two-degree-of-freedom vortex-induced vibration of a...

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Two-degree-of-freedom vortex-inducedvibration of a pivoted cylinder below

critical mass ratio

BY C. M. LEONG AND T. WEI*

Department of Mechanical, Aerospace and Nuclear Engineering,Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA

In this study, we investigated two-degree-of-freedom (2d.f.) vortex-induced vibrations(VIVs) of a circular cylinder with a pinned attachment at its base; it had identical massratios and natural frequencies in both streamwise and transverse directions. The cylinderhad a mass ratio, m� of 0.45, and a mass damping, (m�CCA)z, equal to 0.0841. Laser-induced fluorescence flow visualization and digital particle image velocimetry experimentswere conducted over a Reynolds number range, 820%Re%6050 (corresponding to thereduced velocity range, 1.1%U �%8.3). Measurements and visualization studies were madein a fixed plane at the cylinder mid-height, providing a two-dimensional picture of a highlythree-dimensional system. However, significant insights can be gained from theseexperiments and form the basis of this paper. A large transverse amplitude response,A�

Y w2 (or four diameters peak-to-peak), in the upper branchwas observed. The streamwiseamplitude response exhibits an even higher peak amplitude, A�

X w2:5, which isapproximately 125% of peak A�

Y . Results show that there is no lower branch for thissystem and the transverse upper branch exhibits asymptotic behaviour, i.e. a wide regime ofresonance. For ReO3000, the Strouhal number for the vortex shedding was 0.16 (G9%).Both the transverse cylinder oscillation and vortex-shedding frequencies, fOS,Y and fVS,respectively, were virtually identical throughout this range. While the streamwiseoscillation frequency is typically twice the transverse oscillation frequency for a 2d.f.system, this is not the case at the lowest reduced velocities where oscillations first occur.Under these conditions the streamwise and transverse oscillation frequencies were identical.Finally, we observed that the cylinder wake exhibits both the PCS vortex-shedding modeand a desynchronized vortex pattern, which are uncommon for flow past a cylinderexperiment. Very interestingly, the wide U � range over which resonance occurs isdominated by a desynchronized vortex pattern. These results clearly demonstrate thedifferences that arise in 2d.f. VIV occurring below the critical mass ratio.

Keywords: vortex-induced vibration; fluid–structure interaction; vortex shedding

*A

RecAcc

1. Introduction

Vortex-induced vibrations (VIVs) of a circular cylinder have receivedconsiderable attention over the past three decades from scientists and engineers.This is owing to the particular importance in marine problems such as the

Proc. R. Soc. A (2008) 464, 2907–2927

doi:10.1098/rspa.2007.0166

Published online 17 June 2008

uthor for correspondence ([email protected]).

eived 10 August 2007epted 19 May 2008 2907 This journal is q 2008 The Royal Society

http://rspa.royalsocietypublishing.org/

C. M. Leong and T. Wei2908


vibration of offshore structures, submarine towed array cables and ship mooringsin water as well as atmospheric problems including smokestacks, cellular towersand buildings. Bluff body VIV is a subset of the broader field of fluid–structureinteractions (FSI). Fundamental FSI studies of rigid bodies, both experimentaland computational, often address only a single degree of freedom (1d.f.) due tothe difficulties and complexities associated particularly with VIV.

The present research is focused on fully coupled, two-degree-of-freedom (2d.f.)VIV of a rigid pivoted circular cylinder. That is, a freely oscillating, surface-piercing circular cylinder was mounted in a water tunnel like an invertedpendulum. This is the 2d.f. extension of the 1d.f. work reported in Benaroya &Wei (2000), Dong et al. (2004) and Voorhees et al. (2008). In addition, thecylinder used in this investigation had a mass ratio, m�, below the critical ratio,m�

cr, identified by Govardhan & Williamson (2002); note that most VIV studiesto date have been done on cylinders above m�

cr. It is believed that results for a2d.f. pivoted cylinder below critical mass ratio are shown for the first time.

It should be noted at the outset that a decidedly two-dimensional view hasbeen made of a very three-dimensional system. However, keeping the three-dimensional nature of the problem at the forefront, it is possible to gainsignificant insights into the FSI at subcritical mass ratios. This is particularlytrue for this investigation because the cylinder itself is rigid except for the veryshort pin that anchors the cylinder to its base. In this regard, it is only necessaryto track motions in one plane to capture the entire frequency–amplitude responsecharacteristics. For a detailed analysis of three dimensionalities associated withthese flows, the reader is referred to Voorhees et al. (2008).

The challenge behind studying fully coupled VIV is that the interactionsbetween fluid dynamics and structural motion are typically nonlinear andstochastic. Sarpkaya (1979) aptly put it: ‘VIV is not a small perturbationsuperimposed on a mean steady motion. VIV is inherently nonlinear, self-governed or self-regulated, multi-degree of freedom phenomenon. It presentsunsteady flow characteristics manifested by the existence of large-scalestructures, sandwiched between two equally unsteady shear layers’. Moretraditional decoupled studies have been done either by forcing the structuralmotion and examining vortex dynamics, or mathematically assuming someform of periodic fluid forcing function(s) and computing the structuralresponse. Indeed, much FSI research is still done in the traditional, i.e.decoupled, way.

The canonical two-dimensional VIV experiment is commonly referred to as the‘elastically mounted cylinder’. Except for small-scale three-dimensionalitiessuperimposed on the large-scale Karman vortices (Wei & Smith 1986), flowalong the entire length of the cylinder is effectively two dimensional. Low-pressure regions associated with the Karman vortices induce fluctuating lift anddrag forces on the cylinder. Under the right combination of damping, z, andreduced mass, m�, these fluctuating forces will cause the cylinder to undergoperiodic transverse oscillations. When VIV occurs, the frequency of vortexshedding, fVS, is approximately equal to the oscillation frequency of the cylinder,fOS. This condition has been defined as ‘synchronization’. The range of flowspeeds over which this occurs is called the ‘synchronization regime’. Note thatsynchronization and ‘lock-in’ are used interchangeably throughout this paper.For some excellent and extensive reviews on VIV, the reader is referred to

Proc. R. Soc. A (2008)


2909Two d.f. VIV of a pivoted cylinder


Sarpkaya (1979, 2003), Bearman (1984), Chen (1987), Blevins (1990),Naudascher & Rockwell (1994), Williamson (1996), Zdrakovich (1997, 2003)and Williamson & Govardhan (2004), just to name a few.

(a ) Frequency and amplitude response

The frequency and amplitude response of an elastically mounted rigid circularcylinder undergoing transverse VIV have been well characterized. There exist twomain types of amplitude response depending on the mass-damping parameter,m�z.Here m� is the mass ratio and z is the ratio of critical damping to structuraldamping. Cylinders with high m�z undergoing 1d.f. VIV exhibit two responsebranches, namely the initial and lower branches. This type of response is also oftenreferred to as the classical Feng-type response (Feng 1968). Low m�z cylinders, onthe other hand, display three distinct branches: the initial branch, upper branchand lower branch. This is described in Khalak & Williamson (1999).

The initial and lower branches are common to both types of amplitude response.The initial branch is characterized by a vortex-shedding pattern with two singlecounter-rotating vortices per period. Williamson & Roshko (1988) referred to thisas the ‘2S’ mode. This is followed by the lower branch, characterized by constantoscillation amplitude and frequency across a range of reduced velocity, U �. Incontrast to the initial branch, the upper and lower branches are characterized by a‘2P’ vortex-shedding mode. That is, there are two pairs of counter-rotating vorticesper shedding cycle. The second vortex of each pair in the upper branch is weakerthan the first vortex and decays very rapidly.

Khalak & Williamson (1999) demonstrated that for a fixed value of massdamping, m�z, the extent (i.e. range of U � values) of the synchronization regimeis controlled primarily by m�, whereas peak amplitudes are controlled mainly bym�z. For a complete discussion on frequency and amplitude response ofelastically mounted low m�z cylinders, please refer to Khalak & Williamson(1999). Amplitude and frequency response characteristics of elastically mountedcylinders can be generalized to pivoted cylinders as well and have been examinedby Voorhees et al. (2008).

Two-degree-of-freedom, elastically mounted cylinders, with m�O6 and lowm�z, exhibit transverse amplitude response behaviours similar to those of 1d.f.cylinders oscillating purely in the transverse direction as shown by Jauvtis &Williamson (2004). Responses for transverse cylinders below critical mass ratioare presented by Govardhan & Williamson (2002).

(b ) Two-degree-of-freedom VIV of a circular cylinder

Streamwise and transverse displacements associated with the 2d.f. VIV of acircular cylinder immersed in a uniform free stream can be well represented by

xðtÞZAX sinð26tCqÞ; ð1:1aÞ

yðtÞZAY sinð6tÞ: ð1:1bÞIn these expressions, AX and AY are the oscillation amplitudes in x - andy-directions, respectively. The oscillation frequency, 6, is 2pfOS and q is thephase angle between the two-component displacements. Figure 1 shows the



= 0° = 45° = 90° = 135° = 180° = 225° = 270° = 315°

Figure 1. Characteristic trajectories of a 2d.f. cylinder oscillating in uniform free stream as afunction of phase angle, q. Flow is from left to right with the direction of motion as indicated.



trajectory shapes (Lissajous figures) of 2d.f. cylinder oscillations as a function ofphase angle. Note that the oscillation frequency in the x -direction must be twicethat of oscillations in the y-direction to yield figure-eight type trajectories.

An early experiment on 2d.f. VIV was conducted by Chen & Jendrzejczyk(1979). They worked with a cantilevered circular cylinder in water up to a reducedvelocity of U �z10. They found that the cylinder response was divided intoseveral regions. For U �O4.5, oscillations were predominantly in the transversedirection. They concluded that flow above this regime could be treated as aquasi-1d.f. problem.

Over a decade later, Moe & Wu (1990) carried out an investigation in which theratio of natural frequencies in streamwise and transverse directions (i.e. fn,X/fn,Y)was 2.18. The mass ratios in both directions were also different. In one case, themass ratio in the transverse direction was twice that in the streamwise direction.Owing to these conditions, they did not identify the different response branchesobserved for transversely oscillating cylinders.

Sarpkaya (1995) focused on different natural frequency ratios.When fn,X/fn,YZ1,he showed that the maximum 2d.f. amplitude response was 19% larger than for the1d.f. case. In addition, peak oscillations occurred at a higher U � for the 2d.f. caseversus the transverse oscillating case. This meant that the ability to oscillate in 2d.f.had an impact on the system undergoing VIV. Sarpkaya (1995) also showed anamplitude response plot similar to that of Moe & Wu (1990) which had nodistinct branches. Note, however, that the mass ratios were somewhat different inthose experiments.

Jeon & Gharib (2001) studied cylinders undergoing forced 2d.f. motions.Specific phase angles, qZ08 and K458, were chosen based on the assumption thatfigure-eight type motions (see figure 1) occur most often in nature. Jeon &Gharib (2001) argued that the transverse motions set the vortex-sheddingfrequency and that the streamwise motions determined the phase angles.

Jauvtis & Williamson (2004) investigated elastically mounted 2d.f. circularcylinders with low mass and damping. Their experimental set-up permitted exactmass and natural frequencies in both streamwise and cross-stream directions.They defined a moderate mass ratio (m�O6) and found that the freedom tooscillate in the free-stream direction had surprisingly little effect on thetransverse vibrations. However, there were dramatic changes for mass ratiosless than six (which they called small mass ratio). They observed peak-to-peakamplitudes of three diameters in what they called a ‘super-upper’ branch.





Correspondingly, they discovered a periodic vortex wake mode, comprising atriplet every half cycle which they refer to as the ‘2T’ mode. Working along thelines of Govardhan & Williamson (2002) for critical mass ratio, Jauvtis &Williamson (2004) found a critical mass ratio, m�

critZ0:52, for 2d.f. elasticallymounted cylinders.

Recently, Flemming & Williamson (2005) have studied VIV of a pivotedcylinder in 2d.f. They concluded that for moderate values of inertia damping,I �z, the system exhibited the initial and lower branches, and for low inertiadamping, the initial, upper and lower branches appeared, analogous to results oftransverse-only oscillating cylinders. They also observed a new vortex formationmode, named the ‘2C’ mode, which comprised two co-rotating vortices that shedeach half cycle, for their lightest inertia ratio, I �, of 1.03. The critical mass ratiofor a pivoted cylinder was experimentally determined to be approximately 0.5.

Numerical simulations of cylinders undergoing 2d.f. VIV have been done byBlackburn & Karniadakis (1993) and Newman & Karniadakis (1996). However,their two-dimensional simulations were performed at relatively low Re (i.e.Rez100–200) compared with those investigated experimentally.

This study of 2d.f. VIV is of flow past a subcritical mass ratio circular cylinderwhich is pinned at one end. Thus, this is a system involving three-dimensionaleffects. Three-dimensionality has been thoroughly investigated by Voorhees(2002) and Voorhees et al. (2008) for a transverse oscillating inverted pendulumcase. This experimental system has similar mass ratio and natural frequencies inboth streamwise and cross-stream directions.

(c ) On the existence of a critical mass ratio

The mass ratio m� is defined as the mass of the structure divided by the massof the fluid displaced by the structure. That is, m�Z4m/prD2L, where m is thecylinder mass and D and L are the cylinder diameter and length, respectively.In general, experiments conducted in air necessarily have higher mass ratios,m�ZO(100), like those of Feng (1968). Experiments in water typically havelower mass ratios, O(1)%m�%O(10). Experiments conducted by Williamson’sgroup, e.g. Khalak & Williamson (1999), Govardhan & Williamson (2002) andJauvtis & Williamson (2004) and the present studies, Benaroya & Wei (2000),Dong et al. (2004) and Voorhees et al. (2008), fall into this category.

As has been discussed to this point, the literature clearly shows the richness ofthe dynamics associated across the m� parameter space. For large mass ratios,cylinder oscillations are excited only at harmonics of the structure’s naturalfrequency. For moderate to low mass ratios, however, cylinder oscillations occurat frequencies below the structure’s natural frequency. This is then followed bythe lock-in behaviour at the natural frequency (and higher harmonics) observedfor high mass ratio cylinders. The interesting question that naturally arises iswhether or not there is a critical mass ratio for VIV of a circular cylinder belowwhich cylinder oscillations occur irrespective of the cylinder’s natural frequency.

Govardhan & Williamson (2002) showed that there is, in fact, a critical massratio, m�

cr, for transverse VIV of elastically mounted cylinders. This was foundto occur at low mass-damping values, (m�CCA)z!0.05. In their analysis,Govardhan & Williamson (2002) showed that the cylinder response frequency,defined as f �ZfOS/fn (where fn is the natural frequency in quiescent water), can





be shown to take the form

f � Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim� CCA

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim�CCEA

p : ð1:2Þ

Govardhan & Williamson (2002) defined CA as the potential added masscoefficient and CEA to be the effective added mass coefficient due to wake vortexdynamics. Note that for a circular cylinder, CAZ1. Govardhan & Williamson(2002) deduced an expression for the frequency of the lower branch, at relativelysmall mass-damping values, as

f �lower Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim� C1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim�K0:54

p : ð1:3Þ

The value CEAZK0.54G0.02 was obtained from a curve fit of a range of experi-mental data obtained for low mass damping. Equation (1.3) has very significantimplications. In particular, one can identify a critical mass ratio,m �

crZ0:54, wheref � will become infinitely large.

For 0%m �cr%0:54, Govardhan & Williamson (2002) predicted that the lower

branch can never be reached and ceases to exist. They hypothesized instead thatthe upper branch of synchronization will continue in the limit as U � approachesinfinity. Flemming & Williamson (2005) had experimentally determined that thecritical mass ratio for a 2d.f. VIV of a pivoted cylinder is approximately 0.5. Themain objective here is to study the dynamics related to cylinders below criticalmass ratio.

(d ) Problem statement

In classical cylinder VIV experiments, cylinders oscillate purely in thetransverse direction. However, many real engineering applications, such asoffshore structures, smokestacks and cellular telephone antennae towers, respondin 2d.f. (or higher) and the structures may be pivoted or elastically mounted. Inthis investigation, the cylinder is pivoted at one end and free to move on theother in both streamwise and cross-stream directions to simulate real engineeringapplications. The cylinder has similar mass ratio and natural frequencies inboth directions.

What is salient about this experiment is that the mass ratio of the cylinder iswell below the critical values identified by Williamson’s group. This opens up awhole new paradigm into bluff body VIV research. If there is a dramatic changein the response and body dynamics due to the second degree of freedom below thecritical mass ratio, then there is a need to incorporate this understanding intothe design and analysis of structures operating within this parameter space. Themain goal of this investigation, therefore, is to better understand the dynamics of2d.f. VIV of a circular cylinder, with a subcritical mass ratio, mounted as aninverted pendulum. The critical path to this objective includes the following:

—mapping the frequency and amplitude response of the cylinder over as wide arange of reduced velocities as possible,

— identifying the various vibrational modes arising across the reduced velocityrange, and



free surface

flowdirection

base plate

stainless steelpin

dye slot

Figure 2. Schematic of cylinder model and experimental set-up. The coordinate system is indicated.Note that the drawing is not to scale.



—using laser-induced fluorescence (LIF) flow visualization and digital particleimage velocimetry (DPIV) to characterize the vortex dynamics associatedwith each mode of vibration.

(e ) A note on coordinates and non-dimensional groups

In this study, coordinates were chosen such that x, y and z correspond to thestream, transverse or cross-stream and axial directions, respectively. These areindicated in figure 2. The z -direction points up towards the free surface oppositethe direction of gravity. This coordinate system is consistent with the ones usedwidely in VIV studies.

Table 1 shows the key non-dimensional groups used in this study. Added mass,mA, is given as the drift volumemultiplied by the fluid density (i.e.mAZrfluidVdrift);this was first articulated by Darwin (1953). In this case, the density of fluid is thedensity of water. Added mass can also be defined as mAZCAmd, where CA is theadded mass coefficient andmd is the displaced mass. As noted previously,CAZ1 fora circular cylinder.

Since the study of VIV involves a number of different characteristicfrequencies, it would be expedient to clear up any ambiguities at the outset.The natural frequency of the cylinder in quiescent water has already been definedas fn. Since the natural frequency is independent of direction, i.e. fn,X is identicalto fn,Y, one needs only deal with fn. The cylinder oscillation frequency is definedas fOS, regardless of whether or not there is synchronization. fVS is defined to be



Table 1. Key dimensionless parameters used throughout this paper.

mass ratio m� m=ðprD 2L=4Þdamping ratio z c=ð2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikðmCmAÞ

pÞ

reduced velocity U � U/( fnD)reduced vorticity 6� (6D)/Uamplitude ratios A�

X , A�Y AX/D, AY/D

frequency ratios f �OS;X , f�OS;Y , fOS,X/fn, fOS,Y/fn

Reynolds number Re (rUD)/mStrouhal number St (fVSD)/U



the vortex-shedding frequency of the cylinder in motion. The correspondingStrouhal number (sometimes called the Strouhal frequency), St, refers to thenon-dimensional vortex-shedding frequency when the cylinder is held rigid inthe flow. It is commonly accepted, see Schlichting (1979), that Stz0.21 for flowaround a circular cylinder at ReO1000.

2. Experimental apparatus and methods

(a ) Flow facility

Experiments were conducted in a large free surface water tunnel facility. Thisclosed-loop facility consists of an upstream end tank and settling chamber, two-dimensional contraction, test section, downstream end tank and two pumps. Thetest section is 57.2 cm in width!122 cm in depth!610 cm in length. It isconstructed entirely from 1.91 cm thick glass panels placed in a welded steelI-beam frame. Flow is driven by two pumps operating in parallel. The maximumflow rate of both pumps combined is 1500 l minK1, which is equivalent toapproximately 30.0 cm sK1 when the test section is full. The free-stream velocityis uniform to within G2% across the test section and turbulence intensities areless than 0.1% of the free-stream velocity. Detailed descriptions of this facilitycan be found in Smith (1992) and Grega et al. (1995).

(b ) Cylinder assembly

The circular cylinder used for experiment was designed to allow freedom tooscillate in both x - and y-directions. This cylinder assembly consisted of a baseplate, stainless steel pin, dye injection module and two acrylic cylinder tubesections as shown in figure 2. In this manner, the cylinder was free to oscillatelike an inverted pendulum. For details of the components and assembly, thereader is referred to Leong (2005).

The circular cylinder itself was manufactured entirely from thin-walledextruded acrylic tube owing to its optical clarity. The structure was hollow, rigidand of low mass ratio. The outer diameter was 2.54 cm with an inner diameter of2.22 cm. The overall length was 109.22 cm for an L/D of 43. Each of the two tubesections and dye injection plug, comprising the length of the cylinder, were 50.8and 7.6 cm long, respectively.





The dye injection module consisted of a short section of acrylic tube capped atboth ends by machined plugs. A thin dividing plate was placed in the tube sectionand enabled us to create two separate dye chambers and then perform two-colourflow visualization studies. The module was equipped with two narrow dye injectionslots located 308 on either side of the nominal forward stagnation line.

The cylinder was mounted to a 122 cm long!57.2 cm wide!1.27 cm thickbase plate through a stainless steel pin. The effective length of the pin, i.e. thedistance between the base plate and the bottom end of the cylinder, was 2 cm.For small angle deflections, the pin was assumed to act as a linear flexural springwith a flexural spring constant, k, of 11.0 N m radK1. This was measured andverified by applying static lateral forces to the top of the cylinder and measuringthe deflection while it was mounted in the water tunnel (and the tunnel was filledwith water). The maximum deflection angles caused by VIV in the transverseand streamwise directions were 5.78 and 7.18, respectively.

The cylinder was immersed in a uniform flow of water, 101.6 cm in depth. Thetop of the cylinder protruded through the free surface. The mass ratio of thecylinder m� was 0.45. The natural frequency of the structure in quiescent water, fn,was 1.14 Hz. This was again measured by deflecting and releasing the cylindermounted in the water tunnel with water, but without flow. The damping ratio zwas0.058 and the mass-damping parameter, m�z, was 0.026. It is important to notethat both the mass and natural frequencies are identical in the x - and y-directions.

(c ) Experimental methods

Detailed LIF flow visualization experiments and DPIV measurements wereconducted over the range of reduced velocities, 1.1%U �%8.3. This correspondsto a Reynolds number based on cylinder diameter range of 820%Re%6050. Themaximum flow speed, U �Z8.3, was constrained by the maximum pump speed.All studies were conducted at the mid-height of the cylinder. The distance fromthe free surface was K43.7 cm (or z�Zz/DZK17.2).

In order to identify and characterize the various cylinder vibration modes, therange of flow speeds was subdivided into a large number of increments, DU �. ForU �%4.2, DU � was approximately 0.2. This was done because many of thechanges in vibrational modes occurred at low reduced velocities. The smallincremental steps ensured that every mode would be fully captured. ForU �R4.2, data were collected at increments of DU �z1.0. Each time the speedwas changed, a minimum of 30 min were permitted to elapse beforemeasurements commenced. This ensured that any transient effects were nolonger present. These measurements were performed for approximately 20different cases across the speed range.

Fluorescein (yellow-green) and rhodamine (orange-red) fluorescent dyes wereused for the two-colour LIF experiments. Fifteen-minute-long S-VHS videosequences were recorded for each case. For each DPIV case, two sets of 225 imagepairs (i.e. 15 s per set) were recorded. For lower U � values, the interval betweenimages in a pair, Dt, was 0.005 s. At higher speeds, Dt was 0.001 s. An in-houseDPIV correlation program was used to determine the velocity field data. Thisincluded a two-stage correlation algorithm, zero padding and other features formaximum accuracy. A detailed description of the software, calibration, accuracyand uncertainty of the correlation program can be found in Hsu et al. (2000). For





this study, the coarse and fine correlation interrogation windows were 128!128pixels and 64!64 pixels, respectively. The image resolution was 73.62pixels cmK1.

Cylinder frequency and amplitude response measurements were made using dualphotoelectric sensors, one each for the stream and cross-stream directions. Theseare described in Voorhees (2002) and Leong (2005). Data were sampled at 300 Hz,10 times the DPIV sampling rate. This provided excellent time-resolved responsemeasurements. Two-minute-long time records were recorded for each reducedvelocity. Measurements were made while both incrementing and decrementing theflow speed in order to quantify any hysteresis effects that may be present. It isworth noting that cylinder position measurements were made near the tip of thecylinder, that is, above the free surface. However, the results presented in thefollowing sections have been scaled relative to the cylinder mid-height.

3. Results and discussion

(a ) Frequency and amplitude response

A principal driver for this research was understanding the importance of 2d.f. for apivoted, low mass ratio cylinder. The response of 1d.f. elastically mountedcylinders, such as that of Khalak &Williamson (1999), and pivoted cylinders, as inDong et al. (2004) and Voorhees et al. (2008), has been extensively investigated.Govardhan &Williamson (2002) worked on the response of transversely oscillatingcylinders below critical mass ratio and recently Jauvtis & Williamson (2004) havestudied the response of 2d.f. elastically mounted cylinders.

In this section, the results relevant to a 2d.f. pivoted cylinder with nomin-ally subcritical mass ratio, m�Z0.45, and mass-damping parameter, (m�CCA)zZ0.0841, will be presented and discussed. Both amplitude and frequencyresponse are plotted against reduced velocity, U �, in figure 3. The amplituderesponse from Jauvtis & Williamson (2004) of a 2d.f. elastically mountedcylinder with small mass ratio and from Govardhan & Williamson (2002) for a1d.f. (transverse only) elastically mounted cylinder below critical mass ratio aresuperimposed for comparison. The reduced velocity, U �ZU/fnD, is defined intable 1 for the free vibration experiment along with other key parameters.

For an elastically mounted cylinder with low but super-critical mass ratio, theamplitude response plot can be divided into three regimes. There is the initialbranch (I) at low reduced velocities, characterized by small amplitudeoscillations. This is followed by an upper branch (U) in which the oscillationamplitudes reach a maximum. When the Karman vortex-shedding frequencymatches the cylinder natural frequency, there exists an extended range ofreduced velocities in which the cylinder will oscillate at a frequency close to fn.This is known as lock-in and has come to be known as the lower branch (L).

Observe in figure 3a that for the present 2d.f. subcritical case, there is aninitial branch, which spans the range 1.1!U �!2.6, followed by an upper branchextending beyond U �O2.6. Although there does not appear to be a clearamplitude jump distinguishable from the A�

Y curve, a break in the branch aroundU �z2.6 has been imposed. The reason for this is that there is a small shift ordiscontinuity in the frequency curve f �OS;Y .



2.5

2.0

1.5

1.0

0.5

0

I

U

(a) 3.0

2.5

2.0

1.5

1.0

0.5

0

I

U

(b)

2.01.81.61.41.21.00.80.60.40.2

0 1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

I

U

(c)

Figure 3. Amplitude and frequency response for the 2d.f. pivoted cylinder with mass ratio, m�Z0.45,and mass damping, (m�CCA)zZ0.0841. Amplitude response, (a) A�

Y and (b) A�X , and cross-stream

frequency, (c) f �OS;Y , are plotted against reduced velocity, U �. Dashed line and dot-dashed line in theamplitude response plots represent the results of 2d.f. elastically mounted cylinder with small massratio from Jauvtis &Williamson (2004) and transverse only elastically mounted cylinder below criticalmass ratio from Govardhan & Williamson (2002), respectively. In the frequency response plot,dot-dashed line with open circle and solid line with filled circle represent vortex-shedding frequencyand cylinder oscillation frequency, respectively.



One thing that is particularly noteworthy in figure 3a is the comparativelylarge maximum transverse amplitude in the upper branch. While classicalelastically mounted transverse oscillating cylinders and cylinders mounted likeinverted pendulums have peak amplitudes A�

Y w1, the results of Jauvtis &Williamson (2004) indicated that response characteristics with larger amplitudes(i.e. A�

YO1) should not be too surprising. What is surprising, however, is thevalue of peak amplitude, AYw2, or about four peak-to-peak diameters.Oscillation amplitudes of this magnitude have not yet been observed.

Another important feature of the transverse amplitude response plot, A�Y

versus U �, is the distinct absence of a lower branch. This is consistent with theresults of Govardhan &Williamson (2002) for 2d.f. elastically mounted cylinders.For the low, but super-critical mass ratio VIV case, the amplitude response curvewill contain a ‘break’ (i.e. from either initial or upper branch to lower branch) ata certain reduced velocity. This typically occurs after f �OS;Y zfn. However, for thesubcritical mass ratio, inverted pendulum examined in this study, this is clearlynot the case. There is no lower branch and the transverse amplitude responseapproaches an asymptotic value, A�

Y w2.



4.5

3.5

2.5

1.5

0.5

–0.5

–1.5

–2.5

–3.5

–4.50 10 20 30 40 50 60

time (s)

Figure 4. Time traces of transverse displacement at U �Z8.25. Note the irregularities of cylinderoscillation and the ‘quasi-periodic shift’ in the cylinder mean position at tz15 and 45 s. The periodis, therefore, approximately 30 s.



For the classical cylinder experiment, frequency response in the upper branch isflow driven, while in the lower branch it is structure driven. That is, in the upperbranch, it is the vortex shedding that excites the cylinder motions. By contrast, forthe lower branch, cylinder oscillations at the structure’s natural frequencymodulatethe vortex-shedding mechanisms. In the present case, the cylinder is too ‘light’ todrive the flow. Therefore, the lower branch cannot exist and the upper branch willcontinue with increasing U � indefinitely. The identification of an asymptoticamplitude is entirely consistent with the findings of Govardhan & Williamson(2002). They named this asymptotic amplitude phenomenon ‘resonance forever’.

From the peak of the upper branch, U �z5.2, and beyond, the transverseoscillations become irregular. This can be better understood by examining thesample time trace shown in figure 4 for U �Z8.25. Note in particular the anomaliesat tz15 and 45 s. To the contrary, Jauvtis & Williamson (2004) found that thepeak of super-upper branch showed constant transverse amplitude vibration.The reason for the irregularities observed for the present experiment was traced tothe onset of a ‘quasi-periodic shift’ in cylinder mean position. That is, the cylinderwould oscillate about one deflection angle and then oscillate around a differentdeflection angle, switching back and forth between these positions.

Turning next to the streamwise amplitude response plot, A�X versus U �, in

figure 3b, one can see that the maximum streamwise amplitude is very large,A�

X w2:5. This previously undocumented behaviour is approximately 125% ofpeak A�

Y . In the 2d.f. elastically mounted cylinder case of Jauvtis & Williamson(2004), the maximum amplitude of A�

X across the entire U � range was onlyapproximately 0.3. By comparison with the present experiment, this isonly approximately 20% of A�

Y .





Continuing with the streamwise amplitude response plot shown in figure 3b,the upper branch can be seen to break around U �z4.4 after reaching its peakamplitude. The amplitude branch begins to increase again with increasingreduced velocity. However, owing to limitations in the top speed of the watertunnel, it was not possible to ascertain whether the streamwise amplitude keepsincreasing or if A�

X approaches an asymptote like its transverse counterpart, A�Y .

This streamwise amplitude response is not observed for an elastically mounted2d.f. system above critical mass ratio (Jauvtis & Williamson 2004), principallybecause there is a lower branch. At that branch, the streamwise amplitude A�

X isvery small, albeit visible, and almost constant.

The transverse oscillation frequency response plot shown in figure 3c is verysimilar to that shown by Govardhan & Williamson (2002). The dashed linebetween 0%U �%1.5 indicates the Strouhal frequency, StZ0.21; for vortexshedding from a stationary cylinder, see Schlichting (1979). It may be observedthat the cylinder oscillation frequency data, appearing in figure 3c as solid circles,vary linearly with reduced velocity for U �O2.6. It is expected that this trend willcontinue as U � approaches infinity in light of the absence of a lower branch forsubcritical mass ratio cylinders.

For a 2d.f. system, the oscillation frequency in the streamwise direction isgenerally twice that of the oscillation frequency in the cross-stream direction, i.e.fOS,Xz2fOS,Y. One would therefore expect to see a streamwise resonance whenf �OS;YZ fOS;Y =fnz1=2. However, this is not the case for the present low massratio cylinder.

For the m�Z7 case in Jauvtis & Williamson (2004), the streamwise oscillationamplitude is a maximum for fOS,Xzfn. For their small mass ratio case, m�Z2.6;however, the maximum streamwise oscillations coincide with the maximumtransverse oscillation amplitudes, i.e. for fOS,Yzfn. This is consistent with thepresent findings form�Z0.45. Thus, for lowmass ratios, there is a coupling betweenthe streamwise and transverse motions, which does not occur for large mass ratios.

Figure 3c shows vortex-shedding frequencies across the reduced velocityrange; these appear as open circles. Note that the cylinder transverse oscillationfrequency is virtually identical to the vortex-shedding frequency, i.e. fOS,YzfVS.Recently, Dong et al. (2004) examined the quasi-periodic beating that occurswhen fOS,YzfVS. This phenomenon exists for pivoted or inverted mountedcylinders (Voorhees et al. 2008) and has also been shown by Govardhan &Williamson (2002) for transverse elastically mounted cylinders below criticalmass ratio. This can also be seen in figure 5 in which transverse oscillationamplitude is plotted as a function of frequency ratio, fVS/fOS,Y, and overlaid onthe Williamson & Roshko (1988) map of vortex modes. Because data in figure 5indicate that fVS/fOS,Yz1, one can predict the cylinder oscillation frequencyknowing the vortex-shedding frequency; for a 2d.f. pivoted cylinder belowcritical mass ratio, the non-dimensional vortex-shedding frequency was found tobe StZ0.16G9% for ReO3000 (or U �O4.2).

In summary, then, the most significant response features are the highpeak transverse and streamwise amplitude values and the lack of a lowerbranch. Thus, in some respects, this current research can be thought of as acombination of 2d.f. elastically mounted cylinders (Jauvtis & Williamson 2004),and transverse only elastically mounted cylinders below critical mass ratio



2.5

2.0

1.5

1.0

0.5

0 0.5 1.0 1.5 2.0 2.5 3.0

2P*

P

P+S

C (P+S)

C (2S) 2S

2P

2P+2Sno synchronizedpattern observed

Figure 5. Transverse amplitude response plotted against frequency ratio fVS/fOS,Y, where fVS is theactual cylinder vortex-shedding frequency. The dashed line along fVS/fOS,YZ1 is a reminder thatvortex-shedding frequency is very similar to cylinder oscillation frequency for the present study.



(Govardhan & Williamson 2002). However, it is critically important to keep inmind that the inverted pendulum will become highly three dimensional as thecylinder deflects in the stream direction. The exact nature of these threedimensionalities is beyond the scope of the present study.

(b ) Vortex dynamics

This section contains a description of visual observations made from a detailedLIF flow visualization investigation. Sample images will be shown while acomplete description of these experiments is presented in Leong (2005). To placethis work in context, however, it would be helpful to re-examine the amplituderesponse characteristics in a different way.

An alternative way of plotting transverse and streamwise amplitude responsecharacteristics is to use a different or modified normalized velocity, defined byðU �=f �OS;Y ÞSt. This is analogous to fSt/fOS,Y. Using this independent parameter,the transverse amplitude response plot, shown as figure 5 for the presentexperiment, has been replotted in figure 6 using ðU �=f �OS;Y ÞSt. The Williamson &Roshko (1988) map of vortex-shedding modes is overlaid for comparison.

One advantage of figure 5 is that it shows a more distinguishable break in theinitial and upper response branches at U �z2.6. This provides the justificationfor imposing the break in figure 3 as discussed in §3a. The Williamson & Roshko(1988) map of vortex modes was originally developed for forced transverseoscillations of circular cylinders. And it has been shown to collapse pretty wellwith numerous different systems including transverse freely oscillating elasticallymounted cylinders (Govardhan & Williamson 2000).



0 0.5 1.0 1.5 2.0 2.5 3.0

2.52P*

P

P+S

C (P+S)

C (2S) 2S

2P

2P+2Sno synchronizedpattern observed

2.0

1.5

1.0

0.5

Figure 6. Transverse amplitude response plotted against a modified reduced velocity ðU �=f �OS;Y ÞStfor the 2d.f. pivoted cylinder. Data are superimposed on the Williamson–Roshko map of vortexmodes. Open and filled circles correspond to the initial and upper branches of the responseplot, respectively.



Before proceeding, it may be worthwhile to interject two observations aboutthe Williamson & Roshko (1988) map. First of all, the map may not wellcharacterize the vortex-shedding modes for stationary cylinders because fSt/fOS,Y

is infinite and the map was developed for frequency ratios up to fSt/fOS,YZ3.The second and more important observation is that the vortex-shedding

frequency used for normalization is the shedding frequency of a cylinder at rest(i.e. Strouhal frequency) and not the actual shedding frequency associated withthe VIV. Note that for most systems, the vortex-shedding frequencies of acylinder undergoing VIV are usually not equal to the vortex-shedding frequencyof the same cylinder at rest.

For these reasons, then, the transverse amplitude response was plotted versusthe frequency ratio, fVS/fOS,Y, where fVS is the actual vortex-shedding frequencyof the cylinder. This is shown in figure 5. The key feature of this plot is thatfVS/fOS,Y is, in fact, unity.

For the freely oscillating elastically mounted cylinder described in Govardhan &Williamson (2000), the initial branch is associated with the classical von Karmanvortex street wake. This has come to be known as the 2S mode, with two singlecounter-rotating vortices shed from the cylinder per period. The upper branchshows a weaker 2P mode consisting of two pairs of counter-rotating vortices pershedding cycle. In the 2P mode, the second vortex of each pair is weaker than thefirst. The lower branch exhibits a stronger 2P mode whereby each vortex withineach vortex pair possesses almost equal strength.

For the subcritical mass ratio inverted pendulum used in this study, the initialbranch is initiated with what will be referred to as a coalescence of 2S mode. Thisis essentially a Karman-like vortex street pattern (2S) in which the vortices havejust become strong enough, and the frequency is just close enough to thecylinder’s natural frequency, so that weak oscillations occur.



–2

–1

0

1

2

–2

–3

–1

0

1

0 1 2 3 4 0–1 1 2 3

(a)

(b) ( f )

(e)

4 3 2

22

3 2

3 4 3 2 23

3

22

–2–2–7–5

–2

3

2

4

53

3

–3 –4

–2–3 2 4

–2

–1

0

1

2

0 1 2 3 4

(c)

(d )

5 233 2

456

32

22

32 5 3 4

4

Figure 7. Coalescence of (a,b) 2S mode, (c,d ) in-line mode and (e, f ) 2S mode observed at U �Z1.22,1.54 and 2.37, respectively, through both DPIV and LIF images. Flow is from left to right. Solid anddashed lines on DPIV images represent positive and negative vorticity, respectively. Vorticity contourlevels, 6D/U, are as labelled on the images. The upper and lower shear layers on LIF images areilluminated with fluorescein (green) and rhodamine (orange) dyes, respectively.



At slightly higher reduced velocities, a symmetric vortex-shedding modedevelops. This can be envisioned as the periodic shedding of starting vortex pairs,which, in turn, gives rise to in-line oscillations. This ‘in-line vortex-shedding’mode is not uncommon to 2d.f. and in-line oscillating cylinders but does not existfor transversely vibrating cylinders.

With increasing U �, flow returns to a more typical antisymmetric 2S mode.At the high end of the initial branch, vortex shedding begins to exhibitcharacteristics of a 2P mode. Descriptions of the cylinder motions associatedwith these different vortex-shedding modes are provided in §3c.

Figure 7 shows the coalescence of 2S mode, in-line mode, and 2S mode.Included in the figure are side-by-side comparisons of instantaneous DPIV vectorfields and LIF images. In each image, flow is from left to right. While the imagesin figure 7 are shown in black and white, a complete set in colour may be found inLeong (2005).

Representative vector fields and flow visualization images are also providedfrom experiments in the upper branch. These are shown in figure 8. The upperbranch starts with the 2P mode for U �O2.6. Around the maximum of theamplitude response plot, vortex patterns are best described as a PCS mode. Thisis characterized by a pair of counter-rotating vortices shed from one side of thecylinder followed by and a single vortex shed from the other side. Veryinterestingly, there does not appear to be any synchronized vortex pattern at thepeak of the upper branch.



0

–3

–2

–1

0

–1

0

1

2

3

1 2x/D

3 4

single vortex (S) single vortex (S)

weaker vortex pair (P)stronger vortex pair (P)

0 1 2x/D

3 4

–3

3

3

2

2

38 33

3

5

5

23664

4

2

2

4 8

3

4

–4 2

5 –86

–2

34

53

2

8

74

2

5

3

2

24

23

2

387

3

25

–4

3

2

(a)

(c)

(b)

Figure 8. (a) Weaker and (b) stronger PCS mode observed at U �Z3.57 and 4.19, respectively.(c) The vortex pattern for PCS mode. Vortex patterns in the dashed and the dot-dashedrectangular boxes are the ones observed in (a,b), respectively.



Neither the PCS mode nor the desynchronization of vortices is observed forclassical elastically mounted cylinders. This is probably owing to the muchsmaller amplitude response. A weaker PCS mode, shown in figure 8a, wasobserved at lower values of U �. For these flows, the second vortex of the pairwas notably weaker than the first. At larger values of U �, a stronger PCS modewas observed. In those cases, both vortices within a pair were approximatelyequal in strength. This is shown in figure 8b.

The ‘desynchronization’ of vortices is so named because the Karman vorticesbreak up multiple times (at least in a two-dimensional sense). When this occurs,eddy sizes may appear quite large with seemingly random patterns of vorticitywithin each vortex core. Vorticity magnitudes within these vortices can,however, be very high.

It should be pointed out that the desynchronized vortices occur for VIV(as opposed to forced vibration experiments) when the cylinder oscillationamplitudes are very large, that is, when A�

YO1 and fSt/fOS,Yz1. This occurs





only for small mass ratios where the cylinder does not enter into a lock-in type ofresponse. One can conjecture that this desynchronization phenomenon is relatedto the energy transfer between fluid and structure which drives and sustains thelarge amplitude oscillations.

In summary, the Williamson & Roshko (1988) map of vortex modes, ingeneral, also holds for 2d.f. pivoted cylinders below critical mass ratio. The keyexception is that there does not appear to be a 2P mode in the initial branch. Onemust keep in mind that the inverted pendulum will necessarily induce threedimensionalities in the flow. This may well result in different vortex-sheddingmodes along the span of the cylinder. However, in this study experiments wereconducted only at the cylinder mid-height; the study of three dimensionality isdeferred for later study.

(c ) Flow regimes

The previous two sections were focused on the structural (i.e. frequency andamplitude) response, §3a, and fluid dynamics (i.e. vortex dynamics), §3b, of thelow mass ratio inverted pendulum. In this section, x–y trajectories traced out bythe cylinder are examined. In particular, the phase angle q(t) between theinstantaneous streamwise x(t) and transverse y(t) displacements is examined.Phase angle was defined in equation (1.1a).

Figure 9 shows characteristic trajectory shapes for a number of different reducedvelocities. Each trajectory was developed using approximately 120 s of positionsensor data sampled at 300 Hz. At the lowest speed, U �Z1.54, the sampling periodwas sufficiently long to capture 55 complete cylinder oscillation cycles.

The initial branch exhibits two different types of oscillations: (i) unsteady quasi-in-line oscillation and (ii) figure-C like motions. An interesting feature in figure 9 isthat the onset of figure-eight motions coincides with the beginning of the upperbranch. This result, reported by Leong et al. (2003a,b), provides further justificationfor imposing a break at U �Z2.6 in the amplitude response plot, figure 3. It isimportant to observe, however, that not all figure-eight shapes are identical.

The trajectories shown in figure 9 are notably different from those reported byJauvtis & Williamson (2004) for a 2d.f. elastically mounted cylinder. While theyobserved in-line oscillations at low U �, they also reported figure-eight typemotions in the initial branch. Oscillations in their super-upper branch werealso predominantly figure eights. Cylinder motions at the apex of the super-upper branch, however, were described as a figure-C or crescent shape. Finally,Jauvtis & Williamson (2004) reported that oscillations in the lower branch werequasi-transverse reflecting the very low streamwise oscillation amplitudes.

In order to better understand the relationship between streamwise andtransverse oscillation frequencies, spectra of the cylinder position data werecomputed. A unique feature from the spectral analysis is that for all cases except atlow U � (i.e. U �Z1.54) when the cylinder oscillations just begin, the ratio ofstreamwise to transverse oscillation frequency is unity, i.e. fOS,X/fOS,YZ1. It isgenerally accepted that in 2d.f. systems, the oscillation frequency in the streamwisedirection will be twice the transverse oscillation frequency (i.e. fOS,XZ2fOS,Y). Thisis simply due to the fact that for a figure-eight or C-shape trajectory, the cylindermust move back and forth twice for every transverse oscillation.



2.5

2.0

1.5

1.0

0.5

0 1 2 3 4 5 6 7 8 9

Figure 9. Trajectory shapes (Lissajous figures) superimposed on amplitude response curve for 2d.f.pivoted cylinder with mass ratio, m�Z0.45, and mass damping, (m�CCA)zZ0.0841. Note that theshapes are not drawn to scale with each other.



At U �Z1.54, another significant feature is that there are two competingfrequencies for the cross-stream oscillation: the cylinder oscillation frequency andthe Strouhal frequency. This has not been observed for 2d.f. elastically mountedcylinders. However, Chen & Jendrzejczyk (1979), who worked on a cantileveredcylinder, showed that the streamwise oscillation frequency was similar to cross-stream frequency for their lowest normalized velocity case. They did not,however, show two competing frequencies for that case.

AtU �z1.95, transverse oscillations begin to exhibit quasi-periodic beating. Thisis also a characteristic of initial branch amplitude response for classical elasticallymounted cylinders. At the same time, streamwise oscillations become intermittent.

4. Conclusions

The 2d.f. VIV of a subcritical mass ratio cylinder, attached at one end by a smalldiameter vertical pin, was examined using LIF, DPIV and position sensors. Thegoal of this study was to assess differences between the inherently three-dimensional FSI of the 2d.f. inverted pendulum and the nominally two-dimensionalVIV of an elastically mounted cylinder. Key observations and conclusions drawnfrom this work are summarized below.

First, significantly high transverse amplitude response, A�Y z2, in the upper

branch with an even higher streamwise peak amplitude of A�X z2:5 has been

identified. The present results show that there is no lower branch for this systemand the transverse upper branch extends well beyond the maximum reducedvelocity of this study, if not to U �0N. The vortex-shedding frequency for thissystem matches the transverse oscillation frequency for ReR3000. The corres-ponding Strouhal number is 0.16G9%.





The cylinder wake exhibits PCS and desynchronized vortex patterns, whichare both uncharacteristic of 2d.f. VIV for classical elastically mounted cylinders.Very interestingly, the wide resonance regime is dominated by desynchronizedvortices. In-line vortex shedding, which is characterized by symmetric roll up ofvortices on opposite sides of the cylinder, was also observed at low U �. Ingeneral, the Williamson–Roshko map of vortex-shedding modes described thepinned cylinder. The principal difference may be found in the initial branchwhere one finds only the coalescence of 2S mode, 2S mode and in-line modes.That is, there is no 2P mode.

There are two modes of oscillation in the initial branch: unsteady quasi-in-lineand C-shaped (crescent). Immediately beyond the initial branch, the upperbranch is dominated by figure-8-like motions. The quasi-in-line oscillations occurfor 1.54!U �!1.74, while the C-shaped oscillations were observed in the range1.95!U �!2.16.

In short, there are distinct differences between the pinned cylinder and theclassical elastically mounted cylinder. These differences, both structural and fluiddynamics, must be incorporated when modelling 2d.f. VIV of cylinders withsubcritical mass ratios.

Support for this research from the Office of Naval Research through Dr Thomas Swean is gratefullyacknowledged. The authors would also like to thank Mr John Petrowski for his technical help withthe position sensor instrumentation for this investigation.

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