International Journal of Computational Fluid Dynamics ... › profile › Chunning_JI...two degree-of-freedom mass–spring–damper system. The computing code was veriﬁed against

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/232763785

Numerical investigation on vortex-induced vibration of an elastically

mounted circular cylinder at low Reynolds number using the fictitious domain

method

Article in International Journal of Computational Fluid Dynamics · June 2011

DOI: 10.1080/10618562.2011.577034

CITATIONS

30READS

348

4 authors, including:

Some of the authors of this publication are also working on these related projects:

Vortex-Induced Vibration: Modelling, Simulation & Experiment View project

Numerical Investigation on the Vortex-induced Vibration of Slender Marine Riser View project

Chunning Ji

Tianjin University

87 PUBLICATIONS 691 CITATIONS

SEE PROFILE

Zhong Xiao

Tianjin University


SEE PROFILE

Yuanzhan Wang

Tianjin University


SEE PROFILE

All content following this page was uploaded by Chunning Ji on 29 May 2014.

The user has requested enhancement of the downloaded file.

https://www.researchgate.net/publication/232763785_Numerical_investigation_on_vortex-induced_vibration_of_an_elastically_mounted_circular_cylinder_at_low_Reynolds_number_using_the_fictitious_domain_method?enrichId=rgreq-76722daf5e78645044933a30846977ea-XXX&enrichSource=Y292ZXJQYWdlOzIzMjc2Mzc4NTtBUzoxMDIzMjQzMTU4ODU1ODBAMTQwMTQwNzQyNjYxOA%3D%3D&el=1_x_2&_esc=publicationCoverPdfhttps://www.researchgate.net/publication/232763785_Numerical_investigation_on_vortex-induced_vibration_of_an_elastically_mounted_circular_cylinder_at_low_Reynolds_number_using_the_fictitious_domain_method?enrichId=rgreq-76722daf5e78645044933a30846977ea-XXX&enrichSource=Y292ZXJQYWdlOzIzMjc2Mzc4NTtBUzoxMDIzMjQzMTU4ODU1ODBAMTQwMTQwNzQyNjYxOA%3D%3D&el=1_x_3&_esc=publicationCoverPdfhttps://www.researchgate.net/project/Vortex-Induced-Vibration-Modelling-Simulation-Experiment?enrichId=rgreq-76722daf5e78645044933a30846977ea-XXX&enrichSource=Y292ZXJQYWdlOzIzMjc2Mzc4NTtBUzoxMDIzMjQzMTU4ODU1ODBAMTQwMTQwNzQyNjYxOA%3D%3D&el=1_x_9&_esc=publicationCoverPdfhttps://www.researchgate.net/project/Numerical-Investigation-on-the-Vortex-induced-Vibration-of-Slender-Marine-Riser?enrichId=rgreq-76722daf5e78645044933a30846977ea-XXX&enrichSource=Y292ZXJQYWdlOzIzMjc2Mzc4NTtBUzoxMDIzMjQzMTU4ODU1ODBAMTQwMTQwNzQyNjYxOA%3D%3D&el=1_x_9&_esc=publicationCoverPdfhttps://www.researchgate.net/?enrichId=rgreq-76722daf5e78645044933a30846977ea-XXX&enrichSource=Y292ZXJQYWdlOzIzMjc2Mzc4NTtBUzoxMDIzMjQzMTU4ODU1ODBAMTQwMTQwNzQyNjYxOA%3D%3D&el=1_x_1&_esc=publicationCoverPdfhttps://www.researchgate.net/profile/Chunning-Ji?enrichId=rgreq-76722daf5e78645044933a30846977ea-XXX&enrichSource=Y292ZXJQYWdlOzIzMjc2Mzc4NTtBUzoxMDIzMjQzMTU4ODU1ODBAMTQwMTQwNzQyNjYxOA%3D%3D&el=1_x_4&_esc=publicationCoverPdfhttps://www.researchgate.net/profile/Chunning-Ji?enrichId=rgreq-76722daf5e78645044933a30846977ea-XXX&enrichSource=Y292ZXJQYWdlOzIzMjc2Mzc4NTtBUzoxMDIzMjQzMTU4ODU1ODBAMTQwMTQwNzQyNjYxOA%3D%3D&el=1_x_5&_esc=publicationCoverPdfhttps://www.researchgate.net/institution/Tianjin_University?enrichId=rgreq-76722daf5e78645044933a30846977ea-XXX&enrichSource=Y292ZXJQYWdlOzIzMjc2Mzc4NTtBUzoxMDIzMjQzMTU4ODU1ODBAMTQwMTQwNzQyNjYxOA%3D%3D&el=1_x_6&_esc=publicationCoverPdfhttps://www.researchgate.net/profile/Chunning-Ji?enrichId=rgreq-76722daf5e78645044933a30846977ea-XXX&enrichSource=Y292ZXJQYWdlOzIzMjc2Mzc4NTtBUzoxMDIzMjQzMTU4ODU1ODBAMTQwMTQwNzQyNjYxOA%3D%3D&el=1_x_7&_esc=publicationCoverPdfhttps://www.researchgate.net/profile/Zhong-Xiao?enrichId=rgreq-76722daf5e78645044933a30846977ea-XXX&enrichSource=Y292ZXJQYWdlOzIzMjc2Mzc4NTtBUzoxMDIzMjQzMTU4ODU1ODBAMTQwMTQwNzQyNjYxOA%3D%3D&el=1_x_4&_esc=publicationCoverPdfhttps://www.researchgate.net/profile/Zhong-Xiao?enrichId=rgreq-76722daf5e78645044933a30846977ea-XXX&enrichSource=Y292ZXJQYWdlOzIzMjc2Mzc4NTtBUzoxMDIzMjQzMTU4ODU1ODBAMTQwMTQwNzQyNjYxOA%3D%3D&el=1_x_5&_esc=publicationCoverPdfhttps://www.researchgate.net/institution/Tianjin_University?enrichId=rgreq-76722daf5e78645044933a30846977ea-XXX&enrichSource=Y292ZXJQYWdlOzIzMjc2Mzc4NTtBUzoxMDIzMjQzMTU4ODU1ODBAMTQwMTQwNzQyNjYxOA%3D%3D&el=1_x_6&_esc=publicationCoverPdfhttps://www.researchgate.net/profile/Zhong-Xiao?enrichId=rgreq-76722daf5e78645044933a30846977ea-XXX&enrichSource=Y292ZXJQYWdlOzIzMjc2Mzc4NTtBUzoxMDIzMjQzMTU4ODU1ODBAMTQwMTQwNzQyNjYxOA%3D%3D&el=1_x_7&_esc=publicationCoverPdfhttps://www.researchgate.net/profile/Yuanzhan_Wang?enrichId=rgreq-76722daf5e78645044933a30846977ea-XXX&enrichSource=Y292ZXJQYWdlOzIzMjc2Mzc4NTtBUzoxMDIzMjQzMTU4ODU1ODBAMTQwMTQwNzQyNjYxOA%3D%3D&el=1_x_4&_esc=publicationCoverPdfhttps://www.researchgate.net/profile/Yuanzhan_Wang?enrichId=rgreq-76722daf5e78645044933a30846977ea-XXX&enrichSource=Y292ZXJQYWdlOzIzMjc2Mzc4NTtBUzoxMDIzMjQzMTU4ODU1ODBAMTQwMTQwNzQyNjYxOA%3D%3D&el=1_x_5&_esc=publicationCoverPdfhttps://www.researchgate.net/institution/Tianjin_University?enrichId=rgreq-76722daf5e78645044933a30846977ea-XXX&enrichSource=Y292ZXJQYWdlOzIzMjc2Mzc4NTtBUzoxMDIzMjQzMTU4ODU1ODBAMTQwMTQwNzQyNjYxOA%3D%3D&el=1_x_6&_esc=publicationCoverPdfhttps://www.researchgate.net/profile/Yuanzhan_Wang?enrichId=rgreq-76722daf5e78645044933a30846977ea-XXX&enrichSource=Y292ZXJQYWdlOzIzMjc2Mzc4NTtBUzoxMDIzMjQzMTU4ODU1ODBAMTQwMTQwNzQyNjYxOA%3D%3D&el=1_x_7&_esc=publicationCoverPdfhttps://www.researchgate.net/profile/Chunning-Ji?enrichId=rgreq-76722daf5e78645044933a30846977ea-XXX&enrichSource=Y292ZXJQYWdlOzIzMjc2Mzc4NTtBUzoxMDIzMjQzMTU4ODU1ODBAMTQwMTQwNzQyNjYxOA%3D%3D&el=1_x_10&_esc=publicationCoverPdf

PLEASE SCROLL DOWN FOR ARTICLE

This article was downloaded by: [Ji, Chunning]On: 9 June 2011Access details: Access Details: [subscription number 938506448]Publisher Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

International Journal of Computational Fluid DynamicsPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713455064

Numerical investigation on vortex-induced vibration of an elasticallymounted circular cylinder at low Reynolds number using the fictitiousdomain methodChunning Jiab; Zhong Xiaoa; Yuanzhan Wanga; Huakun Wangaa Tianjin Key Laboratory of Harbor and Ocean Engineering, School of Civil Engineering, TianjinUniversity, Tianjin, P.R., China b Department of Engineering, School of Engineering and MaterialsScience, Queen Mary, University of London, London, UK

Online publication date: 09 June 2011

To cite this Article Ji, Chunning , Xiao, Zhong , Wang, Yuanzhan and Wang, Huakun(2011) 'Numerical investigation onvortex-induced vibration of an elastically mounted circular cylinder at low Reynolds number using the fictitious domainmethod', International Journal of Computational Fluid Dynamics, 25: 4, 207 — 221To link to this Article: DOI: 10.1080/10618562.2011.577034URL: http://dx.doi.org/10.1080/10618562.2011.577034

Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf

This article may be used for research, teaching and private study purposes. Any substantial orsystematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply ordistribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.

http://www.informaworld.com/smpp/title~content=t713455064http://dx.doi.org/10.1080/10618562.2011.577034http://www.informaworld.com/terms-and-conditions-of-access.pdf

Numerical investigation on vortex-induced vibration of an elastically mounted circularcylinder at low Reynolds number using the fictitious domain method

Chunning Jia,b*, Zhong Xiaoa, Yuanzhan Wanga and Huakun Wanga

aTianjin Key Laboratory of Harbor and Ocean Engineering, School of Civil Engineering, Tianjin University, Tianjin 300072,P.R. China; bDepartment of Engineering, School of Engineering and Materials Science, Queen Mary,

University of London, London E1 4NS, UK

(Received 18 August 2010; final version received 11 March 2011)

A direct numerical simulation of two-dimensional (2D) flow past an elastically mounted circular cylinder at lowReynolds number using the fictitious domain method had been undertaken. The cylinder motion was modelled by atwo degree-of-freedom mass–spring–damper system. The computing code was verified against a benchmark problemin which flow past a stationary circular cylinder is simulated. Then, analyses of vortex-induced vibration (VIV)responses, drag and lift forces and the phase and vortex structures were carried out. Results show that the cylinder’snon-dimensional cross-flow response amplitude reaches its summit of 0.572 in the ‘lock-in’ regime. The ‘2S’, insteadof the ‘2P’, vortex shedding mode is dominated in the ‘lower’ branch for this 2D low-Re VIV. A secondaryoscillation is observed in the lift force when ‘lock-in’ occurs. It is shown that this secondary component changes thephase, offset the energy input by the primary component and thus reduces the cylinder responses. Effects of theSkop–Griffin parameter on cylinder responses were also investigated.

Keywords: vortex-induced vibration; fictitious domain method; direct numerical simulation; circular cylinder; lowReynolds number flow

1. Introduction

Vortex-induced vibrations (VIV) are highly non-linearvibrations on bluff bodies induced by periodicalirregularities of an external wake. They occur inmany engineering situations ranging from aerospacesciences to coastal engineering and have been investi-gated for a quite long time. Much of the recentresearch was discussed in the reviews by Sarpkaya(2004) and Williamson and Govardhan (2004).

As we know, the fluid forces, induced by periodicalvortex shedding, are in both the cross-flow (the liftforce) and the stream-wise (the drag force) directions.Such periodical forces can drive an elastically mountedcircular cylinder vibrating in both directions. It shouldbe noted that when the frequency of vortex shedding(also body vibration) is close enough to the naturalfrequency of the structure, vortex-excited bodies canexperience a resonance phenomenon, known as ‘lock-in’ or synchronisation, characterised by wide frequencyrange and large amplitude. The ‘lock-in’ vibration isdangerous because it can cause fatigue failures in manymarine structures, such as marine risers, due to thehigh strain and stress and the long-term continuousaction. To avoid this, we have to know the mechanismof VIV.

VIV, however, is a very complicated process. Asstated by Sarpkaya (2004), it is an inherently non-linear, self-governed or self-regulated, multi-degree-of-freedom phenomenon. Even for the simplest VIVproblem, VIV of elastically mounted two-dimensional(2D) circular cylinder in uniform flows, a completeunderstanding is not yet achieved. VIV encompassessuch many fundamental subjects as turbulent bound-ary layer, free shear flow, vortex dynamics, structurevibrations, complex demodulation analysis and so on.

Although VIV is still an open issue, after severaldecades of investigations performed theoretically,experimentally and numerically, numerous contribu-tions have been added to the knowledge of themechanism of VIV. The experimental study on VIVof a circular cylinder oscillating transversely in air,conducted by Feng (1968), is widely taken as thebeginning of modern study of VIV. In his study, twodistinct cylinder response regimes, ‘initial’ branch and‘lower’ branch, were found for high mass ratio cases inwhich m* ¼ rs/rf * 250 (rs and rf are the densities ofcylinder and fluid, respectively). In the ‘initial’ branch,the vortex structure in the wake comprises a ‘2S’ mode,representing that two single vortexes are formed percycle. However, in the ‘lower’ branch, the wake vortexconsists of a ‘2P’ mode, whereby two vortex pairs are

*Corresponding author. Email: [email protected]

International Journal of Computational Fluid Dynamics

Vol. 25, No. 4, April–May 2011, 207–221

ISSN 1061-8562 print/ISSN 1029-0257 online

� 2011 Taylor & FrancisDOI: 10.1080/10618562.2011.577034

http://www.informaworld.com

Downloaded By: [Ji, Chunning] At: 09:21 9 June 2011

created in a cycle. The non-dimensional cross-flowresponse amplitude A�y ¼ Ay=D (note that we use y/Dto indicate the non-dimensional cross-flow responsehistory) reaches its maximum of 0.6 when the cylin-der’s actual vibration frequency f0, as well as the vortexshedding frequency fV, is close enough to the naturalfrequency fN of the structure, so that the non-dimensional frequency ratio f�0 ¼ f0=fN � 1. After thepioneering work of Feng (1968), another responseregime, the ‘upper’ branch, was found between the‘initial’ and ‘lower’ branches for a circular cylindervibrating in water where mass ratio is relatively lower(m* 5 6) (Williamson 1997). This is of more concernto the offshore industry, because it is just the case forVIV of marine structures. Similar to the ‘lower’branch, the ‘upper’ branch comprises the ‘2P’ vortexmode. However, A�y goes up to 1.0 when the cylinderresponse is in the ‘lock-in’ regime. For a relatively longtime, these results and conclusions were considered asthe footstone of the modern VIV research. However,recent experiment of Williamson and Jauvtis (2004)revealed a novel ‘2T’ vortex mode, in which two vortexgroups, each composed of three vortexes, are formedper cycle. In contrast to the cylinder layout inconventional experiments, the cylinder in this testwas set free in the stream-wise direction. Surprisingly,A�y reaches an unexpected high value of 1.5 in the so-called ‘super-upper’ branch.

According to the description of the flow regimesgiven by Williamson (1996), three-dimensional (3D)vortical structures are observed in the wake of astationary circular cylinder as the Reynolds numberexceeds 180. However, as for oscillating cylinders inthe ‘lock-in’ regime, the vibration substantially corre-lates the vortex shedding along the span-wise directionand renders the wake predominantly 2D. It iscommonly believed that such correlation has an effectof enlarging the response amplitude. However, recentresearch of Hover et al. (2004) on the wake of a flexiblymounted circular cylinder undergoing VIV in auniform flow using a novel force feedback apparatusshowed that the span-wise correlation undergoes asharp reduction near conditions of maximum responseamplitude. Meanwhile, the phase f between the liftforce and cross-flow response transits from 08 to 1808.Similar findings were reported by Lucor et al. (2005)with numerical simulations. As concluded by Hoveret al. (2004) that high mass and damping reinforcecorrelation during such phase transition, whereaslow values admit a correlation loss with the wakepredominantly 3D. The 3D wake undermines thevortex correlation in the span-wise direction, andthus vortex-induced forces offset each other along thecylinder axis due to the phase difference between them.From this point of view, VIV is also ‘self-limited’.

Obviously, the theory and methodology on VIV arefar from complete and perfect. When ‘old’ problemshave been resolved, ‘new’ difficulties appear. In thisstudy, we try to simulate VIV of elastically mounted2D circular cylinder in uniform flows numericallyusing the fictitious domain (FD) method. The FDmethod is a fully coupled fluid–solid interaction (FSI)approach introduced by Glowinski et al. (1994) forparticulate flow problems. In this method, a set of fullycoupled governing equations for the FSI system issolved in a monolithic way and the interaction betweenfluid and solid is modelled by a distributed Lagrangemultiplier (DLM). Readers are suggested to refer to Jiet al. (2007) for details. Due to its preferable capabilityof dealing with large deformation/displacement FSIproblems, the FD method has been widely applied inmany engineering problems, for instance, the fluid flowinside an industrial mixer equipped with intermeshingimpellers (Coesnon et al. 2008), the shape optimisationof a NACA0012 airfoil (Glowinski et al. 2000), the FSIin the aortic valve (de Hart et al. 2000) and the motionof flexible thin beams in Stokes and Navier–Stokesflows (Baaijens 2001). To the best of our knowledge,however, there is few study dedicated on applying theFD method in VIV simulations. The motivation of thisstudy is to investigate the feasibility of this application.

Numerical modelling of VIV, especially VIV ofslender marine structures, is exceedingly difficultbecause it not only involves the FSI problem but alsoconcerns turbulence simulation and related wallfunctions. Also, the simulation should be 3D due tothe 3D nature of the vortex. However, the correlatingeffect of the ‘lock-in’, as mentioned above, justifies theuse of 2D simulations to compute flows past vibratingcylinders at low Reynolds number. Here, we recognisethe limitation of our 2D numerical investigations onVIV of an elastically supported circular cylinder inlaminar flow, but still feel that the results will be ofpractical use in validating our in-house computationalfluid dynamics (CFD) code based on the FD method.It is our long-term goal to develop a practical 3Dparallel computing code with efficient turbulencemodel and this study can be a good start point.

On the other hand, the vast majority of VIV studieshave been performed at high Reynolds number wherethe wake is inherently 3D. Only one experiment(Anagnostopoulos and Bearman 1992, Re ¼ 200) andfew numerical simulations (for example, Singh andMittal 2005, Re ¼ 100; Leontini et al. 2006, Re ¼ 200;Placzek et al. 2009, Re ¼ 200) have been reported.Deeper insight into the mechanics of VIV has beengained as the complicated effects of the three-dimen-sionality are removed. From this point, we can say thatthe 2D low-Re VIV simulation is of the sameimportance of the 3D high-Re VIV simulation.

208 C. Ji et al.


Based on these facts, we do anticipate that this 2Dlow-Re VIV calculation can be used to reasonablypredict response regimes and wake modes of VIV.

2. Numerical methods

2.1. The FSI model based on the FD method

In this model, the fluid flow is governed by the Navier–Stokes equations, and the rigid solid movement iscontrolled by the Newton’s equation. As we know, thedynamic behaviours of a rigid solid are exactly thesame as those of a same (shape and position) fluidbody in which the strain rate tensors are zero. Also, theNewton’s equation can be rewritten in the form of theNavier–Stokes equations on the consideration of zerostrain rate tensors. So, in the context of the FDmethod, the solid is first substituted by the non-deformation ‘fictitious’ fluid. At this stage, the wholedomain is filled with fluid, no matter whether genuineor ‘fictitious’, and the flow can be modelled by theNavier–Stokes equations. Then, to keep the ‘fictitious’fluid behaving as a rigid body (i.e. make sure the strainrate tensors in the ‘fictitious’ fluid are zero), pseudobody forces are exerted on this sub-domain using theDLM. The coupled FSI governing equations aregenerated by combining the Navier–Stokes equationsand pseudo body forces using functional analysis.Finally, the whole governing equations are discretisedwith fixed Eulerian mesh and solved using theoperator-splitting scheme. Of course, the FD methodcan be more general by replacing Newton’s equation ofmotion for the rigid body with the continuum equationfor the elastic material, as in the work of Yu (2005).

The governing equations for the fluid and solid inthe FD method are as the follows.

Let O 2 Rd (d ¼ 2,3) be a space. We suppose that Ois filled with incompressible viscous fluid of density rfand viscosity mf, and it also contains a moving rigidbody B(t) of density rs, as depictured in Figure 1. Here,�0 and �1 indicate the Dirichlet-type and Neumann-type boundaries of the fluid domain, respectively. @B(t)denotes the fluid–solid interface.

The fluid flow is modelled by the Navier–Stokesequations

rf@u

@tþ ðu � rÞu

� �¼ rfgþr � r; in O=BðtÞ ð1Þ

r � u ¼ 0; in O=BðtÞ ð2Þ

uðx; 0Þ ¼ u0ðxÞ; 8x 2 O=Bð0Þ; with r � u0 ¼ 0 ð3Þ

to be completed by

u ¼ g0; on �0;@u

@n¼ g1; on �1 ð4Þ

and the following no-slip boundary condition on @B(t)

uðx; tÞ ¼ VðtÞ þ xðtÞ � XðtÞx~; 8x 2 @BðtÞ ð5Þ

where, X, V and o denote the position, translationalvelocity and rotational velocity of the centroid G of therigid body, respectively. The stress-tensor r verifies

r ¼ �pIþ mfðruþrutÞ ð6Þ

The motion of the rigid solid is controlled by theNewton’s equations,

MdV

dt¼Mgþ Fi þ FeðX;YÞ ð7Þ

Idxdt� Ix� x ¼ Ti þ TeðX;YÞ ð8Þ

dX

dt¼ V ð9Þ

dYdt¼ x ð10Þ

and to be completed by the following initial conditions

Xð0Þ ¼ X0; Yð0Þ ¼ Y0; Vð0Þ ¼ V0; xð0Þ ¼ x0 ð11Þ

In Equations (7) to (11), M and I are the mass andinertia tensor of the rigid body, respectively. Y is therotation angle of the rigid body. Fi and Ti are thehydrodynamic force and torque at G, respectively. Fe

and Te are the external force and torque at G,respectively, and evaluated by a case-by-case analysis.The general strategy will be shown in Section 2.3. It

Figure 1. An example of 2D flow region with one rigidbody.

International Journal of Computational Fluid Dynamics 209


should be noted that this set of governing equations isslightly different from that presented by Glowinskiet al. (2000). The Neumann-type boundary condition isconsidered in Equation (4) for generality. Also, theexternal torque Te is imposed on the solid to reflectthe eccentric effect of the external force Fe (also thelubrication force Fr in the study by Glowinski et al.2000), which generally does not pass through thecentroid G of a 2D non-circular disk or a 3D non-spherical body.

For the sake of conciseness, we directly present thevariational formulation of the fully coupled governingequations based on the FD method for generalincompressible viscous fluid–rigid solid interactionproblems. However, readers can refer the study by Jiet al. (2007) for details.

We define the following functional spaces

Wg0ðtÞ ¼ fvjv 2 ðH1ðOÞÞd; v ¼ g0ðtÞ on �g;

L20ðOÞ ¼ fqjq 2 L2ðOÞ;ZOqdx ¼ 0g;

LðtÞ ¼ ðH1ðBðtÞÞÞd

ð12Þ

The variational formulation of the FSI governingequations with the DLM k 2 L (t) are

For a.e. t 4 0, find {u(t), p(t), X(t), Y(t), V(t), x(t),k(t)} such that

uðtÞ 2Wg0ðtÞ; pðtÞ 2 L20ðOÞ; XðtÞ 2 R

d; YðtÞ 2 R3;

VðtÞ 2 Rd; xðtÞ 2 R3; kðtÞ 2 LðtÞ; ð13Þ

and

rf

ZO

@u

@tþ ðu � rÞu

� �� vdxþ 2mf

ZODðuÞ : DðvÞdx

�ZOpr � vdx� ðk; v� Y� h� Xx~ÞLðtÞ þ 1�

rfrs

� �

MdV

dthYþ I dx

dt� Ix� x

� �� h

� �

� Fe � Y� Te � h ¼ rfZOg � vdxþ 1� rf

rs

� �Mg � Y;

8v 2 ðH10ðOÞÞd; 8Y 2 Rd; 8h 2 R3; ð14Þ

ZOqr � udx ¼ 0; 8q 2 L2ðOÞ ð15Þ

dX

dt¼ V ð16Þ

dYdt¼ x ð17Þ

ðl; uðx; tÞ � VðtÞ � hðtÞ � XðtÞx~ÞLðtÞ ¼ 0; 8m 2 LðtÞð18Þ

uðx; 0Þ ¼ u0ðxÞ; 8x 2 OnBð0Þ;

uðx; 0Þ ¼ V0 þ x0 � X0x~; 8x 2 Bð0Þð19Þ

Xð0Þ ¼ X0; Yð0Þ ¼ Y0; Vð0Þ ¼ V0; xð0Þ ¼ x0 ð20Þ

In the above equations, v, q, Y and h are thevariationals for fluid velocity, fluid pressure, bodytranslation velocity and body angular velocity, respec-tively. k and l are the DLM and its variationalcounterpart.

Obviously, the DLM k is the so-called pseudo bodyforce that acts similarly as the pressure p. Equation(18) represents the rigid body constraint imposed onthe ‘fictitious’ fluid to maintain its rigid bodycharacteristics. The two most natural choices for(�,�)L(t), as suggested by Glowinski et al. (2000), are

ðl; vÞLðtÞ ¼ZBðtÞðl � vþ d2rl : rvÞdx; 8l; v 2 LðtÞ

ð21Þ

ðl; vÞLðtÞ ¼ZBðtÞðl � vþ d2DðlÞ : DðvÞÞdx; 8l; v 2 LðtÞ

ð22Þ

with d as a characteristic length [for example,the maximum length of B(t)].

For 2D VIV problems discussed in this article, therotation of the circular cylinder is restricted and thegravitational field do not influence the oscillation ofthe rigid cylinder in any way. So, we have d ¼ 2,Y ¼ 0, x ¼ 0 and the above general variationalformulation can be simplified as follows with gravityterms neglected.

For a.e. t 4 0, find {u(t), p(t), X(t),V(t), k(t)} suchthat

uðtÞ 2Wg0ðtÞ; pðtÞ 2 L20ðOÞ;

XðtÞ 2 R2; VðtÞ 2 R2; kðtÞ 2 LðtÞð23Þ

and

rf

ZO

@u

@tþ ðu � rÞu

� �� vdxþ 2mf

ZODðuÞ : DðvÞdx

�ZOpr � vdx� ðk; v� YÞLðtÞ þ 1�

rfrs

� �M

dV

dt� Y

� Fe � Y ¼ 0; 8v 2 ðH10ðOÞÞ2; 8Y 2 R2 ð24Þ

210 C. Ji et al.


ZOqr � udx ¼ 0; 8q 2 L2ðOÞ ð25Þ

dX

dt¼ V ð26Þ

ðl; uðtÞ � VðtÞÞLðtÞ ¼ 0; 8l 2 LðtÞ ð27Þ

uðx; 0Þ ¼ u0ðxÞ; 8x 2 O=Bð0Þ; uðx; 0Þ ¼ V0; 8x 2 Bð0Þð28Þ

Xð0Þ ¼ X0; Vð0Þ ¼ V0 ð29Þ

The coupled FSI governing equations are discre-tised using the finite element technique. Time discretisa-tion is achieved by applying operator-splitting method.The linearly constrained quadratic minimisation pro-blem resulted from the coupled governing equations issolved by using the conjugate gradient algorithm.

Three major advantages of the present formulationinclude (i) the unitised governing equations both forfluid and solid help capturing the predominant physicsof interaction phenomena; (ii) the interfacial hydro-dynamic force/displacement between fluid and solidare internal actions for the overall system. Therefore,the stress/velocity consistency conditions on the fluid–solid interface are automatically satisfied in this fullycoupled model; (iii) in the case of the usage of fixedEulerian mesh, it is not necessary to re-mesh thecomputational domain and thus free from meshdistortions.

2.2. Geometry, boundary conditions and meshdiscretisation

The computational domain is presented in Figure 2with a circular cylinder of diameter D. As stated byPlaczek et al. (2009) that VIV responses are verysensitive to the size of the computational domain,particularly when the Reynolds number is relativelysmall. It is suggested by Persillon and Braza (1998) thatthe minimum width–diameter ratio is W/D ¼ 22 forRe ¼ 100 and that the influence of the outflowboundary condition can be neglected for L2/D � 34.For our VIV simulations (Re ¼ 60 * 300), the size ofcomputational domain is chosen as W/D ¼ 24, L1/D ¼ 8 and L2/D ¼ 36.

The boundary conditions adopted are also specifiedin Figure 2. The inflow boundary condition is chosenas a Dirichlet type. The top and bottom boundaryconditions are of slip-wall type with @u/@y ¼ 0, v ¼ 0.A Neumann-type boundary condition is adopted onthe outflow boundary. It should be noticed that noboundary condition is imposed on the fluid–cylinderinterface. The fluid–cylinder interaction is modelled bythe pseudo body force in the fluid region covered bythe circular cylinder, which is considered as one of themajor advantages of the FD method.

The mesh for the fluid domain presented in Figure3a is a set of hybrid structured quadrilateral andunstructured triangular grids. The outline of thestructured mesh is an 8D 6 8D square that surroundsthe circular cylinder and encloses its VIV trajectory, asalso shown in Figure 2. It should be noted that thecentre of structured mesh region does not coincidewith, but is 2D behind, the initial centroid of thecircular cylinder for the consideration of stream-wise

Figure 2. Layout and boundary conditions for VIV of elastically mounted circular cylinder.



shift of the rest position under the mean drag force.The structured quadrilateral grid spacing in bothdirections is chosen as Dx ¼ Dy ¼ D ¼ D/30 to verifyDþ ¼ 0.172 (D/D)Re0.9 � 1. Away from the circularcylinder, the computational domain is discretised withunstructured mesh to keep the number of cells to areasonable amount. The total number of cells in thewhole fluid domain stands at 92,236.

The mesh for the circular cylinder is presented inFigure 3b. Unstructured triangular grids are adoptedfor the purpose of fitting the curved cylinder boundary.Several layers of grids near the boundary are refined toreduce the interpolation error of the fluid velocity andthe pseudo body force between fluid and cylindermeshes. The total number of cells in the cylinderdomain is 5042.

2.3. Cylinder motion

The cylinder motion is modelled by a two degree-of-freedom (2DOF) mass–spring–damper (MSD) systemexcited by the hydrodynamic forces, as seen in Figure 2.

The stream-wise and cross-flow motions are conse-quently governed by the following equations.

m€xþ cx _xþ kxx ¼ Fx ð30Þ

m€yþ cy _yþ kyy ¼ Fy ð31Þ

where, cx and cy denote the structural damping in thestream-wise and cross-flow directions, respectively. kxand ky stand for the spring stiffness in the stream-wiseand cross-flow directions, respectively. Fx and Fy arethe fluid drag and lift forces acting on the circularcylinder, respectively. Note that these forces are thetwo components of the hydrodynamic force Fi inEquation (7). Generally, mass is the same in twodirections. These two equations can be rewritten in amatrix form

M€Xþ C _Xþ KX ¼ Fi ð32Þ

Substituting Equation (32) in Equation (7), we have

Fe ¼ �ðC _Xþ KXÞ ð33Þ

where C ¼h cx 00 cy

iand K ¼

h kx 00 ky

i. Obviously, to

simulate VIV of a circular cylinder, the only thing weneed to do is substituting Equation (33) in the FSIvariational formulation Equations (23) to (29).

3. Results and discussion

VIV of elastically mounted circular cylinder is such acomplicated phenomenon in which many factors areinvolved. However, only two of the most importantfactors are considered here: one is the reduced velocityUr ¼ U?/(fND), where U? is the far-field inflowvelocity, and the other one is the Skop–Griffinparameter SG ¼ 2p3 St2 m*z, where St is the Strouhalnumber and z is the structural damping ratio. BeforeVIV simulations, we first verified our in-house CFDcode against the benchmark problem – flow past a 2Dstationary circular cylinder.

3.1. Verification of the in-house CFD code

The code is verified by applying it to the benchmarkproblem, flow past a 2D stationary circular cylinder,which has been investigated experimentally andnumerically for many years by researchers. Samegeometry, boundary conditions and mesh discretisa-tion as mentioned in Section 2.2, but a fixed cylinder,are adopted. Table 1 shows the comparison of ourresults with experimental data and calculated resultsfrom the literature. The lift force and the Strouhal

Figure 3. Meshes for fluid domain and circular cylinder.

212 C. Ji et al.


number that represents the vortex shedding frequencyare in excellent agreement with the published data. Thedrag force, however, shows a good agreement withthose using non-body-conformal Cartesian grid (Laiand Peskin 2000, Uhlmann 2005), but a perceivablediscrepancy with those using body-conformal curvi-linear grid (Liu et al. 1998). This can be explained asthe result of distribution of the pseudo body force fromthe cylinder mesh to the background fluid mesh usinginterpolation/extrapolation functions. On the fluid–solid interface, these functions tend to smear the bodyforce over several Eulerian grid widths. That is to say,on the non-body-conformal Eulerian grids, the fluid–solid interface is not exactly reconstructed and theboundary conditions are only approximately imposed,although we can use compact interpolation/extrapola-tion functions to reduce these errors. This is a dilemmaof the FD method. A gain on the capability ofhandling the moving boundary is on the cost ofsolution accuracy.

3.2. The influence of the reduced velocity on VIVresponse

A series of numerical simulations on VIV of elasticallymounted circular cylinder were carried out withdifferent reduced velocities Ur from 3.0 to 15.0. Inthese simulations, the mass ratio m* was determined as2.0 and the structural damping cx and cy was both setto zero on the purpose of exciting the maximumpossible response of the cylinder. The spring stiffnesskx and ky was equal to each other to ensure fNx ¼ fNy,where fNx and fNy are the natural frequencies of theMSD system in vacuum. Changes in the reducedvelocity Ur were achieved by altering the far-fieldinflow velocity U?. For this set of simulations, we setRe ¼ 20Ur, such that the Reynolds number variedfrom 60 to 300.

Figure 4 shows the vorticity contours and cylinder’svibration trajectories of some representative results.Figure 5 presents the time histories of drag forcecoefficientCD, lift force coefficientCL and the cylinder’snon-dimensional cross-flow response y/D. Figure 6shows the non-dimensional cross-flow response ampli-tude Ay/D versus the reduced velocity Ur. From thesefigures, we can see that vortexes shed from the cylinderperiodically and are in a ‘2S’ mode at Ur ¼ 3.0 * 4.0when VIV is in ‘initial’ branch regime. Ay/D increasesrapidly and reaches its summit 0.572 at Ur ¼ 4.0. Afterthe peak value, Ay/D decreases gradually and stabilisesat 0.1 approximately when Ur 4 9.0. This stage(Ur 4 4.0) is the so-called ‘lower’ branch. Similar tothe vortex mode in the ‘initial’ branch, the vortexshedding in this stage is also in the ‘2S’ mode.

Figure 6 also shows the comparisons of Ay/Dbetween present results and the results of Leontiniet al. (2006) with m* ¼ 10.0 and z ¼ 0.01 at fixedRe ¼ 200 and the results of Singh and Mittal (2005)with m* ¼ 10.0 and z ¼ 0.0 at fixed Re ¼ 100.Although three results differed from each otherobviously, they do show a similar trend. This dis-crepancy can be due to the different mass ratio used. Asstated by Williamson and Govardhan (2008) that ‘asthe structural mass decreases, so the regime of velocityU* over which there are large-amplitude vibrationsincreases’. Note, U* in their article is denoted as Ur inthis study. In our simulation, m* ¼ 2.0, which is quitelower than those used by Leontini et al. (2006) andSingh and Mittal (2005). This leads to the wider lock-inzone, i.e., 4.0 Ur 8.0 whereAy/D is high, of presentnumerical results. The different Reynolds numbercould be another reason for the discrepancy. Althoughthe peaks of the response amplitude of present resultsand results from Singh and Mittal (2005) appear atdifferent Ur, the maximum values of Ay/D show a goodagreement. The maximum value ofAy/D in the study by

Table 1. Comparison of drag coefficient, lift coefficient and Strouhal number for flow past a 2D stationary circular cylinder.

Re ¼ 100 Re ¼ 150 Re ¼ 200

CD CL St CD CL St CD CL St

Present results 1.442 + 0.011 0.336 0.167 1.434 + 0.029 0.513 0.183 1.422 + 0.047 0.672 0.196Uhlmann 2005 (non-body-conformal Cartesian grid)

1.453 + 0.011 0.339 0.169 – – – – – –

Lai and Peskin 2000 (non-body-conformalCartesian grid)

1.4473+ 0.3299 0.165 – – 0.184 – – 0.190

Liu et al. 1998 (body-conformal curvilinear grid)

1.350 + 0.012 0.339 0.165 1.33 + 0.03 0.53 0.182 1.31 + 0.049 0.69 0.192

Williamson (exp., asreported in Liu et al. 1998)

– – 0.166 – – 0.183 – – 0.197

Roshko (exp., as reported inLiu et al. 1998)

– – 0.164 – – 0.182 – – 0.19



Leontini et al. (2006) is relatively lower due to the effectof the damping.

On closer inspection of Figures 4 and 6, we foundthat the vortex mode in the ‘lower’ branch (Ur � 4.0) is‘2S’ instead of ‘2P’ as described in the literature (Feng1968, Khalak and Williamson 1999) and the ‘upper’branch is not observed. Similar findings were alsoreported in the low-Re VIV numerical simulations of

Leontini et al. (2006), Singh and Mittal (2005), Placzeket al. (2009) and observed in the low-Re VIVexperiments of Anagnostopoulos and Bearman(1992). In our opinion, the low Reynolds number ofthe flow contributes to the absences of the ‘2P’ vortexmodel and the ‘upper’ branch. In the tests of Feng(1968) and Khalak and Williamson (1999), the flowwas turbulent with a relatively high Reynolds number

Figure 4. Contours of vorticity and vibration trajectories of cylinder with different reduced velocities. (a) Ur ¼ 3.0, (b)Ur ¼ 4.0, (c) Ur ¼ 5.0, (d) Ur ¼ 6.1, (e) Ur ¼ 8.0, (f) Ur ¼ 11.0.

214 C. Ji et al.


[Re ¼ O(103 * 104)]. However, in this study, the flowwas in the laminar regime with Re ¼ 60 * 300.

From the cylinder orbitals presented in Figure 4, itis clearly observed that the trajectories change fromforward bending ‘8’ to backward bending ‘8’, whichimplies that the phase between the stream-wise andcross-flow oscillations changed from positive tonegative. The amplitude of stream-wise vibration isan order of magnitude smaller than that of cross-flow

vibration and the frequency of stream-wise vibration istwice as much as that of cross-flow vibration.

Figure 7 shows the phase f between lift force andcylinder’s cross-flow response versus Ur. In the ‘initial’branch (Ur 4.0), f is nearly zero. However, in thelatter part of ‘lower’ branch (Ur 4 8.0), f is almost1808. In the middle (4.0 5 Ur 8.0), f changesrapidly form 08 to 1808. The numerical results ofLeontini et al. (2006) with m* ¼ 10.0 and z ¼ 0.01 at

Figure 4. (Continued).



Figure 5. Time histories of drag force coefficient CD, lift force coefficient CL and cylinder’s non-dimensional cross-flow responsey/D with different reduced velocities. (a) Ur ¼ 3.0, (b) Ur ¼ 4.0, (c) Ur ¼ 5.0, (d) Ur ¼ 6.1, (e) Ur ¼ 8.0, (f) Ur ¼ 11.0.

216 C. Ji et al.


fixed Re ¼ 200 is also presented in Figure 7 for thepurpose of giving a comparative view on the presentresults. Because of the higher mass ratio, the results ofLeontini et al. (2006) show a sharper transition of fwhile the phase in our results changes more gradually.As a whole, two results agree well with each other.

To understand the effects of the phase on theresponse of the cylinder, we relate the changes of fwith the energy transferred between the fluid flow andthe MSD system. From the energic point of view, whenthe response of an MSD system is finally stable, the netenergy input into the system must be zero in one cycle.Furthermore, if we assume that the lift force and res-ponse are both monotone and sinusoidal and the struc-tural damping is zero, f must be either 08 or 1808. Forexample, we assume that the lift force is Fy ¼ sin(2pt/T)and the cross-flow response is y ¼ sin(2pt/T), where t is

time and T is the period. In this case, we have f ¼ 08.The cross-flow velocity of the cylinder is deduced asVy ¼ (2p/T) cos(2p/T). The net transferred energy attime distinct t can be calculated as the integral of theproduct of lift force and cross-flow velocity, i.e., E¼ð2p=TÞ

R t0 sinð2pt=TÞcosð2pt=TÞdt¼ ð1=4Þð1�cosð4pt=

TÞÞ, as shown in Figure 8a. Clearly, the net energy inputinto the MSD system in a cycle is zero. Figure 8b showsthe net transferred energy E ¼ 7(1/4) (1 7 cos (4pt/T))for the case in which the lift force is Fy ¼ sin((2pt/T) 7 p) and the cross-flow response is y ¼ sin(2pt/T). Inthis case, f ¼ 08 and the net energy transferred in a cycleis zero again. However, if 1808 5 f 5 1808, nettransferred energy in a cycle will be accumulated, whichcauses the cylinder response unstable. For example, thenet transfer rate of energy input into the MSD systemreaches its summit when f ¼ 908, as depicted in Figure8c. In this case, the lift force is Fy ¼ sin((2pt/T) 7 (p/2))and the cross-flow response is y ¼ sin(2pt/T). Thus, thetransferred energy is E ¼ (1/4) ((4pt/ T) 7 sin(4pt/T)),which increases with time marching.

However, this seems contradict with the fact thatthe cylinder cross-flow response is stable with08 5 f 5 1808 in the ‘lock-in’ zone, as shown inFigure 7. This is caused by the ‘multi-tone’ (multiplefrequency) character of the lift force when 4.0 5 Ur 8.0, see Figures 5c and 5d. This ‘multi-tone’ characterof the lift force was also reported by Leontini et al.(2006). In this study, we divide the ‘multi-tone’ lift forceinto two components, the primary component and thesecondary component with different phases. The energyinput by the primary component in a cycle is ‘consu-med’ by the secondary component. Thus, although f isneither 08 nor 1808, the net energy transferred fromflow to the MSD system can be still zero. Likewise, thesmall discrepancy that the phase is not exactly equalto 08 or 1808 when Ur 4.0 or Ur 4 8.0 can beconsidered as a result of non-pure sinusoidal lift force,which implies that small secondary vibrations still existin the lift force. The secondary component of lift forceoffset the energy generated by the primary componentand thus reduces the cylinder cross-flow response. Thisfinding presents another point of view from which theself-limited character of VIV can be related.

Upon closer inspection of the lift force time history,we find three distinct destinations of the secondarycomponent of lift force. In the latter part of ‘lower’branch (Ur 4 8.0), the secondary component finallydamped out in the stable stage, although it plays animportant role in the modulating stage as show inFigure 5f. However, this secondary component keepsnon-ignorable throughout the simulation when VIV isin the ‘lock-in’ regime (4.0 5 Ur 8.0), as shown inFigures 5c and 5d. No obvious secondary componentis observed in the ‘initial’ branch (Ur 4.0), as shown

Figure 6. Cylinder’s non-dimensional cross-flow responseamplitude Ay/D versus the reduced velocity Ur.

Figure 7. The phase f between the lift force and cylinder’scross-flow response versus the reduced velocity Ur.



in Figures 5a and 5b. The above difference in thesecondary component can be a reason for the differentstable values of the phase.

Figure 9 shows the relationship between systemcross-flow natural frequency fNy, Strouhal frequencyfSt and the actual cross-flow vibrating frequency f0y(same as the vortex shedding frequency fV) withdifferent reduced velocity. It is clearly shown that, inthe zone 4.0 Ur 8.0, f0y deviates from fSt and fixedon fNy. This is the so-called ‘lock-in’. Inside this zone,the vibration is controlled by the characteristics of theMSD system. Outside this zone, however, the vibrationis dominated by the fluid flow.

3.3. The influence of the Skop–Griffin parameteron VIV response

To investigate the effect of structural damping on VIV,numerical simulations on VIV of elastically mountedcircular cylinder with different values of the Skop–Griffin parameter SG ¼ 2p3 St2 m*z were carried out.In these simulations, similar to those presented by Li(2004), m* was selected as 1.27324 and z(¼ zx ¼ zy)was determined by setting SG ¼ 0.01, 0.1, 1.0 and 10.0.The reduced velocity Ur was set to 5.263 withcorresponding Reynolds number equalled 200. Samespring rigidity was adopted in two directions to makefNx ¼ fNy ¼ fN. The maximum possible response wasexpected to be excited by setting fN/fSt ¼ 1.04 where‘lock-in’ occurred.

Figure 10 shows the time histories of CD, CL andy/D with different values of SG. Clearly, the Skop–Griffin parameter has significant effects on the cylinderresponse. When SG is relatively low, for example SG ¼0.01, the lift force acting on the cylinder experiences acomplex vibration. A secondary component is found

Figure 8. A simple example: the energy transferred by anormalised sinusoidal force on a normalised sinusoidaldisplacement in two cycles. Different phases f between theforce and the cylinder’s cross-flow response are applied. (a)f ¼ 08, (b) f ¼ 1808, (c) f ¼ 908.

Figure 9. The structure cross-flow nature frequency fNy, theStrouhal frequency fSt and the actual cross-flow vibrationfrequency f0y (same to vortex shedding frequency fV) versusthe reduced velocity Ur. ‘Lock-in’ occurs where 4.0 Ur 8.0.

218 C. Ji et al.


with different phase compared with the primarycomponent. This is similar to that of VIV in ‘lock-in’regime without damping, as in Section 3.2. However,this secondary component damps out gradually whenSG increases. As seen in Figure 10d, only a ‘monotone’vibration is left in the lift force time history.

Furthermore, the phase f is also influenced by SG.This is clearly shown in Figure 11, in which f changesfrom 151.48 at SG ¼ 0.01 to 94.28 at SG ¼ 10.0. Asstated in Section 3.2, f is either 08 or 1808 for the zero-damping MSD system. However, when the damping isno longer zero, f have to shift from these two values tosupply the energy consumed by the damper. WhenSG ¼ 0.01 and 0.1, f suffers from both structuraldamping and the secondary component of lift forcethat causes smaller values of f compared with those ofzero-damping MSD system. If the structural dampingis infinity, the lift force should supply the maximum

Figure 10. Time histories of drag force coefficient CD, lift force coefficient CL and cylinder’s non-dimensional cross-flowresponse y/D with different values of the Skop–Griffin parameter SG. (a) SG ¼ 0.01, (b) SG ¼ 0.1, (c) SG ¼ 1.0, (d) SG ¼ 10.0.

Figure 11. Variation of cylinder’s non-dimensional cross-flow response amplitude Ay/D and phase f between lift forceand cylinder’s cross-flow response with the Skop–Griffinparameter SG.



amount of energy it possibly could, i.e. f ¼ 908. Thistrend is clearly shown in Figure 11.

Figure 12 compares non-dimensional response am-plitude Ay/D versus SG of present numerical results withexperimental data and other numerical results from theliterature. It can be seen that the present results showsimilar trends of the experimental data (Williamson andGovardhan 2008) and are in good agreement with thenumerical results of Li (2004). The response amplitude issmaller than those of Newmann and Karniadaki (1995)and Zhou et al. (1999) due to their usage of lower massratio (m* ¼ 1.0). All numerical results tend to under-predict the response amplitude when SG is low andoverpredict the response amplitude when SG is high. Thisdiscrepancy can be due to the effect of Reynolds number.In the experiments, the Reynolds number covers a largerange values from 300 to 106. However, in all numericalstudies, the Reynolds number is only 200. Moreover, inthe experiments, the 3D character of the wake under-mines the total fluid force exerted on the cylinderbecause of the poor span-wise force correlation, thus itleads to smaller response amplitude. However, 2Dnumerical simulation guarantees that the span-wisecorrelation of the fluid force is ideal. So, as a result,the two-dimensionality of numerical simulations couldalso contribute to the overprediction of the responseamplitude when SG is high.

4. Summary and conclusions

We have extended the FD method introduced byGlowinski et al. (1994) for particulate flow problems to

deal with VIV of a 2D elastically mounted circularcylinder. A 2DOF MSD system was adopted to modelthe cylinder responses and an external forcing schemewas then deduced.

The in-house CFD code was first verified againstthe benchmark problem – flow past a 2D stationarycircular cylinder. Then, the effects of two majorfactors, i.e. the reduced velocity and the Skop–Griffinparameter, on the cylinder’s responses were investi-gated. The first test problem showed that in the ‘lock-in’ regime, Ur ¼ 4.0 * 8.0, the actual vibratingfrequency shifted from the Strouhal frequency to thesystem natural frequency, which caused high-ampli-tude oscillations in cylinder response. The maximumnon-dimensional response amplitude Ay/D ¼ 0.572was achieved at Ur ¼ 4.0. The vortex mode was ‘2S’in both initial and lower branches. The trajectorieschanged from forward bending ‘8’ to backwardbending ‘8’. The secondary component of lift forceoffset the energy input by the primary component andthus reduced the cylinder cross-flow response. Thisfinding presents another point of view from which theself-limited character of VIV can be related. Also, thesecondary component played a key role on the stablevalue of the phase between lift force and cylinder’scross-flow response. The ‘upper’ branch and the ‘2P’vortex pattern were both absent from VIV of circularcylinder in low Reynolds number flow. Reasons forthis have been discussed. The second test problemshowed significant effects of the Skop–Griffin para-meter SG on the cylinder responses. With the increas-ing SG, the non-dimensional response amplitudedropped rapidly and the phase changed from 151.48to 908 when SG is infinity. This trend was explainedfrom an ‘energy balance’ point of view.

This study investigated the 2D VIV of elasticallymounted circular cylinder in low Reynolds numberflow numerically using an in-house CFD code based onthe FD method. It can be foreseen that this code couldbe used later on simulating more complex VIV ofslender marine structures, for example marine risers,by coupling with other technologies such as large eddysimulation and parallel computing.

Acknowledgements

This research was supported by the Science Fund forCreative Research Groups of the National Natural ScienceFoundation of China (Grant no. 51021004), the NationalNatural Science Foundation of China (Grant No. 50809047,50979069) and the National Science Foundation forDistinguished Young Scholars of China (Grant No.50725929). This research was supported by a Marie CurieInternational Incoming Fellowship within the 7th EuropeanCommunity Framework Programme (Grant No. PIIF-GA-2009-236457).

Figure 12. Comparison of non-dimensional cross-flowresponse amplitude Ay/D. .: in-water, o: in-air, Williamsonand Govardhan (experimental data, 2008);¤: Newmann andKarniadaki (calculated results forRe ¼ 200,m* ¼ 1.0, 1995);D: Zhou et al. (calculated results for Re ¼ 200, m* ¼ 1.0,1999); r: Li (calculated results for Re ¼ 200, m* ¼ 1.27324,2004); }: present results for Re ¼ 200, m* ¼ 1.27324.

220 C. Ji et al.


References

Anagnostopoulos, P. and Bearman, P.W., 1992. Responsecharacteristics of a vortex-excited cylinder at low Rey-nolds-numbers. Journal of Fluids and Structures, 6, 39–50.

Baaijens, F.P.T., 2001. A fictitious domain/mortar elementmethod for fluid-structure interaction. InternationalJournal for Numerical Methods in Fluids, 35, 743–761.

Coesnon, B., et al., 2008. A fast and robust fictitious domainmethod for modeling viscous flows in complex mixers:the example of propellant make-down. InternationalJournal for Numerical Methods in Fluids, 58 (4), 427–449.

de Hart, J., et al., 2000. A two dimensional fluid-structureinteraction model of the aortic valve. Journal ofBiomechanics, 33, 1079–1088.

Feng, C.C., 1968. The measurement of vortex-induced effectsin flow past stationary and oscillating circular and D-section cylinders. Thesis (M.A.Sc.). University of BritishColumbia, Canada.

Glowinski, R., Pan, T., and Periaux, J., 1994. A fictitiousdomain method for Dirichlet problem and applications.Computer Methods in Applied Mechanics and Engineer-ing, 111 (3–4), 283–303.

Glowinski, R., et al., 2000. A distributed Lagrange multi-plier/fictitious domain method for the simulation of flowaround moving rigid bodies: application to particulateflow. Computer Methods in Applied Mechanics andEngineering, 184 (2–4), 241–267.

Hover, F.S., Davis, J.T., and Triantafyllou, M.S., 2004.Three-dimensionality of mode transition in vortex-induced vibrations of a circular cylinder. EuropeanJournal of Mechanics B/Fluids, 23, 29–40.

Ji, C.N., et al., 2007. Fully coupled fluid-structure interactionmodel based on distributed Lagrange multiplier/fictitiousdomain method. China Ocean Engineering, 21 (3), 439–450.

Khalak, A. andWilliamson, C.H.K., 1999.Motions, forces andmode transitions in vortex-induced vibrations at low mass-damping. Journal of Fluids and Structures, 13, 813–851.

Lai, M.C. and Peskin, C.S., 2000. An immersed boundarymethod with formal second-order accuracy and reducednumerical viscosity. Journal of Computational Physics,160, 705–719.

Leontini, J.S., Thompson, M.C., and Hourigan, K., 2006.The beginning of branching behaviour of vortex-inducedvibration during two-dimensional flow. Journal of Fluidsand Structures, 22, 857–864.

Li, G.W., 2004. A high-accurate method for vortex-inducedvibrations of an elastic circular cylinder. Thesis (M.A.Sc.).Zhejiang University, China.

Liu, C., Zheng, X., and Sung, C.H., 1998. Preconditionedmultigrid methods for unsteady incompressible flows.Journal of Computational Physics, 139, 35–57.

Lucor, D., Foo, J., and Karniadakis, G.E., 2005. Vortexmode selection of a rigid cylinder subject to VIV at lowmass-damping. Journal of Fluids and Structures, 20, 483–503.

Newmann, D.J. and Karniadaki, G.E., 1995. Direct numer-ical simulations of flow over a flexible cable. In: P.W.Bearman, ed. Flow-induced vibration. Rotterdam: Balk-ema, 193–203.

Placzek, A., Sigrist, J.F., and Hamdouni, A., 2009. Numer-ical simulation of an oscillating cylinder in a cross-flow atlow Reynolds number: Forced and free oscillations.Computers & Fluids, 38, 80–100.

Persillon, H. and Braza, M., 1998. Physical analysis of thetransition to turbulence in the wake of a circular cylinderby three-dimensional Navier–Stokes simulation. Journalof Fluid Mechanics, 365, 23–88.

Sarpkaya, T., 2004. A critical review of the intrinsic nature ofvortex-induced vibrations. Journal of Fluids and Struc-tures, 19, 389–447.

Singh, S.P. and Mittal, S., 2005. Vortex-induced oscillationsat low Reynolds numbers: hysteresis and vortex-sheddingmodes. Journal of Fluids and Structures, 20, 1085–1104.

Uhlmann, M., 2005. An immersed boundary method withdirect forcing for the simulation of particulate flows.Journal of Computational Physics, 209, 448–476.

Williamson, C.H.K., 1996. Vortex dynamics in the cylinderwake. Annual Review of Fluid Mechanics, 28, 477–539.

Williamson, C.H.K., 1997. Advances in our understanding ofvortex dynamics in bluff body wakes. Journal of WindEngineering and Industrial Aerodynamics, 69–71, 3–32.

Williamson, C.H.K. and Govardhan, R., 2004. Vortex-induced vibrations. Annual Review of Fluid Mechanics,36, 413–455.

Williamson, C.H.K. and Govardhan, R., 2008. A briefreview of recent results in vortex-induced vibrations.Journal of Wind Engineering and Industrial Aerodynamics,96 (6–7), 713–735.

Williamson, C.H.K. and Jauvtis, N., 2004. A high-amplitude2T mode of vortex-induced vibration for a light body inXY motion. European Journal of Mechanics B/Fluids, 23,107–114.

Yu, Z.S., 2005. A DLM/FD method for fluid/flexible-bodyinteractions. Journal of Computational Physics, 207 (1),1–27.

Zhou, C.Y., So, R.M.C., and Lam, K., 1999. Vortex-inducedvibrations of an elastic circular cylinder. Journal of Fluidsand Structures, 13, 165–189.



View publication statsView publication stats
https://www.researchgate.net/publication/232763785

Documents

International Journal of Computational Fluid Dynamics ... › profile › Chunning_JI...two degree-of-freedom mass–spring–damper system. The computing code was veriﬁed against