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ORIGINAL ARTICLE

Quasisteady theory for the hydrodynamic forces on a circularcylinder undergoing vortex-induced vibration

Yoshiki Nishi Kentaroh Kokubun Kunihiro Hoshino Shotaro Uto

Received: 13 June 2008 / Accepted: 28 October 2008/ Published online: 5 December 2008 JASNAOE 2008

Abstract Vortex-induced vibrations of a rigid circularcylinder were studied by constructing a theory based on awake oscillator model under quasisteady approximations,thereby evaluating vortex-induced hydrodynamic forcesacting on the cylinder. A lock-in limit line representing theboundary for the occurrence of frequency lock-in was alsotheoretically derived. Hydrodynamic forces in forcedoscillation problems estimated by the theory were com-pared with measured ones. Although some discrepancieswere found, particularly in cases with high-frequencyoscillations, good agreement was achieved in most cases.Accordingly, we conclude that the present theory captureswell real phenomena in the wake downstream of a cylindersubjected to a ow.

Keywords Vortex-induced vibration Wake oscillatormodel Riser pipe Hydrodynamic force Lock-in

1 Introduction

Recent rising prices for fossil fuels urgently require us toadvance new resource developments not only on land butalso in the seas. In particular, mining of resources from theseaoor deeper than 2,500 m is attracting a lot of attention

due to the potentially large size of these reserves.

The design and construction of a marine riser pipe isone of the most important technologies to realize this. Itis well recognized that a marine riser pipe subjected toows oscillates under the inuence of vortices generatedand shed downstream of the pipe. These kinds of oscil-lation are generally called vortex-induced vibrations(VIVs) [ 13] and have the characteristic of self-excitedoscillations; thus their amplitude can become remarkablylarge even under weak ow conditions. Therefore, thefatigue accumulated in the pipes structure caused by theVIV is of particular concern, and has to be accuratelyestimated for designing and operating the riser pipesafely.

The VIV occurs in the framework of a coupled uid-structure dynamic system. Thus, we have to address con-currently the structural dynamics of the pipe and thehydrodynamics of the uids surrounding the pipe. To date,several researchers have attempted to simulate the pipesmotion in water numerically by using the nite elementmethod (FEM) [ 4, 5], mode expansion method [ 6, 7], andso on. To develop a practical tool for simulation, theestablishment of a method for estimating hydrodynamicforces acting on the pipe is of considerable signicance. Afew commercial codes such as SHEAR7 have made adatabase composed of experimental data on the forcesmeasured in tank tests. However, in general, the collectionof data applicable to every condition in actual seas is notfeasible because this requires a large test facility. Alter-native to database-based methods, computational uiddynamics (CFD) can compute the forces accurately [ 8, 9] if enough computational times and numbers of grids areeasily available. Nonetheless, considering the inefciencyinvolved in the regridding procedure to handle the complexshape and motion of a long elastic pipe, this does not seemto be a hopeful method.

Y. Nishi ( & )Department of Systems Design for Ocean-Space,Graduate School of Engineering,Yokohama National University, 79-5 Tokiwadai,Hodogaya, Yokohama 2408501, Japane-mail: ynishi@ynu.ac.jp

K. Kokubun K. Hoshino S. UtoNational Maritime Research Institute,6-38-1 Shinkawa, Mitaka 1810004, Japan

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Our strategy is to establish a theory based on modelingof vortex-induced hydrodynamic forces acting on a cir-cular cylinder, and to nd analytical solutions of thistheory. This is because such an approach can provide asubstantially practical tool to estimate the forces withlittle requirement in terms of computing time and mem-ory, if its accuracy is sufciently well validated and iscoupled with a method for simulating the elastic motionof the pipe. Such approaches have scarcely been con-ducted so far.

Birkhoff [ 10] has attempted to determine the vortexshedding frequency of a xed cylinder theoretically byintroducing a model called the wake oscillator. Inspiredby this idea, several previous works have dealt with VIV byanalytical approaches [ 11 14]. However, most of the ana-lytical procedures employed in these works involve severalparameters determined not analytically but empirically,without carrying out sensitivity analyses, leading to a lack of versatility in the theory. In addition, these previousworks have not yet claried whether the external force inthe oscillators equation of motion is proportional to thedisplacement, velocity, or acceleration of the cylinderstranslation motion.

Accordingly, this paper attempts to construct a theoryfor a forced oscillation problem, using laws of dynamics asmuch as possible to estimate the hydrodynamic forcesacting on a rigid circular cylinder, to solve it analytically,and to validate it by conducting experiments.

2 Quasisteady wake oscillator model

Close observation of the wake region has revealed that thisregion exhibits swing-like motion and stretching defor-mation in conjunction with growing and shedding vortices[15]. The present theory is based on the replacement of thelift forces acting on the cylinder with those acting on arigid bar called the wake oscillator moving rotationallywith the wake region (Fig. 1). We should remark that theapproximation using the rigid bar is the lowest-orderapproximation for the wake dynamics. A model with ahigher-order approximation using a exible bar, or severalconnected bars, may be necessary to describe adequatelythe variation of the vortex shedding modes such as 2S, 2P,and 2P ? 2S modes [ 16].

In this study, we will restrict our discussion to transla-tion motions of one degree of freedom transverse to auniform ow. The rotational motion of the wake oscillatoris described using simple rigid-body dynamics andhydrodynamics. A schematic drawing of the wake oscil-lator is shown in Fig. 2. In the following subsections, anequation of motion for the rotating oscillator is built bymodeling each force responsible for the motion of the

oscillator. The denitions of the mathematical notations arelisted in Table 1.

2.1 Inertia and restoring moments

The inertia moment of the oscillator I is

I m d

2 l 2

; 1

Fig. 1 Schematic illustrations for the generation and shedding of vortices at the wake region. The motion of virtual wake oscillators isalso depicted

Fig. 2 Denitions of the sizes, displacement, and angle of rotation of the wake oscillator

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frequency). The mean half-length of the oscillator l isdetermined such that x e equals the well-known vortexshedding frequency of a xed cylinder [ 17].

The most noticeable feature of the VIV is that it is aself-excitation oscillation, and involves a limit cycle.Some researchers have observed temporal variations witha frequency 2 x e in the long-axis length of the wakeregion during one period of the vortex shedding [ 15].This can be simply taken into account in the followingway,

2l 2l l0 sin 2x e t ; 4under the assumption of sinusoidal variation in the oscil-lators rotation with frequency x e .

Substituting Eq. 4 into Eqs. 1 and 2, and making aTaylors expansion around a _a 0; yields an approximaterepresentation for the restoring term normalized by I as:

k I affi

x2e a px ea

2_a ; 5

and p 12ca2f

l0d 2 l

;

where expanded terms higher than second order in a _a areignored. A dot denotes a time derivative with respect todimensional time t . The physical meanings of the right-hand side of Eq. 5 are as follows: the rst term is the linearrestoring moment, and the second term is the nonlineardamping moment forming the limit cycle of the oscillatorsmotion.

2.2 Self-excitation coefcient

A self-excitation term is derived from the lift force origi-nated from vortices shed downstream of the oscillator.Under the quasisteady approximation (Kutta-Joukowskistheorem), the moment of the lift force induced by the shedvortices is represented as

M s qV C sd 2 l : 6

Here, it is assumed that the application point of the liftforce is equivalent to the center of mass of the wake region.By considering the energy gain during one oscillatingperiod, the self-excitation coefcient c can beapproximately written as follows,

c fd

2 ffiffiffi2p p2 l; 7

where the value of f is determined based on [ 18].

2.3 Excitation moments

The oscillators motion is affected by excitation momentscaused by the translation motion of the cylinder. Theexcitation moment proportional to the acceleration of thecylinders motion is incorporated by considering themoment of inertial force m h in a reference frame xed to

the cylinder.Moreover, the excitation moment proportional to the

velocity of the cylinders motion is taken into account byconsidering the variation in an induced angle of attack h,which is a function of the velocity of the translation motiondescribed by

h tan 1_h

V ffi_h

V : 8The introduction of the effective angle of attack a - h

modies the linear restoring term in Eq. 5 into thefollowing form:

x 2e a h ffix 2e a x 2e_h

V : 9

2.4 Equation of motion for the wake oscillatorin a forced oscillation problem

This paper considers a forced oscillation problem, in whichthe translation motion of the cylinder is prescribed as H H sin xs ; 10where H h=d ; H h d ; and s x 0 t : The nondimen-sional frequency x is related to the dimensional onethrough x f =x 0 :

A harmonic resonance solution to the forced oscillationof the form

a a f sin xs u 11is treated in this paper, since our experiment has demon-strated that the harmonic resonance component is the mostdominant among all the frequency components [ 19].Actually, a subharmonic resonance component, which isnot dealt with in this paper, has been also detected by fastFourier transform (FFT) analysis in our experiment [ 20],which will be discussed in our subsequent studies.

The aforementioned descriptions are combined todescribe an equation for the rotating motion of the wakeoscillator, which is represented in dimensionless form as

a00 2ck 1 pa2 a0k2a b H 00 2pS ek H 0; 12

where b d d 2l

; and k x ex 0 :A prime denotes a time derivative with respect to non-

dimensional time s. Equation 12 is a heterogeneousequation for an oscillatory system including self-excitationand nonlinear damping.

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2.5 Lift force coefcient in a forced oscillationproblem

Under the quasisteady approximation, a relationshipbetween the wake oscillators angle of attack and the liftforce coefcient is written as

C L

f a h

;

13

Substitution of Eqs. 8, 10, and 11 into Eq. 13 gives anexplicit representation of the lift force coefcient as

C L f a f cos u sin x t a f sin u 2pS ex

k H cos x t ;

14in which the rst term (proportional to sin x t ) correspondsto the component of added mass (in phase with the cylin-ders acceleration), and the second term (proportional tocos x t ) corresponds to the component of damping (inphase with the cylinders velocity).

Using the synthesis formula for trigonometric functions,Eq. 14 can be recast into the following form

C L f a2f 2pS ex

2

k2 H 2 2

2pS exk

H sin u !12

sin x t u ;u tan 1

a f sin u 2pS e xk H a f cos u

:

8>>>>>>>>>>>>>: 15Note that u * is different from u , and that its value

determines whether energy input to the VIV motion ispositive or negative. Thus, the phase plays an importantrole in the design and control of a marine riser pipeundergoing VIV.

3 Analytical harmonic resonance solution in a forcedoscillation problem

We employ the averaging method [ 21] to solve Eq. 12.After substituting Eqs. 10 and 11 into Eq. 12 , and calcu-lating temporal averages for one oscillation period underthe assumption that variations in H ; H 0; u

during the

period are negligibly small, we obtain temporally averagedequations for the amplitude and phase of the oscillatorsmotion as follows:

2x a0f

a f u 0 a0 ckx 2 12 pa

2

k2 x 2 x H

2pS ek bxbx 2pS ekx cos usin u : 16

Ignoring a column vector including time derivatives of a f ; u (the rst term on the left-hand side) and elimination

of u gives an equation satised by the oscillators responseamplitude under a stationary oscillation state:

X ckx 2 12

pX 2

k2 x 2 " # b2x 2 2pS e

2k2n ox 2 H 2; 17where X a

2f : This is a third-order polynomial equationwith respect to X , which can be solved analytically by

making use of the cubic formula (TartagliaCardanomethod). Among the mathematically obtained solutions, areal and positive one is physically valid. The phase lead uunder the stationary oscillation state is then obtained fromEq. 16 by ignoring terms including time derivatives.

4 Condition for the occurrence of lock-in

Frequency lock-in (or synchronization) associated with theVIV of a cylinder has been experimentally observed bymany researchers [ 22]. Of great signicance is the esti-mation of the hydrodynamic forces in the lock-in state,since the response amplitude of the VIV has its maximumin this state. However, the descriptions above, assuming theoccurrence of lock-in, do not give us any information onwhether lock-in occurs or not. Therefore, a mathematicalrepresentation for its occurrence is considered here basedon the equation of motion and its harmonic resonancesolution derived in the preceding sections.

Regardless of the occurrence of lock-in, the rotationangle of the wake oscillator contains the eigenfrequencycomponent (or eigencomponent, denoted by a e ) as well asthe harmonic resonance component (or forced component),represented asa a f ae : 18

The harmonic resonance component (Eq. 17 ) can berecast under the approximation of weak nonlinearity(e, dened below, is small enough) into

a2f ffib2X 2 2pS e

2n oX 2 H 21 X 2

2 1 X 2

1 X 2 2 e

2 O e4 ( );19

where e c 2 12 pX ; X xk ; and terms higher than

second order in e are ignored.Substitution of Eqs. 18 and 19 into Eq. 12 provides the

following differential equation describing the eigenfre-quency component as

a00e k2a e ek 1 pa f a e2n o a0f a0e : 20

Assuming the form of the solution of Eq. 20 to be a e a e sin ks / ; and employing the averaging method, weobtain temporally averaged equations of this component as

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a0ex ea f / 0 12 a e em 1

p4

a2e 2a2f k2 x 2 : 21Ignoring a column vector including time derivatives of

a e ; / (left-hand side) gives an equation representing arelationship between ae and a f as follows,

0 1 p

4a2

e 2a2

f

n o; 22which expresses the fact that frequency componentsinteract with each other due to the nonlinearity of thedynamic system described by Eq. 12 .

Lock-in corresponds to the evanescence of the eigen-component ae , the condition for which can be writtenmathematically as

a2e 4 p

2a2f 0: 23It follows from Eqs. 19 and 23 that the condition for

lock-in can be described in a plane spanned by the forcedoscillations frequency ( X : horizontal axis) and itsamplitude ( H : vertical axis) as

b2X 2 2pS e 2n o H 2 X 2

2 p

X 2 1 2; 24

in which the equality gives the lock-in limit line in theX ; H plane.

5 Experiment

To validate the present theory, experiments for the forcedoscillation problem were carried out using the two-dimensional water channel at National Maritime ResearchInstitute [ 19]. Its working section is 22.05 m long, 0.5 mwide, and 0.5 m deep (from the free surface to the bottomof the tank). A towing carriage runs at prescribed speeds(monitored with a rotary encoder). A circular cylinder(aluminum and neutral buoyancy) was mounted horizon-tally at depth of 0.2 m, and was linked with a motorthrough a scotch yoke mechanism in order to oscillate itvertically. Acrylic plates were installed at both ends of thecylinder to avoid the end effect on measurements [ 23]. The

vertical displacement of the cylinder was measured with alaser displacement gauge, and forces acting on the cylinderwere measured with a strain gauge attached to the innerwall of the cylinder (Fig. 3).

The measured forces and displacement were Fourieranalyzed at the same time to obtain the amplitude andphase of the forces with the forced oscillation frequency.The number of cycles used in the FFT analyses was 1030,depending on the frequencies of the forced oscillations.Then the forces were separated into their components in

proportion to the acceleration and velocity of the dis-placement to calculate the added mass and dampingcoefcients of the lift forces (denoted by C LA and C LD ,respectively) in the following forms:

C LA F in Mh0x 2f

12 qdLV

2 ; and C LD F out

12 qdLVh 0x f

; 25where F in and F out represent the amplitudes of the forcedcomponent of the lift force in phase and out of phase withthe forced acceleration of the cylinder, respectively, L isthe span length of the cylinder (0.454 m), and M denotesthe mass of the sensitive part of the cylinder (0.221 kg).The inertial force Mh0x 2f was subtracted from F in whencalculating C LA since the measured force in phase with theacceleration contains the inertial force as well as the vor-tex-induced lift force.

6 Results and discussion

In the following description, we dene that positive (neg-ative) values of the damping coefcients (the lift force inphase with velocity) correspond to the growth (deteriora-tion) of the amplitude of the VIV.

6.1 Effects of the eigenfrequency of the wake oscillator

Firstly, a sensitivity analysis of the present theory againstthe variation in the eigenfrequency of the wake oscillator(or Strouhal frequency) was performed since this is deter-mined through Eq. 3 after empirically determining thesizes of the wake region ( s / d and l d ).

The Strouhal number (which is equivalent to S e in ourdenition) varies from 0.15 to 0.30 for Reynolds

Fig. 3 Photograph of the experimental setup for the forced oscilla-tion test using the two-dimensional water channel

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numbers ranging from 1.0 9 10 0 to 1.0 9 10 6 [17].Figure 4 presents: the damping and added mass coef-cients obtained from four cases with different values of S e (0.15, 0.20, 0.25, and 0.30) for the same amplituderatio of H 0:6; and the same ow velocity of 0.4 ms

- 1 . It can be seen that positive peaks of the twocoefcients shift right as S e increases. This is due to thefact that the Strouhal frequency corresponds to the

eigenfrequency of the oscillators motion (Eq. 12 ),around which the phase of the response rapidly varies, asshown experimentally by Bishop and Hassan [ 24] andCarberry et al. [ 25].

Another aspect to be noted is the difference in the fre-quency range over which the damping coefcient ispositive (i.e., where the lift force excites the oscillation of the cylinder). This suggests that a case with higher S e can

a bFig. 4 Sensitivity analysis forthe theoretically obtainedhydrodynamic forces:a damping coefcient, andb added mass coefcient

067.0= 067.0=

100.0=

200.0= 200.0=

100.0=

TheoryExp.

f

H H

H H

H H

a

b

c

Fig. 5 Comparisons of thehydrodynamic forces betweenthe present theory and theexperiment conducted byTanaka and Takahara [ 26]. Left and right panels present thedamping (lift force in phasewith the cylinders velocity) andadded mass (lift force in phasewith the cylinders acceleration)coefcients, respectively. Theamplitude ratio H is a 0.067,b 0.100, and c 0.200. TheReynolds number is 1.0 9 10

5

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have a wider frequency range for excitation. This excita-tion range is particularly important when simulating theoscillatory motion of a full-scale marine riser under actualsea conditions with Reynolds number of the order 10 5 10 6 .Hence, this has to be further investigated by more renedtheory and experiment focusing on the actual scale in thefuture.

Based on this sensitivity analysis, the sizes of the wakeregions were determined so that the Strouhal number wasequivalent to that which applies for the Reynolds numberemployed in this study.

6.2 Comparison with experiments

The damping and added mass coefcients were comparedbetween the theory and experiment in cases of relativelysmall amplitude: H 0:067 ; 0:100 ; and 0 :200 (Fig. 5).The Reynolds number was set to approximately be1.0 9 105 . The data measured are derived from Tanakaand Takahara [ 26]. This experiment exhibits sharp changesin the sign of the coefcients at the forced frequency S f of 0.160.17 both in the damping and added mass

components. The theoretical values are in fairly goodagreement with the experimental ones. In particular,the frequency ranges of the excitation (positive values of the damping coefcient) are favorably reproduced by thepresent theory.

Figure 6 presents the comparisons in higher-amplitudecases: H 0:250 ; 0:500 ; and 0 :750 : These amplitudeswere observed in our tank test for the free oscillation of acylinder. The Reynolds number was 8.0 9 10 3 . In thesecases, the experiment shows that the excitation rangebecomes remarkably narrow; the damping coefcient hasnegative values over most of the frequency range. Thetheory reproduces well this feature observed in theexperiment.

On the other hand, we should note some discrepanciesseen in these comparisons. Firstly, the present theory seemsto produce smaller values of the damping coefcients incases with high S f , approximately greater than 0.25. Thisdifference may be attributed to the employment of thesteady-state formulation for the restoring moment, the self-excitation of the wake oscillator, and the lift force coef-cient. A quasisteady approximation such as that used in the

250.0= 250.0=

500.0= 500.0=

750.0= 750.0=

TheoryExp.

H H

H H

H H

a

b

c

Fig. 6 Comparisons of thehydrodynamic forces betweenthe present theory andexperiment conducted by us.The left and right panels presentthe damping (lift force in phasewith the cylinders velocity) andadded mass (lift force in phasewith the cylinders acceleration)coefcients, respectively. Theamplitude ratio H is a 0.250,b 0.500, and c 0.750. TheReynolds number is 8.0 9 10

3

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present study may be not valid, especially in a high-fre-quency case involving large accelerations and velocities of the wakes swing-like motions. In addition, limitations dueto the assumption of small angle of attack a may also haveto be considered.

We speculate that a solution for some of these issues canbe provided by an updated theory based on unsteady andhigher-order formulations, which will be addressed in oursubsequent study.

Secondly, discrepancies are observed also in low-reduced-frequency cases. This may be caused by theinherent difculty in carrying out FFT analyses for themeasurement of low-frequency cases. Actually, the pre-dominance of a component with a forced oscillationfrequency was not so clear in these cases, thus the exper-imental values for this may be not so accurate. Thus, weshould pay more attention to the frequency analysis fortime histories of forces measured in forced oscillation testswith low frequencies.

6.3 Phase of the vortex-induced lift force

The phase of the lift force relative to the displacementtransverse to a ow determines whether the lift force excitesor damps an oscillation. To examine this clearly, the phaseu * obtained from the theory is shown in the X ; H plane inFig. 7a, in which the shaded area denotes excitation. Thisdemonstrates that the vortex-induced lift force excites theoscillation under the conditions of frequencies near theeigenfrequency, and that the bandwidth of the excitationbecomes narrower as the amplitude H becomes larger.Thesetendencies are also exhibited in the damping coefcientobtained from the experiment (Figs. 5 and 6).

Theoretically, the narrowing of the excitation bandwidthis derived from the incorporation of the induced angle h intoEq. 8. Ignoring the induced angle results in substantiallydifferent results, as shown in Fig. 7b, demonstrating that theexcitation bandwidth becomes wider as the amplitude H becomes larger. This is contrary to the results shown in

a b

H

H

Fig. 7 Contours in X ; H space for the phase of the liftforces with forced frequenciesrelative to that of the forceddisplacement of the cylinder, incases of a with and b withoutconsideration of the inducedangle ( h). The shaded areasrepresent excitation by thevortex-induced lift force. Theunit of phase is degrees; thecontour interval is 30 . Dashed contours represent negativevalues

0.5

0.3

0.1

H H

a bFig. 8 Contours in X ; H space for the amplitude of a theforced component and b theeigencomponent of the wakeoscillators rotation motion.The contour interval is a 0.2and b 0.1

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Fig. 7a. The feature in Fig. 7b has not been explained by theexperiments (Figs. 5 and 6). Hence, the treatment of theinduced angle in this study proves to be appropriate.

6.4 The occurrence of lock-in

Figure 8a presents contours for the amplitude of the forcedcomponent of the oscillators motion a f

: An important

aspect here is the convex downward lines near the pointX ; H 1:0; 0:0 and upper right regions of this point,

meaning that the response of the lift force is maximized atfrequencies X % 1.0 in cases of small H , and that fre-quencies maximizing the response shift toward X 1:0 incases of larger H .

The amplitudes of the eigencomponent a e are drawn inFig. 8b, and the shaded region represents evanescence of the eigencomponent. This shows that conditions for theexistence of the eigencomponent are positioned in thefollowing two regions: small X , and small H regions.

The response amplitude of the eigencomponent is overallsmaller than that of forced component (Fig. 8a, b). However,this does not imply that the eigencomponent can be ignoredwhen simulating a marine riser pipes motion in ows,because this component becomes prominent before or afterlock-in, namely under non-lock-in conditions. Considering afree oscillation state in actual seas, we can guess that theoscillation of a riser pipe that is motionless in an initial stategradually grows under the inuence of excitation by theeigencomponent (small X and H regions in Fig. 8b), then theeigenfrequency of the vortex shedding is entrained into that

of the risers structure, i.e., locked-in, and then unlocked-inagain. The repetition of the lock-in and unlock-in appears inactual seas under the inuence of nonlinear interactionsamong several modes of a long exible pipe. Therefore, theeigencomponent in Fig. 8b has great importance in a time-domain simulation while few previous papers in literaturesearches by us seem to point out this aspect.

Figure 9 shows the lock-in limit line (Eq. 24) obtainedfrom the theory with (solid line) and without (dashed line)taking the induced angle into consideration. The Reynoldsnumber is set to be 3.6 9 103 . The plots denote theexperimental values obtained from Tanaka and Takahara[26] and Stansby [ 27]. The comparison between theseconrms that the theory including the induced angle agreeswell with the experiment except for differences seen incases of high frequencies ( X C 1.8), and that the removalof the induced angle does not result in good results, as forthe discussion on the phase (Fig. 7).

Conversely, we should remark on the different tenden-cies in the high-frequency range. The theory gives concaveupward lines while the experiment exhibits concavedownward plots in this range. We guess that this inaccu-racy of the theory can be attributed to the applicability limitof the quasisteady and small-angle-of-attack approxima-tions used, as in the discussion above on the comparison of the hydrodynamic forces (Figs. 5, 6, 7). Therefore, it fol-lows that we should develop an updated theory to addressthese discrepancies.

7 Conclusion

The present study has theoretically and experimentallyexamined vortex-induced hydrodynamic forces acting on arigid circular cylinder in a forced oscillation problem. Thetheory constructed in this study is based on the wakeoscillator model expressing the swing-like motions of thewake region downstream of a cylinder subjected to ow.The quasisteady approximation was employed to derive theformulation for the lift forces and the lock-in limit in thepresent study. Although we found some discrepancies incases with high-frequency oscillations which may be due tothe applicability of the quasisteady and small-angle-of-attack approximations, overall good agreement with theexperiments has been shown, which conrms the useful-ness of the present theoretical method as a practical tool.

Appendix

According to Kutta-Joukowsikis theorem, the lift forceacting on a at rigid bar subject to a potential ow iswritten as

Exp.

Theory

= 0

H

Fig. 9 Lock-in (synchronization) limit line obtained from the presenttheory ( solid line ), and from the experiment conducted by Tanaka andTakahara [ 26] (closed squares ) and Stansby [ 27] (closed circles ).Reynolds number is 3.6 9 10 3

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L qV C qV 2p lV sin a;

26where the circulation C is determined by the additionalcondition that the velocity of the potential ow must benite at the trailing edge of the bar (Kuttas condition).Under the assumption that the length between anapplication point of the lift force and the center of rotation is d 2 l (Fig. 2), the restoring lift moment actingon the rotation of the wake oscillator can be written as

M pq lV 2 d

2 l sin2 affi2

pq lV 2 d

2 l a k a : 27

Namely, the coefcient of the restoring moment k (Eq. 2) is derived under the assumption of potential ow,quasisteady state, and small angle of attack.

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