Reduction of the loads on a cylinder undergoing harmonic in-line motionOsama A. Marzouk and Ali H. Nayfeh

Citation: Physics of Fluids (1994-present) 21, 083103 (2009); doi: 10.1063/1.3210774 View online: http://dx.doi.org/10.1063/1.3210774 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/21/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The vortex shedding around four circular cylinders in an in-line square configuration Phys. Fluids 26, 024112 (2014); 10.1063/1.4866593 Low-frequency unsteadiness in the vortex formation region of a circular cylinder Phys. Fluids 25, 085109 (2013); 10.1063/1.4818641 Numerical simulation of vortex-induced vibration of two circular cylinders of different diameters at low Reynoldsnumber Phys. Fluids 25, 083601 (2013); 10.1063/1.4816637 A coupled Landau model describing the Strouhal–Reynolds number profile of a three-dimensional circularcylinder wake Phys. Fluids 15, L68 (2003); 10.1063/1.1597471 Numerical studies of flow over a circular cylinder at Re D =3900 Phys. Fluids 12, 403 (2000); 10.1063/1.870318

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Reduction of the loads on a cylinder undergoing harmonic in-line motionOsama A. Marzouka� and Ali H. NayfehDepartment of Engineering Science and Mechanics (Mail Code 0219), Virginia Polytechnic Instituteand State University, Blacksburg, Virginia 24061, USA

�Received 13 February 2009; accepted 24 July 2009; published online 19 August 2009�

We use the finite-difference computational fluid dynamics method to study in detail the flow fieldaround a circular cylinder in a uniform stream while undergoing in-line harmonic motion. For agiven motion amplitude, there exists a critical forcing frequency below which the lift and drag canbe period-n, quasiperiodic, or chaotic. Similarly, for a given frequency, there exists a criticalamplitude below which the lift and drag can be period-n, quasiperiodic, or chaotic. Above thesecritical conditions, the lift and drag are synchronous with the forcing. The lift nearly vanishes andthe mean drag drops and saturates at a value that is independent of the driving frequency, whereasthe oscillatory drag quadratically depends on it. We relate these features to changes in the wake andthe surface-pressure distribution. We examine the influence of the Reynolds number on these criticalfrequency and amplitude. Second- and higher-order spectral analyses show remarkable changes inthe linear and quadratic coupling between the lift and drag when synchronization takes place; itdestroys the two-to-one coupling between them in the cases of no motion and synchronization dueto cross-flow motion. © 2009 American Institute of Physics. �DOI: 10.1063/1.3210774�

I. INTRODUCTION

The development of vortices and unsteady flow motionin the wake of a body in a uniform stream is a classicalproblem in fluid mechanics, which can be observed particu-larly in offshore structures, such as risers, spar platforms,fixed platforms, tension leg platforms, and jack up rigs. If thebody is not fixed, one or more of several motion types canoccur,1 including translational vortex-induced vibrations, tor-sional vortex-induced vibrations, galloping, airfoil flutter,multibody structure wake galloping, and breathing oscilla-tions.

The continuously generated vortices have direct effecton the exerted force on the body. This force can be decom-posed into two components: lift �cross-flow component� anddrag �in-line component�. These components depend on thebody geometry. We consider here a circular cylinder due toits simplicity and its common use in industrial applications.Moreover, the wake behind a circular cylinder is simplerthan those behind noncircular cylinders, which allows moreinformative analysis with less number of geometric param-eters. Furthermore, unlike square cylinders, galloping doesnot occur for a circular cylinder.2,3

The Strouhal number �SK� associated with the vonKármán vortex street is the nondimensional frequency atwhich two counter-rotating vortices are repeatedly shed inthe wake of a bluff body �a cylinder is a special case� re-strained from any type of motion in a uniform stream; it is acharacteristic of the wake of that body. When a cylinder isdriven in the cross-flow direction �perpendicular to the in-coming stream and to its span� with a nondimensional fre-quency SM close to SK, the nondimensional vortex sheddingfrequency SV is synchronized at the forcing frequency SM,4–7

thus shedding is entrained by the cylinder motion. As a con-sequence, the lift is synchronized at SM. Similar to the caseof a fixed cylinder,8–10 the drag frequency is twice the liftfrequency and is synchronized at 2SM. On the other hand,when a cylinder is forced to oscillate in the in-line direction�parallel to the incoming stream� with a nondimensional fre-quency SM around twice SK, shedding is synchronized at12SM.11–13 The synchronization phenomenon also occurswhen a cylinder is constrained to oscillate in either the cross-flow or in-line directions by a spring and a dashpot.14,15

The difference in the range of synchronization frequen-cies in the two cases is explained by the phasing between themoving cylinder and the synchronized shedding.16 For har-monic motion in the in-line direction with two alternatingvortices being shed periodically, either shedding event of thetop and bottom vortex is always promoted when sheddingoccurs at half the motion frequency. However, only everyother shedding event will be promoted if the motion fre-quency is equal to the shedding frequency. The situation isaltered in the case of harmonic motion in the cross-flow di-rection, where shedding is enhanced when it occurs at themotion frequency but is partly opposed if it occurs at half themotion frequency.

Studies of a one-degree-of-freedom moving cylinderusually consider the cross-flow motion with the in-line mo-tion being restrained. Many of these published studies do notindicate the motion direction in the title because it is bydefault the cross-flow direction.17–19 There is disagreementon the importance of the in-line cylinder motion on its wakeand dynamics. Some studies show that the induced in-linemotions are one order of magnitude smaller than the cross-flow motions,20,21 others emphasize the importance of theinduced in-line motions even though they are small, and oth-ers show that they can be comparable to the cross-flow mo-a�Electronic mail: omarzouk@vt.edu.

PHYSICS OF FLUIDS 21, 083103 �2009�

1070-6631/2009/21�8�/083103/13/$25.00 © 2009 American Institute of Physics21, 083103-1

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tions, depending on the cylinder density22,23 or its nondimen-sional natural frequency.24

Tanida et al.12 carried out experiments on a one-degree-of-freedom circular cylinder oscillating harmonically in thein-line direction at the Reynolds numbers Re=80 and 4000.The first part of their study considered a single circular cyl-inder oscillating in a uniform stream, whereas the secondpart was dedicated to the case of a circular cylinder oscillat-ing in the wake of another cylinder �i.e., tandem arrange-ment�. In both parts, they measured the lift and drag on theoscillating cylinder. In the first case, the cylinder was madeto oscillate sinusoidally with a prescribed amplitude that is14% of its diameter �the amplitude was 4.2 mm and thediameter was 30 mm� and frequencies varying from below toabove 2SK. Each experiment corresponded to a single fre-quency. Individual experiments lasted for at least 20 s, whichwas long enough for the steady state to be achieved. Theworking liquid was oil for Re=80 and water for Re=4000.They observed synchronization for both configurations. Theyalso reported that the fluctuating lift “vanishes” occasionallyat Re=4000, where its magnitude drops to very small levelswhen the forcing frequency is around twice SK. They relatedthis lift suppression to a positive aerodynamic damping in-duced by the cylinder motion, which they described as being“stable” in this condition, and to the work done by the dragforce. We show this lift suppression numerically at otherReynolds numbers. We then enhance their explanation byexamining variations of multiple flow variables and theircharacteristics, such as the vortex structure, the lift and dragharmonics and their coupling, and the phase between thesynchronous drag and motion. We consider a broad range ofSM that is not limited to the synchronous range.

Kim and Williams25 measured the lift and drag on acylinder undergoing harmonic motion in air at Re=15 200and studied the nonlinear coupling between them. The cylin-der diameter was 50.8 mm and its length was 610 mm. Aprinted circuit motor connected to a Scotch-yoke mechanismwas used to control the motion. The main part of the experi-ment was conducted at a constant motion amplitude of 3.5%of the diameter, which is very small. Also, the motion fre-quency was 0.8SK, which is far below the synchronizationrange. They revealed interesting facts about the nonlinearinteraction between the lift and drag and explained the struc-ture of their measured power spectra.

In this study, we consider a circular cylinder undergoinga one-degree-of-freedom harmonic motion in the in-line di-rection in a uniform stream at different Re. The motion in-duces a mechanical perturbation in the shear layer. We limitourselves to the influence of the forcing frequency on thedynamics of the flow and the exerted forces. We compute thevelocities and pressure and pay special attention to the syn-chronization that occurs near twice SK. Several differencesexist between this synchronization and the one that occurswhen the cylinder is fixed or driven in the cross-flow direc-tion around SK. We use different analysis techniques to ex-amine and highlight these differences and relate them tochanges in the vortex shedding and the surface pressures.

II. APPROACH

All variables are made nondimensional using the cylin-

der diameter D as a reference length, the incoming far-field

velocity U� as a reference velocity, and U� / D as a referencetime. For the cylinder motion, we let

��t� = Ax sin�2�SMt� , �1�

where � is the displacement and Ax and SM are the nondi-mensional amplitude and cyclic frequency of the cylindermotion, respectively.

The velocity and pressure fields are governed by thetwo-dimensional, unsteady Navier–Stokes equations,

�u

�x+

�v�y

= 0,

�u

�t= − �u − ug�

�u

�x− �v − vg�

�u

�y−

�p

�x+

1

Re� �2u

�x2 +�2u

�y2� ,

�2�

�v�t

= − �u − ug��v�x

− �v − vg��v�y

−�p

�y+

1

Re� �2v

�x2 +�2v�y2� ,

where u and v are the x-component and y-component of thefluid velocity, ug and vg are the x-component andy-component of the grid speed, and p is the pressure. Equa-tions �2� are first Reynolds-averaged and the Baldwin–Bartheddy-viscosity model is used to account for the unresolvedturbulent scales. The resulting equations are then rewritten inbody-fitted coordinates � and �, instead of x and y, which arealigned with the boundaries.26,27 The differential equationsare discretized using central second-order differences for theviscous terms, an upwind flux-difference splitting scheme forthe convective terms, and a second-order implicit backwardEuler scheme for the local derivatives. The numerical fluxesare calculated at the midcell location. A colocated arrange-ment of the primitive variables is used, where the discretizedvelocities and pressure fields are defined at the same gridpoints. For efficient calculations, the orthogonal viscousterms are treated implicitly, whereas the nonorthogonal terms�which arise due to nonorthogonality in the grid lines� aretreated explicitly by moving them to the right-hand side ofthe algebraic system. For a fixed cylinder, the grid is or-thogonal and the nonorthogonal viscous terms are analyti-cally equal to zero. On the other hand, the in-line motioncauses deformations in the grid, but the nonorthogonality issmall as can be seen in Fig. 1. Therefore, these nonorthogo-nal terms are not significant and their explicit treatment is aminor issue. We performed grid sensitivity checks and con-firmed that higher resolutions do not cause importantchanges in the simulation results. A no-slip boundary condi-tion is applied at the cylinder surface. At the far-field inflowboundary, a uniform horizontal stream is applied for the ve-locities �i.e., u=U�=1 and v=V�=0�, whereas the pressureis extrapolated. At the far-field outflow boundary, the pres-sure is specified �p= P�=1�, whereas the velocities are ex-trapolated. Figure 1 shows a portion of the cylinder at itsfarthest downstream and upstream locations. The outer shape

083103-2 O. A. Marzouk and A. H. Nayfeh Phys. Fluids 21, 083103 �2009�

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of the computational region is a nondeforming circle with anondimensional radius equal to 25. The equations are solvediteratively, and the iteration is stopped when the maximumabsolute residual of the continuity equation over all gridpoints is less than 10−4 or after 12 iterations. The problem issolved on a Silicon Graphics, Inc. �SGI� machine with 1.6GHz processor speed and 512 GB memory. Each time steptakes about 1.18 s of CPU time. The nondimensional timestep is 0.05.

We start by comparing our simulations with the reportedexperimental data of Tanida et al.12 The experiments wereconducted in oil at Re=80, with an in-line motion amplitudeequal to 14% of the cylinder diameter. To reduce three-dimensional effects, they took measurements at the centralsection of the test cylinder. In Fig. 2, the ratio of the shed-ding frequency to the forcing frequency SV /SM is plotted as afunction of SM. In addition to the good agreement betweenthe simulations and the measurements, this figure shows alsoone type of synchronization at this Re and Ax, in which SV

�which is also the fundamental lift frequency� is equal to12SM. Further analysis of the simulation results of the syn-chronous lift and drag showed that CD is synchronous at SM.Therefore, there is a quadratic coupling between the synchro-nous lift and drag as in the case of no motion and the case ofsynchronization due to cross-flow motion. We focus in thisstudy on another type of synchronization due to in-line mo-tion in which the lift is synchronous at SM and its amplitudeis reduced to very low levels. Figure 3 shows the componentof the drag coefficient CD in phase with the velocity of thein-line motion, which we denote by CD,V, as a function ofSM. Again, there is good agreement between our simulationsand the measurements. Figure 4 compares variations of theroot mean square �rms� of the calculated lift coefficient CL

with SM to the measured one. As mentioned before, the syn-chronous CL is not reduced at this very low Re.

S M

SM

/SV

0 0.1 0.2 0.3 0.40

1

2

3ExperimentSimulation

FIG. 2. �Color online� Comparison between our simulations and the mea-surements of Tanida et al. �Ref. 12� for the variation of the forcing-to-shedding frequency ratio SM /SV with the nondimensional forcing frequencySM for a cylinder oscillating in-line at Re=80 with nondimensional ampli-tude Ax=0.14.

S M

CD

,V

0 0.1 0.2 0.3 0.4-1.5

-1

-0.5

0

ExperimentSimulation

FIG. 3. �Color online� Comparison between our simulations and the mea-surements of Tanida et al. �Ref. 12� for the variation of the drag componentin phase with velocity CD,V with the nondimensional forcing frequency SM

for a cylinder oscillating in-line at Re=80 with Ax=0.14.0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

x

y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

x

y

(b)

(a)

FIG. 1. Partial views of the cylinder and grid at the instants of �i� maximumdisplacement and �ii� minimum displacement.

S M

RM

SC

L

0 0.1 0.2 0.3 0.40

0.5

1ExperimentSimulation

FIG. 4. �Color online� Comparison between our simulations and the mea-surements of Tanida et al. �Ref. 12� for the variation of rms CL with thenondimensional forcing frequency SM for a cylinder oscillating in-line atRe=80 with Ax=0.14.

083103-3 Reduction of the loads on a cylinder Phys. Fluids 21, 083103 �2009�

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III. RESULTS AND DISCUSSION

A. Lift and drag

We investigate several flow properties over a wide rangeof mechanical frequencies SM while fixing the amplitude ofmotion Ax at 0.20. This value allows us to compare the flowfeatures for this case with those obtained for a cylinder withcross-flow motion at the same amplitude and Re.28 The ma-jority of the results correspond to Re=500; the others corre-spond to Re=300. Synchronization is illustrated in Fig. 5,which shows the ratio SV /SK as a function of the ratio SM /SK.When SM �2SK, vortex shedding is synchronized at SM.Hence the graph of SV /SK with SM /SK is a straight line withunity slope; it starts at the critical value SM /SK=1.81 andextends to SM /SK=3.33. Beyond this value, the synchronousshedding bifurcates and becomes nonsynchronous with lessregular pattern and with an asymmetric lift. We refer to thisrange of SM as postsynchronization compared to presynchro-nization when SM is below the critical value. This range is incontrast to the range 0.81�SM /SK�1.05 for the case ofcross-flow motion at the same amplitude and Re. So, thesynchronization range here is not just shifted, but it is alsobroadened. It should be mentioned that the postsynchroniza-tion range is preceded by a region in the presynchronizationrange where the shedding occurs at half SM. Therefore, onecan describe vortex shedding as being synchronized at 1

2SM

as reported in other studies.12,32 However, because we want

to focus on the situation in which vortex shedding occurs atSM, we use the terms “synchronization” and “postsynchroni-zation” to refer to this situation.

Variation of the rms CL with SM is shown in Fig. 6. Theresults are presented in terms of relative values, thus the rmsCL is presented relative to its value in the case of Ax=0.Similarly, we use the ratio SM /SK to express changes in SM.The lift reduction is clear in this figure, the relative rms CL isreduced by two orders of magnitude from 1.696 at SM /SK

=1.80 to 0.0087 �a reduction of 99.5%� at the criticalSM /SK=1.81, which corresponds to the beginning of syn-chronization.

We decompose the steady-state CD�t� into a constantmean component �CD and an oscillatory component CD,osc

and analyze each one separately. Variation of the relativemean CD with SM /SK is shown in Fig. 7. Synchronization ofthe drag causes a reduction in its mean value from 1.32 to0.77 �a reduction of 42%�. In the postsynchronization cases,�CD is independent of SM, in contrast with CD,osc whose rmsvalue grows monotonically with SM as shown in Fig. 8. Thisgrowth can be represented by a quadratic function, as indi-cated in Fig. 9.

B. Mechanical work and drag phase

The nondimensional mechanical work done by the cyl-inder on the flow per motion cycle TM �starting from anarbitrary time to� is

S M / S K

Rel

ativ

eR

MS

CL

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

FIG. 6. �Color online� Variation of the relative rms CL with the forcing-to-Strouhal frequency ratio SM /SK.

S M / S K

Rel

ativ

e<

CD>

0 0.5 1 1.5 2 2.5 30.6

0.8

1

1.2

1.4

FIG. 7. �Color online� Variation of the relative mean drag coefficient �CDwith the forcing-to-Strouhal frequency ratio SM /SK.

S M / S K

SV

/SK

0 0.5 1 1.5 2 2.5 30

1

2

3SimulationSlope 1.0Slope 0.5

FIG. 5. �Color online� Variation of the shedding-to-Strouhal frequency ratioSV /SK with the forcing-to-Strouhal frequency ratio SM /SK. Postsynchroniza-tion region in the simulation results is indicated by a slope of unity.

S M / S K

Rel

ativ

eR

MS

CD

,osc

0 0.5 1 1.5 2 2.5 30

15

30

45

60

FIG. 8. �Color online� Variation of the relative rms CD,osc with the forcing-to-Strouhal frequency ratio SM /SK.

083103-4 O. A. Marzouk and A. H. Nayfeh Phys. Fluids 21, 083103 �2009�

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Wcyc = − Ax2�SMto

to+TM

CD�t�cos�2�SMt�dt . �3�

The minus sign on the right-hand side of Eq. �3� ensures thecorrect sign for Wcyc so that, when the velocity of the cylin-der and the drag force are in the same direction, Wcyc isnegative and the work is actually done by the flow on thecylinder. We carry out the above integration numerically us-ing the trapezoidal rule. The results of Wcyc as a function ofSM /SK are given in Fig. 10. The sign of Wcyc is always posi-tive, indicating that work is being done by the cylinder onthe flow. This constitutes another difference from the case ofcross-flow motion, where Wcyc takes on both negative andpositive values, depending on SM.

To first order, the steady-state synchronous drag can beapproximated as

CD�t� = �CD + �CD,osc�sin�2�SMt + �� , �4�

where � . � indicates an amplitude and � indicates the phaseangle by which CD leads �. Substituting Eqs. �4� and �1� intoEq. �3� leads to the following expression:

Wcyc = − �Ax�CD,osc�sin��� . �5�

Equation �5� implies that � must be negative in order to havepositive Wcyc. We computed � for the synchronous-drag

cases and found that it is always negative varying from�0.262 ��15°� to �0.175 ��10°�, as shown in Fig. 11. Forcross-flow motion, a sudden change in the phase between thesynchronous lift and the motion was reported in differentstudies;28–31 we also found it to occur with the current Reand motion amplitude. Such a feature does not occur with theexamined in-line motion.

C. Dissipation

We compute the average dissipated power in the flow ateach mechanical frequency. Its dimensional expression is

Pave =1

t2 − t1

t1

t2Fx�U� − x�dt , �6�

where Fx is the dimensional drag force, t1 and t2 are arbitrary

�but appropriate for statistical analysis�, and �U�− x� is therelative velocity between the cylinder and free-stream fluid

in the Fx direction. Combining Eqs. �1� and �6� yields thefollowing nondimensional expression for the average dissi-pated power:

Pave =1

t2 − t1

t1

t2

CD�t��1 − Ax2�SM cos�2�SMt��dt . �7�

We use at least 40 motion periods to evaluate the integrandin Eq. �7�. When Ax=0, Pave reduces to �CD. In Fig. 12,variation of the relative Pave with SM /SK is shown. Over the

S M / S K

Rel

ativ

eR

MS

CD

,osc

1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.410

20

30

40

50

60

70

SimulationFitting

y = 8.178 x 2 + 8.000 x - 3.726

FIG. 9. �Color online� Variation of the calculated and fitted relative synchro-nous rms CD with the forcing-to-Strouhal frequency ratio SM /SK.

S M / S K

Wcy

c

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

FIG. 10. �Color online� Variation of the mechanical work Wcyc with theforcing-to-Strouhal frequency ratio SM /SK.

S M / S K

βo

1.8 2.2 2.6 3 3.4-20

-15

-10

-5

0

FIG. 11. �Color online� Variation of the phase angle � of the synchronousCD relative to the nondimensional displacement � with the forcing-to-Strouhal frequency ratio SM /SK.

S M / S K

Rel

ativ

eP

ave

0 0.5 1 1.5 2 2.5 30

2

4

6

8

FIG. 12. �Color online� Variation of the relative average dissipated powerPavg with the forcing-to-Strouhal frequency ratio SM /SK.

083103-5 Reduction of the loads on a cylinder Phys. Fluids 21, 083103 �2009�

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presynchronization frequencies, Pave increases slowly,whereas it increases quickly with a cubic profile over thepostsynchronization frequencies, as indicated by the polyno-mial fit in Fig. 13. This cubic profile can be explained asfollows. The first term in the integrand in Eq. �7� gives �CD,which remains unchanged with SM for postsynchronizationcases, as shown in Fig. 7. Therefore, Pave is controlled by thesecond term in the integrand, in which CD�t� is multiplied bySM. Recalling the approximation in Eq. �4�, we find that thisterm becomes −�2� sin���AxSMrmsCD,osc, which is propor-tional to the product of rms CD,osc and SM. The angle � variesweakly with SM, as shown in Fig. 11, and rms CD,osc in-creases quadratically with SM as shown in Fig. 9; hence Pave

increases in a cubic fashion with SM.

D. Synchronization map

We found that increasing the motion amplitude decreasesthe critical frequency at which the lift reduction starts. Thereis a threshold of this amplitude below, which the lift does notexhibit this feature for any mechanical frequency. We foundthat the threshold here is Ax=0.157. Figure 14 shows thelocus of the critical conditions in the Ax−SM /SK plane, whichseparates synchronization and nonsynchronization. Similarvariations of the lift and drag with SM take place at other Re.The loci of the critical conditions at Re=500 and 300 arecompared in Fig. 15. The locus of the critical conditionswhen Re=300 is nearly a straight line with a negative slope,

and the threshold Ax is increased to 0.2. The need for a largermotion amplitude for lift reduction at lower Re can be ex-plained by the higher viscous dissipation. This also explainsthe absence of such a feature at Re=80 in the experiments ofTanida et al.12 and our simulations.

E. Pre- and postsynchronization modes

In the following part, we present several lift and dragresponse modes that occur before and after the onset of syn-chronization. These modes are very different quantitativelyand qualitatively. Before the critical frequency, we found liftand drag responses that are either periodic with large period�period-n�, quasiperiodic, or chaotic. The best method to dis-tinguish among these responses is Poincaré sections. In Fig.16, we show representative nonsynchronous cases showingthe Poincaré sections of CL for �i� a quasiperiodic responsewith SM /SK=0.4, �ii� a period-6 response with SM /SK=0.72,�iii� a chaotic response with SM /SK=0.73, and �iv� a period-2response with SM /SK=1.7. The corresponding Poincaré sec-tions of CD are shown in Fig. 17, which clearly indicate thatboth of the lift and drag have the same response type for eachcase.

S M / S K

Rel

ativ

eP

ave

1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.41

3

5

7

9

11

SimulationFitting

y = 1.083 x 3 - 4.721 x 2 + 8.397 x - 4.693

FIG. 13. �Color online� Variation of the calculated and fitted relative aver-age dissipated power Pavg with the forcing-to-Strouhal frequency ratioSM /SK for synchronous cases.

NonsynchronousThreshold

Synchronous

S M / S K

Ax

1.56 1.62 1.68 1.74 1.8 1.86 1.920.15

0.18

0.21

0.24

0.27

0.3

FIG. 14. �Color online� Locus of the critical forcing frequency in the Ax

−SM /SK plane at Re=500.

S M / S K

Ax

1.5 1.6 1.7 1.8 1.9 20.15

0.18

0.21

0.24

0.27

0.3

FIG. 15. �Color online� Loci of the critical forcing frequency in the Ax

−SM /SK plane at Re=500 �solid line� and Re=300 �dashed line�. The thresh-old Ax is indicated by a solid circle.

C L

CL

-2 -1 0 1 2-2

-1

0

1

2

.

C L

CL

-2 -1 0 1 2-2

-1

0

1

2

.

C L

CL

-2 -1 0 1 2-2

-1

0

1

2

.

C L

CL

-2 -1 0 1 2-2

-1

0

1

2

.

(b)(a)

(d)(c)

FIG. 16. �Color online� Poincaré sections of nonsynchronous CL at �i�SM /SK=0.4, �ii� SM /SK=0.72, �iii� SM /SK=0.73, and �iv� SM /SK=1.7.

083103-6 O. A. Marzouk and A. H. Nayfeh Phys. Fluids 21, 083103 �2009�

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The period-2 mode of the lift and drag takes place for awide range of SM /SK �from 1.15 to 1.8�. It is followed bysynchronization, which is accompanied by large changes inCL and CD, as can be seen from comparing the amplitudesand patterns of the time histories for the period-2 case atSM /SK=1.7 in Fig. 18 to those for the synchronous mode atSM /SK=1.81 in Fig. 19. The synchronous lift and drag arecharacterized by a single point in their Poincaré sections, asin Fig. 20. We pay attention to changes that occur in the liftand drag and their coupling due to synchronization. Toachieve this, we compare their spectra and cross bicoherenceto those we found for the period-2 case at SM /SK=1.7 �thePoincaré sections of CL and CD for this case were shownalready in the last plots of Figs. 16 and 17, respectively� andalso to those we found for another synchronous case atSM /SK=1 for a cross-flow motion. The Poincaré sections ofCL and CD for the latter case are shown in Fig. 21. Whereaseach of these sections contains a single point, as was the casein the synchronous cases due to the current in-line motion,we show below that the synchronous CL is period-1 for bothcases and the synchronous CD is period-1 in case of in-line

motion but period-12 in case of cross-flow motion. Because

one cannot distinguish between the two cases using thePoincaré sections for CD, we use second- and third-orderspectral analyses to differentiate between them.

The spectra of CL and CD for the period-2 case atSM /SK=1.7 are shown in Fig. 22. The frequency S is scaledwith the mechanical frequency SM to better indicate the po-sitions of the fundamental components and their superhar-monics and subharmonics. The fundamental component ofCL is at 1

2SM, whereas the fundamental component of CD is atSM. There are odd and even superharmonics in the spectrumof CL, but the odd harmonics are stronger. There are frac-tional superharmonics in the spectrum of CD, but the integerones are stronger. The spectra of CL and CD for the synchro-nous case at SM /SK=1.81 are shown in Fig. 23. The funda-mental component of CL is now at SM and the fundamentalcomponent of CD is still at SM. Whereas there are still evenand odd superharmonics in the spectrum of CL �as in thepresynchronization case�, their amplitudes decay monotoni-cally and there is no bias toward the odd superharmonics.The fractional superharmonics of CD have disappeared. Thespectra of CL and CD for the synchronous case at SM /SK=1for cross-flow motion are shown in Fig. 24. One of the maindifferences between this mode of synchronization and theone due to in-line motion is that CD is synchronous at 2SM inthe case of cross-flow motion rather than at SM. Also, thespectrum of CL consists mainly of odd superharmonics, andthe spectrum of CD consists mainly of even ones. It should

C D

CD

1 1.1 1.2 1.3 1.4 1.5-0.6

-0.4

-0.2

0

0.2

0.4

0.6

.

C D

CD

1.3 1.4 1.5 1.6 1.70

0.2

0.4

0.6

.

C D

CD

0.6 0.8 1 1.2 1.4 1.6-0.2

0

0.2

0.4

0.6

0.8

1

.

C D

CD

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

.

(b)(a)

(c) (d)

FIG. 17. �Color online� Poincaré sections of nonsynchronous CD at �i�SM /SK=0.4, �ii� SM /SK=0.72, �iii� SM /SK=0.73, and �iv� SM /SK=1.7.

t

CL

270 280 290 300-2

0

2

CD

0

2

4

FIG. 18. �Color online� Steady-state time histories of CL and CD for theperiod-2 mode at a forcing-to-Strouhal frequency ratio SM /SK=1.7.

CD

-1

0

1

2

3

t

CL

270 280 290 300-0.01

0

0.01

FIG. 19. �Color online� Steady-state time histories of CL and CD for thesynchronous mode at a forcing-to-Strouhal frequency ratio SM /SK=1.81.

C L

CL

-0.1 -0.05 0 0.05 0.1-0.1

-0.05

0

0.05

0.1

.

C D

CD

0.8 1 1.2 1.4 1.6 1.8 2-5

-4

-3

-2

.

(b)(a)

FIG. 20. �Color online� Poincaré sections of CL and CD for the synchronousmode due to in-line motion at a forcing-to-Strouhal frequency ratio SM /SK

=1.81.

083103-7 Reduction of the loads on a cylinder Phys. Fluids 21, 083103 �2009�

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be mentioned here that, in the fixed cylinder case, the spectraof CL and CD are qualitatively very similar to those in Fig. 24�after scaling S with SK instead of SM�.

If we define a total-force coefficient CT as �CL2 +CD

2 andits angular orientation �T as arctan�CL /CD�, which is mea-sured in the counterclockwise direction from the positivex-axis, then the near-harmonic profiles of both CT and �T

found in the case of synchronization due to cross-stream mo-tion �as shown in Fig. 25 for SM /SK=1� or in the absence ofmotion are totally altered in the case of synchronization dueto in-line motion as shown in Fig. 26 for SM /SK=1.81. In theformer cases, the frequency of CT is equal to 2SM �or 2SK�,which is twice the frequency of �T. Also, the angle �T islimited to the first and fourth quadrants, thus CD is alwayspositive. With in-line synchronization, the profile of �T be-comes nearly a step function with values equal to 0° or�180°. This effect is a consequence of the extreme reductionin CL. The interval when ��T��180° corresponds to negativeCD.

F. Higher-order spectral analysis

To examine the nonlinear coupling between CL and CD,we calculate the magnitude-squared cross-bicoherencebLLD

2 �S1 ,S2� as

bLLD2 �S1,S2� =

�MLLD�S1,S2��2

MLL�S1�MLL�S2�MDD�S1 + S2�, �8�

where

MLLD�S1,S2� = E�L��S1�L��S2�D�S1 + S2�� �9�

is the cross bispectrum, L�S� and D�S� are the discrete Fou-rier transforms of CL�t� and CD�t�, respectively, E indicates

C L

CL

-2 -1 0 1 2-2

-1

0

1

2

.

C D

CD

1.6 1.7 1.8 1.9 2-0.5

0

0.5

1

.

(b)(a)

FIG. 21. �Color online� Poincaré sections of CL and CD for the synchronousmode due to cross-flow motion at a forcing-to-Strouhal frequency ratioSM /SK=1.

S / S M

CL

Spe

ctru

m

0 0.5 1 1.5 2 2.5 3 3.5 410-4

10-3

10-2

10-1

100

101

S / S M

CD

Spe

ctru

m

0 0.5 1 1.5 2 2.5 3 3.5 410-4

10-3

10-2

10-1

100

101

(b)

(a)

FIG. 22. �Color online� Magnitude power spectra of CL and CD for theperiod-2 mode due to in-line motion at a forcing-to-Strouhal frequency ratioSM /SK=1.7.

S / S M

CL

Spe

ctru

m

0 0.5 1 1.5 2 2.5 3 3.5 410-5

10-4

10-3

10-2

10-1

S / S M

CD

Spe

ctru

m

0 0.5 1 1.5 2 2.5 3 3.5 410-4

10-3

10-2

10-1

100

101

(b)

(a)

FIG. 23. �Color online� Magnitude power spectra of CL and CD for thesynchronous mode due to in-line motion at a forcing-to-Strouhal frequencyratio SM /SK=1.81.

S / S M

CL

Spe

ctru

m

0 0.5 1 1.5 2 2.5 3 3.5 410-4

10-3

10-2

10-1

100

S / S M

CD

Spe

ctru

m

0 0.5 1 1.5 2 2.5 3 3.5 410-4

10-3

10-2

10-1

100

101

(b)

(a)

FIG. 24. �Color online� Magnitude power spectra of CD and CL for thesynchronous mode due to cross-flow motion at a forcing-to-Strouhal fre-quency ratio SM /SK=1.

083103-8 O. A. Marzouk and A. H. Nayfeh Phys. Fluids 21, 083103 �2009�

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the expected value �or time average�, and the superscript �

indicates a complex conjugate. The autopower spectraMLL�f� and MDD�f� of CL and CD are given by

MLL�S� = E�L��S�L�S�� , �10�

MDD�S� = E�D��S�D�S�� . �11�

The corresponding magnitude-squared cross bicoherencefor the CL and CD spectra in Fig. 22 is shown in Fig. 27. Thepresence of many quadratically interacting lift components isnoticeable. A fractional subharmonic or superharmonic in CD

at 12mSM is formed by quadratic coupling between the CL

components at �m+k�SM and −�k+ 12m�SM, where k0 is an

integer. These coherence points are located in the differenceregion of bLLD

2 . They are in addition to other couplings be-tween the CL components at � 1

2m−k�SM and kSM, where k0 is an integer. These coherence points are located in thesum region of bLLD

2 . The number of coherence points is re-duced in the synchronized cases, as shown in Fig. 28, whichcorresponds to the spectra of CL and CD in Fig. 23. This isbecause there are phase-coherent components at fractions ofSM. The fundamental frequency of CD at SM is formed by theinteraction of the components of CL at kSM and −�k+1�SM,where k0 is an integer. Similarly, the superharmonic in thespectrum of CD at 2SM is formed by the interaction of thecomponents of CL at kSM and −�k+2�SM in addition to self-

interacting CL component at SM. A similar structure occursfor the higher superharmonics in CD. The bicoherence plot iseven simplified further for synchronization cases due tocross-flow motion, as shown in Fig. 29, which correspondsto the CL and CD spectra in Fig. 24. This is because of thepresence of half the number of significant superharmonics inthe CL �odd ones� and CD �even ones� spectra. The funda-mental component of CD at 2SM is formed by self-interactingCL component at SM in addition to the interaction of thecomponents of CL at 3SM and −SM. The small CD subhar-monic at SM is formed by the interaction of the componentsof CL at 2SM and −SM.

The bicoherence analysis provides information about thequadratic coupling of the CL components in the CD compo-nents. To examine the linear correlation between the CL andCD components, we use the �linear� cross-power spectrumMLD�f�, defined as

MLD�S� = E�L��S�D�S�� . �12�

Figures 30–32 show the absolute value of the cross-powerspectrum �MLD�S�� corresponding to the CL and CD spectra inFigs. 22–24, respectively. For the presynchronization case,�MLD�S�� in Fig. 30 indicates that all CL components havelinear coupling with the respective CD components. The cou-plings at 1

2SM and SM are both one order of magnitude largerthan the one at the subsequent superharmonic at 3

2SM. For thesynchronized cases with either in-line or cross-flow motion,

CT

1

1.5

2

t

βT

270 280 290 300-30

-15

0

15

30

(deg

rees

)

FIG. 25. �Color online� Steady-state time histories of CT and �T for thesynchronous mode due to cross-flow motion at a forcing-to-Strouhal fre-quency ratio SM /SK=1.

CT

0

1

2

3

t

βT

270 280 290 300-180

-90

0

90

180

(deg

rees

)

FIG. 26. �Color online� Steady-state time histories of CT and �T for thesynchronous mode due to in-line motion at a forcing-to-Strouhal frequencyratio SM /SK=1.81.

S / S M

S/S

M

0 0.5 1 1.5 2 2.5 3 3.5 4-4

-3

-2

-1

0

1

2

3

4

b2LLD1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1sss

FIG. 27. �Color online� Magnitude-squared cross bicoherence for theperiod-2 mode due to in-line motion at a forcing-to-Strouhal frequency ratioSM /SK=1.7.

FIG. 28. �Color online� Magnitude-squared cross bicoherence for the syn-chronous mode due to in-line motion at a forcing-to-Strouhal frequencyratio SM /SK=1.81.

083103-9 Reduction of the loads on a cylinder Phys. Fluids 21, 083103 �2009�

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linear couplings occur at SM and its integer superharmonics.The decay of �MLD�S�� at higher superharmonics is faster inthe case of in-line motion.

G. Wake structure

In the remaining part of this study, we relate the largedifferences in the postsynchronization cases, including thesignificant lift reduction and the saturation of the mean drag,to changes in the vortex shedding and the pressure distribu-tion on the surface of the cylinder. In Fig. 33, vorticity con-tours in the near field for two postsynchronization cases �atSM /SK=1.81 and 3� are shown at the instant when ��t�=0

and increasing �from negative to positive�, and therefore � ismaximum. The typical von Kármán vortex street with 2Smode is replaced by two parallel 1S streets: one with positivevortices located behind the bottom point of the surface andthe other with negative vortices located behind its top point.This instantaneous symmetry in the wake is what causes thereduction in CL because the lift force is a result of the instan-taneous imbalance in the surface pressure �which is related tothe vortex strength at the surface� between the top and bot-tom parts of the surface. As SM increases, the vortex shed-ding frequency also increases and the shed vortices alongeach street become closer and those being shed from thesurface become stronger �higher vorticity levels�. So, we ex-pect more negative pressure at the locations of these vorticeson the surface of the cylinder.

In the classical vortex shedding mechanism for a fixedcylinder, two contrarotating vortices are periodically formedand shed downstream in the wake. This is due to the shear-layer instability, which causes vortex roll-up and leads to thegeneration of an upper vortex and lower vortex in a periodicand staggered pattern. The lift frequency is equal to the shed-ding frequency but the drag frequency is equal to twice theshedding frequency because both upper and lower vorticescontribute to the drag through the changes they induce in thepressure field. This shedding mechanism is preserved in thecase of a synchronous wake due to cross-flow oscillation,which is expected because the wake structure remains similarto the one taking place in the case of a fixed cylinder, and theeffect of the shear-layer instability is not inhibited by thecross-flow oscillation. Consequently, either the upper orlower vortex still contributes to the drag, keeping its fre-quency twice the frequency of the shedding and lift. On theother hand, the in-line oscillation influences the locations andmotions of the formed vortices, which tend to oppose thecylinder motion due to the added-mass effect. When the cyl-inder velocity is strong enough, which corresponds to a highfrequency or high amplitude, this disturbing influence of theoscillation overpowers the effect of the shear-layer instabilityand becomes the dominant mechanism that controls the vor-tex formation and shedding. This results in a sudden changein the wake state, causing it to be instantaneously symmetricand consequently reduces the lift dramatically. The vorticesare no longer staggered, but they are convected parallel to

FIG. 29. �Color online� Magnitude-squared cross bicoherence bLLD2 for the

synchronous mode due to cross-flow motion at a forcing-to-Strouhal fre-quency ratio SM /SK=1.

S / S M

|ML

D|

0 0.5 1 1.5 2 2.5 3 3.5 410-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

FIG. 30. �Color online� Absolute of cross-power spectrum �MLD� for theperiod-2 mode due to in-line motion at a forcing-to-Strouhal frequency ratioSM /SK=1.7.

S / S M

|ML

D|

0 0.5 1 1.5 2 2.5 3 3.5 410-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

FIG. 31. �Color online� Absolute of cross-power spectrum �MLD� for thesynchronous mode due to in-line motion at a forcing-to-Strouhal frequencyratio SM /SK=1.81.

S / S M

|ML

D|

0 0.5 1 1.5 2 2.5 3 3.5 410-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

FIG. 32. �Color online� Absolute of cross-power spectrum �MLD� for thesynchronous mode due to cross-flow motion at a forcing-to-Strouhal fre-quency ratio SM /SK=1.

083103-10 O. A. Marzouk and A. H. Nayfeh Phys. Fluids 21, 083103 �2009�

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each other, which causes the drag frequency to be equal tothe shedding frequency because both vortices contribute si-multaneously to the drag. This explanation is supported bythe four vorticity contours in Fig. 34 that are taken over onecycle of in-line motion, separated in time by a quarter cycle,for the postsynchronization case at SM /SK=1.81. The corre-sponding surface distributions of the pressure coefficient areshown and discussed in Sec. III H, which are in support ofthe mentioned change in the drag frequency. This explana-tion is also justified by our finding that the critical frequency,at which the wake state changes, decreases with the oscilla-tion amplitude as shown in the synchronization maps in Figs.14 and 15. Therefore, the combined effect of the frequencyand amplitude of the cylinder motion is what drives thechange in the wake state. Barbi et al.32 experimentally per-turbed the inflow velocity rather than the in-line motion of

the cylinder. Therefore, the constant velocity U� of the in-coming stream is replaced by an oscillating one, which has anonzero mean value. Although the perturbation mechanismof the shedding is different from the one we implement here,the Reynolds number is higher �3000–40 000�, the aspectratios of the cylinders used are low �including a length-to-diameter ratio of 2.5�, and the perturbation could not bemade purely harmonic, they found cases where shedding oc-curs at the pulsation frequency. It is interesting that, for thiscase, the lift and drag have the same frequencies, the lift isremarkably reduced, and two vortices are symmetricallyshed every perturbation cycle. These features are similar tothe ones we found in our study, which covered a wider rangeof frequencies

H. Surface pressure

Figure 35 shows distributions of the mean pressure co-efficient �CP at the cylinder surface for the same two post-synchronization cases shown in Fig. 33. The distributions inthis figure are almost symmetric about the base point �wherethe angular coordinate � is 180°�, which causes the mean CL

to be zero. Whereas this figure cannot reveal much about thelarge reduction in the rms CL due to synchronization, we useit to interpret the reduction and saturation behavior of themean CD due to synchronization. The value of �CD ismainly due to the imbalance in �CP between the upstreamand downstream parts of the surface. The upstream �CP isclose to unity, whereas the downstream �CP is negative. Werecall that the part of the surface near the base point ��=180°� is nearly isolated from shed vortices in the postsyn-chronization cases, as shown in Fig. 33. This is reflected in

the “bump” in the downstream �CP at SM /SK=1.81; it ex-plains the 42% reduction in �CD once synchronization oc-curs. To interpret the saturation behavior, we compare thetwo surface distributions of �CP at SM /SK=1.81 and 3. Thedownstream bump is stronger in the latter case, which shouldresult in a reduced �CD for this high-frequency synchronized

(b)(a)

FIG. 33. �Color online� Vorticity contours when �=0 for synchronous cases:�i� SM /SK=1.81 and �ii� SM /SK=3. Positive �counterclockwise� vortex isbeing shed from the bottom surface.

(a)

(b)

(c)

(d)

FIG. 34. �Color online� Vorticity contours over one motion cycle for

SM /SK=1.81. The motion sequence is �i� �=0 and � is maximum, �ii� � is

maximum and �=0, �iii� �=0 and � is minimum, and �iv� � is minimum and

�=0. Positive �counterclockwise� vortex is being shed from the bottomsurface.

θ

<C

P>

0 60 120 180 240 300 360

-2.0

-1.0

0.0

1.0

θ

<C

P>

0 60 120 180 240 300 360

-2.0

-1.0

0.0

1.0

(b)(a)

FIG. 35. �Color online� Surface distribution of �CP for synchronous cases:�i� SM /SK=1.81 and �ii� SM /SK=3. The angle � is 0 at the stagnation pointand 90° at the top point.

083103-11 Reduction of the loads on a cylinder Phys. Fluids 21, 083103 �2009�

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case. However, this is counteracted to a large extent by areduction �more negative� in �CP over �=30° –90° and270°–360°. Therefore, variations in �CD for the postsyn-chronization cases are minimal.

To support and augment the above discussion on CL, CD,and surface �CP, we examine the surface distributions of CP

over one motion cycle for the postsynchronization case atSM /SK=1.81 in Fig. 36. The surface CP is shown at fourequally spaced instants of time. The surface CP exhibits in-stantaneous symmetry about the x-axis. This is due to theinstantaneous symmetry of the magnitude of the vorticity atthe surface. Because the peaks and valleys in CL�t� aremainly due to the difference between CP at the top and bot-tom parts of the surface, these peaks and valleys are reducedin the postsynchronization cases due to the strong reductionin the CP difference that contributes to �CL�. These CP snap-shots also explain the increase in rms CD,osc with SM for thepostsynchronization cases even though �CD remains un-changed. Because the surface distribution of �CP becomesmore distorted as SM increases �as in Fig. 35�, the distortionin the instantaneous CP distributions is strengthened also.The rms CD,osc depends on the instantaneous CP imbalancebetween the upstream and downstream parts of the surface.This imbalance increases steadily with SM as a result of theintensified vorticity at the surface.

IV. CONCLUSIONS

We studied the dynamics of the loads on a cylinder un-dergoing harmonic in-line motion in a uniform stream at dif-ferent nondimensional mechanical frequencies SM and ampli-tudes Ax using the computational fluid dynamics �CFD�method. For a given Reynolds number Re, there exists acurve in the Ax−SM plane above which synchronization oc-curs. In the absence of synchronization, the lift and drag canbe periodic with large period, quasiperiodic, or chaotic. Syn-chronization occurs at lower values of SM when either Ax orRe increases. When synchronization takes place, the lift anddrag are both synchronous with SM, the lift has almost zeroamplitude, the mean drag drops and saturates at a constantvalue regardless of SM, its rms grows quadratically with SM,

the wake structure and shedding change and become instan-taneously symmetric, and the coupling between the lift anddrag changes. Whereas the linear coupling between the syn-chronous lift and drag is similar for the in-line and cross-flow motions, their quadratic coupling is different. The draghas an exciting effect for all cases, synchronous or not, andthe mechanical work due to the drag and the motion is doneon the flow and not on the cylinder. The instantaneous sym-metry in the vortex structure affects the surface distributionof the pressure, which in turn explains the qualitative andquantitative changes in the lift and drag when they becomesynchronous.

1G. V. Parkinson, “Mathematical models of flow-induced vibrations ofbluff bodies,” IUTAM/IAHK Symposium on Flow-Induced Structural Vi-brations, Karlsruhe, Germany, 1972, pp. 81–127.

2G. Di Silvio, F. Angrilli, and A. Zanardo, “Fluidelastic vibrations: Math-ematical model and experimental result,” Meccanica 10, 269 �1975�.

3T. Sarpkaya, “Fluid forces on oscillating cylinders,” J. Wtrwy., Port,Coast., and Oc. Div. 104, 275 �1978�.

4G. H. Koopmann, “The vortex wakes of vibrating cylinders at low Rey-nolds numbers,” J. Fluid Mech. 28, 501 �1967�.

5S. Krishnamoorthy, S. J. Price, and M. P. Païdoussis, “Cross-flow past anoscillating circular cylinder: Synchronization phenomena in the nearwake,” J. Fluids Struct. 15, 955 �2001�.

6O. A. Marzouk and A. H. Nayfeh, “A study of the forces on an oscillatingcylinder,” Proceedings of the 26th International Conference on OffshoreMechanics and Arctic Engineering, San Diego, CA, 2007, Vol. 3,pp. 741–752.

7O. A. Marzouk and A. H. Nayfeh, “Loads on a harmonically oscillatingcylinder,” Proceedings of ASME International Design Engineering Tech-nical Conferences and Computers and Information in Engineering Confer-ence, Las Vegas, NV, 2007.

8H. Persillon and M. Braza, “Physical analysis of the transition to turbu-lence in the wake of a circular cylinder by three-dimensional Navier–Stokes simulation,” J. Fluid Mech. 365, 23 �1998�.

9A. H. Nayfeh, F. Owis, and M. R. Hajj, “A model for the coupled lift anddrag on a circular cylinder,” Proceedings of ASME International DesignEngineering Technical Conferences and Computers and Information inEngineering Conference, Chicago, IL, 2003.

10A. H. Nayfeh, O. A. Marzouk, H. N. Arafat, and I. Akhtar, “Modeling thetransient and steady-state flow over a stationary cylinder,” Proceedings ofASME International Design Engineering Technical Conferences and Com-puters and Information in Engineering Conference, Long Beach, CA,2005.

11M. Tatsuno, “Vortex street behind a circular cylinder oscillating in thestream direction,” Bulletin of Research Institute for Applied Mechanics,Kyushu University 36, 25 �1972�.

12Y. Tanida, A. Okajima, and Y. Watanabe, “Stability of a circular cylinderoscillating in uniform flow or in a wake,” J. Fluid Mech. 61, 769 �1973�.

13O. M. Griffin and S. E. Ramberg, “Vortex shedding from a cylinder vi-brating in line with an incident uniform flow,” J. Fluid Mech. 75, 257�1976�.

14A. Leonard and A. Roshko, “Aspects of flow-induced vibration,” J. FluidsStruct. 15, 415 �2001�.

15R. D. Gabbai and H. Benaroya, “An overview of modeling and experi-ments of vortex-induced vibration of circular cylinders,” J. Sound Vib.282, 575 �2005�.

16L. E. Ericsson, “Karman vortex shedding and the effect of body motion,”AIAA J. 18, 935 �1980�.

17A. Ongoren and D. Rockwell, “Flow structure from an oscillating cylinderPart 1. Mechanisms of phase shift and recovery in the near wake,” J. FluidMech. 191, 197 �1988�.

18M. Gaydon and D. Rockwell, “Vortices incident upon an oscillating cyl-inder: Flow structure and loading,” J. Fluids Struct. 13, 709 �1999�.

19Y. Constantinides and O. H. Oakley, Jr., “Numerical prediction of bare andstraked cylinder VIV,” Proceedings of the 25th International Conferenceon Offshore Mechanics and Arctic Engineering, Hamburg, Germany,2006.

20J. K. Vandiver, “The relationship between in-line and cross-flow vortex-induced vibration in cylinder,” J. Fluids Struct. 1, 381 �1987�.

θ

CP

0 60 120 180 240 300 360-5.0

-3.0

-1.0

1.0

3.0

θ

CP

0 60 120 180 240 300 360-5.0

-3.0

-1.0

1.0

3.0

θ

CP

0 60 120 180 240 300 360-5.0

-3.0

-1.0

1.0

3.0

θC

P0 60 120 180 240 300 360

-5.0

-3.0

-1.0

1.0

3.0

(c) (d)

(b)(a)

FIG. 36. �Color online� Surface distribution of CP over one motion cycle for

SM /SK=1.81. The motion sequence is �i� �=0 and � is maximum, �ii� � is

maximum and �=0, �iii� �=0 and � is minimum, and �iv� � is minimum and

�=0.

083103-12 O. A. Marzouk and A. H. Nayfeh Phys. Fluids 21, 083103 �2009�

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21A. Torum, G. Moe, J. V. Johansen, and E. Katla, “On current inducedin-line motion in pipelines free-spans,” Proceedings of the 15th Interna-tional Conference on Offshore Mechanics and Arctic Engineering,Florence, Italy, 1996, Vol. 5, pp. 459– 469.

22R. Korpus, P. Jones, O. Oakley, and L. Imas, “Prediction of viscous forceson oscillating cylinders by Reynolds-averaged Navier–Stokes solver,”Proceedings of the 10th International Offshore and Polar EngineeringConference, Seattle, WA, 2000.

23N. Jauvtis and C. H. K. Williamson, “Vortex-induced vibration of acylinder with two degrees of freedom,” J. Fluids Struct. 17, 1035 �2003�.

24L. Lee, D. Allen, J. P. Pontaza, F. Kopp, and V. Jhingran, “In-line motionof subsea pipeline span models experiencing vortex-shedding,” Proceed-ings of the 28th International Conference on Ocean, Offshore Arctic En-gineering, Honolulu, Hawaii, 31 May–5 June, 2009.

25B.-H. Kim and D. R. Williams, “Nonlinear coupling of fluctuating dragand lift on cylinders undergoing forced oscillations,” J. Fluid Mech. 559,335 �2006�.

26J. F. Thompson, Z. U. A. Warsi, and C. W. Mastin, Numerical Grid Gen-eration �Elsevier Science Publishers, New York, 1985�.

27K. A. Hoffmann and S. T. Chiang, Computational Fluid Dynamics forEngineers, 4th ed. �Engineering Education System, Wichita, KS, 2000�,Vol. 2.

28O. A. Marzouk and A. H. Nayfeh, “Detailed characteristics of the reso-nating and non-resonating flows past a moving cylinder,” Proceedings ofthe 16th AIAA/ASME/AHS Adaptive Structures Conference, Schaum-burg, IL, 2008.

29M. M. Zdravkovich, “Modification of vortex shedding in the synchroniza-tion range,” ASME J. Fluids Eng. 104, 513 �1982�.

30T. Staubli, “Calculation of the vibration of an elastically mounted cylinderusing experimental data from forced vibration,” ASME J. Fluids Eng.105, 225 �1983�.

31X. Lu and C. Dalton, “Calculation of the timing of vortex formationfrom an oscillating circular cylinder,” J. Fluids Struct. 10, 527�1996�.

32C. Barbi, D. P. Favier, C. A. Maresca, and D. P. Telionis, “Vortex sheddingand lock-on of a circular cylinder in oscillatory flow,” J. Fluid Mech. 170,527 �1986�.

083103-13 Reduction of the loads on a cylinder Phys. Fluids 21, 083103 �2009�

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