around 0.40D, while the amplitude is nearly zero throughout all the reduced velocity tested for the cylinder fitted with round-sectioned strakes. Moreover, it can be found that the round-sectioned strake has compatible IL displacement suppression effectiveness as the square-sectioned strake does on the vertical flexible cylinder. However, the increase of yaw angle results in a sharp rise in the max RMS IL VIV amplitude of the inclined cylinder fitted with round-sectioned strake, even causes a larger amplitude than the smooth cylinder does in certain conditions. The round-sectioned strake maintains better IL VIV suppression effectiveness compared with the square-sectioned strake at a=15° and 30°, as the displacement of the round-sectioned straked cylinder is smaller than the corresponding part of the square-sectioned straked cylinder. When the yaw angle increases to 45° and the reduced velocity exceeds 3.75, it can be observed that the cylinder model with round-sectioned strake shows more drastic movement in contrast to the smooth cylinder in the IL direction. Fig. 4 shows the variation of the CF VIV suppression efficiencies of the inclined cylinder by mounting two kinds of helical strakes with the reduced velocity. In this paper, the VIV suppression efficiency of the helical strakes is defined as: where the variables γ and γ strake can be defined as any VIV response indicators of the smooth and straked cylinders, such as vibration amplitudes, dominant modes and dominant frequencies.
Here the variable refers to the vibration amplitudes. It can be seen from Fig. 4 that the CF VIV suppression efficiency of round-sectioned strake on the vertical cylinder remains in a high level around 90%, but as the reduced velocity exceeds 20, the efficiency drops to a valley value of about 40%. In the case of a=15°, the CF VIV suppression efficiency of the round-sectioned strake fluctuates around 70% and reaches a valley around 40% at a reduced velocity of 8.0, as for the square-sectioned strake is stabilized at around 80% with relatively low mean deviation. Also, when the inclination is 30°, the CF VIV suppression efficiencies of round-and square-sectioned strakes are 75.82% and 76.87% respectively. At much larger yaw angle  XU Wan-hai et al. China Ocean Eng., 2018, Vol. 32, No. 4, P. 413-421 (a=45°), the CF VIV suppression efficiencies of two strakes are in good agreement over all the reduced velocity, where the highest and lowest values are around 60% and 0% respectively. This means that the round-sectioned strake could achieve the comparable effect in CF VIV suppression effectiveness as the square-sectioned strake does at large inclination. Moreover, as the yaw angle increases from 0° to 45°, the averaged CF VIV suppression efficiency of the round-sectioned strake decreases from 90% to 41.5%. A similar trend has been observed for the inclined cylinder attached with square-sectioned strakes. Fig. 5 shows the IL VIV suppression efficiencies of the inclined cylinder by using different sectioned strakes versus the reduced velocity. It can be observed that the round-sectioned strake reaches a satisfying average suppression efficiency of around 95% for the vertical cylinder (a=0°). For the yaw angles a= 15°, 30° and 45°, the averaged IL VIV suppressing efficiencies are 90%, 86%, and -2%, respectively. In contrast, the corresponding results of square-sectioned strake are 82%, 76%, and 19%, respectively. For a=15° and 30°, the round-sectioned strake shows better IL VIV suppression effectiveness than square-sectioned one in the range of V r =10-20. However, when the cylinder model is 45° inclined, the IL VIV suppression efficiency of roundsectioned strake is in negative values at some reduced velocities, which means the helical strakes even enhance the VIV in the IL direction. This result is coincident with that of the rigid cylinder experiment carried out by Zeinoddini et al. (2015).
Figs. 6 and 7 show the dimensionless dominant frequency in the CF and IL directions versus the reduced velocity. Note that the CF and IL dominant frequencies, f y and f x  were taken as the largest peak in the spectra plot obtained by the FFT of the CF and IL time-varying displacements. Linear fits to the CF frequencies, which is equal to the Strouhal number, by using the least square method were also superimposed in Fig. 6, together with lines with doubled slopes in Fig. 7. It is clear that the dominant frequencies of the IL and CF displacement responses roughly follow a doubled relationship for the inclined smooth cylinder. And the Strouhal number varies from 0.163 to 0.171 as the yaw angle increases from 0° to 45°. For the vertical round-sectioned straked cylinder, the dimensionless dominant frequency does not increase linearly with the reduced velocity in both the CF and IL directions. There is a relatively high level of agreement between the cylinders fitted with round-and square-sectioned strakes. For the yawed case, the deviation of the CF dimensionless dominant frequency between smooth and round-sectioned straked cylinder is gradually weakened as the yaw angle rises from 15°t o 45°. It can be seen from Fig. 6 that the CF dimensionless frequency of the round-sectioned straked cylinder is very close to that of the smooth and square-sectioned straked cylinders with a=15°, 30° and 45°. These trends illustrate that neither round-nor square-sectioned strakes can obviously reduce regular vortex shedding from the inclined flexible cylinder. This means that the strake with either round-or square-section may fail in suppressing the oscillation frequency in the CF direction for the inclined cylinder. As the inclination becomes larger, it can be observed from Fig. 7 that the IL frequencies of the round-sectioned straked cylinder increase with the reduced velocity again but the variations are no longer so sensitive to the yaw angle in contrast to the CF frequency. In other words, the IL frequency of the round-sectioned straked cylinder under inclination of 15°, 30° and 45° shows significant deviation from the data of the vertical straked cylinder. Besides, the variation of IL frequencies of the square-sectioned cylinder under different yaw angles basically overlap with that of the round-sectioned one, which implies that both two shapes of helical strakes can slightly mitigate the IL frequency on the inclined flexible cylinder. Fig. 8 shows the CF dominant modes versus the reduced velocity. It should be pointed out that the dominant mode is the mode which has the largest modal weight from Eq. (10). For the vertical cylinder (a=0°), the VIV of the flexible cylinder with round-sectioned strakes is mainly dominated by the 1st-order mode, except that 3rd-order mode dominates the vibration of the flexible cylinder when the reduced velocity varies from 18.8 to 20. A similar trend has been observed for the vertical cylinder with square-sectioned strakes. By contrast, the response of the smooth cylinder subjected to VIV ranged from the 1st-order mode to the 4th-order mode with the increasing reduced velocity. These results indicate that both strakes with two types of cross-section can suppress the mode of vibration of the vertical cylinder in the CF direction. On the other hand, when a=15°, 30° and 45°, both CF dominant modes of the cylinder with round-and square-sectioned strakes are in good agreement with that of the smooth cylinder, going up from the 1st-order to the 3rd-order with the increasing reduced velocity. It can be found that the strakes' suppression effectiveness in CF dominant mode gradually deteriorates on the inclined cylinder with increasing yaw angle. Fig. 9 shows the IL dominant modes versus the reduced velocity. For the vertical case, the IL dominant mode of the round-sectioned straked cylinder remains the 2nd-order throughout nearly all the reduced velocity. While the mode of the vertical smooth cylinder increases with the reduced velocity and reaches the maximum of the 6th-order. At larger inclination (a=15°, 30° and 45°), the IL dominant mode of the round-sectioned straked cylinder no longer maintains an unchanging value and gradually increases with the reduced velocity. The mode of vibration ranges from the 1st- order to the 3rd-order, but it is still lower than the counterparts of the smooth cylinder. By contrast, the square-sectioned strake shows slightly better mode suppressing effect on the vertical cylinder when the reduced velocity was smaller than 10, also on 15° and 30° inclined cylinder and the reduced velocity is in the range of 8-15. However, when a=45°, the IL mode variation curves of two types straked cylinders are in good agreement with each other.

Conclusions
A series of experimental tests were conducted to study the effectiveness of round-sectioned strake for passively suppressing VIV of an inclined flexible cylinder with four different yaw angles (a=0°, 15°, 30° and 45°), and the following conclusions can be drawn.
(1) Round-sectioned strake shows satisfactory results in suppressing VIV amplitudes, frequencies and modes as the square-sectioned strake does for the vertical flexible cylinder, which further proves the high suppression efficiency of three-stars strake with the pitch of 17.5D and height of 0.25D, as illustrated in Trim et al. (2005) and Gao et al. (2015).
(2) Round-sectioned strake has higher IL VIV suppressing efficiency on 15° and 30° inclined cylinders, but appeared to be less effective in mitigating VIV in the CF direction when the reduced velocity is higher than 20. Also, as the yaw angle becomes larger, the suppression efficiency of round-sectioned strake deteriorates and even could enhance the VIV in the IL direction by 60% when the yaw angle is 45°. A similar trend has been observed in our previous experimental tests of VIV reduction of the inclined cylinder fitted with square-sectioned strakes by Xu et al. (2017b).
(3) On the whole, the VIV suppression effectiveness of round-sectioned strake is consistent with that of square-sectioned strake. This trend means that the cross-section of strake has insignificant effect on the VIV reduction.These findings are very meaningful for the VIV suppression of flexible cylinders (such as risers, cables, and free spanning pipelines) in offshore engineering.