Experimental study of vortex-induced vibrations of a tethered cylinder

Journal of Fluids and Structures 34 (2012) 51–67

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Experimental study of vortex-induced vibrations of a tethered cylinder X.K. Wang a,n, B.Y. Su b,1, S.K. Tan c a

Maritime Research Centre, Nanyang Technological University, 639798 Singapore, Singapore DHI-NTU Centre, Nanyang Technological University, 639798 Singapore, Singapore Nanyang Environment and Water Research Institute and School of Civil and Environmental Engineering, Nanyang Technological University, 639798 Singapore, Singapore

b c

a r t i c l e i n f o

abstract

Article history: Received 8 July 2011 Accepted 14 April 2012 Available online 9 May 2012

This paper presents an experimental study of the motions, forces and flow patterns of a positively buoyant tethered cylinder (mn o 1) in uniform flow undergoing vortexinduced vibration (VIV). The flow fields have been measured using digital Particle Image Velocimetry (PIV) technique, in conjunction with a piezoelectric load cell for direct measurement of drag and lift forces acting on the tethered cylinder. The effects of varying mass ratio and Reynolds number over the range 0.61 rmn r0.92 and 4000 r Re r12 000 are examined. Results of a fixed (or stationary) cylinder at the same Reynolds numbers are provided to serve as the benchmark reference. The peak amplitude of oscillation, ymax =yD , generally increases with Re and deceases with mn. Similar to previous studies, the results reveal the existence of a critical mass ratio mncrit  0:7, below which large-amplitude oscillations would take place when Re is high enough, with the largest peak amplitude of ymax =yD ¼ 0:9 observed for the case of mn ¼ 0.61 and Re ¼12 000. Thus two distinct states of oscillation are categorized, namely, the small- and large-amplitude oscillation states. The distinction between the two states is also vivid in the mean and root-mean-square (r.m.s.) force coefficients (including C D , C 0D and C 0L ). The frequency of vortex shedding (fV) from the tethered cylinder is always synchronized with the cylinder’s oscillation frequency (fosc), regardless of the oscillation state. A time series of instantaneous vorticity fields illustrate that vortex shedding from the tethered cylinder undergoing VIV maintains the 2S mode, but at an inclined angle to the free stream, which is most obvious in the large-amplitude oscillation state. This leads to an asymmetry in the shear layers separated from opposite sides of the cylinder, as shown by the distribution of ensemble-averaged Reynolds stress. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Vortex-induced vibration Tethered cylinder Vortex shedding

1. Introduction The practical significance of vortex-induced vibration (VIV) has received considerable attention, see for example the extensive reviews by Bearman (1984), Sarpkaya (2004), Williamson and Govardhan (2004, 2008) and references cited

n

Corresponding author. Tel.: þ65 67906619; fax: þ65 67906620. E-mail address: [email protected] (X.K. Wang). 1 Present address: Biofluid Mechanics Research Laboratory, Department of Bioengineering, National University of Singapore, 117576 Singapore, Singapore. 0889-9746/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfluidstructs.2012.04.009

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therein. Most of the previous studies are focused on an elastically mounted (or spring supported), rigid cylinder placed in uniform cross-flow. The principal vorticity dynamics giving rise to VIV of a cylinder, which could be a combination of transverse and in-line oscillations, are the spanwise von Ka´rma´n vortex pairs alternately shed from the two sides of the cylinder. It is found that as the frequency of oscillation (f) approaches the natural structural frequency (fN), largeamplitude (which can be up to the order of the cylinder diameter) oscillations occur, which is referred to as the ‘lock-in’ or ‘synchronization’ phenomenon. The amplitude and frequency response exhibits distinct branches as a function of reduced velocity and mass–damping ratio, namely the ‘initial’, ‘upper’ and ‘lower’ branches, which would correspond to different vortex modes, referred to as 2S, 2P and PþS (in the terminology originally proposed by Williamson and Roshko (1988)). Another main finding in recent years on cylinder VIV is the existence of a critical mass ratio (  0.54), below which a largeamplitude response will persist up to infinite normalized velocity (see reviews of Williamson and Govardhan (2004)). In the present study, we consider the case of a tethered cylinder vibrating in uniform flow. The cylinder is positively buoyant with relative density mn o1, where mn ¼m/mwater (m is the mass of cylinder, and mwater is the mass of displaced fluid medium). The cylinder is restricted to oscillate along the path of normalized radius (Ln ¼L/D, which is the tether length to cylinder diameter ratio, or abbreviated as the tether length ratio) with the axis of cylinder kept horizontal, where L is the tether length and D is the cylinder diameter (see Fig. 1). This flow configuration, first investigated by Williamson and Govardhan (1997) on a tethered sphere, is one of the most basic fluid–structure interaction problems, for example the underwater tethered buoyant bodies or risers. Another non-dimensional governing parameter is the Reynolds number (Re ¼UD/n), where U is the uniform free-stream velocity and n is the kinematic viscosity of the fluid. Depending on the value of U (and hence Re), the tethered cylinder will move downstream and downward, forming an angle (y) with respect to the vertical y-axis. The tethered cylinder is considered as an extension of the widely studied elastically mounted cylinder, but with two new features (Ryan et al., 2007). The first is that the cylinder oscillation has an imposed curvature on its path, which is inversely proportional to the tether length; the second is that the cylinder is oscillating at a mean (time-averaged) layover angle, y, to the free stream. Here, we follow the work of Ryan et al. (2007) and introduce the equation of motion of the tethered cylinder as J y€ ¼ F y L,

ð1Þ

where J is the polar moment of inertia and Fy is the force acting on the cylinder in the direction of motion (or in tangential direction). Both y and Fy contain the mean and fluctuating (or time-varying) components and can be written as y ¼ y þ y~ and F y ¼ F y þ F~ y , respectively. The other forces acting on the cylinder are also shown in Fig. 1, namely FD ¼drag force; FL ¼lift force; T¼tension force acting through the tether; B¼ buoyancy force that has already taken into account the gravitational force. Thus, Fy can be expressed as a function of drag, lift and buoyancy forces acting on the cylinder as F y ¼ F D cos yðF L þBÞsin y:

ð2Þ

By assuming small-amplitude oscillations, such that sin y~  y~ and cos y~  1, Fy may be decomposed with its mean and fluctuating components as F y ¼ y~ ðT þ T~ Þ þF osc ,

ð3Þ

where T ¼ F D sin y þF L cos y,

ð4Þ

Fig. 1. Schematic diagram of the tethered cylinder system.

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T~ ¼ F~ D sin y þ F~ L cos y,

ð5Þ

F osc ¼ F~ D cos yF~ L sin y:

ð6Þ

Fosc may be considered as the driving force, and is a function of fluctuating drag/lift forces and mean layover angle. In contrast to an elastically mounted cylinder, the tethered cylinder has no mechanical restoring force, but it does have a ‘fluid/buoyancy’ restoring force due to the force balance between the mean fluid forces (drag and lift) and the buoyancy force. When Ln-N, the motion of a tethered cylinder will approach that of an elastically mounted cylinder at infinite velocity. However, it should be noted that the tethered cylinder always oscillates at a mean layover angle ðyÞ to the free stream. The value of y is dependent on the mass ratio of the cylinder, the length ratio of the tether, as well as the Froude number of the flow (Ryan et al., 2007). For tethered systems, the natural frequency (fN) cannot be determined a-priori, as it is dependent on the instantaneous fluid forces acting on the body. Carberry and Sheridan (2007) showed that assuming a zero net lift force (i.e., F L ¼ 0), fN is analogous to that of a pendulum and can be calculated using sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð1mn Þg fN ¼ , ð7Þ 2p ðmn þ C A ÞL cos y where CA is the potential flow added mass coefficient. However, in this case, CA cannot be assumed to be unity (as is the case for a freely vibrating cylinder whose motion is normal to the flow direction) due to the curvature of the cylinder path, as discussed in details by Newman (1977). This leads to CA o1 with the two limiting cases: as Ln-N, CA-1 and as Ln-0, CA-0. Thus, the approximation of CA E1 is only valid for sufficiently large tether lengths. Under the same assumption of F L ¼0, the mean drag coefficient C D can be calculated by CD ¼

ð1mn ÞpDg tan y 2U 2

:

ð8Þ

Most of the previous studies on tethered bodies are focused on spheres, see for example Govardhan and Williamson (2005), who identified multiple oscillation modes with respect to the reduced velocity Un ¼ U/fND. Recently, van Hout et al. (2010) conducted time resolved PIV measurements (2000 fps) of the instantaneous wake of a tethered sphere, and revealed unique vortex shedding patterns corresponding to different bifurcation regions. To date, however, there are only a handful of publications, almost exclusively by a group of researchers from Monash University, on the VIV phenomenon of a tethered cylinder, namely, Ryan et al. (2004, 2005, 2007), Carberry and Sheridan (2007), and Ryan (2011). Ryan and coauthors conducted a series of numerical simulations of a tethered cylinder with a wide range of mass ratios (mn ¼0.1–0.97) and tether length ratios (Ln ¼1–10) at relatively low Reynolds numbers (up to Re ¼200). It has been shown that y increases with Un. Depending on the value of y , three modes of oscillation are identified and referred to as: the ‘in-line’ mode (for y o451), where oscillations are predominantly in line with the free stream; the ‘transverse’ mode (for y 4 451), where oscillations are predominantly transverse to the free stream; and the ‘transition’ mode between the in-line and transverse modes. Ryan et al. (2007) reported that for a majority of mean layover angles, the oscillations of the tethered cylinder induce a negative mean lift due to the asymmetry in the wake. In analogy with the elastically mounted cylinder, there also exists a critical mass ratio below which the cylinder ‘jumps’ to large-amplitude oscillations at higher velocities. Previous studies on the elastically mounted cylinder have indicated the dependence of cylinder VIV on Re. As shown by Bearman (2011) on a freely vibrating cylinder at low mass and damping, the maximum transverse amplitude of oscillation keeps roughly constant in the laminar regime (up to Re ¼200), but continues to rise almost linearly in the subcritical regime (Re ¼103–104). In the case of the tethered cylinder, besides the above-mentioned low-Re numerical studies by Ryan and co-authors, Carberry and Sheridan (2007) provided an experimental investigation for mn ¼0.54–0.97 with a fixed tether length Ln ¼4.6 in the subcritical regime, namely, Re ¼1000–7000. The flow fields were measured using particle image velocimetry (PIV), while the cylinder’s motion was recorded by a video camera. The critical mass ratio, mn crit , was found to be about 0.74, considerably higher than the predicted value of 0.4 by Ryan et al. (2007) at Re¼200. This observation is consistent with the finding of Ryan et al. (2005) that the value of mncrit varies with Reynolds number. On the other hand, both Ryan et al. (2007) and Carberry and Sheridan (2007) showed an excellent collapse of the response data (especially the mean layover angle y ) when plotted against the buoyancy Froude number, Fr0 ¼U/((1  mn)gD)0.5, which is defined as the inertial force to buoyancy force ratio. The present study is motivated by the lack of detailed quantitative information on the VIV characteristics of a tethered cylinder. Therefore, particle image velocimetry (PIV) has been employed in this study, since it is an effective technique to study cylinder VIV in providing whole-field, instantaneous velocity and vorticity distributions (e.g., Lam et al., 2010). To date, the only experimental study on tethered cylinder is Carberry and Sheridan (2007). However, they did not measure the hydrodynamic forces (lift and drag) of the cylinder, yet according to Eq. (6) we know that the driving force is a function of fluctuating force components. To fill this gap, concurrent direct force measurements have been also performed with a piezoelectric load cell. It is hoped that this study could provide further insights into the underlying physics of fluid– structure interaction of a tethered cylinder, as well as experimental database for validation of computational fluid dynamics (CFD) codes. Effects of varying mass ratio and Reynolds number on the cylinder response (amplitude and frequency of oscillations), vortex shedding patterns and drag/lift coefficients are examined in details.

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2. Experimental setup and methodology 2.1. Flow facility and apparatus The experiments were performed in the re-circulating open channel at Maritime Research Centre, Nangyang Technological University. To ensure smooth entrance of the flow into the test section, the settling chamber upstream of the contraction was fitted with perforated steel plates and honeycomb-screen arrangements. The test section had a total length of 5 m, a width of 0.3 m and a height of 0.45 m. The bottom wall and the two side walls of the test section were made of glass to enable optical access. The flow rate was controlled using a centrifugal pump fitted with a variable speed controller, and the flow velocity in the test section could be adjusted to any value between 0.02 and 0.65 m/s. The freestream velocity was uniform to within 1.5% across the test section, and the turbulence intensity in the free stream was below 2%. The tethered cylinder was made of smooth, hollow perspex tube. The diameter of the cylinder was fixed at D ¼20 mm, while its mass was varied, yielding 5 different mass ratios, namely, mn ¼0.61, 0.68, 0.76, 0.84 and 0.92. The cylinder was mounted in the test section with its axis kept horizontal and aligned in the spanwise direction under free end conditions at both ends. The length (or span, H) of the cylinder is 260 mm, resulting in an aspect ratio (H/D) of 13. This experimental model was built to minimize the 3-dimensional (3D) effects of the flow, which could be attributed to non-uniformities that exist in the flow and along the body span, and to the particular end constraints imposed in the experimental arrangement. At each end, there was a gap of 20 mm (or 1D) from the channel’s side wall to the cylinder, such that the cylinder was well beyond the near-wall region (outside the boundary layer developed on the side walls), since the boundary layer would further complicate the flow (Wang et al., 2006). In the literature, several studies have been conducted on a stationary cylinder to investigate the influence of aspect ratio, such as that by Park and Lee (2000) with three aspect ratios (H/D ¼6, 10 and 13) at a fixed Reynolds number of Re ¼20 000. It was shown that the end effects are limited to the region near the free end. In the case of H/D ¼10, for example, beyond a spanwise length of 1.17D from the free end, the pressure distribution has nearly recovered to that of 2D cylinder. In this regard, the present model with H/D ¼13 was considered long enough to ensure a nominally 2D flow along a majority of the cylinder span, particularly in the near-wake region. Thus, all the velocity measurements were carried out in the mid-span plane of the channel (or cylinder). The cylinder was tethered at both ends of the span with a thin perspex rod. The tether was 100 mm in length, yielding a tether ratio of Ln ¼5, similar to Ln ¼4.6 in Carberry and Sheridan (2007). The cylinder and the tethers were rigid and there was no relative motion between them. The other end of the tethers was pivoted on a rigid, aluminum frame with bearings and the cylinder was free to rotate about the pivot point (Fig. 1). A fixed (or stationary) cylinder with D ¼20 mm was also measured under the same experimental conditions to serve as the benchmark reference. The coordinate system is also shown in Fig. 1. The x-axis (streamwise direction) is pointed downstream, the y-axis (transverse direction) is vertical, and the z-axis (spanwise direction) is horizontal and perpendicular to the flow direction. For each mn, four different free-stream velocities were considered, namely, U¼0.2, 0.3, 0.4 and 0.6 m/s, corresponding to Re¼4000, 6000, 8000 and 12 000. Varying U (or Re) alters the ratio of the inertial force to the buoyancy force, whereas varying mn directly alters the buoyancy force of the cylinder. The Reynolds number range is about twice as high as that of Re ¼1000–7000 in Carberry and Sheridan (2007). The calculated gravity Froude number, Fr ¼U/(gD)0.5, is varied over the range of Fr ¼0.45–1.35. 2.2. Measurement techniques Velocity measurements were performed using a PIV system (LaVision model). The flow field was illuminated with a double cavity Nd:YAG laser light sheet at 532 nm wavelength (Litron model, power 135 mJ per pulse, duration  5 ns). The particle images were recorded using a 12-bit charge-coupled device (CCD) camera with a resolution of 1.6 K  1.2 K pixels and a frame rate of 15 Hz. Neutrally buoyant hollow glass spheres with a mean diameter of 13 mm were seeded in the flow as the tracer particles, which offered good traceability and scattering efficiency. The LaVision Davis PIV package was used to process the particle images and determine the flow vector field. Particle displacement was calculated using the fast-Fourier-transform (FFT) based cross-correlation algorithm with the standard Gaussian sub-pixel fit structured as an iterative multi-grid method. The processing procedure included two passes, starting with a grid size of 64  64 pixels, stepping down to 32  32 pixels overlapping by 50%, which resulted in a set of 7500 vectors (100  75) for a typical field. In between passes, the vector maps were filtered by using a 3  3 median filter in order to remove possible outliers. If the center vector (surrounded by the eight interrogation windows) differed from the median vector by more than 5 times the root-mean-square (r.m.s.) value, the center vector was replaced by the averaged vector obtained from the neighboring interrogation windows. The final vector maps were smoothed with a 3  3 average filter. The number of particles in a 32  32 pixel window was of the order of 10–15, which was sufficient to yield strong correlations. The field of view was 190 mm  142 mm, therefore the spatial resolution for the present setup was 1.9 mm  1.9 mm (i.e., 0.095D  0.095D). For each case, a series of 840 instantaneous flow fields were acquired at a frequency of 15 Hz (or 56 s recordings), in order to achieve a reasonably converged statistics of the measured quantities, such as Reynolds shear stress. It should be noted that all the PIV recordings were performed after the cylinder had reached the state of temporally periodic oscillations. The duration of the recordings covered about 110–220 cycles of vortex

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shedding, the frequency of which was about 2–4 Hz depending on Re. The uncertainty in the instantaneous velocities (u and v) was estimated to be about 2%. Based on the velocity vector distribution, the instantaneous spanwise vorticity (oz ¼ Dv/Dx  Du/Dy) was calculated using the least operations extrapolation scheme. The uncertainty in oz was about 10%. A more detailed description of the PIV post-processing procedure and uncertainty analysis is given in Wang and Tan (2008). The motions of the cylinder, which could be determined by analyzing the time series of the original PIV images sampled at 15 Hz, were also recorded by a Canon video camera at a sampling rate of 25 Hz. Force measurements were performed using a three-component piezoelectric load cell (Kistler Model 9317B). This type of load cell has the advantage of high response, resolution and stiffness, and hence has been widely used in the research community (e.g., Lam et al., 2003). The load cell was installed between the aluminum frame and a carriage platform that was fixed above the channel, and hence the integral forces (drag and lift) on the cylinder could be acquired. The amplified output was captured with a National Instruments data acquisition card at a sampling rate of 1 KHz (which was 2 orders of magnitude higher than the cylinder’s oscillation frequency). The duration of each record was of the order of O(102) seconds, which was sufficiently long to obtain converged statistics of the measured data. Then, the mean and r.m.s. drag and lift coefficients (C D , C L , C0 D and C0 L) of the cylinder could be calculated in the similar form as C D ¼ 2F D =rU 2 D, where F D ¼mean drag force per unit span. The vortex shedding frequency (and hence the Strouhal number) was obtained by power spectral analysis of the lift fluctuations using FFT algorithm. Based on repeated measurements of a stationary cylinder, the uncertainty in the mean drag coefficient was determined to be within 3% and that in Strouhal number was 1.5%. 3. Results and discussion 3.1. Mean layover angle, oscillation amplitude Fig. 2 presents the measurement results of the mean layover angle ðy Þ as a function of gravity Froude number, Fr ¼ U/(gD)0.5, for different mass ratios. The experimental results by Carberry and Sheridan (2007) for mn ¼0.59 and 0.87 are also included. In general, y increases with Fr but decreases with mn as anticipated. Also, a dotted line is added in the figure to delineate the oscillation mode to be ‘in-line’ (for y o 451) or ‘transverse’ (for y 4 451). Most cases are in transverse mode, except for very small mn cases at low Fr. For a given mn, y generally increases smoothly with Fr. The mn ¼0.59 curve in Carberry and Sheridan (2007) exhibits an abrupt increase at Fr E1.0. This abrupt increase, referred to as a ‘jump’ in the literature, is likely to be an inherent phenomenon for a tethered cylinder below critical mass ratio (mn crit  0:74 in Carberry and Sheridan (2007); mn crit  0:4 in Ryan et al. (2007)). However, the present data do not show such a jump because of a lack of data in between the two highest Froude numbers (i.e., Fr ¼0.9–1.35, corresponding to Re ¼8000–12 000).

Fig. 2. Mean layover angle as a function of gravity Froude number.

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It is evident in Fig. 2 that the data do not collapse when plotted against Fr (or Re, which also varies linearly with U), a phenomenon already reported by Ryan et al. (2007) and Carberry and Sheridan (2007). Instead, these authors showed that the response data, particularly the mean layover angle y , collapse well when plotted versus the buoyancy Froude number, Fr0 ¼U/((1  mn)gD)0.5. Therefore, the variation of y as a function of Fr0 is given in Fig. 3, which shows a reasonably good collapse of the data into a single exponential growth curve, which is defined as 

y ¼ 83:8133:1 exp 

 Fr0 : 0:992

ð9Þ

The discontinuous jump in y for cases below mncrit at high Fr0 as reported by Ryan et al. (2007) and Carberry and Sheridan (2007) is, however, not observed. In the following figures, all the data for the tethered cylinder are plotted against both Fr (upper horizontal axis) as well as Re (lower horizontal axis) in order to make comparison with those for the stationary cylinder. At the lowest Reynolds number Re¼4000, the cylinder is essentially stationary and the oscillation amplitude is very small. When ReZ6000, the cylinder oscillates with appreciable amplitude. For a given mn, the amplitude of oscillation gradually increases with Re as anticipated. In the case of mn ¼ 0.61 and 0.68, however, when Re is increased to 12 000, the cylinder oscillates rigorously with considerably large amplitude that can be as high as of the order of one cylinder diameter. This demonstrates a jump of response to the large-amplitude oscillation state. Fig. 4(a) and (b) show a time history of the tether angle, y(t), for the case of mn ¼0.61 at Re¼8000 and 12 000, respectively. It is evident that the oscillations become much more intense at Re¼12 000. This abrupt jump between the small- and large-amplitude states is quantitatively depicted in the peak angular amplitude of oscillation, ymax =yD (Fig. 5), where ymax is the maximum variation in the cylinder motion (or half the peak-topeak value) and yD is the angle subtended by the cylinder diameter. The value of ymax =yD ¼ 0:5 to distinguish the small- and large-amplitudes is based on the results reported by Carberry and Sheridan (2007) on a tethered cylinder and by Govardhan and Williamson (2000) on an elastically mounted cylinder at low mass and damping (with mass ratio mn ¼1.19 and damping coefficient z ¼0.00502). For mn ¼0.61 and 0.68 at the lowest Reynolds number Re¼4000, ymax =yD is about 2.6% and 2% respectively, and becomes extremely small (0.5%) for larger mass ratios (mn Z0.76). For mn ¼0.61 and 0.68 at Re¼4000, the mean layover angle y is about 221 (see Fig. 2), and thus the small-amplitude oscillations are predominantly in the ‘in-line’ mode. An overall trend is that ymax =yD for each mn case monotonically increases with Re, but when Re is fixed, ymax =yD decreases with mn. For all the cases considered, ymax =yD generally falls in the range of 0–0.4, with the exception of the two smallest mass ratios (mn ¼0.61 and 0.68) at the highest Re, where ymax =yD can be as high as 0.5–0.9, which is of the same order as that of about 0.6–0.7 in Carberry and Sheridan (2007). This remarkable jump in ymax =yD indicates the occurrence of large-amplitude oscillations, which are in transverse mode since y is about 651 in both cases. This jump in ymax =yD corresponds to a sharp increase in the hydrodynamic force coefficients (including C D , C 0D and C 0L ), as well as a different vortex shedding pattern in the cylinder wake, as will be shown in Section 3.4. These properties indicate the existence of two distinct states for small mn cases, i.e., the small- and large-amplitude states, taking place before and after the abrupt jump, respectively. The critical mass ratio, mncrit , is about 0.7 (in the range 0.68omncrit o0.76), which agrees satisfactorily with the reported value of mncrit ¼ 0:74 by Carberry and Sheridan (2007).

Fig. 3. Mean layover angle as a function of buoyancy Froude number.

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Fig. 4. Time history of the instantaneous tether angle for mn ¼0.61 at (a) Re ¼8000 and (b) 12 000.

Fig. 5. Peak amplitude of oscillation as a function of Reynolds number (lower horizontal axis) and gravity Froude number (upper horizontal axis).

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3.2. Drag and lift The hydrodynamic forces (drag and lift) acting on the tethered cylinder have been measured directly using a load cell, together with the case of a stationary cylinder as a benchmark reference for comparison. The mean drag coefficient (C D ), r.m.s. drag and lift coefficients (C 0D and C 0L ) for the stationary cylinder at different Reynolds numbers are given in Fig. 6. The mean drag coefficient decreases slightly with respect to Re, from C D ¼ 1:185 at Re¼ 4000 to C D  1:0 at Re¼12 000. These values are in good agreement with the experimental result of C D ¼ 1:186 by Gopalkrishnan (1993) and the direct numerical simulation (DNS) result of C D ¼ 1:143 by Dong and Karniadakis (2005) at a comparable Reynolds number of Re ¼10 000. The variations of C 0D and C 0L with Re are similar, decreasing from about 0.2 to about 0.06 over the range Re ¼4000–12 000. It is noted that the C 0L values are substantially lower than C 0L ¼ 0:384 by Gopalkrishnan (1993) and C 0L ¼ 0:448 by Dong and Karniadakis (2005). This discrepancy, however, is not surprising, considering that there is a wide scatter of documented values on a stationary cylinder, with C 0L varying from 0.045 to 0.47 over the range Re ¼1600–20 000, as summarized by Norberg (2003). This wide scatter of data is attributed partially to large variation in the flow conditions (such as Reynolds number, free-stream turbulence level, aspect ratio and blockage ratio of the cylinder, etc.), and partially to difference in measurement techniques among those studies—for example, most of the early studies deduced the lift by integrating the measured surface pressure. It is believed that the present method of direct force measurement using a high-precision load cell is more reliable and accurate. Fig. 7 presents the variation of mean drag coefficient (C D ) with Re for the tethered cylinder at different mass ratios. A general trend is that each curve generally decreases with Re, similar to the case of stationary cylinder, except for the case of mn ¼0.61 and 0.68 at the highest Reynolds number Re ¼12 000. When Re is fixed, C D generally deceases with mn. At the lowest Reynolds number Re ¼4000, the value of C D for the two smallest mn cases (i.e., mn ¼0.61 and 0.68) is 1.5–1.7, which is remarkably higher than that of C D ¼1.185 for a stationary cylinder. The discrepancy is attributed to the small-amplitude oscillations (which are in-line mode with ymax =yD ¼2–3%) for these two cases, as shown in Fig. 5. This is reminiscent of reported studies on cylinder VIV, for example the simulation of Watanabe and Kondo (2006) on a circular cylinder at Re O(104) undergoing in-line oscillations with an amplitude of about 1.5%. It was shown that under such conditions, C D is about 1.77, which is about 18% higher than that of 1.515 for a stationary cylinder. For mn ¼0.76, 0.84 or 0.92, C D monotonically decreases with Re over the range considered; in the two smallest mn cases n (m ¼0.61 and 0.68), on the other hand, C D decreases with Re until Re ¼8000, but thereafter the curve increases sharply to C D ¼ 1:321:5 at Re ¼12 000. These two points, as highlighted by the black arrows in Fig. 7, are in the large-amplitude oscillation state. This sharp increase in C D corresponds to the abrupt rise in ymax =yD as shown in Fig. 5. The large-amplitude oscillations would result in a remarkably larger ‘effective diameter’ of the cylinder and hence a sharp increase in C D , a phenomenon that is generic to the canonical case of elastically mounted cylinder. Similar values are reported by Ryan et al. (2007) on a tethered cylinder at Ln ¼5 and Re ¼200 over the range mn ¼0.1–0.8. Further, the present findings are compared with the experimental data by Carberry and Sheridan (2007) at subcritical Reynolds numbers (Re ¼1000–7000), in which C D is calculated from the measured y using Eq. (8). It was reported that after the jump C D is about 1.3–1.4 that is comparable to the present study, but before the jump C D is approximately constant at 0.9, which is unreasonably lower than the universally accepted value of C D  1:0 for a stationary cylinder. Similarly, we have used Eq. (8) to predict C D using the measured value of y . As compared to measurement results, the predicted value is significantly lower: in the case of mn ¼0.61, for instance, the calculated C D decreases from 1.14 to 0.88

Fig. 6. Hydrodynamic force coefficients (mean drag coefficient C D , r.m.s drag coefficient C 0D and r.m.s. lift coefficient C 0L ) of the stationary cylinder as a function of Reynolds number.

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Fig. 7. Mean drag coefficient of the tethered cylinder as a function of Reynolds number and gravity Froude number.

Fig. 8. Mean drag coefficient of the tethered cylinder as a function of buoyancy Froude number.

over the range of Re ¼4000–12 000. This suggests that this method based on the zero lift assumption may underestimate the mean drag coefficient of a tethered cylinder particularly in the small-amplitude oscillation state. The foregoing section shows that the mean layover angles collapse pretty well when plotted against the buoyancy Froude number Fr0 . To examine whether the mean drag coefficient displays the similar property, the data are re-plotted as a function of Fr0 , as shown in Fig. 8. Except for the two abrupt jump points that lie in the large-amplitude oscillation state, C D gradually decreases with increasing Fr0 . A striking feature is that all the data points falling in the small-amplitude state can be satisfactorily fitted with a single exponential decay curve, which is indicated by the thick black line defined as   Fr0 : ð10Þ C D ¼ 1:087 þ1:726 exp  0:613 This indicates the existence of scaling laws if proper parameter(s) have been selected for normalization. As commented by Carberry and Sheridan (2007), for tethered systems, the buoyancy Froude number (which is defined as the inertial force to buoyancy force ratio) is an important physical parameter. In Carberry and Sheridan (2007), the curve of either the mean layover angle or oscillation amplitude versus Fr0 is split into two segments that correspond to the small- and largeamplitude states. However, our results indicate that the mean layover angles can be fitted by a single curve when plotted against Fr0 , irrespective of the oscillation state. On the other hand, the mean drag coefficients in the small-amplitude state

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collapse into a single exponential fit curve (Fig. 8), whereas those in the large-amplitude state show a significant departure from the curve (they may themselves collapse into another curve, but it is merely a conjecture because of insufficient number of data points). The distinction between the small- and large-amplitude states is also obvious in the r.m.s. drag and lift coefficients (C 0D and C 0L ), as shown in Fig. 9. The two points corresponding to large-amplitude state exhibit distinctly high values of C 0D and C 0L , whereas C 0D and C 0L in the small-amplitude state show a satisfactory collapse and decrease with Re over the range considered. The jump from small- to large-amplitude states occurs at about Fr0 ¼2.0, which is considerably higher than the corresponding value of Fr0 ¼0.8–1.0 in Ryan et al. (2007) and Fr0  1.3 in Carberry and Sheridan (2007). This discrepancy is probably due to the difference in Re among these studies. As shown in Eq. (6), the fluctuating drag and lift components are also factors determining Fosc. Therefore, a time history of the instantaneous drag and lift coefficients, CD(t) and CL(t), for the case of mn ¼0.61 at Re ¼8000 and 12 000 is shown in Fig. 10(a) and (b), representative of the small- and large-amplitude states, respectively. Substantial difference can be observed between these two states. In the small-amplitude state (Fig. 10(a)), besides the primary, low-frequency oscillations corresponding to that of von Ka´rma´n vortex shedding, both CD(t) and CL(t) signals show the presence of highfrequency oscillation component, which is likely due to the small-scale, Kelvin–Helmeholtz (K–H) type vortices in the shear layers separated from opposite sides of the cylinder. In the large-amplitude state, the signals become much more organized with a dominant frequency of about 4 Hz, which is the same as the oscillation frequency of the cylinder as shown in Fig. 4(b). Three observations can be made from Fig. 10(b). First of all, CD(t) and CL(t) are in-phase with each other. The second observation is that neither CD(t) nor CL(t) signal resembles the simple sinusoidal variation about the mean value; rather, it shows a modulated sine with a flat lower half. The modulation is likely due to the asymmetry in the vortices shed from the two sides of the cylinder, as will be discussed later in Section 3.4. Thirdly, both CD(t) and CL(t)

Fig. 9. R.m.s. drag and lift coefficients of the tethered cylinder as a function of Reynolds number and gravity Froude number.

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Fig. 10. Time history of the instantaneous drag and lift coefficients for mn ¼0.61 at (a) Re ¼8000 and (b) 12 000.

signals are not exactly periodic; instead, occasional large departures are distinguishable as ‘glitches’, one of which is most evident at t ¼5.5–6 s. This type of glitches, which is also appreciable in the time history of the tether angle (Fig. 4), implies that the cylinder oscillates in an intermittent manner instead of an exactly periodic manner. 3.3. Frequencies The various types of frequency terms in the tethered system, namely, the vortex shedding frequency (fV), the oscillation frequency of cylinder (fosc), and the structural natural frequency (fN), are discussed herein. In the present study, fN is calculated using Eq. (7) and fV is obtained by spectral analysis of a time history of signal (such as velocity or hydrodynamic force) using Fourier transform. Similarly, the cylinder’s motion and hence fosc can be determined by analyzing the time series of the original images recorded by the video camera or PIV CCD camera. We find that for all the cases considered, the vortex shedding frequency always matches the cylinder’s oscillation frequency, i.e., fV  fosc. The variation of St for the frequency of vortex shedding from the stationary cylinder (fV0) with Re is shown in Fig. 11. All the data fall within the narrow range of St ¼0.195–0.2, and are in satisfactory agreement with the empirical formula:   19:7 , ð11Þ St ¼ f V0 D=U ¼ 0:198 1 Re which generally holds true over a broad range of Re ¼250–2  105. In the case of a tethered cylinder, on the other hand, the vortex shedding pattern depends on the cylinder’s oscillation response (such as amplitude and frequency of oscillation), which is controlled by various dynamic and geometric parameters including Reynolds number Re, mass ratio mn, and tether length ratio Ln. The calculated Strouhal number for

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Fig. 11. Strouhal number for vortex shedding frequency (fV0) of the stationary cylinder and the oscillation frequency (fosc) of the tethered cylinder undergoing VIV.

Fig. 12. Variation of frequency ratio (fV0/fosc) of the tethered cylinder as a function of Reynolds number and gravity Froude number.

the oscillation frequency of the cylinder, St ¼foscD/U, is also shown in Fig. 11. For a given mn, St decreases approximately linearly with Re, which is in contrast to the case of stationary cylinder that is basically Re-invariant. It is noted that for mn ¼0.61, 0.68 and 0.74 at Re ¼4000 (small-amplitude oscillation state, in-line mode), St remains close to 0.2, indicating that the cylinder wake is dominated by the classical von Ka´rma´n vortex street, similar to the case of stationary cylinder. All the five curves satisfactorily agree with one another and decrease monotonically with Re, from StE 0.18–0.2 at Re ¼4000, to St E0.12–0.13 at Re ¼12 000. A similar trend was reported by Carberry and Sheridan (2007) where St decreases from about 0.17 to 0.115 over the range Re ¼1000–7000. The following is an attempt to establish some relationship between the various frequency terms in the tethered cylinder system. Although Carberry and Sheridan (2007) showed that fN ofosc o fV0, there is no further information about relationship among them. Fig. 12 presents the curve for the frequency ratio fV0/fosc as a function of Re. It can be seen that fV0/fosc increases (or fosc/fV0 decreases) linearly with respect to Re at an almost identical slope for all mn cases considered. A linear fit of the curves indicates that a value of fV0/fosc ¼1 will be achieved when Re is slightly below 4000. This is consistent with the present observation that at Re ¼4000 the tethered cylinder exhibits very small-amplitude oscillations, and a further decrease in Re (and hence a smaller inertial force) would result in the cylinder oscillations to be negligibly small, or approaching the case of stationary cylinder (which can be considered as a limiting case of tethered cylinder).

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Fig. 13. Variation of the normalized frequency fn ¼fosc/fN as a function of reduced velocity Un. Open symbols are the data of Govardhan and Williamson (2000) on an elastically mounted cylinder at low mass and damping (mn ¼1.19, z ¼0.00502).

Reported studies on the elastically mounted cylinder have widely used the fn–Un plot to illustrate the frequency response, therefore, a similar plot is given (see Fig. 13) for the tethered cylinder, given the strong similarities between these two systems. Note that in the case of tethered cylinder, fn ¼fosc/fN and Un ¼U/fND. The straight line of St¼ 0.2 stands for the vortex shedding from a stationary cylinder, namely, f¼fV0. Obviously, the tethered cylinder does not exhibit the socalled ‘lock-in’ or ‘synchronization’ phenomenon that the cylinder oscillates at a frequency close to fN (or fn E1). Overall, all data points fall within the angular region prescribed by the lines of St ¼0.2 and fn ¼1, quantitatively validating the relationship fN ofosc ofV0. The values of fn, which are dependent on both Un and mn, depart remarkably from unity, lying in the range 2.5 ofn o4. A closer look at the curves shows that fn increases with Un over the low-Un range, but thereafter remains roughly constant irrespective of further increase in Un. This type of synchronization at a frequency different from fN is especially obvious for the three largest mn cases (mn ¼0.76, 0.84 and 0.92), where fn is approximately constant at about 3.9 beyond Un E20. A similar departure of fn from unity (fn E1.8) has been reported by Govardhan and Williamson (2000) on an elastically mounted cylinder at low mass and damping (mn ¼1.19, z ¼0.00502), as shown in Fig. 13. Williamson and Govardhan (2008) showed that such a departure is likely to be an intrinsic nature for VIVs of very light bodies, e.g., mn ¼O(1)–O(10). It is noted that the reduced velocity range of the present study is much higher than that of Govardhan and Williamson (2000), namely, 13oUn o32 versus 3oUn o18. However, each curve, especially when mn is large enough (namely, mn ¼0.76, 0.84 and 0.92), resembles that of the elastically mounted cylinder: fn increases almost linearly with respect to Un over the low-Un range, and then remains nearly constant when Un is high enough. This implies that the motion of the tethered cylinder under the conditions of large-mn and high-Un will approach that of an elastically mounted cylinder. This conclusion is valid intuitively, considering that under such conditions the mean layover angle of the tethered cylinder is large and the oscillations will be predominantly transverse to the free stream, similar to the case of the elastically mounted cylinder. 3.4. Flow patterns The mean flow quantities were obtained by ensemble averaging the 840 instantaneous velocity fields recorded for each experimental condition. Fig. 14 shows the contour plots of the normalized Reynolds shear stress, u0 v0 =U 2 , for both the stationary cylinder and the tethered cylinder (mn ¼0.61 and 0.68) at different Reynolds numbers. In the case of the stationary cylinder, the distributions of u0 v0 =U 2 are essentially symmetric about the cylinder centerline; as Re increases, the entire pattern moves slightly upstream towards the cylinder base. It is noteworthy that besides the two large-scale clusters of u0 v0 =U 2 , there are two additional small-scale clusters of oppositely signed vorticity located immediately upstream of the main shear layer. This feature agrees with the DNS and PIV results of Dong et al. (2006) on a cylinder at comparable Reynolds numbers (Re¼4000 and 10 000). For the tethered cylinder undergoing vibrations, on the other hand, this feature does not exist and only the small-cluster of negative u0 v0 =U 2 is visible at Re ¼6000 and 8000. It is clear that for the tethered cylinder, the region of significant u0 v0 =U 2 is larger in both the streamwise and transverse directions as compared to the stationary cylinder at the same Re, indicating that the cylinder wake increases in size due to the cylinder oscillations. An asymmetry in the wake flow is also observable for the case of the tethered cylinder. At relatively low Reynolds numbers (i.e., Rer8000, corresponding to the small-amplitude oscillation state), the upper shear layer (denoted by negative u0 v0 =U 2 ) is larger in size than the lower counterpart but with a roughly similar magnitude. However, for

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Fig. 14. Contour plot of the normalized Reynolds shear stress u0 v0 =U 2 of the stationary cylinder (left) and the tethered cylinder for mn ¼0.61 (middle) and mn ¼ 0.68 (right) at different Reynolds numbers. Positive: solid lines; negative: dashed lines. 9u0 v0 =U 2 9min ¼ 0:01 and Dðu0 v0 =U 2 Þ ¼ 0:01.

mn ¼0.61 and 0.68 at Re¼ 12 000 (in the large-amplitude oscillation state), the upper shear layer is significantly intensified, as indicated by a much higher magnitude of Reynolds stress. Also of note is that the upper shear layer is appreciably slanted upward, and the entire pattern of Reynolds stress moves substantially upstream to the cylinder base as compared to the other cases. The difference in the Reynolds stress distribution is due to the vortex shedding patterns, as shown below. Fig. 15(a)–(d) shows some representative snapshots of the instantaneous vorticity distribution (ozD/U) around the tethered cylinder for mn ¼0.61 at Re ¼4000, 6000, 8000 and 12 000. The mean layover angle for these four cases is y ¼ 20.81, 39.21, 511 and 69.31. As shown earlier, the oscillations could be in-line with or transverse to the free stream, depending on the value of y . Therefore, the four cases would correspond to different modes of oscillation, namely, Re ¼4000: in-line mode with very small amplitude ðymax =yD ¼ 0:023Þ; Re ¼6000: in-line mode with relatively small amplitude ðymax =yD ¼ 0:24Þ; Re ¼8000: transverse mode with relatively small amplitude ðymax =yD ¼ 0:33Þ; Re ¼12 000: transverse mode with large amplitude ðymax =yD ¼ 0:9Þ. Correspondingly, different wake vortex modes would be expected, as clearly depicted in these snapshots. At Re ¼4000 (Fig. 15(a)), it is essentially the classical von Ka´rma´n vortex street pattern. It is known as the 2S mode, with two single counter-rotating vortices alternately shed from the cylinder per shedding cycle. At Re ¼6000 (Fig. 15(b)) and 8000 (Fig. 15(c)), the vortex shedding is still the 2S mode, but the oscillations with magnitude of ymax =yD ¼ 0:220:3 do appreciably affect the vortex pattern depending on oscillation mode—in-line or transverse. At Re ¼6000, the vortices shed from opposite sides of the cylinder are essentially symmetric and a twin vortex street is formed in the near wake (about 1–2 diameters downstream of the cylinder trailing edge) due to the in-line oscillations. At Re ¼8000, on the contrary, the oscillations are transverse to the free stream, resulting in a significantly wider distribution of vorticity (or a wider wake as

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Fig. 15. A representative snapshot of the normalized instantaneous spanwise vorticity ozD/U for mn ¼0.61 at (a) Re ¼ 4000; (b) 6000; (c) 8000; and (d) 12 000. The cylinder is at the top of its cycle, or at the most anticlockwise position. Right column shows a time series of consecutive snapshots following that of (d) with an interval of 1/15 s, namely, at (e) t¼ 1/15 s; (f) 2/15 s; (g) 3/15 s; and (h) 4/15 s. The cylinder’s trajectory and the corresponding position at each instant are highlighted in the inset graph (not drawn to scale). Positive: solid lines; negative: dashed lines. 9oz D=U9min ¼ 1:0 and D(ozD/U)¼ 1.0.

shown in Fig. 14) than that of Re ¼4000 and 6000. At Re ¼12 000 that belongs to the large-amplitude state, however, the wake is significantly asymmetric about the cylinder centerline with a distinct upward angle, as shown in Fig. 15(d). Considering that the vortex shedding is periodic and reproducible, a time series of instantaneous flow fields can be considered to be a good representation of the dynamic process. As such, four subsequent snapshots following that of Fig. 15(d) are given in Fig. 15(e)–(h) at about quarter-cycle intervals. At t ¼0 (Fig. 15(d)), the cylinder is at its most anticlockwise position and the wake is characterized by a pair of counterrotating vortices (denoted as A and B). The subsequent four snapshots (Fig. 15(e)–(h)) clearly illustrate a process that with vortices A and B gradually propagating outside the field of view, a new pair of vortices (denoted as A0 and B0 ) roll up at 1/15 s (Fig. 15(e)), shed into discrete vortices and subsequently move downstream (Fig. 15(f)–(h)). Two main observations can be made herein. The first is that the vortex shedding mode is still 2S, although there are some small regions with

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concentrated vorticity (which are more likely due to the stretching and broken-down of vortices). But it is a modified 2S mode, with the vortices shed at an upward angle with respect to the free stream. This is consistent with the observation in Fig. 10(b) that the time history of either CD or CL shows a modulated sine with a flat lower half, rather than the simple sinusoidal variation about the mean value. This oblique 2S mode of vortex shedding in large-amplitude oscillation state is different from the 2P mode (with two counter-rotating vortex pairs per oscillation cycle) as proposed by Carberry and Sheridan (2007), but is in accordance with the numerical results of Ryan et al. (2007), who attributed the negative mean lift to the asymmetry in vortex shedding. Another observation is that at t ¼4/15 s (Fig. 15(h)), the cylinder has just returned to the most anticlockwise position—the same as in Fig. 15(d), indicating that this set of data represents a full cycle of oscillation. In the meantime, the vortex shedding experiences a full cycle as well, since the positions of A and A0 (as well as those of B and B0 ) are almost exactly the same in Fig. 15(d) and (h). In other words, the vortex shedding frequency is synchronized with the oscillations of the cylinder, i.e., fV fosc, as discussed in Section 3.3.

4. Conclusions This paper presents the findings of an experimental study on the vortex-induced vibrations (VIVs) of a buoyant tethered cylinder in uniform flow over a range of mass ratios (mn ¼0.61–0.92) and Reynolds numbers (Re ¼4000–12 000). The tether length ratio is fixed at Ln ¼5. For most cases considered in the parameter space, the cylinder oscillates in the smallamplitude state with the peak angular amplitude ymax =yD o0:4. However, the results illustrate the existence of a critical mass ratio (mncrit ) below which the cylinder ‘jumps’ to large-amplitude oscillations at sufficiently high Re. The peak amplitude ymax =yD ¼ 0:520:9 (or 1.0–1.8 cylinder diameters peak-to-peak) is observed for the two smallest mn cases (mn ¼0.61 and 0.68) at the highest Reynolds number Re¼ 12 000. The value of mncrit is about 0.7, which is in satisfactory agreement with that of mncrit ¼ 0:74 reported by Carberry and Sheridan (2007) on a tethered cylinder with a similar tether length ratio. The mean and r.m.s. drag/lift coefficients (C D , C 0D and C 0L ) also exhibit clear distinction between the small- and large-amplitude states, with remarkably higher values in the latter. When plotted against the buoyancy Froude number Fr0 , the mean layover angle ðy Þ data for all the cases collapse into a well-defined curve regardless of the oscillation state. Similarly, the mean drag coefficients ðC D Þ can be fitted by a single exponential curve, but is only valid for the smallamplitude state. The frequency response is also analyzed to establish the relationship among the various frequency terms associated with a tethered cylinder system, namely, the vortex shedding frequency (fV), the oscillation frequency of cylinder (fosc), and the structural natural frequency (fN). The vortex shedding frequency is at all times synchronized with the oscillations of the cylinder, i.e., fV fosc. This type of synchronization at a frequency different from the natural frequency fN is in qualitative agreement with the finding of Govardhan and Williamson (2000) on an elastically mounted cylinder at low mass and damping. The values of fosc are about 2.5–4 times fN, but are always lower than the shedding frequency for the stationary cylinder fV0. The results illustrate appreciable difference in the vortex shedding mode from the tethered cylinder depending on the cylinder’s oscillation characteristics. In the small-amplitude state, it is basically the 2S mode. Nevertheless, the oscillation amplitude (ymax =yD can be varied from O(1) to O(10) percent) and oscillation mode (which can be either in-line with or transverse to the free stream depending on the mean layover angle), do appreciably affect the vortex shedding process, resulting in the somewhat modified 2S mode as compared to the classical von Ka´rma´n vortex patterns. In the largeamplitude state, vortex shedding is still in 2S mode, but is highly asymmetric about the cylinder centerline. The shed vortices travel downstream at a significant upward angle, resulting in that the upper shear layer is much stronger than its lower counterpart, as quantitatively depicted by the distributions of normalized Reynolds shear stress. References Bearman, P.W., 1984. Vortex shedding from oscillating bluff bodies. Annual Review of Fluid Mechanics 16, 195–222. Bearman, P.W., 2011. Circular cylinder wakes and vortex-induced vibrations. Journal of Fluids and Structures 27, 648–658. Carberry, J., Sheridan, J., 2007. Wake states of a tethered cylinder. Journal of Fluid Mechanics 592, 1–21. Dong, S., Karniadakis, G.E., 2005. DNS of flow past a stationary and oscillating cylinder at Re ¼ 10 000. Journal of Fluids and Structures 20, 519–531. Dong, S., Karniadakis, G.E., Ekmekci, A., Rockwell, D., 2006. 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