Two-degree-of-freedom VIV of circular cylinder with variable naturalfrequency ratio: Experimental and numerical investigations

Narakorn Srinil n, Hossein Zanganeh, Alexander DayDepartment of Naval Architecture and Marine Engineering, University of Strathclyde, Henry Dyer Building, Glasgow G4 0LZ, Scotland, UK

a r t i c l e i n f o

Article history:Received 21 May 2013Accepted 31 July 2013Available online 4 September 2013

Keywords:Vortex-induced vibrationCross-ow/in-line motionCircular cylinderExperimental investigationNumerical prediction

a b s t r a c t

Slender offshore structures possess multiple natural frequencies in different directions which can lead todifferent resonance conditions when undergoing vortex-induced vibration (VIV). This paper presents anexperimental and numerical investigation of a two-degree-of-freedom VIV of a exibly mounted circularcylinder with variable in-line-to-cross-ow natural frequency ratio. A mechanical spring-cylindersystem, achieving a low equivalent mass ratio in both in-line and cross-ow directions, is tested in awater towing tank and subject to a uniform steady ow in a sub-critical Reynolds number range of about21035104. A generalized numerical prediction model is based on the calibrated Dufng-van der Pol(structure-wake) oscillators which can capture the structural geometrical coupling and uid-structureinteraction effects through system cubic and quadratic nonlinearities. Experimental results for sixmeasurement datasets are compared with numerical results in terms of response amplitudes, lock-inranges, oscillation frequencies, time-varying trajectories and phase differences of cross-ow/in-linemotions. Some good qualitative agreements are found which encourage the use of the implementednumerical model subject to calibration and the sensitivity analysis of empirical coefcients. Moreover,comparisons of the newly-obtained and published experimental results are carried out and discussed,highlighting a good correspondence in both amplitude and frequency responses. Various patterns ofgure-of-eight orbital motions associated with dual two-to-one resonances are observed experimentallyas well as numerically: the forms of trajectories are noticed to depend on the system mass ratio, dampingratio, reduced velocity parameter and natural frequency ratio of the two-dimensional oscillating cylinder.

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1. Introduction

Vortex-induced vibration (VIV) has received a considerableamount of interest over the years due to the variety of nonlinearphenomena governed by the uid mechanics, structural dynamicsand uid-structure interactions. In many ocean and offshoreengineering applications, VIV continues to be of great concern inthe context of fatigue analysis, design and operation of deep waterstructures exposed to ocean currents. From a theoretical andpractical viewpoint, both experimental tests and numerical pre-diction models capable of capturing VIV occurrences andbehaviors in a wide range of both the hydrodynamics and thestructural parameters are important. However, in spite of manypublished studies, the vast majority of the research literature hasfocused on one-dimensional cross-ow VIV of a circular cylinderfor which the transverse response is typically observed to bethe largest (Bearman, 2011; Sarpkaya, 2004; Williamson andGovardhan, 2004), and on the related semi-empirical modeling

of a 1-degree-of-freedom (DOF) cross-ow-only VIV (Gabbai andBenaroya, 2005). While there are some recent computational uiddynamics and ow visualization studies to advance the compre-hension of VIV phenomena (Bao et al., 2012; Jauvtis andWilliamson, 2003; Jeon and Gharib, 2001; Williamson andJauvtis, 2004), experimental investigations and comparisons withnumerical prediction results for two-dimensional in-line (X) andcross-ow (Y) or 2-DOF VIV are still rather limited (Hansen et al.,2002; Pesce and Fujarra, 2005; Stappenbelt and ONeill, 2007) andtherefore needed to be further addressed comprehensively.

In this study, new experimental VIV results of a 2-DOF circularcylinder with equivalent mass ratio in both XY directions(mnx mny) and variable in-line-to-cross-ow natural frequencyratio (fn fnx/fny) are presented and compared with the associatednumerical outcomes predicted by new nonlinear structure-wakeoscillators (Srinil and Zanganeh, 2012). Some insightful VIVaspects are also discussed in the light of other published experi-mental results with variable fn but mnxam

ny (Dahl et al., 2006).

Note that the condition of mnx mny is more relevant in practicethan that of mnxam

ny to real cylindrical offshore structures includ-

ing risers, mooring cables and pipelines. The fn variation is also ofpractical relevance because such a distributed-parameter system

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0029-8018/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.oceaneng.2013.07.024

n Corresponding author. Tel.: +44 1415483463; fax: +44 1415522879.E-mail address: narakorn.srinil@strath.ac.uk (N. Srinil).

Ocean Engineering 73 (2013) 179194

contains an innite number of natural frequencies in differentdirections entailing various fn (Srinil and Rega, 2007; Srinil et al.,2007). These can result in different lock-in or resonant conditionswith the vortex shedding frequencies of the uctuating lift anddrag forces. As the drag oscillation has double the frequency of thelift oscillation, a perfect two-dimensional resonance case mightoccur when fn2. This circumstance could lead to a large-amplitude response for a system with low mass and damping.

Recent experimental studies have highlighted some interestingfeatures of 2-DOF VIV of circular cylinders and meaningful con-tributions from the in-line VIV to the overall dynamics, dependingon several control parameters. In general, the freedom of thesystem to oscillate in the in-line direction can cause an increase ofthe cross-ow response amplitude and widen the lock-in range(Moe and Wu, 1990; Sarpkaya, 1995); it has been suggested thatthese effects may result from an enhanced correlation of thetransverse force along the cylinder span (Moe and Wu, 1990).With respect to the ow eld visualization, a new 2 T (two ofvortex triplets) wake mode has been observed for the cylinderwith signicant combined XY motion (Williamson and Jauvtis,2004) in addition to the typical 2 S (two single vortices) and 2 P(two vortex pairs) modes dened in the Y-only cylinder motioncase (Khalak and Williamson, 1999). In the framework of 2-DOFforced vibration where the amplitudes and oscillation frequencies(fox and foy) of the cylinder are specied a priori, Jeon and Gharib(2001) found that, in the case of fox/foy2, a small amount of in-line motion can inhibit the vortex formation of the 2 P mode. Theyalso suggested a possible change in the relative phases betweenthe lift and drag forces: this implies the possible energy transferbetween the oscillating body and the wake forces in the freevibration case. Recently, Laneville (2006) proposed that the levelof XY motion depends on a ratio of the time derivatives betweenin-line and cross-ow responses.

With mnx mny and fn1, Jauvtis and Williamson (2004) showedthat there is a slight inuence on the cross-ow response of thecylinder with mn46 when comparing the results obtainedbetween 1- and 2-DOF models. When mno6, there is a super-upper branch in the cross-ow response with the peak amplitudeAy/DE1.5 coexisting with the in-line response with the peakamplitude Ax/DE0.3, along with response jump and hysteresis

phenomena. Similar nonlinear responses and ranges of maximumAx/D and Ay/D have been experimentally reported by Stappenbeltet al. (2007) and Blevins and Coughran (2009), and numericallycaptured by Zhao and Cheng, 2011. A two-dimensional lock-inrange is found to be mainly inuenced by the variation of the massratio (Stappenbelt et al., 2007). However, both mass (Stappenbeltet al., 2007) and damping (Blevins and Coughran, 2009) para-meters can inuence on 2-DOF peak amplitudes as in the 1-DOFcases (Khalak and Williamson, 1999).

With mnxamny and f

na1, different qualitative and quantitativefeatures of 2-DOF VIV responses appear. In particular, a two-peakcross-ow response has been noticed by Sarpkaya (1995) and Dahlet al. (2006) with fn2 and 1.9, respectively. Dahl et al. (2010)further highlighted various gure-of-eight patterns in differentsubcritical and supercritical Reynolds number (Re) ranges(1.5104oReo6104 and 3.2105oReo7.1105) anddescribed a gure-eight occurrence as a representation of dualresonance. Under this dual resonance, the frequencies of theunsteady drag and lift forces are resonantly tuned with fox and foy,respectively, such that fox/foyE2. In addition, a large third harmo-nic component of the lift force was observed although the maincross-ow response was primarily associated with the rst-harmonic lift force. This is in agreement with a direct numericalsimulation work by Lucor and Triantafyllou (2008) and a labexperiment of a exible cylinder by Trim et al. (2005).

In spite of the above-mentioned studies, the most practical caseof mnx mny and variable fn has not been thoroughly investigated.For a particular ow with very low Re150 and zero structuraldamping, Bao et al. (2012) recently have performed direct numer-ical simulations of a circular cylinder with fn1, 1.25, 1.5, 1.75 and2. They showed dual resonances in all fn cases and illustrated howthe oscillating drag component is maximized when fn2 with theappearance of the P+S vortex wake mode associated with themaximum in-line response. Nevertheless, more experimental andnumerical investigations in a higher Re range are still neededalong with the improvement of relevant prediction models.

The main objectives of the present study are to (i) experimen-tally investigate 2-DOF VIV of a exibly mounted circular cylinderwith mnx mny and variable fn; (ii) compare the obtained experi-mental results with numerical prediction outcomes in order to

Nomenclature

Ax/D, Ay/D Dimensionless in-line and cross-ow amplitudesAxm/D, Aym/D Dimensionless maximum attainable amplitudesCD, CL Fluctuating drag and lift coefcients of an oscillating

cylinderCD0, CL0 Fluctuating drag and lift coefcients of a stationary

cylinderCM Potential added mass coefcientD Cylinder diameterfn Cylinder in-line to cross-ow natural frequency ratioFD, FL Fluctuating drag and lift forcesFx, Fy Hydrodynamic forces in streamwise and transverse

directionsfnx, fny Natural frequency in still water of cylinderfox, foy Dominant frequency of the oscillating cylinderLc (Lp) Cylinder submerged (pendulum) lengthLc/D Aspect ratioMD, ML System mass parametersmf Fluid added massms Cylinder massp, q Reduced vortex drag and lift variablesRe Reynolds number

St Strouhal numberSGX, SGY Skop-Grifn mass-damping parametert Dimensionless timeT Dimensional timeV Uniform ow velocityVr Reduced velocity parameter~Y Dimensional transverse displacementX, Y Streamwise and transverse coordinatesx, y Dimensionless in-line and cross-ow displacementsx, y, x, y Dimensionless geometrically-nonlinear

coefcients Stall parameterx, y Wake oscillator coefcients Direction of effective lift (drag) force measured from Y

(X) axisx, y Phase anglesx, y Combined uid-structural damping termsx, y Wake-cylinder coupling coefcients, mn Mass ratiosx, y, Structural reduced damping or damping ratios Fluid density Ratio of vortex-shedding to cylinder cross-ow

natural frequencies

N. Srinil et al. / Ocean Engineering 73 (2013) 179194180

improve the newly-proposed coupled oscillators (Srinil andZanganeh, 2012) with a proper choice of system coefcients; and(iii) compare various observations from the present experimentalcampaign with other studies. To calibrate model empirical coef-cients, particular attention is placed on the determination ofcylinder maximum attainable amplitudes, associated lock-inranges (both the onset and the end of synchronization), two-dimensional orbital XY motions and oscillation frequencies, bycomparing various cases of fna1 and fn1. These analysis out-comes based on a 2-DOF rigid cylinder could be practically usefulin the improvement of VIV prediction tools and design guidelinesfor cross-ow/in-line VIV of exible cylinders with multi DOF andvarious fn, as conducted, for instance, by Srinil (2010, 2011) forcross-ow-only VIV cases.

This paper is structured as follows. In Section 2, details of thenew experimental arrangement used for the 2-DOF VIV study of aexibly mounted circular cylinder are presented along with thetest matrix. The associated numerical prediction model andgoverning equations are explained in Section 3. Depending onsystem parameters, comparisons of experimental and numericalprediction results of six measurement datasets are made inSection 4 which also demonstrates the sensitivity analysis andthe inuence of key geometrical parameters on VIV predictions. InSection 5, various new and published experimental results arecompared and discussed. Some insightful aspects from a mathe-matical modeling, numerical prediction and experimental view-point are summarized in Section 6. The paper ends with theconclusions in Section 7.

2. Experimental arrangement and test matrix

A new experimental test rig for the study of 2-DOF VIV of aexibly mounted, smooth and rigid circular cylinder subject to a

uniform steady ow has been developed for use in the towing tankat the Kelvin Hydrodynamics Laboratory (KHL) of the University ofStrathclyde, Glasgow, UK. The design of this rig was motivated by arecent collaborative work conducted at the University of Sao Paulo,Brazil (Assi et al., 2012). The KHL tank has dimensions of 76 m longby 4.57 m wide; water depth can be varied from 0.52.3 m. Thetank is equipped with a self-propelled towing carriage on whichthe experimental apparatus can be rmly installed, and a variety ofdamping systems to calm the water rapidly between runs.

Fig. 1 displays the experimental set-up where the test cylinderis mounted vertically and connected at its upper end to a longaluminum pendulum with total length of about 4.1 m (Lp). Thependulum is attached to the supporting framework via a high-precision universal joint at the top of the frames. The test cylinderadopted in the present study is made of thick-walled cast nylontube, having an outer diameter (D) of 114 mm and a fullysubmerged length (Lc) of 1.037 m. The lower end of the cylinderis located 50 mm from the bottom of the tank, and the upper endis located 50 mm beneath the static free surface. Such lower endcondition was deemed to produce a negligible effect on the peakamplitudes (Morse et al., 2008). It should be noted that thependulum effect on the uniformity of the local ow eld isbelieved to be insignicant since the maximum roll and pitchangles of the cylinder about the universal joint were found to beonly about 21 in all tests. The blockage is about 2.5% and the aspectratio (Lc/D) of the cylinder is about 9 being comparable to Lc/D insome recent studies (Jauvtis and Williamson, 2004; Sanchis et al.,2008; Stappenbelt et al., 2007).

The mechanical system is restrained to allow the cylinder tooscillate freely with arbitrary amplitudes in both in-line (X) andcross-ow (Y) directions by using two pairs of coil springs (withlengths of about 50 cm) rearranged perpendicularly in the hor-izontal XY plane. Each spring obeys Hooke's law (i.e. with alinear constant stiffness K); nevertheless, as the cylinder oscillates

Fig. 1. Experimental model of a exibly mounted circular cylinder undergoing 2-DOF VIV.

N. Srinil et al. / Ocean Engineering 73 (2013) 179194 181

two-directionally due to VIV, the assembly creates the geometri-cally nonlinear coupling of cross-ow/in-line displacements. Thesenon-linear effects are accounted for in the numerical predictionmodel (see Section 3). Measurement of cylinder motions wascarried out using a Qualisys optical motion capture system witha xed sampling frequency of 137 Hz. Four infrared cameras wereused to identify and optimize the three-dimensional positions ofseveral reective markers mounted on the pendulum, and calibra-tion was performed with an average residual across all cameras ofless than 0.3 mm. Key outputs were the roll and pitch angles witha degree resolution of 0.001. In contrast with traditional displace-ment measurement instruments, the non-contact nature of thissystem ensures that no unwanted additional damping or restoringforces are applied to the pendulum. Note that establishing thesystematic uncertainty of measurements of the optical motion-tracking systems is more challenging than the contact-basedmeasurements since this uncertainty depends upon the positionand orientation of both the cameras and the reective markers.Procedures for formal assessment of uncertainty for use in futurecampaigns are currently being developed. The acquisition time foreach steady-state response was about 2 min and the waiting timebetween each two consecutive measurements was about 5 min. Atrailing wheel of very accurately dened circumference wasattached to the carriage, and the angular velocity of the wheelwas determined using a high-precision magnetic encoder and acounter-time which outputs the velocity signal representing thecarriage speed.

When the cylinder is towed, the mean drag causes a mean in-line displacement of the cylinder in the ow direction; however,only the uctuating displacements are of main interest. In orderthat the mean position of the cylinder for the measurements isvertical in the in-line direction to avoid the possible cylinderinclination effect on VIV, this displacement was initially adjustedby pre-tensioning the upstream in-line spring such that thecylinder mean position, as measured using the Qualisys system,remains nearly vertical during the VIV test. While the pre-tensionis meaningful in the evaluation of hydrodynamic forces which isbeyond the scope of this study, overall VIV measurements wereconducted after such an adjusted equilibrium position wasachieved. A thorough treatment of a similar spring arrangementis contained in Stappenbelt (2010).

As actual slender structures have multiple natural frequenciesin different directions (Srinil et al., 2007), attention in the presentstudy is placed on the cylinder model with varying ratios betweenin-line (fnx) and cross-ow (fny) natural frequencies in still water(fn fnx/fny). This was achieved in practice by using springs withdiffering stiffness. The reported experimental cross-ow (Ay/D)and in-line (Ax/D) amplitudes normalized by the cylinder diameterare referred to as the maximum displacements at the bottom tip ofthe cylinder. Based on a free decay test in air, the experimentalapparatus with and without the cylinder-spring system was foundto be lightly damped at around 0.5% and 0.2% of the criticaldamping, respectively. A series of free decay tests in calm waterwere performed to identify fnx, fny and the associated dampingratios (x, y) in both X and Y directions. Small initial displace-ments were assigned independently in each X or Y direction toensure that no geometric nonlinear coupling took place: fnx and fnywere obtained from the free damped responses whose maximumamplitudes were about 0.1 of the diameter in all datasets. Therepresentative averaged x and y values have been evaluated bysubtracting the uid damping component from the total dampingof the system (Sumer and Fredsoe, 2006).

Table 1 summarizes a test matrix of 6 datasets (labeled asKHL1-KHL6) in which two mass ratios (mnx mny mn1.4 and 3.5)of the cylinder are considered. These mn were considered to below, being less than 6 (Jauvtis and Williamson, 2004), to

encourage the effect of in-line VIV and the overall large-amplitude responses. Due to the amplitude-dependence natureof the structural and uid-added damping in water, variable x andy values (between 15%) are reported. The combined mass-damping values are in the range of 0.014omno0.081. Themn1.4 case (KHL1-KHL5) corresponds to the initial apparatussetup, whilst in later tests mn was increased by adding lumpmasses to the rig system such that mn3.5 (KHL6). Such anincreased mn case allows us to evaluate the prediction model(Section 4) whose empirical coefcients have been calibratedbased on the experiments with varying mn (Stappenbelt et al.,2007). For mn1.4, ve tests with different fnE1.0, 1.3, 1.6 and1.9 were performed to justify the occurrence of a dual 2:1resonance regardless of fn and as the drag uctuation has doublethe frequency of the lift uctuation. In all datasets, the reducedvelocity Vr range in which VrV/fnyD was about 0oVro20,corresponding to 2103oReo5104 of the sub-critical owsand the ow speed V of 0.020.6 m/s. This considered rangeencompassed a Vr value at which the peak amplitude occurred.Some tests were repeated in the neighborhood of peaks andresponse jumps.

With the aim of comparing our experimental results with otherpublished studies by also focusing on the variation of fn, theexperimental model performed at the MIT towing tank (Dahlet al., 2006) is herein considered. Their test matrix, comprising6 datasets (labeled as MIT1-MIT6) with Lc/D of 26,0.041omno0.353, and 11103oReo6104, is given inTable 2 in comparison with KHL datasets in Table 1. It is worthnoting that both experiments have similar x and y values in therange of about 16%. The role of damping will be again discussedin Section 6. Apart from being different in the experimentalarrangement and procedure, in Lc/D, and variable and fn values,the main distinction between KHL and MIT datasets is due to thespecied mass ratios: mnx mny in this study whereas mnxamny inDahl et al. (2006). This aspect along with other observations willbe taken into account in the comparison of results in Section 5.

3. Numerical model with nonlinear coupledstructure-wake oscillators

The capability to reasonably model and predict the VIV struc-tural response excited by the unsteady ow eld has been a majorchallenge to modelers and engineers for many years. Recently, a

Table 1KHL experimental data with variable mn, and fn.

Dataset fny (Hz) fnx (Hz) y (%) x (%) mny mnx f

n

KHL1 0.312 0.316 1.0 4.7 1.4 1.4 1.01KHL2 0.218 0.281 1.5 1.0 1.4 1.4 1.29KHL3 0.262 0.419 1.6 1.0 1.4 1.4 1.60KHL4 0.203 0.376 1.8 1.2 1.4 1.4 1.85KHL5 0.192 0.192 2.0 3.1 1.4 1.4 1.00KHL6 0.223 0.223 1.5 2.3 3.5 3.5 1.00

Table 2MIT experimental data with variable mn, and fn.

Dataset fny (Hz) fnx (Hz) y (%) x (%) mny mnx f

n

MIT1 0.715 0.715 2.2 2.2 3.8 3.3 1.00MIT2 0.799 0.975 1.3 1.7 3.9 3.8 1.22MIT3 0.894 1.225 1.1 2.5 3.9 3.7 1.37MIT4 0.977 1.485 1.6 3.2 4.0 3.6 1.52MIT5 0.698 1.166 2.6 2.9 5.5 5.3 1.67MIT6 0.704 1.338 6.2 2.5 5.7 5.0 1.90

N. Srinil et al. / Ocean Engineering 73 (2013) 179194182

new semi-empirical model for predicting 2-DOF VIV of a spring-mounted circular cylinder in a uniform steady ow has beendeveloped and calibrated with some published experimentalresults in the cases of mnx mny and fn1 (Srinil and Zanganeh,2012). As an extended study, the same model is improved andutilized to predict cross-ow/in-line VIV responses in the cases ofvariable fn, based on calibration with new in-house experimentalresults. Some modied empirical coefcients will be then sug-gested in Section 4 for a future use.

A schematic model of the cylinder restrained by two pairs ofsprings to oscillate in X and Y directions is displayed in Fig. 2a. Thekey aspect in the formulation of system equations of motions is tocapture the quadratic relationship between in-line and cross-owdisplacements (Vandiver and Jong, 1987). Following Wang et al.(2003), the two-directional unsteady uid forces are exerted onthe oscillating cylinder as opposed to the stationary one, by alsoaccounting for the relative velocities between the incoming owand the cylinder in-line motion. As a result, the instantaneous lift(FL) and drag (FD) forces coincide with an arbitrary plane makingup an angle of with respect to the Y and X axes, respectively. Twocases can be realized depending on whether is counterclockwise(Fig. 2b) or clockwise (Fig. 2d). From our numerical simulationexperience, it has been discovered that such direction plays a keyrole in the ensuing phase difference between cross-ow and in-line oscillations and, correspondingly, the gure-of-eight appear-ing shape. In general, the orbital plot exhibits a gure-eighttrajectory with tips pointing upstream with a counterclockwise model (e.g. Fig. 2c) or downstream with a clockwise model (e.g.Fig. 2e). As both cases have been experimentally observed in theliterature depending on the system parameters, they are hereinaccounted for in the improved model formulation. Note that onlythe counterclockwise model was proposed in (Srinil andZanganeh, 2012).

Consequently, by assuming a small , the unsteady hydrody-namic forces Fx and Fy may be simplied and expressed afterresolving FL and FD into the X and Y directions as

Fx FD cos 8FL sin FD8FL _~Y=V ; 1

Fy FL cos 7FD sin FL7FD _~Y=V ; 2

where ~Y is the dimensional transverse displacement, a dot denotesdifferentiation with respect to the dimensional time T,FD DV2CD=2; FL DV2CL=2; is the uid density, CD and CLare the time-varying drag and lift coefcients, the minus (positive)and positive (minus) sign in Eq. (1) (Eq. (2)) corresponds to thecase of counterclockwise and clockwise , respectively.

By assigning the uid vortex variables as p2CD/CD0 andq2CL/CL0 (Facchinetti et al., 2004) in which CD0 and CL0 are theassociated oscillating drag and lift coefcients of a stationarycylinder (assumed as CD00.2 (Currie and Turnbull, 1987) andCL00.3 (Blevins, 1990)), the time variation of p and q may beassumed to follow the self-excitation and -limiting mechanism ofthe van der Pol wake oscillator (Bishop and Hassan, 1964). Byintroducing the dimensionless time tnyT and normalizing thedisplacements with respect to D, the nonlinearly coupled equa-tions describing the in-line (x) and cross-ow (y) oscillations ofthe cylinder subject to the uctuating uid force components (p, q)are expressed in dimensionless forms as

x x _x f n2x xx3 xxy2 MD2p82ML2q_y=Vr; 3

p 2xp21 _p 42p x x; 4

y y _y y yy3 yyx2 ML2q72MD2p_y=Vr; 5

q yq21_q2q y y; 6in which MD CD0=162St2, ML CL0=162St2, ms mf =D2; x 2xf n =, y 2y =; StVr, mfD2CM/4, msis the cylinder mass, mf the uid added mass, CM the potential addedmass coefcient assumed to be unity for a circular cylinder (Blevins,1990), St the Strouhal number, the stall parameter which is directlyrelated to the sectional mean drag coefcient and assumed to be aconstant equal to 0.8 (Facchinetti et al., 2004), and co-subscripts xand y identify properties in these directions. Note that the massratio denition in the literature is variable but the widely recognizedone with mn 4=CM is herein considered (Williamson andGovardhan, 2004).

In contrast to typical VIV models which consider a linearstructural oscillator to describe the cylinder displacement(Gabbai and Benaroya, 2005), Eqs. (3) and (5) account for theeffects of geometric and hydrodynamic nonlinearities on theoscillating cylinder. These equations are so-called Dufng oscilla-tors (Nayfeh, 1993). It is also worth mentioning the following keypoints.

i. Cubic nonlinear terms capture the effect of nonlinear stretch-ing (x3, y3) and physical coupling of cross-ow and in-linedisplacements (xy2, x2y), depending on the geometrical para-meters (x, y, x, y).

ii. Quadratic nonlinear terms q_y; p_y capture the effect of wake-cylinder interaction; these have been found to be responsiblefor the gure-of-eight appearance associated with a dual 2:1resonance (Srinil and Zanganeh, 2012).

X

YV

D

Fx

FyF DF

L

Fx

Fy

FD

FL

Fig. 2. A schematic numerical model of a spring-supported circular cylinder undergoing 2-DOF VIV (a) due to effective lift/drag forces exerted on the oscillating cylinder (b)or (d): sample gure-of-eight trajectories (c) and (e) based on model in (b) and (d), respectively.

N. Srinil et al. / Ocean Engineering 73 (2013) 179194 183

iii. In Eqs. (3) and (5), the maximum cross-ow/in-line amplitudesare unaffected by the choice of since the associated velocitiesare trivial, making q_y=Vr p_y=Vr 0.

iv. In qualitative agreement with a 1-DOF VIV study of Facchinettiet al. (2004), the linear coupling terms based on the cylinderaccelerations x x;y y in Eqs. (4) and (6) have been found toproduce a better 2-DOF VIV prediction than the displacementxx;yy and velocity x _x;y _y models (Zanganeh and Srinil,2012).

v. The mean drag force component is omitted from Eqs. (1) and(3) by assuming that it does not affect the uctuating displace-ment components as considered by Kim and Perkins (2002).However, the nonlinear terms in Eq. (3) can generate the in-line static drift (Nayfeh, 1993) of the cylinder; this drift isdisregarded from the numerical simulations as attention is

placed on the evaluation of the oscillating amplitudecomponents.

The analysis of coupled cross-ow/in-line VIV depends onseveral empirical coefcients (x, y, x, y) and geometricalparameters (x, y, x, y). In this study, x, y, x and y are alsotreated as empirical coefcients to account for the time- andamplitude-dependent uncertainties during the experiment such asthe spatial correlation of vortices along the cylinder span, theuid-added damping, the free surface effect and the wake-cylinderinteraction. Based on calibration with experimental results(Stappenbelt et al., 2007) with fn1 and varying mn, it may beassumed that (Srinil and Zanganeh, 2012, 2013)

y 0:00234e0:228mn: 7

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2.5 5 7.5 10 12.5 15 17.5 200

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Vr

A y /D

A y /D

A y /D

0 2.5 5 7.5 10 12.5 15 17.5 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Vr

A x /D

A x /D

A x /D

KHL1 KHL1

KHL5 KHL5

KHL6 KHL6

Fig. 3. Comparison of numerical (lines) and experimental (symbols) cross-ow and in-line amplitude responses based on KHL data with fn1: blue lines and squares (pinklines and circles) denote maximum (RMS) values; dashed lines denote numerical response jumps. (For interpretation of the references to color in this gure legend, thereader is referred to the web version of this article.)

N. Srinil et al. / Ocean Engineering 73 (2013) 179194184

To reduce the time-consuming task involving the tuning ofindividual coefcients, x0.3, xy12, and xyxy0.7 are initially assumed in the fn1 cases following Sriniland Zanganeh (2012). However, some of these values will bemodied in Section 4 based on the comparisons with experimen-tal results in the fna1 cases. Nonlinear coupled Eqs. (3)(6) can benumerically solved by using a fourth-order RungeKutta schemewith an adaptive time step enabling solution convergence andstability, and with assigned initial conditions at t0 of xy0,pq2 and zero velocities. In all numerical simulation cases, Vr isincreasingly varied in steps of 0.1 and at least 60 cycles of thevortex shedding frequency are accounted for in the evaluation ofsteady-state responses.

4. Comparisons of experimental and numericalprediction results

Experimental and numerical prediction results are nowcompared based on KHL data in Table 1. As typical gure-of-eight orbital motions with tips pointing downstream (e.g.Fig. 2e) are mostly observed in the present experiments, thesystem equations of motions (Eqs. 36) used in numericalsimulations are based on the model conguration shown inFig. 2d. To facilitate the comparison and discussion, two sets ofresults are classied depending on fn: (i) fn1 (KHL1, KHL5,KHL6) and (ii) variable fn with fn1.29 (KHL2), 1.60 (KHL3) and1.85 (KHL4). Both maximum and root-mean-squared (RMS)values of cross-ow (Ay/D) and in-line (Ax/D) amplitudes areevaluated.

Results in the fn1 cases are plotted in Fig. 3 which illustrates afairly good qualitative comparison of numerical (lines) and experi-mental (symbols) responses. From the experiments, pure in-lineresponses are observed in a marginal range of about 2oVro4(Fig. 3b, d and f) whereas coexisting cross-ow/in-line VIVresponses take place in the range of about 4oVro17.5 (Fig. 3aand c) or 4oVro12.5 (Fig. 3e), depending on mn. As expectedfrom both a numerical and experimental viewpoint, both KHL1and KHL5 datasets with the lower mn1.4 exhibit a widersynchronization region. With increasing Vr, some jumps of peakamplitudes from upper to lower branches (Fig. 3cf) are experi-mentally as well as numerically (denoted by vertical dashed lines)observed. These jumps are in agreement with several recentlypublished experimental results of 2-DOF VIV with fn1 (Blevinsand Coughran, 2009; Jauvtis and Williamson, 2004; Stappenbeltet al., 2007).

In view of quantitative comparisons, the highest values ofexperimental and numerical RMS amplitudes are found to becomparable in the range of about 0.91.25 for Ay/D (Fig. 3a, c ande) and 0.10.3 for Ax/D (Fig. 3b, d and f), depending on the systemmass and damping. As regards the maximum attainable responses(Aym/D, Axm/D), Fig. 3 shows a better comparison in the cross-owVIV than in the in-line VIV, with both experimental and numericalresponses providing 1.4oAym/Do1.75. The numerical modelapparently underestimates Axm/D although it predicts well theassociated RMS values. These outcomes could be inuenced by thetemporal modulation of Ay/D and Ax/D. To exemplify this aspect,experimental (dashed blue lines) and numerical (solid pink lines)time histories of y and x responses of KHL1 data with Vr10.9(Fig. 3a and b) and KHL5 data with Vr11.7 (Fig. 3c and d) areplotted in Fig. 4ab and cd, respectively. It is found that, in spiteof the nearly-zero mean values of the time-varying x (about 0.046in Fig.4b and 0.013 in Fig. 4d), experimental in-line responses areseen to have a higher modulation when compared to the asso-ciated numerical ones. In contrast, both experimental and

numerical y responses (Fig. 4a and c) are comparable, exhibitinga much less uctuating signal.

In the case of fna1, experimental and numerical comparisonsof Ay/D and Ax/D are shown in Fig. 5. To also demonstrate the effectof empirical coefcients, two sets of numerical results are plotted:one based on xy12 (solid lines) and the other based onxy15 (dashed lines), while keeping other parametersunchanged. This change in x and y has been motivated by apossible variation of both lock-in ranges and ensuing amplitudes(Srinil and Zanganeh, 2012). With increasing fnsome VIV behaviorsare noticed experimentally. First, the in-line-only responses seemto disappear with increasing fn1.6 (Fig. 5d) and fn1.85 (Fig. 5f).This is in agreement with the numerical prediction. Secondly, bothcross-ow and in-line responses in Fig. 5c and d (fn1.6) andFig. 5e and f (fn1.85) reveal the attening slopes of their upperbranches with amplitudes starting from VrE2.5 and ending atVrE12.5. These amplitude proles are qualitatively similar to theexperimental results of Assi et al. (2009) with fn1.93.

Nevertheless, overall experimental results show Aym/DE1.5and Axm/DE0.5, and the associated excitation ranges are quitecomparable, in all fn cases. Given the similar values of mn, theseimply the negligible effect of varying fn on the maximum responseoutcomes based on this pendulum-spring-cylinder system. Withrespect to numerical comparisons, the predicted Aym/D and Axm/Dare found to be overestimated and the associated upper branchesshow higher slopes being typical for resonance diagrams. Thesereect the difculty in matching numerical and experimentalresults in which several coefcients control the dynamic responsesand some of the inuential parameters are variable, i.e. xay.

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T (s)Fig. 4. Comparison of numerical (pink solid lines) and experimental (blue dashedlines) cross-ow (a, c) and in-line (b, d) time histories: KHL1 data with Vr10.9(a, b) and KHL5 data with Vr11.7 (c, d). (For interpretation of the references tocolor in this gure legend, the reader is referred to the web version of this article.)

N. Srinil et al. / Ocean Engineering 73 (2013) 179194 185

However, with a demonstrated small increment of x and y, thequalitative prediction of lock-in ranges appears to be satisfactorilyimproved. Hence, values of xy15 are hereafter considered.

Next, it is of practical importance to carry out a sensitivitystudy on the numerical model in order to understand the inu-ence of varying parameters on the 2-DOF VIV prediction and thedependence of the latter on fn. To also capture possible qualitativeand quantitative changes, the sensitivity analysis should be per-formed with respect to the parameters related to the greater yresponse (Srinil and Zanganeh, 2012). By ways of examples, thegeometrical coefcient y or y is varied in the numerical simula-tions with fn1.3, 1.6 and 2. In each fn case, mnx mny 1:4 and theaveraged x1.6% and y1% (based on KHL2-4 datasets) areassigned. Contour plots of Ay/D and Ax/D are displayed inFigs. 6 and 7 in the varying y and y cases, respectively.

For each fn, it is seen in Fig. 6 that Aym/D increases (Fig. 6a-c)whilst Axm/D decreases (Fig. 6df) as y increases, with the

associated peaks locating at higher Vr values. These reect boththe quantitative and qualitative inuence of the cubic nonlinearstretching termwhich results in the bent-to-right response as yy3

becomes greater. On the other hand, it is found in Fig. 7 that, as yincreases, both Aym/D (Fig. 7ac) and Axm/D (Fig. 7df) decrease;the associated peaks are slightly bent for lower fn (Fig. a and d) ornearly vertical for higher fn (Fig. 7bc and e-f). These show themostly quantitative effect of the geometric coupling yyx2 term.Based on the above observations, the similar experimentalresponse patterns with comparable y and x in Fig. 5 might bemore inuenced by the displacement coupling terms than thestretching nonlinearities. For this reason, a suitable new xed yvalue (e.g. 1.5) based on Fig. 7 may be suggested to improve thenumerical quantitative comparison with experimental results inFig. 5 whose Aym/DE1.5 and Axm/DE0.5 in all fn cases. Thesesample contour plots might be applicable to a future predictionanalysis once exact geometrical parameters are practically known.

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KHL2 KHL2

KHL3 KHL3

KHL4 KHL4

Fig. 5. Comparison of numerical (lines) and experimental (symbols) cross-ow and in-line amplitude responses based on KHL data with fna1: blue lines and squares (pinklines and circles) denote maximum (RMS) values; dashed (solid) lines with xy15 (12). (For interpretation of the references to color in this gure legend, the reader isreferred to the web version of this article.)

N. Srinil et al. / Ocean Engineering 73 (2013) 179194186

Now, it is interesting to perform numerical and experimentalcomparisons of the time-varying orbital xy motions as well asphase angles because this information could shed some light onhow the uid-cylinder interaction affects the resulting vortex-shedding modes. Corresponding to KHL1-6 results in Figs. 3 and 5,the xy trajectory plots within several cycles of the oscillation aredisplayed in Fig. 8a with some chosen Vr. The normalized xyphase differences (x2y)/ of KHL3 and KHL4 datasets are alsoexemplied in Fig. 8b. Depending on fn, mn and (Table 1) andinitial conditions in both numerical simulations and experiments,various characteristics of gure-of-eight trajectories appear withvariable phase differences between x and y motions. In particular,the crescent shapes are evidenced in the experiments (see bluelines in Fig. 8a) with their tips pointing mostly downstream (allKHL datasets) and occasionally upstream (KHL3 and KHL4 forVro10). The former case justies the use of system equationsbased on the model conguration in Fig. 2d. Similar orbitalmotions have been found in recent 2-DOF VIV experiments ofrigid circular cylinders (Blevins and Coughran, 2009; Dahl et al.,2006, 2010; Flemming and Williamson, 2005; Jauvtis andWilliamson, 2004), and the present study conrms these studieswith both experimental and numerical results.

It is worth noting that experimental orbital motions exhibit ahigh modulation feature of amplitudes whereby the oscillatingcylinder does not follow the same path from cycle to cycle. Thissuggests a strong uid-structure interaction effect associated witha 2:1 resonance during the test. On the contrary, numerical orbital

motions are perfectly repeatable which justify the limit cyclecharacter of the two pairs of coupled Dufng and van der Poloscillators for which stable periodic solutions are attained. Thenumerical model is found to predict quite well overall qualitativebehaviors of the gure-eight appearance due to the associatedquadratic nonlinearities (Srinil and Zanganeh, 2012; Vandiver andJong, 1987). The experimental (squares) and numerical (circles)comparisons of phase differences in Fig. 8b also reveal their goodagreement in the range of about 8oVro14 where responseamplitudes are maximized (Fig. 5). By following the cylindermovement at the top of the gure of eight (Dahl et al., 2007),several gures of eight of KHL3 and KHL4 datasets can be denedas counterclockwise (0ox2yo/2 and 3/2ox2yo2)or clockwise (/2ox2yo3/2) paths. Yet, in the smaller-amplitude ranges of Vro8 or Vr414, a qualitative differencebetween experimental and numerical phase angles is still seen;this suggests possible use of the alternative conguration in Fig. 2bfor the numerical model.

Experimental results in Fig. 8a suggest similar vortex formationpatterns for KHL1, KHL5 and KHL6 with fn1 since the associatedgures of eight are qualitatively similar in all Vr cases. Whenincreasing fn to be about 1.3 (KHL2), 1.6 (KHL3) and 1.9 (KHL4), thegure-eight orbits corresponding to some particular Vr cases arenoticed to be modied and these imply the possible change in thevortex formation patterns (Bao et al., 2012).

A comparison of experimental and numerical oscillation fre-quencies (foy, fox) obtained from the amplitude responses within

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Fig. 6. Sensitivity analysis showing the inuence of geometrical parameter y on cross-ow (a-c) and in-line (df) amplitude responses: fn1.3 (a,d), fn1.6 (b,e), fn2 (c,f).

N. Srinil et al. / Ocean Engineering 73 (2013) 179194 187

the main excitation ranges (Figs. 3 and 5) and normalized by theassociated fny is exemplied in Fig. 9 based on the selected KHL1(fnE1), KHL3 (fn1.6) and KHL4 (fnE1.9) data. Overall goodqualitative agreement is appreciated, with x (Fig. 9b, d and f)and y (Fig. 9a, c and e) frequency responses exhibiting their dual2:1 resonances irrespective of the specied fn. For the testedcylinder with lowmn1.4, the oscillation frequencies of all datasetincrease with increasing Vr due to the decreasing value of thehydrodynamic added mass. This justies the fundamentalmechanism of 2-DOF VIV (Sarpkaya, 2004; Williamson andGovardhan, 2004).

5. Experimental comparisons with other studies

It is of considerable theoretical and practical importance tounderstand the extent to which 2-DOF VIV results are sensitive tothe various arrangements of test rigs and measurement proce-dures. A comparison is now presented between the results of thepresent study and those obtained by Dahl et al. (2006) at MITusing a very different test rig. The comparison between KHL(Table 1) and MIT (Table 2) experimental data are considered bycategorizing the results into four groups depending on thecomparable values of fn as follows:

a) KHL1, KHL5, KHL6 vs. MIT1, all with fnE1,

b) KHL2 vs. MIT2 and MIT3, all with the averaged fnE1.3,c) KHL3 vs. MIT4 and MIT5, all with the averaged fnE1.6,d) KHL4 vs. MIT6, all with fnE1.9.

Comparisons are made in terms of Ay/D and Ax/D diagrams(Fig. 10), the associated oscillation-to-natural frequency ratiosfoy/fny and fox/fny (Fig. 11) and the Grifn plots (Fig. 12) of peakamplitudes vs. the SkopGrifn parameter SGX 23St2mnxx andSGY 23St2mnyy (Skop and Balasubramanian, 1997). Note that thevalue of Aym/D with MIT apparatus was limited to 1.35 (Dahl et al.,2006). fox and foy are the dominant oscillation frequencies obtainedfrom the fast Fourier transform analysis of relevant response timehistories.

With fnE1, overall response amplitudes of KHL1, KHL5, KHL6and MIT1 data show a variation of Aym/D in the range of about1.351.75 (Fig. 10a) and Axm/D in the range of about 0.40.8(Fig. 10b). This disparity of peak responses may in part be due tothe inuence of variable y and x whose values are mostly yax(except MIT1). The KHL6 and MIT1 results with comparable mn

(3.3-3.8) provide a good qualitative agreement with a similarlock-in range of 4oVro12 in which both Ay/D and Ax/D aresimultaneously excited. Good qualitative agreements are alsoappreciated by the comparison of KHL1 and KHL5 data. In theselower mn1.4 cases, the lock-in range is noticed to be broader(4oVro18). This inuence of varying mn on the 2-DOF lock-inrange has recently been highlighted by the experiments of

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Fig. 7. Sensitivity analysis showing the inuence of geometrical parameter y on cross-ow (a-c) and in-line (df) amplitude responses: fn1.3 (a,d), fn1.6 (b, e), fn2 (c,f).

N. Srinil et al. / Ocean Engineering 73 (2013) 179194188

Stappenbelt et al. (2007) and the numerical predictions of Sriniland Zanganeh (2012).

With the averaged fnE1.3 and fnE1.6, the comparison of KHL2,MIT2 and MIT3 data (Fig. 10c and d) and that of KHL3, MIT4 andMIT5 data (Fig. 10e and f) reveal their good qualitative agreementof Aym/D and Axm/D in a small range of about 1.351.5 and 0.40.6,respectively. As previously mentioned, a difference in the lock-inrange between KHL and MIT results is possibly due to theirdifferent mn values, apart from assigning whether mnx mny (KHL)or mnxam

ny (MIT). The effect of variable dampingwhich has been

found to control the response amplitude rather than the lock-inrange (Blevins and Coughran, 2009)might in part again beresponsible for the difference in response peaks as in the previouscase of fn1.

Both qualitative and quantitative differences are now realizedwhen considering the results with the averaged fnE1.9. For theMIT6 data with mnxam

ny and yax, results reveal a two-peak

cross-ow response (Fig. 10g)similar to those reported inSarpkaya (1995) with fn2 (although therein mn and were notreported) with the two Aym/DE1 and 1.1 taking place at VrE5and 8, respectively. Note that the MIT6 in-line response stillexhibits a single peak of Ax/DE0.3 at VrE8 (Fig. 10h). Theseobservations are in contrast with KHL4 results with mnx mny andyax which show single-peak responses in both cross-ow andin-line responses. Recent numerical studies by Lucor andTriantafyllou (2008) have also found only single-peak responseswith mnx mny and yx. Owing to the lower mn and of KHL4data, the associated Aym/DE1.3 (Fig. 10g) and Axm/DE0.5(Fig. 10h) are greater and the associated lock-in range is wider ofabout 4 oVro18. These qualitatively justify the present experi-mental results.

In Fig. 11, overall comparisons of foy/fny (a, c, e and g) and fox/fny(b, d, f and h) plots highlight good correspondence between KHLand MIT results. In general, foy/fny values vary from 0.5 to 2 and

fox/fny values vary from 1 to 3, with increasing Vr. These imply thevariation of hydrodynamic added mass caused by VIV; that is, itsvalue is rst positive when foy/fnyo1 and fox/fnyo2, being zero atfoy/fnyE1 and fox/fnyE2, and then becoming negative when foy/fny41 and fox/fny42. Regardless of the assigned fn, the fox/foyvalues in Fig. 11 are nearly commensurable to 2:1 ratios in variousVr cases. These conrm the existence of dual resonance conditions(Bao et al., 2012; Dahl et al., 2006, 2010) and demonstrate theintrinsic quadratic relationships between in-line and cross-owresponses (Vandiver and Jong, 1987) corresponding to the variousgure-of-eight orbital motions traced out in Fig. 8a in comparisonwith numerical prediction results. These outcomes also conrmother recent experimental results of circular cylinders undergoing2-DOF VIV with fn1 (Blevins and Coughran, 2009; Jauvtis andWilliamson, 2004; Sanchis et al., 2008).

Comparisons of various experimental 2-DOF VIV results(Blevins and Coughran, 2009; Dahl et al., 2006, 2010;Stappenbelt et al., 2007) with Aym/D vs. SGY and Axm/D vs. SGX arenow discussed through the Grifn plots in Fig. 12. Numericalprediction results with specied mnx mny mn 1:4 (lowest valuefrom KHL data) and 5.7 (highest value from MIT data), and fn1and 2 in each of these cases are also given. The numerical variationof SGY and SGX values (from 0.01 to 1) is performed by varying yand x, respectively, with a small increment. A general qualitativeagreement can be seen in Fig. 12 where both Aym/D and Axm/Ddecrease as SGY and SGX increase. However, for a specic SGYSGX,experimental results with different values of mn, and fn arescattered. The Re range might also play a role (Govardhan andWilliamson, 2006) although the majority of the experimentalresults in Fig. 12 were based on the subcritical Re ows, exceptfor some results in a supercritical Re range of Dahl et al. (2010).These imply how the combined mass-damping parameter fails tocollapse different experimental 2-DOF VIV data. Numerical resultsalso capture these quantitative effects, by also providing theapproximate ranges of peak amplitudes and highlighting a possi-bly inuential role of fn. From a prediction viewpoint, both Aym/Dand Axm/D increase as mn decreases; however, for a higher mn5.7value, the variation of Aym/D is slightly inuenced by the varying fn.This observation is reminiscent of the experimental study ofJauvtis and Williamson (2004) where there was a slight inuenceon the cross-ow response with mn46 when comparing theresults obtained between 1- and 2-DOF models. In contrast,numerical Axm/D values are found to be susceptible to any changeof system mn, and fn parameters. The above-mentioned discus-sion and comparisons deserve further experimental explorationsbefore we could draw a rm conclusion on whether and how eachor the combination of these parameters actually governs the 2-DOF VIV of circular cylinders.

6. Discussion

As 2-DOF VIV of a exibly mounted circular cylinder dependson several system uid-structure parameters in both cross-owand in-line directions, it is a very challenging task to quantitativelymatch numerical prediction results to experimental measure-ments. One of the main reasons suggested is that some of thekey variables were not assessed with sufcient condence duringthe testing campaign. In particular, values of the structural damp-ing in water used as one of the inputs in the numerical model were found to be sensitive to the initial displacement condition,the change of springs' stiffness and the apparatus arrangementleading to some repeatability issues in determining total and uid-added damping from the free decay tests in water. This observa-tion was in contrast to the measurements made in air for whichthe estimated damping was highly repeatable. As a consequence,

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Fig. 8. (a) Comparison of numerical (red lines) and experimental (blue lines) xytrajectories based on KHL datasets with variable fn; (b) comparison of numerical(circles) and experimental (squares) xy phase differences for KHL3 (lled symbols)and KHL4 (open symbols) dataset. (For interpretation of the references to color inthis gure legend, the reader is referred to the web version of this article.)

N. Srinil et al. / Ocean Engineering 73 (2013) 179194 189

the damping ratios x and y appeared variable and xay whencomparing and calibrating all datasets with different fn. To over-come this constraint, an improved means to assess and controlequivalent damping values in both directions (Klamo, 2009) or asystematic approach to measure system uncertainties (Hughes andHase, 2010; Taylor, 1997) should be implemented.

Another aspect deals with the difference between the numer-ical model idealization and the real experimental setup. For thesake of generality, the cylinder is theoretically postulated to beinnitely long such that the ow eld might be approximated tobe two-dimensional. However, during the experiment, the three-dimensional ow eld along with the free surface around theoscillating cylinder with a nite length could play an inuentialrole. Although the structurally geometric coupling terms asso-ciated with the movement of two pairs of springs have beenaccounted for in the numerical model, exact geometrical x, y, xand y values of the present experimental pendulum-spring-cylinder model are presently unknown. Some of these have beenshown through a sensitivity study to produce a qualitative and

quantitative effect on the prediction of both peak amplitudes andlock-in ranges (Figs. 6 and 7).

Nevertheless, after substantial parametric studies involvingcalibration and tuning of model empirical coefcients and geo-metric parameters with various experimental results, the numer-ical prediction model based on double Dufng-van der Poloscillators is capable of predicting quite well several importantqualitative features observed in the experimental 2-DOF VIV.These include (i) the pure in-line VIV lock-in ranges in the caseof fn1 (Fig. 3) and their disappearances in the case of higher fn(Fig. 5), (ii) the two-dimensional lock-in ranges in all fn cases(Figs. 3 and 5), (iii) the response amplitude jump phenomenacaptured by system cubic nonlinearities (Fig. 3), (iv) various gure-of-eight trajectories representing periodically coupled x-y motionsassociated with dual 2:1 resonances and captured by systemquadratic nonlinearities (Figs. 8 and 9) and (v) the independenteffect of mn, and fn (Fig. 12). The hysteresis effect and theoccurrence of critical mass whereby maximum cross-ow ampli-tudes exhibit an unbounded lock-in scenario have recently been

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Fig. 9. Comparison of experimental (circles) and numerical (squares) cross-ow/in-line oscillation frequencies as function of Vr for selected KHL datasets with variable fn.

N. Srinil et al. / Ocean Engineering 73 (2013) 179194190

shown in Srinil and Zanganeh (2012). With suitably speciedcoefcients, the possible occurrence of two-peak cross-owresponse in the case of fn2 and mnxamny has also been foundduring the parametric trials. Other insights into the effect of(displacement, velocity and acceleration) coupling terms in thewake oscillators, the appearance of smaller-amplitude higherharmonics in xy motions and the occurrence of chaotic VIVhave been presented in Zanganeh and Srinil (2012). As faras model empirical coefcients in Eqs. (3)(6) are concerned,one may preliminarily suggest y based on Eq. (7), x0.3,12oxyo15, and xyxy0.7, based on calibration

with experimental results in the present study. Of course, thesensitivity analysis should also be performed, by limiting thenumber of considered cases with the only variation of y-relatedparameters.

It is of practical importance to ascertain whether the variable fn

inuences the maximum attainable cross-ow/in-line amplitudes.Based on the range of mn, and Re considered, our experimentalresults reveal the small or even negligible effect of varying fn onthe maximum attainable amplitudes of 2-DOF VIV. This suggeststhat the model developed for a rigid cylinder with 2 DOF might beapplicable to the analysis of a exible cylinder vibrating in two

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Fig. 10. Experimental comparisons of cross-ow and in-line amplitudes between KHL and MIT data with variable fn.

N. Srinil et al. / Ocean Engineering 73 (2013) 179194 191

directions with multi DOF and various fn through, e.g., a nonlinearmodal expansion approach (Srinil et al., 2009, Srinil, 2010, 2011).However, the variation of fn can inuence the phase differencebetween cross-ow and in-line motions (Fig. 8b) which in turnmay result in the change in the vortex shedding formation in thewake due to the uid-structure coupling mechanism (Bao et al.,2012). These KHL observations are in good qualitative agreementwith MIT results (Dahl et al., 2006) since both studies consider thesimilar ranges of mn, , fn and Re. It is noticed that the lock-inranges of KHL data are greater than those of MIT data owing to thelower mn of the former (Fig. 10). Recent 2-D numerical simulationresults of Bao et al. (2012) where mnx mny 2:55, xy0showed comparable Aym/D and Axm/D with fn increasing from1 to 1.5. However, due to the very low Re150, their reported

Aym/D and Axm/D values are quite small, being only about 0.60.9and 0.050.25, respectively, when compared to KHL/MIT data inFig. 10 and others in Fig. 12. Such difference in peak amplitudes indifferent Re data suggests a possible inuence of Re on 2-DOF VIV,similar to what has been observed in the eld tests of exiblecylinders (Swithenbank et al., 2008).

As a nal remark, due to a nite length of the tank, it ispresently unfeasible to perform a perfect sweeping test where atowing speed (i.e. V) is altered during a single run of the carriage.As this change (whether increasing or decreasing V) would requirethe time for the transient dynamics to die out in order to achievethe steady state responses (especially for large-amplitude 2-DOFvibrations in the neighborhood of the hysteresis), the carriagewould reach the end and be terminated before the cylinder

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Fig. 11. Experimental comparisons of normalized cross-ow and in-line oscillation frequencies between KHL and MIT data with variable fn.

N. Srinil et al. / Ocean Engineering 73 (2013) 179194192

steady-state response takes place. Hence, possible coexistingresponses associated with the jump and hysteresis as usuallyobserved in a water ume facility (Jauvtis and Williamson, 2004) were not ascertained in our towing tank tests although theproposed numerical model can capture such important behaviors(Srinil and Zanganeh, 2012). Depending on system parameters andinitial conditions, the jump and hysteresis phenomena should befurther experimentally investigated in the framework of a 2-DOFVIV of circular cylinder with variable fn and in a higher Re range.

7. Conclusions

Experimental investigations of 2-DOF VIV of a exibly mountedcircular cylinder with a low equivalent mass ratio (mn1.4 and3.5) and variable in-line-to-cross-ow natural frequency ratio(fnE1, 1.3, 1.6, 1.9) have been performed in a water towing tank.The VIV experiments cover a sub-critical Re range of about21035104. A generalized numerical prediction model hasalso been investigated based on double Dufng-van der Pol(structure-wake) oscillators which can capture the structurallygeometrical coupling and uid-structure interaction effects

through system cubic and quadratic nonlinearities. The modelempirical coefcients have been calibrated based on new experi-mental results and parametric investigations, and their valueshave been suggested. Some important aspects in the 2-DOF VIVhave been numerically captured which are in good qualitativeagreement with experimental observations.

With low values of mn1.4 equally in both directions, the two-dimensional VIV excitation ranges have been experimentallyfound to be in a broad range of the reduced velocity parameter,4oVro17.5, with maximum attainable cross-ow and in-lineamplitudes achieving high values of about 1.251.6 and 0.50.7,respectively, depending on the level and combination of the xystructural damping ratios in all fn cases. This damping parameteralong with the two-directionally geometrical coupling coefcientsmight in part be responsible for the disparity of response ampli-tudes and the quantitative differences between experimental andnumerical results, apart from the fact that actual three-dimensional features of the ow around the nite cylinder cannotbe presently captured by the numerical model. As regards experi-mental comparisons, the present measurement results and MITpublished data based on similar Re and mass-damping ratioranges exhibit fairly good agreement with comparable responseamplitudes, lock-in ranges and oscillation frequencies. However,there is no appearance of two-peak cross-ow response found inthe present testing campaign as a result of the equivalentmn in thetwo motion directions. Regardless of the specied fn and overallhydro-geometric nonlinearities, various features of gure-of-eightorbital motions have been experimentally as well as numericallyobserved in a wide Vr range. These evidence the fundamentalcharacteristics of dual 2:1 resonances of coupled in-line/cross-owVIV responses. The proposed numerical model is able to capturethese dual resonances associated with quadratic nonlinearities inaddition to the reasonable estimation of response amplitudes,lock-in ranges and oscillation frequencies.

More experimental and computational uid dynamics studieswhich assess and control the equivalence of system uid-structureparameters in different directions with reduced uncertainty areneeded to improve the model empirical coefcients and capabilityin effectively matching numerical predictions to experimentaloutcomes. These should be furnished by the identication ofvortex formation patterns in the cylinder wake using the owvisualization technique such as the particle image velocimetry.

Acknowledgments

The authors are grateful to the SORSAS and Early CareerResearcher International Exchange Awards supported by the Uni-versity of Strathclyde and Scottish Funding Council (via GRPe-SFC).They also wish to thank technicians at the Kelvin HydrodynamicsLaboratory for the experimental setup and measurement support,as well as the reviewers for their helpful comments.

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10-2 10-1 1000

0.4

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SGY

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Two-degree-of-freedom VIV of circular cylinder with variable natural frequency ratio: Experimental and numerical...IntroductionExperimental arrangement and test matrixNumerical model with nonlinear coupled structure-wake oscillatorsComparisons of experimental and numerical prediction resultsExperimental comparisons with other studiesDiscussionConclusionsAcknowledgmentsReferences