-
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.
& 2013 Elsevier Ltd. All rights reserved.
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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Srinil).
Ocean Engineering 73 (2013) 179194
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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
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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
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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
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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
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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.
50 55 60 65 70 75 80 85 90 95 100-2
-1
0
1
2
50 55 60 65 70 75 80 85 90 95 100-1
-0.5
0
0.5
1
50 55 60 65 70 75 80 85 90 95 100-2
-1
0
1
2
50 55 60 65 70 75 80 85 90 95 100-1
-0.5
0
0.5
1
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.
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
A y /D
A y /D
A y /D
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
A x /D
A x /D
A x /D
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
0.8
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
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
0.8
Vr
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
y
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Vr
y
4 8 12 16 200
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Vr
4 8 12 16 20Vr
4 8 12 16 20
0.1
0.2
0.3
0.4
0.50.60.7
0.8
0.91
1.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.2
0.4
0.60.8
1
1.2
1.4
1.61.8
2
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
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
y
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Vr Vr Vr
y
4 8 12 16 200
0.25
0.5
0.75
1
1.25
1.5
1.75
2
4 8 12 16 20 4 8 12 16 20
0.10.20.30.40.50.60.70.80.9
11.11.21.31.4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.1
0.2
0.3
0.4
0.5
0.6
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
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,
2 4 6 8 10 12 14 16 18 200
1
2
3
4
5
6
7D
atas
et
2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2
Vr
( x-2
y)/
f*=1.85
f*=1.60
f*=1.29
f*=1.01
f*=1.00
f *=1.00
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
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
0
0.5
1
1.5
2
2.5
3
3.5
4
0 4 8 12 160
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Vr Vr
f oy/f n
y
f ox/f n
yf ox
/f ny
f ox/f n
y
f oy/f n
yf oy
/f ny
0 4 8 12 160
0.5
1
1.5
2
2.5
3
3.5
4
KHL1 KHL1
KHL3 KHL3
KHL4 KHL4
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
0
0.5
1
1.5
2
A y /D
A x /D
A x /D
A x /D
A x /D
A y /D
A y /D
A y /D
0
0.2
0.4
0.6
0.8
0
0.5
1
1.5
2
0
0.2
0.4
0.6
0.8
0
0.5
1
1.5
2
0
0.2
0.4
0.6
0.8
0 4 8 12 16 200
0.5
1
1.5
2
Vr
0 4 8 12 16 200
0.2
0.4
0.6
0.8
Vr
MIT6KHL4
MIT4MIT5KHL3
MIT2MIT3KHL2
MIT1KHL1KHL5KHL6
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
0.2
0.5
0.8
1.1
1.4
1.7
2
f oy /f n
yf oy
/f n
yf oy
/f n
yf oy
/f n
y
f ox /f n
yf ox
/f n
yf ox
/f n
yf ox
/f n
y
1
1.4
1.8
2.2
2.6
3
3.4
0.2
0.5
0.8
1.1
1.4
1.7
2
1
1.4
1.8
2.2
2.6
3
3.4
0.2
0.5
0.8
1.1
1.4
1.7
2
1
1.4
1.8
2.2
2.6
3
3.4
0 4 8 12 16 200.2
0.5
0.8
1.1
1.4
1.7
2
Vr0 4 8 12 16 20
1
1.4
1.8
2.2
2.6
3
3.4
Vr
MIT6KHL4
MIT4MIT5KHL3
MIT2MIT3KHL2
MIT1KHL1KHL5KHL6
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.
References
Assi, G.R.S., Bearman, P.W., Kitney, N., 2009. Low drag
solutions for suppressingvortex-induced vibration of circular
cylinders. Journal of Fluids and Structures25 (4), 666675.
Assi, G.R.S., Srinil, N., Freire, C.M., Korkischko, I., 2012.
Experimental investigation ofthe vortex-induced vibration of a
curved cylinder. In: Proceedings of the 31stInternational
Conference on Ocean, Offshore and Arctic
Engineering.OMAE2012-83016.
Bao, Y., Huang, C., Zhou, D., Tu, J., Han, Z., 2012.
Two-degree-of-freedom ow-induced vibrations on isolated and tandem
cylinders with varying naturalfrequency ratios. Journal of Fluids
and Structures 35, 5075.
Bearman, P.W., 2011. Circular cylinder wakes and vortex-induced
vibrations. Journalof Fluids and Structures 27 (56), 648658.
Bishop, R.E.D., Hassan, A.Y., 1964. The lift and drag forces on
a circular cylinderoscillating in a owing uid. Proceedings of the
Royal Society of London, 5175.
10-2 10-1 1000
0.4
0.8
1.2
1.6
2
2.4
SGY
A ym
/D
10-2 10-1 1000
0.2
0.4
0.6
0.8
1
1.2
A xm
/D
SGX
Fig. 12. Grifn plots of maximum attainable cross-ow and in-line
amplitudesbased on several 2-DOF VIV experimental (symbols) and
numerical prediction(lines) results.
N. Srinil et al. / Ocean Engineering 73 (2013) 179194 193
-
Blevins, R.D., 1990. Flow-Induced Vibrations. Van Nostrand
Reinhold, New York.Blevins, R.D., Coughran, C.S., 2009.
Experimental investigation of vortex-induced
vibration in one and two dimensions with variable mass, damping,
andReynolds number. Journal of Fluids Engineering 131 (10),
101202101207.
Currie, I.G., Turnbull, D.H., 1987. Streamwise oscillations of
cylinders near thecritical Reynolds number. Journal of Fluids and
Structures 1 (2), 185196.
Dahl, J.M., Hover, F.S., Triantafyllou, M.S., 2006.
Two-degree-of-freedom vortex-induced vibrations using a force
assisted apparatus. Journal of Fluids andStructures 22 (6-7),
807818.
Dahl, J.M., Hover, F.S., Triantafyllou, M.S., Dong, S.,
Karniadakis, G.E., 2007. Resonantvibrations of bluff bodies cause
multivortex shedding and high frequencyforces. Physical Review
Letters 99 (14), 144503.
Dahl, J.M., Hover, F.S., Triantafyllou, M.S., Oakley, O.H.,
2010. Dual resonance invortex-induced vibrations at subcritical and
supercritical Reynolds numbers.Journal of Fluid Mechanics,
395424.
Facchinetti, M.L., de Langre, E., Biolley, F., 2004. Coupling of
structure and wakeoscillators in vortex-induced vibrations. Journal
of Fluids and Structures 19 (2),123140.
Flemming, F., Williamson, C.H.K., 2005. Vortex-induced
vibrations of a pivotedcylinder. Journal of Fluid Mechanics 522,
215252.
Gabbai, R.D., Benaroya, H., 2005. An overview of modeling and
experiments ofvortex-induced vibration of circular cylinders.
Journal of Sound and Vibration282 (35), 575616.
Govardhan, R., Williamson, C., 2006. Dening the modied Grifn
plot in vortex-induced vibration: revealing the effect of Reynolds
number using controlleddamping. Journal of Fluid Mechanics 561,
147180.
Hansen, E., Bryndum, M., Mayer, S., 2002. Interaction of in-line
and cross-owvortex-induced vibrations in risers. In Proceedings of
the 21st InternationalConference on Offshore Mechanics and Arctic
Engineering. Oslo. OMAE2002-28303.
Hughes, I.G., Hase, T.P.A., 2010. Measurements and their
Uncertainties: A PracticalGuide to Modern Error Analysis.
Oxford.
Jauvtis, N., Williamson, C.H.K., 2003. Vortex-induced vibration
of a cylinder withtwo degrees of freedom. Journal of Fluids and
Structures 17 (7), 10351042.
Jauvtis, N., Williamson, C.H.K., 2004. The effect of two degrees
of freedom onvortex-induced vibration at low mass and damping.
Journal of Fluid Mechanics509, 2362.
Jeon, D., Gharib, M., 2001. On circular cylinders undergoing
two-degree-of-freedomforced motions. Journal of Fluids and
Structures 15 (34), 533541.
Khalak, A., Williamson, C.H.K., 1999. Motions, forces and mode
transitions invortex-induced vibrations at low mass-damping.
Journal of Fluids and Struc-tures 13 (78), 813851.
Kim, W.J., Perkins, N.C., 2002. Two-dimensional vortex-induced
vibration of cablesuspensions. Journal of Fluids and Structures 16
(2), 229245.
Klamo, J., 2009. The application of controlled variable magnetic
eddy currentdamping to the study of vortex-induced vibrations.
Experiments in Fluids 47(2), 357367.
Laneville, A., 2006. On vortex-induced vibrations of cylinders
describing XYtrajectories. Journal of Fluids and Structures 22
(67), 773782.
Lucor, D., Triantafyllou, M.S., 2008. Parametric study of a two
degree-of-freedomcylinder subject to vortex-induced vibrations.
Journal of Fluids and Structures24 (8), 12841293.
Moe, G., Wu, Z.J., 1990. The lift force on a cylinder vibrating
in a current. Journal ofOffshore Mechanics and Arctic Engineering
112 (4), 297303.
Morse, T.L., Govardhan, R.N., Williamson, C.H.K., 2008. The
effect of end conditionson the vortex-induced vibration of
cylinders. Journal of Fluids and Structures 24(8), 12271239.
Nayfeh, A.H., 1993. Introduction to Perturbation Techniques.
Wiley, New York.Pesce, C., Fujarra, A., 2005. The super-upper
branch VIV response of exible
cylinders. In: Proceedings of the International Conference on
Bluff Body Wakesand Vortex-Induced Vibrations. Santorini.
Sanchis, A., Slevik, G., Grue, J., 2008. Two-degree-of-freedom
vortex-inducedvibrations of a spring-mounted rigid cylinder with
low mass ratio. Journal ofFluids and Structures 24 (6), 907919.
Sarpkaya, T., 1995. Hydrodynamic damping, ow-induced
oscillations, and bihar-monic response. Journal of Offshore
Mechanics and Arctic Engineering 117 (4),232238.
Sarpkaya, T., 2004. A critical review of the intrinsic nature of
vortex-inducedvibrations. Journal of Fluids and Structures 19 (4),
389447.
Skop, R.A., Balasubramanian, S., 1997. A new twist on an old
model for vortex-excited vibrations. Journal of Fluids and
Structures 11 (4), 395412.
Srinil, N., 2010. Multi-mode interactions in vortex-induced
vibrations of exiblecurved/straight structures with geometric
nonlinearities. Journal of Fluids andStructures 26 (78),
10981122.
Srinil, N., 2011. Analysis and prediction of vortex-induced
vibrations of variable-tension vertical risers in linearly sheared
currents. Applied Ocean Research 33(1), 4153.
Srinil, N., Rega, G., 2007. Two-to-one resonant multi-modal
dynamics of horizontal/inclined cables. Part II: Internal resonance
activation, reduced-order modelsand nonlinear normal modes.
Nonlinear Dynamics 48 (3), 253274.
Srinil, N., Rega, G., Chucheepsakul, S., 2007. Two-to-one
resonant multi-modaldynamics of horizontal/inclined cables. Part I:
Theoretical formulation andmodel validation. Nonlinear Dynamics 48
(3), 231252.
Srinil, N., Wiercigroch, M., O'Brien, P., 2009. Reduced-order
modelling of vortex-induced vibration of catenary riser. Ocean
Engineering 36 (1718), 14041414.
Srinil, N., Zanganeh, H., 2012. Modelling of coupled
cross-ow/in-line vortex-induced vibrations using double Dufng and
van der Pol oscillators. OceanEngineering 53, 8397.
Srinil, N., Zanganeh, H., 2013. Corrigendum to Modelling of
coupled cross-ow/in-line vortex-induced vibrations using double
Dufng and van der Pol oscillators[Ocean Eng. 53 (2012) 8397]. Ocean
Engineering 57 (0), 256.
Stappenbelt, B., 2010. Vortex-Induced Motion of Nonlinear
Offshore Structures: AStudy of Catenary Moored System FluidElastic
Instability. Lambert AcademicPublishing, Germany.
Stappenbelt, B., Lalji, F., Tan, G., 2007. Low mass ratio
vortex-induced motion. In:Proceedings of the 16th Australasian
Fluid Mechanics Conference. Gold Coast,Australia. pp. 14911497.
Stappenbelt, B., ONeill, L., 2007. Vortex-induced vibration of
cylindrical structureswith low mass ratio. In: Proceedings of the
17th International Offshore andPolar Engineering Conference.
Lisbon.
Sumer, B.M., Fredsoe, J., 2006. Hydrodynamics Around Cylindrical
Structures. WorldScientic, Singapore.
Swithenbank, S.B., Vandiver, J.K., Larsen, C.M., Lie, H., 2008.
Reynolds numberdependence of exible cylinder VIV response data. In:
Proceedings of the 27thInternational Conference on Offshore
Mechanics and Arctic Engineering.OMAE2008-57045.
Taylor, J.R., 1997. An Introduction to Error Analysis: The Study
of Uncertainties inPhysical Measurements. University Science Books,
USA.
Trim, A.D., Braaten, H., Lie, H., Tognarelli, M.A., 2005.
Experimental investigation ofvortex-induced vibration of long
marine risers. Journal of Fluids and Structures21 (3), 335361.
Vandiver, J.K., Jong, J.Y., 1987. The relationship between
in-line and cross-owvortex-induced vibration of cylinders. Journal
of Fluids and Structures 1 (4),381399.
Wang, X.Q., So, R.M.C., Chan, K.T., 2003. A non-linear uid force
model for vortex-induced vibration of an elastic cylinder. Journal
of Sound and Vibration 260 (2),287305.
Williamson, C.H.K., Govardhan, R., 2004. Vortex-induced
vibrations. Annual Reviewof Fluid Mechanics 36, 413455.
Williamson, C.H.K., Jauvtis, N., 2004. A high-amplitude 2T mode
of vortex-inducedvibration for a light body in XY motion. European
Journal of MechanicsB/Fluids 23 (1), 107114.
Zanganeh, H., Srinil, N., 2012. Interaction of wake and
structure in two-dimensionalvortex-induced vibrations. In:
Proceedings of the International Conference onAdvances and
Challenges in Marine Noise and Vibration. Glasgow. pp. 177188.
Zhao, M., Cheng, L., 2011. Numerical simulation of
two-degree-of-freedom vortex-induced vibration of a circular
cylinder close to a plane boundary. Journal ofFluids and Structures
27 (7), 10971110.
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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