Vortex-induced vibrations of a neutrally buoyant circular cylinder near a plane wall

Vortex-induced vibrations of a neutrally buoyant circular cylinder near a plane wall

Journal of Fluids and Structures 39 (2013) 188–204 Contents lists available at SciVerse ScienceDirect Journal of Fluids and Structures journal homep...

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Journal of Fluids and Structures 39 (2013) 188–204

Contents lists available at SciVerse ScienceDirect

Journal of Fluids and Structures journal homepage: www.elsevier.com/locate/jfs

Vortex-induced vibrations of a neutrally buoyant circular cylinder near a plane wall X.K. Wang a,n, Z. Hao b, S.K. Tan c a

Maritime Research Centre, Nanyang Technological University, Singapore 639798, Singapore College of Logistics Engineering, Shanghai Maritime University, Shanghai 201306, China Nanyang Environment and Water Research Institute, School of Civil and Environmental Engineering, Nanyang Technological University, Singapore 639798, Singapore b c

a r t i c l e in f o

abstract

Article history: Received 31 December 2011 Accepted 11 February 2013 Available online 22 March 2013

This paper presents an experimental study of the motions, drag force and vortex shedding patterns of an elastically mounted circular cylinder, which is held at various heights above a plane wall and is subject to vortex-induced vibration (VIV) in the transverse direction. The cylinder is neutrally buoyant with a mass ratio mn ¼ 1:0 and has a low damping ratio z ¼ 0.0173. Effects of the gap ratio (S/D) ranged from 0.05 to 2.5 and the free-stream velocity (U) ranged from 0.15 to 0.65 m/s (corresponding to 3000 r Re r 13 000, and 1.53 r U n r 6.62) are examined. The flow around the cylinder has been measured using particle image velocimetry (PIV), in conjunction with direct measurements of the dynamic drag force on the cylinder using a piezoelectric load cell. Results of the vibrating cylinder under unbounded (or free-standing) condition, as well as those of a near-wall stationary cylinder at the same gap ratios, are also provided. For the free-standing cylinder, the transition from the initial branch to the upper branch is characterized by a switch of vortex pattern from the classical 2S mode to the newlydiscovered 2PO mode by Morse and Williamson (2009). The nearby wall not only affects the amplitude and frequency of vibration, but also leads to non-linearities in the cylinder response as evidenced by the presence of super-harmonics in the drag force spectrum. In contrast to the case of a stationary cylinder that vortex shedding is suppressed below a critical gap ratio (S/DE 0.3), the elastically mounted cylinder always vibrates even at the smallest gap ratio S/D ¼0.05. Due to the proximity of the plane wall, the vortices shed from the vibrating cylinder that would otherwise be in a double-sided vortex street pattern (either 2S or 2PO mode) under free-standing condition are arranged into a singlesided pattern. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Vortex-induced vibration (VIV) Vortex shedding Near-wall cylinder Low mass-damping Neutrally buoyant

1. Introduction Vortex-induced vibration (VIV) of cylindrical structures is a challenge in many branches of engineering, for example marine pipelines, offshore risers, bridge piers, and so on. The practical significance of cylinder VIV has received much attention over the past several decades, see for example the reviews by Sarpkaya (2004), Williamson and Govardhan (2004, 2008) and Bearman (2011) regarding the progress, current state and debate, as well as some unresolved problems.

n

Corresponding author. Tel.: þ 65 67906619; fax: þ65 67906620. E-mail address: [email protected] (X.K. Wang).

0889-9746/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfluidstructs.2013.02.012

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Most previous studies are focused on the paradigm of a freely vibrating, elastically mounted rigid cylinder placed in uniform cross-flow under unbounded (or free-standing) condition. The principal dynamics giving rise to VIV of a cylinder, which could be a combination of transverse and in-line vibrations (or oscillations), are the spanwise von Ka´rma´n vortex pairs alternately shed from two sides of the cylinder. The resulting unsteady force on the cylinder can lead to ‘lock-in’ (or ‘synchronization’) phenomenon that is characterized by an amplification of cylinder’s vibration amplitude, which can be up to the order of the cylinder diameter. It has been shown that the VIV characteristics depend not only on the combined parameter—the mass–damping ratio (mn z, where the mass ratio mn is defined as the ratio of cylinder mass to the displaced mass of fluid, and z is the ratio of damping coefficient to critical damping coefficient), but also on mn and z individually. Khalak and Williamson (1999) found that the cylinder response, depending on the magnitude of mn z, may be of two different types. For low mn z, the response curve composes three distinct branches when plotted against the reduced velocity (U n ¼ U=f N D, where U is the free-stream velocity, f N is the natural structural frequency in still fluid and D is the cylinder diameter), namely: the ‘initial’, ‘upper’ and ‘lower’ branches. For high mn z, however, the upper branch does not exist at all. The classical definition n of lock-in, namely the cylinder’s vibration frequency (f osc ) is close to the natural frequency (f N ), or f ( ¼ f osc =f N )E1.0, is n valid only for heavy structures with m ¼O(100), since most of the early experiments were conducted in air. On the other hand, most of recent research studies use water as the working medium (due to practical applications in marine n engineering), resulting in much smaller mass ratios mn ¼O(10)–O(1). In this case, f may reach much higher values than unity. Therefore, Khalak and Williamson (1999) suggested that for light structures, the matching between the vortex shedding frequency (f V ) and the body’s vibration frequency (f osc ) is a more suitable definition of lock-in. The distinct branches in the response curve may be associated with different vortex shedding modes in the wake of the vibrating cylinder, for example the ‘2S’ mode (two single vortices per cycle, i.e., the familiar von Ka´rma´n vortex street), ‘2P’ mode (two pairs of vortices per cycle), and asymmetric ‘PþS’ mode (a pair of vortices and a single vortex per cycle), as originally proposed by Williamson and Roshko (1988). Recently, Morse and Williamson (2009) reported results of controlled vibration experiments of a cylinder with extremely fine resolution at two constant Reynolds numbers, Re ¼4000 and 12 000. Besides the PþS, 2S and 2P modes, a new mode responsible for peak amplitude vibration has been identified, n which is defined as the ‘‘2POVERLAP’’ or ‘‘2PO’’ mode because its regime in the normalized amplitude–wavelength (An l ) plane overlaps with other regions. This mode comprises two pairs of vortices in each cycle (similar to 2P mode), but the second vortex of each pair is distinctly smaller than the first vortex. The present study considers the case that the cylinder is placed near a plane boundary – a flow configuration widely used in engineering practice as schematically shown in Fig. 1 – for example a submarine pipeline near seabed and cablelaying. VIV has long been recognized as one of the main causes of structural fatigue damage for this type of application (Bearman and Zdravkovich, 1978), and may also induce dynamic coupling with soil scour beneath a pipeline (e.g., Chiew, 1990; Yang et al., 2008). As compared to the case of a free-standing cylinder, the flow around the cylinder in proximity to a plane wall becomes more complex, since it involves the development of three shear layers, i.e., the two separated from the upper and lower sides of the cylinder, as well as the wall boundary layer. According to the previous findings on a near-wall stationary (fixed) cylinder (e.g., Lei et al., 1999; Wang and Tan, 2008), the flow depends mainly on three parameters: the Reynolds number (Re ¼UD/v, where v is the kinematic viscosity of fluid), the boundary layer thickness d , and the height of gap S between the cylinder and the wall. Previous studies, mostly conducted in the subcritical regime (Re ¼103–105), have shown that the gap height to cylinder diameter ratio (S/D, abbreviated hereafter as the gap ratio) is the predominant parameter. When the cylinder is sufficiently close to the wall, the lower shear layer is interfered with the wall boundary layer and thus its vorticity strength becomes weak. Therefore, the periodic vortex shedding would be suppressed below a critical gap ratio (about S/D¼ 0.3 according to previous studies).

Fig. 1. Side view of the near-wall cylinder that is elastically mounted and subject to VIV in the transverse direction.

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However, when a cylinder is elastically mounted in proximity to a plane wall, the question arises whether the structural vibration can be suppressed, in a way similar to that of vortex shedding from a near-wall stationary cylinder. To date, there are only a handful of reported (experimental) studies on the VIV phenomena of a near-wall cylinder, e.g., Jacobsen et al. (1984), Fredsøe et al. (1985) and Yang et al. (2009). These studies showed that the nearby wall greatly affects the cylinder response. Jacobsen et al. (1984) reported that the cylinder still vibrates when placed very near a plane wall, although no regular vortices are shed in the wake. The influence of a nearby wall (S/D ¼0–1.0) was shown by Fredsøe et al. (1985) that the vibration frequency of the cylinder undergoing VIV is noticeably different from the vortex shedding frequency for a stationary cylinder. Recently, Yang et al. (2009) investigated this flow configuration based on velocity measurement using hot film and flow visualization using hydrogen bubbles. It was revealed that the cylinder response depends on many parameters, including reduced velocity, gap ratio, stability parameter and mass ratio. The above review indicates that there is still a dearth of detailed information on a near-wall cylinder subject to VIV, particularly the important vortex dynamics leading to the structural response, because the problem of VIV is ‘‘a fascinating feedback between body motion and vortex motion’’ (Williamson and Govardhan, 2008). Therefore, the present study employed digital particle image velocimetry (PIV) as the measurement technique, in order to capture the instantaneous, whole-field velocity and vorticity distributions of the flow. In addition, the dynamic drag force on the cylinder was measured directly with a piezoelectric load cell, based on which the mean and root-mean-square (r.m.s.) drag coefficients could be obtained. It is envisaged that this study would provide further insight into the underlying physics for the VIV problem of a near-wall cylinder, as well as benchmark data for validating computational fluid dynamics (CFD) code, since in recent years numerical simulations become more feasible with the development of numerical methods and the advance of computational power. For example, by using unsteady Reynolds-averaged Navier–Stokes (URANS) simulation and detached-eddy simulation (DES), Nishino et al. (2008) simulated the flow around a stationary cylinder placed close to a moving wall, and successfully predicted the suppression of vortex shedding and the resultant critical drag reduction of the cylinder due to the proximity of the wall. Zhao and Cheng (2011) numerically studied the effects of a nearby plane wall on the VIV characteristics of a cylinder under two gap ratios (i.e., S/D ¼0.002 and 0.3). It was shown that the cylinder may be bounced back when reaching the plane boundary, and the amplitude of vibration depends on the bouncing coefficient. It is noteworthy that both studies were carried out at Re ¼103–104 (in the subcritical regime), which is about the same order of magnitude as achievable in typical laboratory experiments. The proximity of the wall imposes a constraint on the amplitude of vibration for the elastically mounted cylinder, which is otherwise a linear support system under free-standing condition. Zhao and Cheng (2011) showed that the cylinder’s bouncing process is associated with corresponding change in vortex shedding patterns. More recently, Bourdier and Chaplin (2012) experimentally examined the effects of restricting the displacement of a low mass–damping cylinder undergoing VIV by means of stiff mechanical end-stops (which is somewhat similar to the constraint imposed by the plane wall in the present study). It was shown that the cylinder response exhibits some degree of chaos due to the constraints imposed by the end-stops. Fig. 1 shows the sketch of the elastically mounted, near-wall cylinder that is free to vibrate in the transverse direction (perpendicular to the free stream). During the experiments, the cylinder has been kept neutrally buoyant (mn ¼1.0). The effects of varying both the gap ratio (S/D) and the free-stream velocity (U) on the cylinder’s VIV characteristics are examined in terms of response amplitude and frequency of vibration, drag force, Strouhal number and pertinent vortex dynamics.

2. Experimental set-up and methodology The experiments were performed in the re-circulating open channel located at Maritime Research Centre, Nanyang Technological University, with a test section of 5 m  0.3 m  0.45 m (length  width  height). The channel floor and the two side walls of the test section were made of glass to ensure 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 from 0.02 m/s up to 0.7 m/s. The free-stream velocity was uniform to within 1.5% across the test section, and the turbulence intensity in the free stream was below 2%. The objective of the experimental design is to replicate, as far as practical, an ideal 2-dimensional (2D) rigid cylinder that is elastically mounted and deeply submerged at various heights above a plane wall and displaces perpendicular to the free stream, meanwhile allowing for simultaneous force and velocity measurements. Fig. 2 shows the cylinder model and the schematic diagram of the experimental set-up. During the course of experiments, the water depth in the test section of the channel was maintained at 220 mm (measured from the channel floor). The test cylinder was made of smooth, hollow, acrylic tube with an outer diameter of D ¼20 mm. It was mounted in the test section with its axis horizontal and aligned in the spanwise direction. Each end of the cylinder was attached to an acrylic rod 2 mm in diameter that could slide in (but never touch) the pre-manufactured slot on each vertical acrylic plate (10 mm in thickness). The rods were fixed on one end of the oscillating beam, which was made of 2 mm thick acrylic plate, 4 mm in height and 200 mm long. The cylinder, the rod and the beam were all rigid and there was no relative motion between them. The cylinder assembly was carefully machined to ensure it to be neutrally buoyant in water (i.e., mn ¼1.0). The other end of the beam was held at the pivot point with precision and thus the cylinder is free to rotate about the pivot point.

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Fig. 2. Sketch of the experimental set-up and apparatus.

Table 1 Cases conducted on the free-standing cylinder in this study. U (m/s) Re (  103) Un

0.15 3 1.53

0.2 4 2.04

0.25 5 2.55

0.3 6 3.05

0.35 7 3.56

0.4 8 4.07

0.45 9 4.58

0.5 10 5.1

0.55 11 5.6

0.65 13 6.62

The experimental set-up was designed to reduce as far as possible all sources of damping except for that on the cylinder itself. The relatively small size of the oscillating beam’s cross section (2 mm  4 mm) was aimed to minimize the additional damping during rotation. On the other hand, the relatively long oscillating beam resulted in that the motion of the cylinder was almost vertical. For a vertical displacement of one diameter (20 mm), the horizontal displacement of the cylinder was only about 1 mm. The plane wall, made of smooth acrylic plate (1.2 m long, 10 mm thick and spanned the channel width) and placed 40 mm above the channel floor had a sharpened leading edge, thus allowing for the flow to be separated and re-developed into a wall boundary layer. The cylinder was located at 500 mm (or 25D) downstream of the leading edge (as shown in Fig. 1), where the boundary layer was fully developed with a thickness of d E0.4D when the cylinder was absent. The gap height S was varied from 1 mm to 50 mm, yielding a range of gap ratios S/D ¼0.05–2.5. It should be noted that when a cylinder is inserted into a wall boundary layer, there is a positive mean lift force acting on the cylinder that tends to push it away from the wall (Nishino et al., 2007). Therefore, the gap height was the timeaveraged value measured in situ under flowing condition, instead of in still water. A pair of springs at each end of the cylinder provided the restoration force in the transverse (y) direction. Through free decay tests in still water, the natural frequency of the system was determined to be fN ¼4.91 Hz, and the damping ratio z ¼0.0173. The span of the cylinder (L) was 280 mm, resulting in an aspect ratio (L/D) of 14. This aspect ratio was considered large enough to ensure a nominally 2D flow in the near wake of the cylinder along a majority of the span. Thus, the PIV measurements were carried out in the mid-span plane of the cylinder, as shown in Fig. 2. The origin of the coordinate system is set at the intersection between the cylinder’s vertical centerline and the wall, as shown in Fig. 1. The coordinates, x, y and z, denote the streamwise, transverse and spanwise directions, respectively. Two sets of experiments were conducted. The first set aimed to examine the effects of varying free-stream velocity on the cylinder dynamics under free-standing condition. Prior studies on the near-wall cylinder indicated that the effects from the wall or the free surface become negligible when the separation distance is larger than 2–2.5D. As such, the cylinder was located approximately halfway between the free surface and the plane wall (the water depth above the plane wall was 170 mm), so the distance from the cylinder to either boundary was 3.75D. In this case, the coordinate system is originated from the centre of the cylinder. The free-stream velocity was varied between U ¼0.15 and 0.65 m/s, resulting in Re ¼3000–13 000 and U n ¼ 1:5326:62, as listed in Table 1. In the second set of experiments, referred to as the near-wall case, the cylinder was placed near the wall with varying gap heights (S/D ¼0.05–2.5) under two constant free-stream velocities: U ¼0.3 and 0.5 m/s, corresponding to Re¼ 6000 and 10 000, or U n ¼ 3:05 and 5.1, respectively. Velocity measurements were performed using a digital 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

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Table 2 Key parameters used in this paper. Parameter name

Symbol

Amplitude ratio Drag coefficient

A CD

Cylinder diameter Natural frequency Vibration (oscillation) frequency Vortex shedding frequency Frequency ratio

D fN f osc fV

Mass ratio

n

n

f mn

Reynolds number Gap ratio Strouhal number Reduced velocity Damping ratio

Re S/D St Un

Normalized wave length

ln

z

Definition

Value

A=D

Dimensionless Dimensionless

2F D =ðrU 2 DLÞ 20 4.91

f osc =f N 4m=ðprD2 LÞ UD=n f D=U U=ðf N DÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c=ð2 kðm þ mA ÞÞ l=D ¼ U=f osc D

Units

mm Hz Hz Hz Dimensionless

1.0

Dimensionless

3000–13 000 0.05–2.5

Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless

1.53–6.62 0.0173

Dimensionless

 5 ns). Sphericels 110P8 hollow glass spheres (neutrally buoyant with a mean diameter of 13 mm) were seeded in the flow as tracer particles, which offered good traceability and scattering efficiency. The particle images were recorded using a 12-bit charge-coupled device (CCD) camera, which had a resolution of 1600  1200 pixels. The sampling rate of the PIV measurements was 15 Hz, which was well beyond double the vortex shedding frequency (its maximum value was about 7 Hz), therefore, satisfying the Nyquist sampling theorem. LaVision Davis software (Version 7.2) was used to process the particle images and determine the velocity vectors. Particle displacement was calculated using the fast-Fourier-transform (FFT) based cross-correlation algorithm with 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 centre vector (surrounded by the eight interrogation windows) differed from the median vector by more than five times the root-mean-square (r.m.s.) value, the centre vector was replaced by the averaged vector obtained from the neighboring interrogation windows. 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 set-up was 1.9 mm  1.9 mm (i.e., 0.095D  0.095D). For each case, a series of 1260 instantaneous flow fields were acquired at the sampling frequency of 15 Hz (or 84 s recordings). It should be noted that all the PIV recordings were performed after the cylinder had reached the state of temporally periodic vibrations. The uncertainty in the instantaneous velocities (u and v) was estimated to be about 2% for the present set-up. Based on the velocity vector distribution, the instantaneous spanwise vorticity (oz ¼ Dv=DxDu=Dy) was calculated using the least squares extrapolation scheme. The uncertainty in oz was about 10%. By analyzing the velocity data obtained by PIV, some important parameters of the flow, such as the vortex shedding frequency (f V ), can be obtained. We first looked at a time series of PIV snapshots to confirm whether discrete vortices were indeed shed periodically in the wake. Based on the PIV snapshots, we found that the location (x,y)¼(4D,S þ0:5D) was a good point for detection of vortex shedding. In the case of the free-standing cylinder, it was also located on the cylinder centerline with the same downstream distance of 4D. Then, the transverse velocity at that particular location was retrieved for spectral analysis using FFT. The motions of the cylinder displacement were recorded by a Canon video camera at a sampling rate of 25 Hz. The normalized amplitude (An ), defined as the peak displacement away from its mean position (A) normalized by the cylinder diameter (D), was calculated by taking the average of 10% highest peaks recorded for each case. The duration for the displacement measurements was 200 s, which covered at least 300 cycles of vibration and was thus sufficiently long to get a reliable statistics. In addition, the dynamic drag force (F D ) on the cylinder (integrated over the whole span of the cylinder) was measured using a piezoelectric load cell (Kistler Model 9317B). The amplified output was captured with a National Instruments data acquisition card at a sampling rate of 100 Hz (which was at least 1 order higher than the vortex shedding frequency). Then, the mean and r.m.s. values of the drag coefficients were calculated. Through a number of repeated measurements of a stationary cylinder, the uncertainty in the mean drag coefficient (C D ) was determined to be within 2%. Some pertinent parameters used in this paper are listed in Table 2.

3. Results and discussion 3.1. Free-standing cylinder: cylinder response, drag force and Strouhal number Fig. 3 shows the response amplitude and frequency of the free-standing cylinder as a function of reduced velocity U n ¼ U=f N D. The data reported by Govardhan and Williamson (2000) on a similarly low mass-damping cylinder (i.e., mn ¼ 1:19 and

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Fig. 3. VIV response of the free-standing cylinder (mn ¼ 1:0, z ¼ 0.0173) as a function of reduce velocity, together with that reported by Govardhan and Williamson (2000) on a free-standing cylinder (mn ¼ 1:19, z ¼ 0.00502): (a) amplitude and (b) frequency. Also included in the frequency response plot are the results of Yang et al. (2009) for an elastically mounted cylinder (mn ¼ 3:87, z ¼0.0152) placed far away from a plane wall (S/D ¼ 4.9).

z ¼0.00502 versus mn ¼ 1:0 and z ¼0.0173 in the present study) are also included for comparison. Fig. 3(a) illustrates that the two sets of data collapse perfectly, with the present data points belonging to the initial branch (3.05rU n r4.58) and the upper branch (5.1rU n r6.62) in the response curve. The value of the reduced velocity to distinguish the initial and upper branches is U n E5 (which is consistent with reported findings), as indicated by the sharp jump in amplitude of vibration, namely, from An ¼ 0:53 at U n E4.58 to An ¼0.84 at U n E5.1. The maximum amplitude is An max E1.0, which agrees well with that reported by Govardhan and Williamson (2000). The frequency response data are shown in Fig. 3(b), which are normalized by the natural n frequency of the system, i.e., f ¼ f osc =f N . The thick straight line f osc ¼ f St (or St¼0.19) represents the vortex shedding from a n stationary cylinder. Except for U n ¼3.05 (the lowest reduced velocity in the vibrating state), the values of f in the present study are slightly lower (variation within 10%) than those in Govardhan and Williamson (2000), but the overall trend of the two sets of n data is in good agreement. In particular, f increases with U n over the measurement range (almost linearly with a slope of n St¼0.167), and passes across the line of f osc ¼ f N (f ¼ 1). Owing to the limitations in the top speed of the water channel, we were n unable to predict its behavior at even higher U . For example, it was not possible to ascertain whether there exists the lower n branch as shown in Govardhan and Williamson (2000), which occurs at U n ¼ 1118 with a nearly constant frequency f E1.8. However, it can be confirmed that the cylinder will vibrate at a frequency other than f N if U n is further increased. This is consistent with the conclusion of Govardhan and Williamson (2000) on light structures. It is also noteworthy that when the velocity is very low (U n r2.55), the cylinder is essentially stationary (An E0). At n U n ¼3.05 that corresponds to the beginning of the initial branch, f deviates significantly from the f osc ¼ f St line and can be

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as high as 0.83. In other words, f osc approaches f N . A similar phenomenon was reported in Yang et al. (2009) over the lowU n range (2.6oU n o 4), as shown in Fig. 3(b). This suggests that the initial branch may be initiated with small-amplitude vibrations at a frequency close to the natural frequency of the system. In the present study, U n is varied by means of variation in free-stream velocity (not the spring stiffness or the cylinder diameter), which in turn induces a change in Re. Therefore, the drag coefficients and the Strouhal numbers are plotted in a double x-axis form, but our main discussion is presented with reference to U n , which is considered to be a more important 0 VIV-controlling parameter than Re. The mean and r.m.s. drag coefficients (C D and C D ) on the free-standing cylinder are 0 n presented in Fig. 4(a). The variation trend of either C D or C D with U is similar to that of An . As compared to a stationary cylinder, the cylinder vibration leads to significant amplification in the drag coefficients, with C D increasing from 1.05 to 0 2.5 and C D from 0.04 to 0.6. This is in accordance with Sarpkaya (2004) that the mean drag on a vibrating cylinder at n A ¼ 1:0 (similar to the present study) is about three times that of a stationary cylinder (the universally accepted value of C D for a stationary cylinder is about 1.0 in the subcritical regime). The Strouhal numbers (St ¼ f D=U) for the vortex shedding frequency (f V ) and the vibration frequency (f osc ) are presented in Fig. 4(b). In this study, f V is obtained by spectral analysis of a time history of signal (either the transverse velocity at a particular location in the wake or the drag force on the cylinder) with FFT. Due to the relatively long period of sampling (which is 84 s for PIV measurement and 200 s for force measurement), the discrepancy in the calculated values using these two methods is very small (variation within 5%). It should be noted that in some cases, such as when the freestream velocity is very small, there are no obvious peaks in the force spectrum. Under those circumstances, f V is obtained

Fig. 4. Plot of measurement results of the free-standing cylinder in a double x-axis form (lower: Re; upper: U n ): (a) mean and r.m.s. drag coefficients 0 (C D and C D ) and (b) Strouhal number (St).

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solely based on the velocity data. It is clearly shown that in the stationary state (Reo 6000 or U n o3.05, at which no obvious motion is found for the cylinder), f V remains essentially constant with St E0.187, which is in good agreement with the previously reported values of St¼ 0.18–0.21 in the literature for von Ka´rma´n vortex shedding. With the increase in U n , the Strouhal number for f V decreases until U n ¼ 4.58, and thereafter remains nearly constant at St ¼0.167 for U n Z5.1 (in the upper branch). The vibration frequency of the cylinder (f osc ) is observed to be considerably higher than the vortex shedding frequency (f V ) at the beginning of the initial branch (U n ¼ 3:05), where St for f osc can be as high as 0.27. Beyond that, however, the two frequencies always approximately match (or lock-in) with each other, i.e., f osc  f V . This kind of lock-in is, however, different from the classical definition that f osc is close to f N . This is a consistent phenomenon for very ‘‘light’’ structures, see the review by Williamson and Govardhan (2004). 3.2. Near-wall cylinder at two constant velocities: effects of gap ratio This section aims to investigate the effects of the proximity of the plane wall on the cylinder response under two purposefully selected free-stream velocities, U ¼0.3 and 0.5 m/s. The resultant reduced velocities are U n ¼3.05 and 5.1, which correspond to the beginning of the initial and upper branches, respectively (as shown in Fig. 3). Fig. 5 displays the time histories of the normalized displacement of the cylinder under different flow conditions. It is obvious that the cylinder motion is strongly dependent on both U n and S/D in terms of the maximum amplitude, predominant frequency and periodicity of vibration. In the case of U n ¼ 3:05 (Fig. 5(a)), there exists an optimum gap ratio around S/D E0.4, at which the motion is more regular with a greater amplitude than the rest gap ratios. At very large gap ratios (e.g., S/D Z1.5 when the wall effects are negligible), the displacement patterns become irregular with a much smaller amplitude (An E 0.06). In the case of U n ¼ 5:1 (Fig. 5(b)), on the other hand, both the periodicity and the amplitude of vibration increase monotonically with S/D until S/D E1.5, and thereafter the displacement is nearly sinusoidal with constant frequency and amplitude. In both cases, the predominant frequency of vibration decreases with S/D over the small-S/D range (about S/D o0.5), and then remains constant with further increase in S/D. Moreover, when S/D is very small (S/D¼ 0.05–0.15), the lower side of the cylinder will hit the wall below and be bounced back. Similar observation was reported by Yang et al. (2009) and Zhao and Cheng (2011) on a near-wall cylinder at S/D ¼0.06. It is evident that the cylinder vibrates even at the smallest gap ratios (S/D ¼0.05), which is in contrast to the case of a near-wall stationary cylinder where periodic vortex shedding is suppressed below a critical gap ratio (S/D E0.3). The motion of the cylinder exhibits different characteristics depending on the values of U n and S/D. When S/D is very small, the displacement patterns generally become more complex due to the presence of the plane wall. In the case of U n ¼ 3:05 and S/D¼0.1, the cylinder hits the wall at every cycle of vibration, leading to appreciable increase in displacements in the opposite

Fig. 5. Time histories of displacement at different gap ratios for the near-wall cylinder undergoing VIV at: (a) U n ¼ 3:05 and (b) 5.1.

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Fig. 6. Time histories of drag coefficient (C D ) and corresponding spectrum at: (a) U n ¼ 5:1 and S/D ¼0.35; (b) U n ¼ 3:05 and S/D ¼0.3.

(larger-y) direction. At U n ¼ 5:1, however, the bouncing phenomenon occurs once every 1–2 cycles for S/D¼0.15, and 2–3 cycles for S/D¼0.35; in between two bouncing cycles, the amplitude in the lower direction is obviously smaller. Similar phenomena have recently been reported by Bourdier and Chaplin (2012), who studied the effects of restricting the displacement of a vibrating cylinder by means of stiff mechanical end-stops, which is in some way similar to the constraint imposed by the plane wall in the present study. They showed that the cylinder response (which is otherwise a linear elastic system under free-standing condition) exhibits some degree of chaos when the motion of the cylinder is more and more confined. Similarly, the response of the nearwall cylinder exhibits some type of non-linearities, as shown in Fig. 6 for time traces of the drag force and the corresponding power spectra for two selected cases, namely: U n ¼ 5:1 and S/D¼0.35; U n ¼ 3:05 and S/D¼0.3. The sharpness of the peaks in the spectrum clearly indicates that drag force is indeed integer multiples (up to four or five times, or super-harmonics) of the dominant vortex shedding frequency (about 5.6 Hz and 4.2 Hz, respectively, for these two cases). These super-harmonics, probably due to the synchronized vortex shedding from the multiple shear layers as will be shown later in the PIV results, cause the cylinder motion to be not sinusoidal. Fig. 7 presents the response amplitude, Strouhal number and drag coefficients of the near-wall cylinder as a function of S/D under a constant velocity U n ¼ 3:05. Fig. 7(a) shows the existence of a narrow band 0.05 rS=D r0.75, at which the vibration amplitude is considerably higher than the asymptotic value of An E0.06 at large enough gap ratios (S/D Z1.0). The maximum amplitude, Anmax ¼ 0:38, is attained at S/D ¼0.4. The results at this reduced velocity (corresponding to the beginning of the initial branch) show that the wall proximity actually promotes the cylinder VIV, rather than suppressing it as may be expected. The calculated Strouhal numbers for the cylinder vibration frequency (f osc ), as well as the vortex shedding frequencies from the vibrating (f V ) and stationary (f V 0 ) cylinders, are plotted in Fig. 7(b). In the case of the stationary cylinder, the values of St for f V 0 remain approximately constant at about 0.187 over the range 0.3oS/D r2.5. When S/D is very small (S/D r0.3), vortex shedding is suppressed and hence there are no data points in this range. In the case of the vibrating cylinder, on the contrary, the cylinder always vibrates irrespective of the values of S/D. The maximum value of St for f osc is about 0.3, taking place at the smallest gap ratio (S/D ¼0.05). Then it monotonically decreases with S/D until S/D ¼0.5, and thereafter asymptotes to about St¼ 0.27 (equal to that of the free-standing cylinder at U n ¼ 3:05 as shown in Fig. 4(b)). The vortex shedding frequency from the vibrating cylinder (f V ) is seen to synchronize with the vibration frequency (f osc ), i.e., f V ¼ f osc , within the range S/Dr0.4. Beyond this range, f V abruptly departs from f osc (or de-synchronization), and switches to f V0 with a much lower value of St¼ 0.187. The synchronization of f V with either f osc or f V0 depending on the value of S/D is an interesting phenomenon, which has not been reported elsewhere in the literature. It is also noted that the largest gap ratio for synchronization between f V and f osc is S/D¼0.4, which coincides with the peak response amplitude (Fig. 7(a)). The abrupt jump of f V at S/D ¼0.4 brings to mind that the extraordinarily high f osc for the free-standing cylinder also n takes place at the same velocity of U n ¼3.05 as shown in Fig. 3(b), where f is as high as 0.83 (or St ¼0.27 as shown in Fig. 4(b)). When the wall effects are negligible (see for example at S/D ¼1.5 as shown in Fig. 5(a)), the amplitude of vibration is very small (An E 0.1 or less). Furthermore, the displacement is quite irregular in that the amplitude and

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Fig. 7. Measurement results of the near-wall cylinder as a function of gap ratio for U n ¼ 3:05: (a) vibration amplitude (An ), mean drag (C D ) and r.m.s. drag 0 (C D ) coefficients; (b) Strouhal number of the cylinder’s vibration frequency (f osc ), together with vortex shedding frequencies from both the vibrating (f V ) and stationary (f V0 ) cylinders.

frequency of vibration vary with time, suggesting the co-existence of multiple frequency components in the signal. This is also supported by the fact that in this case, f osc is shifted towards f N , whereas f V keeps close to f V0 . A plausible explanation is that the vibration at relatively small amplitude (An E 0.1 or less) is not sufficient to exert appreciable effects on the vortex shedding process. When the cylinder is located in close proximity to the plane wall (see for example at S/D ¼0.4 as shown in Fig. 5(a)) or lies in the upper branch, on the contrary, An become much larger (An Z 0.2), implying a higher energy transfer between fluid and structure that is sufficient to drive and sustain VIV. Consequently, the vortex shedding and the cylinder motion are synchronized at the same frequency, namely f V ¼ f osc , as evidenced by the collapse of the two curves over the small-S/D range (S/D r0.4) in Fig. 7(b) for U n ¼ 3:05 and the complete-S/D range in Fig. 8(b) for U n ¼ 5:1. 0 The variations of the mean and r.m.s. drag coefficients (C D and C D ) with S/D are also shown in Fig. 7(a). As expected, 0 n both curves exhibit a trend similar to that of A , with a local maxima, ðC D Þmax E1.5 and ðC D Þmax E0.2, both occurred at S/ 0 D ¼0.3–0.4. When S/D 41.0, C D and C D approach the asymptotic values of about 1.1 and 0.04, respectively. Note that the asymptotic values are the same as those for the free-standing cylinder, suggesting that the wall effects become negligible when S/D 41.0–1.5. This observation is supported by the amplitude and frequency response as well. The results for the case of U n ¼ 5:1 are provided in Fig. 8. Similarly, the wall effects are limited to a certain gap ratio range (up to S/D ¼1.0–1.5). However, the response data exhibit a behavior distinctly different from that of U n ¼ 3:05. At this reduced velocity (lying in the upper branch), An increases monotonically with S/D before leveling off to an asymptotic value of An E0.8 at S/D Z1.0, which is in contrast to the case of U n ¼ 3:05 that has a local maxima at S/D E0.4. Moreover, f osc always synchronizes with f V , which decreases monotonically with S/D, from a maximum of 0.26 at S/D ¼0.05, until approaching an asymptotic value of 0.17 at S/D Z2.0 (Fig. 8(b)). The sustained synchronization is likely due to a higher

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Fig. 8. Measurement results of the near-wall cylinder as a function of gap ratio for U n ¼ 5:1. For caption, see Fig. 7.

energy transfer between fluid and structure at relatively larger amplitude, as compared to the case of U n ¼ 3:05 that lies in the small-amplitude initial branch. Fig. 8(a) also shows that although the vibration amplitude increases almost linearly 0 with S/D over the small-S/D range, both C D and C D exhibit a peak, albeit not as sharp and distinct as that of U n ¼ 3:05. However, the local maxima of the mean and r.m.s. coefficients take place at different gap ratios, namely at S/D E0.75 for 0 C D and S=D ¼ 0.3–0.4 for C D . 3.3. Vortex shedding patterns In this section, the vortex shedding patterns for the free-standing cylinder will be discussed as a function of reduced velocity. Subsequently, the results of the near-wall cylinder at different gap ratios are presented to illustrate the influence of the wall proximity on the vortex patterns under the two constant velocities. Fig. 9 shows the phased-averaged flow field in the near wake of the free-standing cylinder at U n ¼ 2:5526:62 when it is at the mid-point of the upward stroke during the vibration cycle. The vector maps have been Galilean decomposed by subtracting the convection velocity of U c ¼0.7U (which has been estimated by trial and error) in order to visualize the vortex motions clearly together with the contours of the normalized spanwise vorticity. Periodic vortex shedding can be observed for all cases, but the flow structure varies appreciably with the change in U n . One apparent trend is that the vortex formation length, which is defined as the streamwise distance from the axis of the cylinder to the core of fully formed vortex (Krishnamoorthy et al., 2001), decreases with U n . At low reduced velocities (U n ¼ 2:55 and 3:56), the wake is characterized as the classical von Ka´rma´n vortex street, or the so-called 2S mode, representing two single vortices per cycle. This is consistent with the finding of Govardhan and Williamson (2000) that the initial branch corresponds to the ‘2S’ vortex mode. At U n ¼ 4:58 (i.e., the last point of the initial branch), however, the vortex pattern is somewhat modified

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Fig. 9. An example of the phase-averaged flow field measured by PIV at different reduced velocities: (a) U n ¼ 2:55; (b) 3.56; (c) 4.58; (d) 6.62. Each plot is a mean of 20 snapshots corresponding to the correct phase 7 51. Velocity vector plot is with a reference-frame velocity of 0.7U, superimposed with flood contours of the normalized spanwise vorticity oz D=U: blue, clockwise vorticity; red, anticlockwise vorticity. The cylinder’s position and movement are highlighted, with the dashed lines indicating the extent of the transverse motion. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

in that the transverse spacing between the opposite-signed vortices becomes noticeably larger. The increase in transverse spacing is more obvious at even higher reduced velocities, causing the vortex mode in the upper branch to be completely different from the 2S mode. For example at U n ¼ 6:62, as shown in Fig. 9(d), the wake is composed of two rows of oppositesigned vortices that are widely separated in the transverse (y) direction with a spacing of the order of 3D. This wake mode is actually the newly-discovered ‘2PO’ mode by Morse and Williamson (2009), as will be illustrated below. Morse and Williamson (2009) showed that the initial and upper branches are associated with 2S and 2PO vortex modes, respectively. As shown in Fig. 10, the 2PO mode is similar to the 2P mode in that two pairs of vortices are shed per cycle of cylinder vibration, but the secondary vortex of each vortex pair is significantly weaker. The present study indicates that the secondary vortices are indeed rather weak and incoherent, such that they may be evident in some instantaneous snapshots, but are invisible in the phase-averaged flow field. This mode is equivalent to the ‘‘intermediate wake state’’ identified by Carberry et al. (2003) occurred at vibration frequencies between the previously observed low- and highfrequency states. In the case of 2PO mode, the maximum value of spanwise vorticity is observed in the immediate vicinity of the cylinder, thereafter the vortices decay rapidly with downstream distance. This trend is opposed to the 2S mode, where the vortices are shed after a certain formation length (about 4.8D as suggested by Jeon and Gharib, 2004), with their strength generally kept constant over a substantial streamwise distance. n Fig. 11 plots the measurement results of the free-standing cylinder in the normalized amplitude–wavelength (An –l ) plane, overlaid on the vortex mode map proposed by Morse and Williamson (2009). As U n is increased within the vibrating n n state (U n Z3.05), An increases monotonically and sharply, but l increases slightly (l E5.4–6). The first four cases (i.e., n 3.05rU r4.58, in the initial branch) fall in the 2S mode regime, whilst the last three cases (i.e., 5.09 rU n r6.62, in the upper branch) in the 2PO mode regime. This is consistent with the PIV results. The distinction between the 2S and 2PO modes is vivid in terms of length- and velocity-scales of the shed vortices, and is also reflected by quantitative characterization of near-wake flow statistics, including the mean velocity profiles and Reynolds shear stress distribution, which are not shown herein for brevity.

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Fig. 10. Sketch of the vortex modes: (a) 2S; (b) 2PO. Adapted from Morse and Williamson (2009).

Fig. 11. Locations of the fee-standing cylinder undergoing VIV in the normalized amplitude–wavelength plane, superimposed on the map of regimes for vortex shedding modes of a cylinder under controlled vibrations at Re ¼12 000 by Morse and Williamson (2009).

Fig. 12 shows a typical snapshot of the instantaneous flow fields for both the near-wall stationary (left column) and vibrating (right column) cylinders at four different gap ratios under the free-stream velocity of U ¼0.3 m/s. In the case of the stationary cylinder, the flow is relatively insensitive to the free-stream velocity (Reynolds number), but is strongly dependent on the gap ratio. The wake flow patterns can be categorized into three regimes as a function of S/D: (i) vortexshedding suppression regime (S/D o0.3); (ii) intermediate regime (0.3 rS/D o1.0) where the wall effects are significant resulting in the flow asymmetry about the cylinder centerline; and (iii) wall-effect-free regime (S/D Z1.0) where the flow resembles that of a free-standing cylinder. A more detailed description of the flow structure was presented in our earlier paper of Wang and Tan (2008). In the case of the vibrating cylinder, however, the vortex shedding patterns are dictated by both the flow velocity and the gap ratio. In general, the cylinder VIV appears to increase the vortex strength, resulting in vortex shedding to be more regular and organized as compared to the stationary cylinder at the same gap ratios. Even at the smallest gap ratio (S/D ¼0.15), the upper shear layer still curls up into a row of discrete vortices that periodically travel downstream, while the lower shear layer is relatively weak and does not form into discrete vortices. Similar single-rowed vortex street was also reported in the simulation study of Zhao and Cheng (2011) on a vibrating cylinder at U n ¼3.5 and S/D ¼0.002. The single-rowed vortex shedding is more obvious when S/D is increased to 0.4, at which the peak amplitude occurs. At S/ D ¼0.75 or 1.5, on the other hand, the lower shear layer is already strong enough to shed as discrete vortices. Therefore, the wake is characterized as the double-rowed vortex street, which is similar to that for a stationary cylinder at the same gap ratio. This vortex shedding pattern is the 2S mode—with two single counter-rotating vortices shed from the cylinder per shedding cycle, although not exactly symmetrical about the cylinder centerline at moderate gap ratios (e.g., S/D ¼0.75). This is reminiscent of the vortex shedding frequency (f V ) for this case as shown in Fig. 7(b). When the gap ratio is relatively small (S/D r0.4), f V collapses with f osc ; but when S/D 40.4, it switches to f V0 instead. A closer examination of the PIV results indicates that when the gap ratio is small (say S/D r0.4), the single-rowed vortex shedding in the upper shear layer destabilizes the wall boundary layer, causing the latter to separate in the form of clockwise vortices, which periodically convect downstream at the same frequency as that of vortex shedding. This type of synchronization of vortex shedding in multiple shear layers is believed to account for the super-harmonics in the drag force spectrum.

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Fig. 12. Representative snapshot of the instantaneous flow fields for both the stationary (left column) and vibrating (right column) cylinders at U¼ 0.3 m/s. The instantaneous vector maps are subtracted by 0.7U (Galilean decomposition), superimposed with flood contours of the normalized spanwise vorticity.

The vortex shedding patterns for the vibrating cylinder at U ¼0.5 m/s are shown in Fig. 13. Comparison with Fig. 12 shows that the two cases share certain similarities in their variation trend with S/D, such as changing from a single-rowed vortex street pattern at small S/D, to a double-rowed pattern at large S/D. However, the onset of double-rowed vortex shedding appears to be slightly delayed in this case. At S/D ¼0.75, the positive-signed vortices (lower row) are still apparently weaker than their negative-signed counterparts (upper row), which is opposed to the case of U¼0.3 m/s where a satisfactory symmetry of the flow has already been achieved at this gap ratio. Also, the vortex shedding pattern at S/D ¼1.5 and beyond is distinctly different from the classical 2S mode. Instead, it is the 2PO mode as shown earlier for the free-standing cylinder in the upper branch. 4. Concluding remarks This study investigates the structural dynamics and flow patterns of a low damped, neutrally buoyant (mn ¼ 1.0, z ¼0.0173) cylinder that is elastically mounted at various heights above a plane wall (0.05rS/D r2.5) and is subject to VIV in the transverse direction. While the structural properties of the cylinder are kept constant, the free-stream velocity (U) is varied from 0.15 to 0.65 m/s, which corresponds to a reduced velocity range of 1.53rU n r6.62, and a Reynolds number range of 3000rRe r13 000. The results reveal that the motions, Strouhal numbers and drag coefficients of the near-wall cylinder are dependent on both U and S/D. Moreover, velocity and vorticity field measurements using high-resolution PIV make it possible for quantitative visualization of the vortex shedding patterns in the cylinder wake. The effects of varying free-stream velocity (or reduced velocity) on the cylinder response under free-standing condition have been examined first, and the results are in close agreement with published data on similarly low mass-damping cylinders, mostly notably Govardhan and Williamson (2000). The range covered in the present study is from the initial branch to the upper 0 branch. Within this range, the vibration amplitude (An ), the mean and the r.m.s. drag coefficients (C D and C D ) increase

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Fig. 13. Representative snapshot of the instantaneous flow fields for both the stationary and vibrating cylinders at U¼ 0.5 m/s. For caption see Fig. 12.

monotonically with U n . However, there is a switchover of vortex shedding frequency for the cylinder undergoing VIV, that is, from StE0.187 when the cylinder is essentially stationary, to StE0.167 when the amplitude is considerably large (An ¼ 0.8–1.0). Except for the limiting case of U n ¼ 3.05 at which the cylinder just begins to vibrate, the vibration frequency, f osc , always synchronizes with f V , instead of with f N . This kind of synchronization is an inherent feature of light structures, as discussed in the literature review. The vortex formation mode in the initial branch is the same as that of a stationary cylinder (i.e., the 2S mode), whereas that in the upper branch is the 2PO mode recently proposed by Morse and Williamson (2009). As compared to the 2S mode, the 2PO mode is characterized by: (a) a much wider transverse spacing between the two rows of opposite-signed vortices; and (b) a much shorter vortex formation length in the streamwise direction. The effects of the plane wall on the VIV characteristics of the cylinder are examined with respect to the gap ratio (S/D) under two constant velocities, i.e., U¼ 0.3 and 0.5 m/s. The resultant reduced velocities are U n ¼3.05 and 5.1, corresponding to the beginning of the initial and upper branches, respectively. For the case of the stationary cylinder, the flow is rather insensitive to flow velocity in the subcritical regime and vortex shedding is suppressed below a critical gap ratio (S/ D E0.3). For the elastically mounted cylinder, on the other hand, the flow patterns (and also the cylinder motions) depend on both the flow velocity and the gap ratio. The presence of the nearby wall poses a constraint on the cylinder displacements, exerting complicated effects on the cylinder response in terms of amplitude and frequency modulation. VIV tends to persist at all gap ratios considered. When the gap ratio is very small (S/Dr0.15), the cylinder will hit the wall and be bounced back, causing appreciable increase in displacements in the opposite direction. It is also shown that the proximity of the wall leads to some type of nonlinearities in the cylinder response as evidenced by the super-harmonics (up to four or five times the vortex shedding frequency) in the drag force spectrum.

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An interesting phenomenon is that the presence of the nearby wall may not necessarily suppress VIV; instead it may promote VIV under some circumstances. For the case of U n ¼3.05, the cylinder exhibits large-amplitude vibrations over the range 0 oS/D r0.75, with a peak amplitude Anmax ¼0.38 occurring at S/DE 0.4. It is noteworthy that S/D ¼0.4 is exactly the last point where f V still synchronizes with f osc , and thereafter (S/D 40.4) it abruptly switches to f V 0 . For the case of U n ¼5.1, on the other hand, the variation of cylinder response with S/D is monotonic, with An and f osc respectively kept on increasing and decreasing, until asymptotic values are achieved at large enough gap ratios. At this velocity, f osc always synchronizes with f V , indicating a higher energy transfer between fluid and structure that tends to drive and sustain VIV as compared to the case of U n ¼3.05. The wall effects seem to be limited to a certain range of gap ratios, namely S/D r1.0–1.5, which is, however, independent of the flow velocity. Unlike the case of a near-wall stationary cylinder that vortex shedding is suppressed below a critical gap ratio (S/D E0.3), the elastically mounted cylinder always exhibits VIV and periodic vortex shedding irrespective of the values of S/D. When S/D is relatively small, vortex shedding occurs only in the upper shear layer, forming a single-rowed vortex street that is periodically shed and travels downstream. The single-rowed vortex shedding destabilizes the wall boundary layer (which is in close proximity to the cylinder), causing it to separate and shed as clockwise vortices at the same frequency as that of vortex shedding. This type of synchronization of vortex shedding in multiple shear layers is probably responsible for the existence of super-harmonics in the drag force spectrum. When S/D is large enough (i.e., the wall effects become negligible), the wake pattern is characterized as the double-rowed vortex street but with obvious difference depending on the values of reduced velocity, which is the 2S mode for U n ¼3.05 (in the initial branch) and 2PO mode for U n ¼5.1 (in the upper branch). As shown in this paper, the presence of a nearby wall exerts complicated effects on the structural dynamics and vortex patterns of a low mass-damping cylinder on linear elastic supports. Further investigations are required to fully resolve this issue. To overcome the upper limit of reduced velocity handicapped by the facility used in this study, we are planning to conduct another set of experiments in a larger-scale towing tank or water channel such that a data set for a complete three-branch (initial, upper and lower) response curve could be obtained.

Acknowledgment The funding support from the Singapore National Research Foundation (NRF) through the Competitive Research Programme (CRP, NRF-CRP5-2009-01) on this project is gratefully acknowledged. We would also like to acknowledge the support from Science and Technology Commission of Shanghai Municipality (Pujiang Program, Grant no. 10PJ1404700), Shanghai Education Committee (Program Grant nos. 12ZZ149 and J50604), and Shanghai Maritime University (Science & Technology Program, Grant no. 20100089). References Bearman, P.W., 2011. Circular cylinder wakes and vortex-induced vibrations. Journal of Fluids and Structures 27, 648–658. Bearman, P.W., Zdravkovich, M.M., 1978. Flow around a circular cylinder near a plane boundary. Journal of Fluid Mechanics 89, 33–47. Bourdier, S., Chaplin, J.R., 2012. Vortex-induced vibrations of a rigid cylinder on elastic supports with end-stops, Part 1: experimental results. Journal of Fluids and Structures 29, 62–78. Carberry, J., Sheridan, J., Rockwell, D., 2003. Controlled oscillations of a cylinder: a new wake state. Journal of Fluids and Structures 17, 337–343. Chiew, Y.M., 1990. Mechanics of local scour around submarine pipelines. Journal of Hydraulic Engineering 116, 515–529. Fredsøe, J., Sumer, B.M., Andersen, J., Hansen, E.A., 1985. Transverse vibration of a cylinder very close to a plane wall. In: Proceedings of the International Offshore Mechanics and Arctic Engineering Symposium, vol. 1, pp. 601–609. Govardhan, R., Williamson, C.H.K., 2000. Modes of vortex formation and frequency response of a freely vibrating cylinder. Journal of Fluid Mechanics 420, 85–130. Jacobsen, V., Bryndum, M.B., Nielsen, R., Fines, S., 1984. Cross-flow vibrations of pipe close to a rigid boundary. Journal of Energy Resources Technology, Transactions of the ASME 106, 451–457. Jeon, D., Gharib, M., 2004. On the relationship between the vortex formation process and cylinder wake vortex patterns. Journal of Fluid Mechanics 519, 161–181. Khalak, A., Williamson, C.H.K., 1999. Motions, forces and mode transitions in vortex-induced vibrations at low mass-damping. Journal of Fluids and Structures 13, 813–851. Krishnamoorthy, S., Price, S.J., Paı¨doussis, M.P., 2001. Cross-flow past an oscillating circular cylinder: synchronization phenomena in the near wake. Journal of Fluids and Structures 15, 955–980. Lei, C., Cheng, L., Kavanagh, K., 1999. Re-examination of the effect of a plane boundary on force and vortex shedding of a circular cylinder. Journal of Wind Engineering and Industrial Aerodynamics 80, 263–286. Morse, T.L., Williamson, C.H.K., 2009. Prediction of vortex-induced vibration response by employing controlled motion. Journal of Fluid Mechanics 634, 5–39. Nishino, T., Roberts, G.T., Zhang, X., 2007. Vortex shedding from a circular cylinder near a moving ground. Physics of Fluids 19, 025103. Nishino, T., Roberts, G.T., Zhang, X., 2008. Unsteady RANS and detached-eddy simulations of flow around a circular cylinder in ground effect. Journal of Fluids and Structures 24, 18–33. Sarpkaya, T., 2004. A critical review of the intrinsic nature of vortex-induced vibrations. Journal of Fluids and Structures 19, 389–447. Wang, X.K., Tan, S.K., 2008. Near-wake flow characteristics of circular cylinder close to a wall. Journal of Fluids and Structures 24, 605–627. Williamson, C.H.K., Govardhan, R., 2004. Vortex-induced vibrations. Annual Review of Fluid Mechanics 36, 413–455. Williamson, C.H.K., Govardhan, R., 2008. A brief review of recent results in vortex-induced vibrations. Journal of Wind Engineering and Industrial Aerodynamics 96, 713–735. Williamson, C.H.K., Roshko, A., 1988. Vortex formation in the wake of an oscillating cylinder. Journal of Fluids and Structures 2, 355–381.

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