The effect of spacing on the vortex-induced vibrations of two tandem flexible cylinders

26 Pages·2017·3.17 MB·English

The effect of spacing on the vortex-induced vibrations of two tandem flexible
cylinders
Enhao Wang, , Qing Xiao, , Qiang Zhu, and , and Atilla Incecik
Citation: Physics of Fluids 29, 077103 (2017); doi: 10.1063/1.4995463
View online: http://dx.doi.org/10.1063/1.4995463
View Table of Contents: http://aip.scitation.org/toc/phf/29/7
Published by the American Institute of Physics
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PHYSICSOFFLUIDS29,077103(2017)
The effect of spacing on the vortex-induced vibrations of two
tandem flexible cylinders
EnhaoWang,1 QingXiao,1,a) QiangZhu,2 andAtillaIncecik1
1DepartmentofNavalArchitecture,OceanandMarineEngineering,UniversityofStrathclyde,
Glasgow,ScotlandG40LZ,UnitedKingdom
2DepartmentofStructuralEngineering,UniversityofCaliforniaSanDiego,LaJolla,
California92093-0085,USA
(Received12February2017;accepted8July2017;publishedonline31July2017)
Vortex-induced vibrations (VIVs) of two flexible cylinders arranged in tandem are studied using
atwo-wayfluid-structureinteraction(FSI)methodwithdifferentspacingratios(S /D)atReynolds
x
numberRe=500usingatwo-wayfluid-structureinteraction(FSI)method.Themainobjectiveofthis
studyistoinvestigatetheeffectofspacingonthehydrodynamicinteractionsandtheVIVresponses
ofthesecylinders.Theresponsesofthetwoflexiblecylindersarefoundtobesimilartotheclassical
VIVresponsesatsmallS /D.OnceS /Dislargeenoughforthevorticestobecomedetachedfromthe
x x
upstreamcylinder,theresponseoftheupstreamcylinderissimilartothetypicalVIVresponsewhereas
thedownstreamcylinderundergoeswake-inducedvibration.Thecharacteristicsoftheresponseofthe
downstreamcylinderinthepresentstudyaresimilartothoseofthefirsttworesponseregimesclas-
sifiedbypreviousresearchers.Thethirdregimeisnotobservedfortheflexibledownstreamcylinder
withbothendsfixed.Thetwochangesinthephaserelationbetweenthecross-flowdisplacementsof
thetwotandemflexiblecylindersarediscoveredtobelinkedwiththeinitial-upperbranchtransition
andtheupper-lowerbranchtransition,respectively.Thecorrelationlengthsofthetwoflexiblecylin-
dersdecreasesignificantlyinthetransitionrangebetweentheupperandlowerbranches.Threemodes
ofvortexshedding(2S,P+S,and2P)havebeenidentifiedinthepresentstudy.Theupper-branch2P
modeisfoundtobeassociatedwithlarge-amplitudevibrationoftheupstreamcylinderandtheP+S
mode is observed to be related to large-amplitude vibration of the downstream cylinder for
S /D = 3.5 and 5. On the other hand, the lower-branch 2P mode leads to small-amplitude vibra-
x
tion of the downstream cylinder in the post-lock-in range at S /D = 2.5. The relative phase shifts
x
of the sectional lift coefficients on different spanwise cross sections can be attributed to the varia-
tion of the vortex shedding flow along the flexible cylinders, and these phase shifts result in poor
phasing between the forces and the displacements that is related to the decrease of the correlation
lengths.PublishedbyAIPPublishing.[http://dx.doi.org/10.1063/1.4995463]
I. INTRODUCTION thespacingbetweenthetwocylinders.11,12Fortwostationary
cylindersintandemarrangement,threemajortypesofflowpat-
Vortex-induced vibration (VIV) of a single cylinder has
ternshavebeenidentified.Whenthetwocylindersareplaced
beenextensivelystudiedbyvariousresearchersoverthepast
inverycloseproximity,thetwocylindersbehaveasasingle
few decades and comprehensive reviews can be found in
body.Theseparatedshearlayersfromtheupstreamcylinder
Blevins,1Sarpkaya,2Bearman,3WilliamsonandGovardhan,4
wraparoundthedownstreamcylinderwithoutreattachingonto
Gabbai and Benaroya,5 Sumer and Fredsøe,6 Bearman,7 Wu
itssurfaceandformasingleKa´rma´nvortexstreetinthewake.
etal.,8 Pa¨ıdoussisetal.,9 andTriantafyllouetal.10 However,
With the increase in the spacing, the separated shear layers
therearemanyengineeringproblemsthatinvolveclustersof
from the upstream cylinder reattach onto the surface of the
cylindricalstructuresbeingarrangedintandemconfigurations.
downstreamcylinder.Whenthespacingbetweenthecylinders
Theexamplesincludeheatexchangertubes,chimneystacks,
increasesbeyondacriticalspacing,vortexsheddingoccursin
offshorestructures,andtransmissionlines.Duetoitspractical
the gap region between the two cylinders so that the wake
significance,moreandmorestudiesarecarriedoutontheVIV
behindthecylindersisacombinationofthewakesofthetwo
ofmultipletandemcylinders.
cylinders.
Comparedtoasinglecylinder,theflowpasttwotandem
Someexperimentalstudieshavebeenconductedtoinves-
cylindersismuchmorecomplexduetotheintroductionofa
tigate the interference between two rigid circular cylinders
newgeometricparameter,i.e.,thespacingbetweenthecylin-
undergoingVIV.HoverandTriantafyllou13andAssietal.14,15
ders (S ). There are a number of flow regimes depending on
x considered the effect of the upstream cylinder wake on the
response of the downstream cylinder. In their investigations,
a)Authortowhomcorrespondenceshouldbeaddressed:qing.xiao@strath. the upstream cylinder was stationary whilst the downstream
ac.uk.Tel.:[phone]. cylinder was elastically mounted and free to vibrate in the
1070-6631/2017/29(7)/077103/25/$30.00 29,077103-1 PublishedbyAIPPublishing.
077103-2 Wangetal. Phys.Fluids29,077103(2017)
cross-flow direction only. In Hover and Triantafyllou,13 the S /D,theresponseofthedownstreamcylinderexhibitednon-
x
downstreamcylinderwasplacedat4.75Dbehindtheupstream classicalVIVresonancewithlargeamplitudesathighreduced
cylinderandtheReynoldsnumberwasRe=3×104.Large- velocities.
amplitude galloping response was observed for the down- Apartfromtheexperiments,therehavealsobeenanumber
streamcylinder.Theresultsalsoindicatedthatthefrequency ofnumericalstudiesontheVIVoftwotandemcylinders.The
lock-inbeganatalowreducedvelocity(V ),whichwasnearly majorityofexistingnumericalstudiesweretwo-dimensional
r
the same as a single cylinder and its range extended to a V (2D) and focussed on one-degree-of-freedom (1DOF) cross-
r
of at least 17. A phase change in the lift force, which was flowmotionsofrigidcylinders.Carmoetal.21 conducted2D
typically associated with the frequency lock-in, occurred at numericalsimulationsoftheflowaroundtwotandemcircular
higherV .Theforcespectrasuggestedthatthesheddingfrom cylinders at Re = 150. The upstream cylinder was fixed and
r
the upstream cylinder was not affected by motions of the thedownstreamcylinderwasfreetovibrateinthetransverse
downstream cylinder. Assi et al.14 elaborated the excitation direction. S /D was varied from 1.5 to 8. Compared with an
x
mechanism of wake-induced vibration (WIV) of the down- isolatedcylinder,thedownstreamcylinderwasfoundtohave
streamcylinder.TheReynoldsnumberintheirstudywasup higher maximum amplitudes and wider lock-in ranges. The
to3×104.TheysuggestedthattheWIVofthedownstream vibration amplitudes for higher V beyond the lock-in range
r
cylinderwasexcitedbytheunsteadyvortex-structureinterac- wereverysignificant.Carmoetal.22 investigatedtheVIVof
tions between the body and the upstream wake. In a recent acylinderthatwascompletelyfreetomoveinthecross-flow
research by Assi et al.,15 the authors investigated how the direction(i.e.,withnospringordamperattachedtoit)subject
cylinder responded to the vortex-structure interactions exci- tothewakeofanidenticalstationarycylinder.Afixedspacing
tation. They introduced the concept of wake stiffness and ratioS /D=4wasconsideredandRevariedfrom100to645.
x
concluded that it was the wake stiffness phenomenon that Threedifferentregimeswereidentifiedintheir2Dsimulations.
defined the character of the WIV response. Zdravkovich16 A monotonically decreasing amplitude with increasing Re
studied the VIV of two elastically mounted rigid cylinders wasobservedinthefirstregime(90≤Re≤165).Thesecond
in tandem arrangement with different centre-to-centre spac- regimewascharacterisedbysignificantscatterinthevibration
ingratios(S /D)intheRerangeof104–105.Itwasobserved amplitudeanditsrangeisfromRe≈180toRe=360.Gradual
x
that for a very small S /D up to 1.1, the two cylinders acted transitionbetweenthesecondandthirdregimestookplacefor
x
asasinglebody.WithaslightincreaseinS /Dupto1.6,the 360≤Re≤405.ThethirdregimeextendeduptothehighestRe
x
shear layers from the upstream cylinder reattached onto the tested.Theamplitudewasfoundtogrowmonotonicallywith
downstreamcylinder.AbistableregimewasfoundwhenS /D Re.Zhao23presentedtheresultsofVIVoftworigidlycoupled
x
varied from 2.5 to 4, which indicated the minimum spacing circular cylinders in tandem arrangement with four spacing
requiredfortheupstreamcylindertohaveregularvortexshed- ratiosrangingfrom1.5to6atRe=150.Theresultsshowed
ding.Inthisregime,theoscillationamplitudesoftheupstream thatthegapbetweenthetwocylindershadasignificanteffect
cylinder were larger than those of the downstream cylinder. ontheresponse.Whentworigidlycoupledtandemcylinders
For S /D larger than 4, the response of the upstream cylin- weresubjecttoVIV,thecriticalspacingforvortexshedding
x
derwassmallerandlessregularthanthatofthedownstream fromtheupstreamcylinderwassignificantlysmallerthanthat
cylinder. fortwotandemstationarycylinders.Thevortexsheddingfrom
The interference has also been considered for vibrating the upstream cylinder took place at S /D = 2 in the lock-in
x
flexible cylinders. Brika and Laneville17,18 investigated the range,andthelock-inrangewasfoundtobenarrowerthanthat
VIVofalongflexiblecircularcylinderimmersedinthewake ofasinglecylinderforS /D =1.4and2andwiderforS /D
x x
ofanidenticalstationarycylinder.S /Dbetweenthetwocylin- =4and6.Dingetal.24used2DunsteadyReynolds-averaged
x
dersrangedfrom7to25andRerangedfrom5000to27000. Navier-Stokes (RANS) equations with the Spalart-Allmaras
Itwasfoundthattheresponseoftheflexiblecylinderwasno turbulence model to study the effect of tandem spacing on
longer hysteretic and showed a single branch with a wider the VIV of two cylinders with passive turbulence control in
synchronizationregionthatdecreasedwiththeincreaseinthe the Re range of 30000–100000. The spacing between the
spacingbetweenthetwocylinders.Theonsetofthesynchro- cylinders varied from 2D to 6D. The numerical simulation
nizationshiftedtohigherV comparedwiththecaseofaniso- successfully predicted all the ranges of responses including
r
latedcylinder.BrikaandLaneville17alsoconsideredthecase VIVandgalloping.Furthermore,theresultsagreedwellwith
inwhichbothcylinderswereallowedtovibrate.Inthatcase, the experimental measurements. The influence of the down-
the response of the downstream cylinder became hysteretic. streamcylinderontheamplitudeandfrequencyresponsesof
Huera-HuarteandBearman19andHuera-HuarteandGharib20 the upstream cylinder was found to be negligible when the
experimentally studied the vortex- and wake-induced vibra- spacingislargerthan2D.Arisingtrendofthevibrationampli-
tionsoftwotandemflexiblecylinderswithnearandfarwake tudeofthedownstreamcylinderwasobservedinallthecases
interferencesinaRerangeupto12000.Itwasfoundthatthe when2×104 <Re<3×105.Thegallopingbranchmerged
responses of both cylinders showed classical VIV resonance with the VIV upper-branch for spacing larger than 3D. Vor-
when V was close to the typical lock-in reduced velocities. tex structures showed significant variation in different flow
r
WhenS /Dissmall,themaximumvibrationamplitudeofthe regimes.
x
upstreamcylinderishigherthanthatofthedownstreamcylin- Two-degree-of-freedom(2DOF)VIVoftwotandemrigid
der and the maximum vibration amplitude of the upstream cylinders has also been studied using 2D numerical simula-
cylinder increased with the decrease in spacing. For a large tions.Papaioannouetal.25studiedtheeffectofspacingonthe
077103-3 Wangetal. Phys.Fluids29,077103(2017)
VIVoftworigidcylindersintandemarrangement.Thecom- method in conjunction with a chimera domain decomposi-
putationswerecarriedoutforS /Dintherangeof2.5–5atRe tion approach with overset grids. The in-line and cross-flow
x
=160.Itwasobservedthattherangeoftheresponseregionof responsesoftheriserswerecalculatedusingatensionedbeam
theupstreamcylinderbecamewiderwiththedecreaseofS /D. motion equation. For two vertical risers in tandem arrange-
x
The synchronization curve shifted on the V axis depending ment,thevortexstreetwassuppressedbetweentherisersand
r
onthespacing.Therewasanincreaseinthemaximumvibra- therewasareattachmentoftheshearlayerstothedownstream
tionamplitudeofthedownstreamcylinderwhenthecylinders riser when S /D = 3. When S /D increased to 4, a vortex
x x
werebroughttothespacingcorrespondingtothereattachment streetstartedtoappearinthegapbetweentheriserswithinter-
regimeinthestationarysystem.PrasanthandMittal26investi- mittentreattachmentofshearlayersonthedownstreamriser
gated the VIV of two circular cylinders in tandem arrange- surface.Ingeneral,thecomputedvortexpatternsanddynamic
ment with S /D = 5.5 at Re = 100 using a 2D stabilised responsesoftheriserswereingoodagreementwiththeexper-
x
finite element method (FEM). V ranged from 2 to 15. The imental data. Gonza´lez et al.29 presented the results of their
r
downstream cylinder was found to undergo large-amplitude numericalsimulationattemptsfortheexperimentalcampaigns
vibrationsinboththein-lineandcross-flowdirections.Lock- onatandemarrangementofflexiblecylindersbyHuera-Huarte
in and hysteresis were observed for both the upstream and etal.30atRe=16000.Thenumericalsimulationqualitatively
downstream cylinders. The large-amplitude vibrations of the reproduced the physical phenomena. However, the numeri-
downstreamcylinderweremaintainedevenbeyondthelock-in cal model oversimplified the mechanical complexity of the
range.Thephasedifferencebetweenthecross-flowdisplace- flexible cylinder and the application of the axial tensions
(cid:176)
ment and the lift force went through an 180 jump in the caused the numerical instability of the computation. There-
middleofthesynchronizationregimesofbothcylinders.The fore,theauthorssuggestedthatfutureresearchwasrequiredin
flow regime was divided into five sub-regimes based on the ordertomakeaquantitativecomparisonwiththeexperimental
phase difference and the flow patterns. Bao et al.27 studied data.
the2DOFVIVoftwotandemcylinderswithvaryingnatural Overall, VIV of two elastically mounted rigid cylin-
frequency ratios at Re = 150 by solving the 2D incompress- ders in tandem arrangement has been widely studied both
ible Navier-Stokes equations using a characteristic-based- experimentally and numerically. Due to the complexity of
split FEM. The spacing between the two cylinders was 5D. the problem, there have been fewer studies on the VIV
They found that the in-line response of the downstream of two tandem flexible cylinders and most of the existing
cylinder was more sensitive to the natural frequency ratio publications are experiments. Fully 3D fluid-structure inter-
than that in the transverse direction. As dual resonance was action (FSI) simulations of VIV of two tandem flexible
excited, the vortex shedding of the upstream cylinder in cylindersthatcanaccuratelypredicttheresponsesofthecylin-
the tandem arrangement might show a P+S pattern, which ders as well as provide the instantaneous 3D flow structures
strongly suppressed the vortex shedding of the downstream that are difficult to obtain in the experiments are still quite
cylinder. limited.
Compared with the popularity of 2D numerical studies, Inthisstudy,cross-flowVIVoftwoflexiblecylindersin
three-dimensional(3D)numericalsimulationsofVIVoftwo tandem arrangement at Re = 500 is investigated using a 3D
tandemcylindershavereceivedlessattention.Carmoetal.21 numerical method. The choice of Re stems from the consid-
compared the 3D numerical simulation results of VIV of a erationthattheflowaroundthecylindersis3D,anditcanbe
rigidcylinderexposedtoanupstreamstationarycylinderwake modelledbydirectlysolvingthe3DNavier-Stokesequations.
at Re = 300 with their 2D simulation results at Re = 150. WefocussedontheV rangeofV =4–10,inwhichthecylin-
r r
Foralltheconfigurationsinvestigated,theyobservedthatthe derswillmainlyvibrateinthefirstmode.Themainobjective
3Dresultsfollowthesametrendasthe2Dresults.However, ofthepresentworkistonumericallystudytheeffectofspac-
theflowfieldwasstronglythree-dimensionalformostofthe ingonthehydrodynamicinteractionsandVIVresponsesofthe
configurations.Therefore,theyconcludedthat3Dsimulations twotandemflexiblecylinders.Particularattentionispaidtothe
were strictly necessary in order to obtain accurate values of aspectsthathavenotbeenfullyaddressedbypreviousstudies
forces or structural response. Carmo et al.22 carried out 3D suchasthecorrelationlengthsandthetime-dependent3Dflow
numerical simulations with the same parameters as their 2D structures.
calculationsinthesamestudy.Theresultsshowedthatthevari-
ationofamplitudewithintheRerangetestedwasverysmall.
Theoscillationfrequencyofthedownstreamcylinderandthe
II. NUMERICALMETHODSANDPROBLEM
shedding frequency of the upstream cylinder were constant
DESCRIPTION
and had the same value throughout the entire Re range. The
amplitudesobservedinthe3Dresultsaresmallerthanthose The FSI simulation in the present study is carried out
observedinthe2DresultsforthesameRe,which,according using the ANSYS MFX multi-field solver.31 The governing
totheauthors,wascausedbytheweakeningofthespanwise equationsfortheflowaroundtheflexiblecylindersarethe3D
vortices. unsteadyincompressibleNavier-Stokesequations.TheArbi-
There are even fewer numerical studies on the VIV of traryLagrangian-Eulerian(ALE)schemeisemployedtodeal
two tandem flexible cylinders. Chen et al.28 simulated VIV withthemovingboundariesoftheflexiblecylinders.TheALE
and WIV for two vertical risers in tandem and side-by-side form of the governing equations in the Cartesian coordinate
arrangementsatRe=8400usinganunsteadyRANSnumerical systemcanbeexpressedas
077103-4 Wangetal. Phys.Fluids29,077103(2017)
∂u
i =0, (1) lengthofthecomputationaldomainis40Dinthestreamwise
∂xi direction (x-direction) with the upstream cylinder located at
∂u (cid:16) (cid:17) ∂u 1 ∂p ∂2u 10Ddownstreamtheinletboundary.Thewidthofthedomain
i + u −uˆ i =− +ν i, (2)
∂t j j ∂xj ρ∂xi ∂x2 inthecross-flowdirection(y-direction)is20Dandthecom-
j putationaldomainsizeinthespanwisedirection(z-direction)
wherexi representstheCartesiancoordinatewith(x1,x2,x3) issettothecylinderlength,whichis12D.Thecomputational
=(x,y,z),uiisthevelocitycomponentinthexi-direction,uˆiis mesh for Sx/D = 5 used in the CFD simulation is shown in
thegridvelocitycomponentinthexi-direction,pisthepres- Fig.1(b).Theboundaryconditionsforthegoverningequations
sure,tisthetime, ρisthefluiddensity,andνisthekinematic are as follows. The surfaces of the cylinders are assumed to
viscosityofthefluid. besmooth,whereno-slipboundaryconditionsareemployed.
The Navier-Stokes equations are discretised with an Apartfromtheno-slipboundaryconditions,thecylindersur-
element-based finite volume method (FVM).32 Rhie-Chow facesarealsosettobefluid-solidinterfaceswhereforceand
interpolation33isusedtoobtainpressure-velocitycouplingon displacement data are transferred. The inlet velocity bound-
collocated grids. A second-order backward Euler scheme is aryconditionissettobethesameasthefreestreamvelocity.
adoptedforthetemporaldiscretisationandahighresolution Attheoutflowboundary,thegradientsofthefluidvelocityin
scheme is used as the convection scheme that can be cast in thestreamwisedirectionaresettozero.Onthetwotransverse
thefollowingform: boundaries,thevelocityinthedirectionnormaltothebound-
ary is zero and a periodic boundary condition is imposed on
ϕ = ϕ + β∇ϕ·∆r, (3)
ip up
thetopandbottomboundaries.Asforthepressureboundary
where ϕ isthevalueattheupwindnode,(cid:79)ϕ isthecontrol conditions,thenormalgradientofpressureiszeroonthecylin-
up
volumegradientsevaluatedbyGauss’divergencetheoremand dersurfaces,theinletboundary,andthetransverseboundaries.
∆risthevectorfromtheupwindnodetotheintegrationpoint Thepressureattheoutflowboundaryisgivenareferencevalue
ip. In the high resolution scheme, a nonlinear recipe for β ofzero.
basedontheboundednessprinciplesbyBarthandJespersen34 In the present study, the two flexible cylinders are mod-
isused. elledasbeamsfixedatthetwoendsandtheyarefreetovibrate
Figure1(a)showsthecomputationaldomainusedinthe inthetransversedirection.Thevibrationsofthecylinderssat-
computationalfluiddynamics(CFD)simulation.Theoriginof isfytheEuler-Bernoullibeamtheory.Thetransversemotions
theCartesiancoordinatesystemislocatedatthecentreofthe ofthetwoflexiblecylinderscanbedescribedas
(cid:34) (cid:35)
bottomendoftheupstreamcylinderasshowninFig.1(a).The
∂2 ∂2y ∂2y ∂y
EI +m +c =F , (4)
∂z2 ∂z2 ∂t2 ∂t y
whereE isYoung’smodulus,I isthemomentofinertia,mis
themassperunitlength,c isthestructuraldamping,z isthe
undeflectedcylinderaxialcoordinate,y isthetransversedis-
placement,andF isthehydrodynamicforceinthetransverse
y
direction.TheboundaryconditionsforEq.(4)inthecaseofa
beamfixedatthetwoendsare
y(0,t)=0, y(L,t)=0 ∀t,
∂y(0,t) ∂y(L,t) (5)
=0, =0 ∀t.
∂z ∂z
A finite element method is used to discretise the finite
elementanalysis(FEA)modelsandthecorrespondingglobal
equationsystemisgivenby
Mq¨ +Cq˙ +Kq=F, (6)
where q is the nodal displacement vector and a dot denotes
differentiation with respect to time. The global mass, damp-
ing, and stiffness matrices (M, C, and K) are the collective
effectsoftheindividualelements’mass,damping,andstiffness
matrices(Me=∫∫∫NTρsNdxdydz,Ce=∫∫∫NTcNdxdydz,and
Ke =∫∫∫BTEBdxdydz,whereNisthematrixofshapefunc-
tions, ρ isthestructuraldensity,Bisthestrain-displacement
s
matrix,andEistheelasticitymatrix)derivedfromtheprinci-
pleofvirtualwork.Fisthehydrodynamicforcevector.The
governingequationissolvedusingtheHilber-Hughes-Taylor
(HHT)method35withasecond-orderaccuracy.
A 3D 20-node solid element that exhibits quadratic dis-
FIG.1. (a)Computationaldomainand(b)computationalmeshintheCFD
simulation. placementbehaviourisusedforthediscretisationoftheFEA
077103-5 Wangetal. Phys.Fluids29,077103(2017)
FIG.2. ComputationalmeshintheFEAsimulationatSx/D=5:(a)initialconfigurationand(b)meshafterdeflections.
models of the flexible cylinders. The element is defined by beginswithsolvingtheflowequationsontheupdatedmesh.
20 nodes having three degrees of freedom per node: trans- Such an FSI loop is repeated until the last time step of the
lations in the nodal x-, y-, and z-directions.36 Figures 2(a) simulation.
and 2(b) show the finite elements of the two tandem flexi- VIV of two tandem flexible cylinders is investigated
blecylindersforSx/D=5intheinitialconfigurationandafter numerically. Two identical flexible cylinders with diameter
deflections,respectively.Inthesoliddomain,thesurfacesof D and length L are aligned in the direction of the flow in
the cylinders are also set to be fluid-solid interfaces for data theirrestpositions.Tosimplifytheproblem,thevibrationsof
transfer. thecylindersareconfinedtothecross-flowdirection.Similar
Toaccommodatethemotionsoftheflexiblecylinders,the simplifications have been adopted by previous researchers
displacementdiffusionmodel37–39formeshmotionisadopted. suchasEvangelinosandKarniadakis,40 Evangelinosetal.,41
Thedisplacementsofthemeshpointsarecalculatedbasedon and Xie et al.42 to study the VIV of a flexible cylinder. The
thefollowingequation: length-to-diameterratioisL/D=12,whichallowsthediscreti-
(cid:16) (cid:17)
∇· Γ∇S =0, (7) sationofthecylinderspanwithafinemeshtoresolvethechar-
y
acteristics of the flow with acceptable computational efforts.
whereS representsthedisplacementsofthenodalpointsin
y Moreover,L/D=12isalsocomparabletoL/Dvaluesusedby
thecross-flowdirection,Γisthemeshstiffness.Inthisstudy,in
Evangelinosetal.41andXieetal.42intheirnumericalstudies.
ordertoavoidexcessivedeformationofthenear-wallelements,
Amoderatemassratiom*=4m/ρπD2=10isconsidered.The
the parameter Γ is set to be Γ = 1/∀2 where ∀ is the control
influenceofthestructuraldampingismainlyreflectedinthe
volumesize.
maximum vibration amplitude and in order to maximize the
Atwo-wayexplicitapproachisutilizedfortheFSIsimu-
vortex-induced responses of the flexible cylinders, the struc-
lationinthepresentstudy.Detailedsolutionproceduresofthis
turaldampingissettozero.TheReynoldsnumberisdefined
approachcanbedescribedasfollows.Atthebeginningofeach
asRe=VD/νwhereVisthefreestreamvelocityandaconstant
time step, the flow equations are solved to obtain the forces
ReynoldsnumberRe=500isadoptedinthepresentsimula-
on the flexible cylinders. Then the forces are interpolated to
tion. The reduced velocity V = V/f D ranges from 4 to 10.
the FEA mesh using the conservative interpolation, and the r 1
Here, f is the fundamental natural frequency (i.e., the first
global equation system for structural dynamics is solved to 1
eigenfrequency) of the flexible cylinder and for a beam with
opblatcaeinmtehnetsqauraenitnittieerspoolfacteydlintodethremCoFtiDonms.eAshftweritthhathtethperodfiisle- two fixed ends, f1 = 21π(cid:16)1+ π2(cid:17)2(cid:113)mELI4. Since V is fixed in
preservinginterpolation,andthepositionsofthemeshpoints the present study, the bending stiffness EI is varied in each
are calculated and updated using Eq. (7). The next time step simulation to obtain the corresponding f1 of the desired
077103-6 Wangetal. Phys.Fluids29,077103(2017)
V . In order to study the effect of spacing on the VIV
r
of two tandem flexible cylinders, three different centre-to-
centre spacing ratios are selected, i.e., S /D = 2.5, 3.5, and
x
5 as in Papaioannou et al.25 These three spacing ratios are
based on different regimes in the stationary system. S /D
x
= 2.5 belongs to the reattachment regime where the shear
layers from the upstream cylinder reattach onto the sur-
face of the downstream cylinder; S /D = 5 represents the
x
binary-vortexregimewheretheseparatedshearlayersofthe
upstream cylinder roll up in the gap region and a binary
vortex streetis formedbehind the downstreamcylinder con-
sisting of the interacting wakes of the two cylinders and the
spacing S /D = 3.5 is very close to the critical spacing of
x
transition from the reattachment regime to the binary-vortex
FIG.3. Comparison of spanwise mean drag coefficients of two tandem
regime. stationary cylinders at Re = 500: blue open circles—upstream cylinder
To confirm the flow regimes for the three spacing ratios (Papaioannouetal.);blueopensquares—downstreamcylinder(Papaioannou
etal.);redfilledcircles—upstreamcylinder(presentsimulation);redfilled
predicted by the present methods, flow past two tandem sta-
squares—downstreamcylinder(presentsimulation).
tionary cylinders at Re = 500 was simulated and the span-
wisemeandragcoefficients(C )werecomparedwiththe
Dmean
numerical results of Papaioannouet al.43 in Fig. 3. It can be positive. Figures 4 and 5 show the 3D flow structures and
seenthatthepresentresultsareingoodagreementwiththose thevortex sheddingatz/L =0.5 aroundstationarycylinders.
of Papaioannou et al.43 In the reattachment regime, C The present results agree with the descriptions of the dif-
Dmean
of the upstream cylinder decreases with the increase in the ferent flow patterns for two tandem stationary cylinders by
spacing and the value of C experiences a jump at the Igarashi,12Igarashi,44Zdravkovich,11Papaioannou,45Xuand
Dmean
critical spacing. This jump is more pronounced on CDmean Zhou,46 ZhouandYiu,47 Carmoetal.,48 Carmoetal.,49 and
ofthedownstreamcylinder,whichchangesfromnegativeto Sumner.50
(cid:12)
FIG.4. Iso-surfacesoftheeigenvalueλ2= 0.1withthecontoursofthespanwisevorticityωzontheiso-surfacesforstationarycylinders:(a)Sx/D=2.5,(b)
Sx/D=3.5,(c)Sx/D=5,and(d)singlecylinder.
077103-7 Wangetal. Phys.Fluids29,077103(2017)
FIG.5. Spanwisevorticitycontoursωzforstationarycylindersatz/L=0.5:(a)Sx/D=2.5,(b)Sx/D=3.5,(c)Sx/D=5,and(d)singlecylinder.
III. VALIDATIONANDMESHDEPENDENCYSTUDY thecylinderN ,theminimummeshsizenexttothecylinder
c
surface in the radial direction ∆r, and the mesh size in the
TheFSIsolverhasbeenusedinourpreviousstudy38 to
spanwisedirectionofthecylinder∆z.Similarly,theimportant
investigatetheVIVofaverticalriserwithL/D=481.5inuni-
meshparameterintheFEAofthepresentFSIsimulationsis
form and linearly sheared currents and the numerical results
thenumberofsegmentsalongthecylinderspanN .Inorderto
werecomparedwiththeExxonMobilverticalrisermodeltest z
evaluatethedependencyofthenumericalresultsontheCFD
resultsobtainedattheMARINTEKbyLehn.51Thenumerical
meshparameters,simulationswerecarriedoutfortheVIVofa
resultswereingoodagreementwiththeexperimentaldata.In
singleflexiblecylinderwithm*=10andL/D=12atV =6and
thepresentstudy,furtherinvestigationsonthecross-flowVIV r
Re=500withthreedifferentCFDmeshsystemswhileusing
ofasingleflexiblecylinderwithm* =10,L/D=12,andtwo
thesameFEAmeshwithN =30.Theresultscomputedusing
fixed ends at Re = 1000 were conducted using the present z
thedifferentCFDmeshesaretabulatedinTableI.Asshown
numerical methods and the results are compared with the
inthetable,themaximumdifferenceof3.36%occursinthe
numericalresultsbyXieetal.42inFig.6.Itdemonstratesthat
value of the spanwise root mean square (rms) lift coefficient
thepresentnumericalmethodsarecapableofaccuratelypre-
C betweenCFDM1andCFDM2,whereasthemaximum
dictingVIVresponsesofflexiblecylindersatearlysubcritical Lrms
difference between CFDM2 and CFDM3 reduces to 1.4%.
Reynoldsnumbers.
It can be concluded that the combination of N = 120, ∆r
According to a series of publications on 3D numerical c
= 0.001D, and ∆z = 0.1D is sufficient for the CFD simula-
simulations of VIV of an elastically mounted circular cylin-
tion of VIV of flexible cylinders in the present study. Then,
der,37,52,53 the important mesh parameters in the CFD sim-
CFDM2 was used to study the dependency of the numeri-
ulation are the number of nodes along the circumference of
cal results on the FEA mesh parameter. Three FEA meshes
are generated with N =15, 30, and 60. The comparison of
z
the results computed with different FEA mesh parameters is
showninTableII.ThemaximumdifferencebetweenFEAM1
andFEAM2isobservedinthespanwisermsdragcoefficient
C and takes the value of 1.14%. In contrast, the maxi-
Drms
mumdifferencebetweenFEAM2andFEAM3is0.55%,which
appears in C . Based on the FEA mesh dependency test
Lrms
results,N =30isusedinthepresentstudy.
z
TABLEI. CFDmeshdependencytestresults.
CFDM Nc ∆r/D ∆z/D Ay/D CDmean CDrms CLrms
1 80 0.002 0.2 0.7443 1.5975 0.2501 0.1717
FIG.6. Comparisonofvibrationamplitudesofasingleflexiblecylinderwith 2 120 0.001 0.1 0.7480 1.6127 0.2449 0.1661
m*=10,L/D=12atRe=1000:bluecircles—Xieetal.;redfilledcircles— 3 160 0.0005 0.05 0.7524 1.6039 0.2482 0.1638
presentsimulation.
077103-8 Wangetal. Phys.Fluids29,077103(2017)
TABLEII. FEAmeshdependencytestresults. B. Responseamplitudes
FEAM Nz Ay/D CDmean CDrms CLrms In Huera-Huarte and Bearman,19 the envelopes of the
1 15 0.7515 1.5964 0.2421 0.1673 displacements are calculated using the analytical signals
2 30 0.7480 1.6127 0.2449 0.1661 described by Pikovsky et al.55 based on the use of Hilbert
3 60 0.7480 1.6076 0.2453 0.1652 transforms, and the mean values of the envelopes are used
torepresentthevibrationamplitudes(A /D).Thesametech-
y
niqueisadoptedinthepresentstudytodeterminethevibration
IV. RESULTSANDDISCUSSIONS
amplitudesoftheflexiblecylinders.Figure10showsthetime
NumericalsimulationwasperformedfortheVIVoftwo histories of the nondimensional displacements (y/D) on the
flexiblecylindersintandemarrangementwiththreedifferent different cross sections along the two tandem flexible cylin-
spacingratios(S /D =2.5,3.5,and5)atanearlysubcritical ders when S /D = 5 and V = 6. The solid red lines are the
x x r
Reynolds number Re = 500 for a reduced velocity range V envelopesofthesignalsandthedashedredlinesrepresentthe
r
=4–10.Theresultsarecomparedwiththoseofasingleflexible mean values of the envelopes inside the selected time win-
cylinder to investigate the effect of spacing on the hydrody- dow by which the sectional vibration amplitudes [A (z)/D]
y
namicinteractionsandtheVIVresponsesofthetwotandem are denoted. A /D is determined by the maximum value of
y
flexiblecylinders. A (z)/Dalongeachcylinder.Figure11showsthevariationof
y
A /D with V for the two flexible cylinders arranged in tan-
y r
A. Dominantmodes dematdifferentS /D.Theresultsforasingleflexiblecylinder
x
are also included for comparison. As the dominant mode in
In order to investigate the dominant modes of the flex-
most of the cases considered in the present study is the first
ible cylinders and the contributions of each vibration mode
mode,themajorityofA /Disobservedatz/L=0.5.Theexcep-
to the overall dynamic responses, the modal amplitudes are y
tions appear at V = 10 when the contribution of the second
computedusingthesamemodaldecompositionmethodasin r
Huera-HuarteandBearman.19,54 Themodalanalysisisbased mode is of a similar order of magnitude to that of the first
mode. As shown in Fig. 9, for two flexible cylinders in tan-
onthefactthattheriserresponsecanbeexpressedinthematrix
demarrangement,A /Dappearsaroundz/L=0.4anditshifts
formasalinearcombinationofitsmodes, y
to z/L ≈ 0.3 in the case of a single flexible cylinder where
y(z,t)=Φ(z)y (t), (8)
M the second mode vibration dominates. In general, the A /D
y
whereΦ=[φ ,φ ,...,φ ]isthedisplacementmodal-shape response curves in Fig. 11 are similar to the modal ampli-
1 2 n
matrix, which is built by stacking the nondimensional mode tude response curves of the first mode in Fig. 7 due to the
shapesthatarenormalizedtobe1atitsmaximumascolumns. fact that the first mode is the dominant mode in most cases.
y(z, t) = [y (z , t), y (z , t), ..., y (z , t)] is the measured
1 1 2 2 n n
displacementmatrixwhoserowvectorsarethetimeseriesof The maximum vibration amplitudes (Aym/D) of the
the measured displacements along the cylinder. yM(t) is the upstream cylinder for Sx/D=3.5 and 5 are observed at
modalcontributionmatrixcontainingthetimeseriesofeach an identical reduced velocity Vr=6 with a magnitude of
mode’scontributiontotheoverallresponseastherowvectors Aym/D≈0.76, which is similar to Aym/D of a single flexible
yM(t)=[yM1(t),yM2(t),...,yMn(t)].Wehave cylinder.AsthespacingdecreasestoSx/D=2.5,Aym/Dofthe
upstream cylinder increases to 0.87. This increase in A /D
y (t)=Φ−1(z)y(z,t). (9) ym
M of the upstream cylinder with the decrease of S /D was also
x
Figure7illustratesthevariationofthemodalamplitudes observedbyHuera-HuarteandBearman19intheirexperimen-
ofthefirstthreemodeswithV .Itcanbeobservedfromthefig- tal study with Re up to 12 000 and by Papaioannou et al.25
r
urethatthedominantmodeinmostcasesisthefirstmode.The in their 2D numerical simulation at Re = 160. Compared
contributionofthesecondmodeisrelativelysmallcompared with the upstream cylinder, the effect of the hydrodynamic
withthatofthefirstmodeatlowV ,butthisincreasesgradu- interactions between the two tandem flexible cylinders on
r
allywiththeincreaseinV .AtV =10,themodalamplitude the vibration amplitude responses of the downstream cylin-
r r
ofthesecondmodeovertakesthatofthefirstmodeforasingle der is more obvious. The maximum vibration amplitudes of
flexiblecylinder.Thecontributionofthethirdmoderemains the downstream cylinder for Sx/D=3.5 and 5 are found to
smallthroughouttheVr rangeconsidered.Figure8showsthe be similar (Aym/D≈1). This is different from the results of
instantaneousnondimensionaldeflectionsoftheflexiblecylin- Papaioannouetal.25 inwhichthemaximumvibrationampli-
dersatVr =6.AsshowninFig.8,allthevibrationsatVr =6 tude of the downstream cylinder in the case Sx/D = 3.5 is
demonstrateeasilyidentifiablefirstmodecharacteristics.The similartothatatSx/D=2.5.Itisbelievedthatthisdifference
instantaneousnondimensionaldeflectionsoftheflexiblecylin- canbeattributedtothedifferenceinRe.Papaioannouetal.25
dersatV =10arepresentedinFig.9.Thevibrationofasingle considered a relatively low Reynolds number Re = 160. In
r
flexible cylinder can be seen to be dominated by the second contrast,theReynoldsnumberinthepresentsimulationisRe
mode. Meanwhile, the vibrations of the two tandem flexible =500.AccordingtoSumner,50thecriticalspacing(Sx/D)cr is
cylindersatV =10arenotasregularasthoseatV =6.The particularlysensitivetotheReynoldsnumber.Ljungkronaand
r r
irregularity in the vibrations of the two flexible cylinders is Sunde´n56reportedthat(Sx/D)cr atRe=160isapproximately
due to the fact that contributions of the first and the second 4.54anddecreasesto(Sx/D)cr≈3.7whenReincreasesto500.
modesarecomparableatVr =10. Therefore,Sx/D=3.5isfurtherbelowthecriticalspacingratio
077103-9 Wangetal. Phys.Fluids29,077103(2017)
inthestudyofPapaioannouetal.25 thaninthepresentstudy classifiedtheresponseofanelasticallymounteddownstream
resultinginA /DatS /D=3.5intheirstudybeingsimilarto cylinderintothreeregimes:(i)beforethevortexsheddingfre-
ym x
thatatS /D=2.5. quencyf =thenaturalfrequencyofthecylinderf ,whenthe
x v n
In the present study, when S /D falls to 2.5, A /D of StrouhalnumberSt=0.2isapproaching f ,theA /Dresponse
x ym n y
the downstream cylinder increases to 1.1 and it shifts to a resembles the typical VIV response; (ii) the second regime,
higherreducedvelocityV =7duetotheshieldingeffectof between f =f and the equivalent natural frequency of the
r v n
the upstream cylinder. When V = 8–10, for the three spac- wakestiffnessf =f ,ismarkedbyasteepslopeintheA /D
r w n y
ing ratios considered, the upstream cylinder shows a typical response curve; and (iii) the third regime beyond f =f is
w n
VIV response with very small vibration amplitudes at high characterised by a change of the slope in the response curve
V .Intermsofthedownstreamcylinder,alower-branchVIV of A /D. According to their descriptions, the A /D response
r y y
response is observed for S /D = 2.5 and 3.5 due to the fact of the flexible downstream cylinder at S /D=5 in Fig. 11 is
x x
that the vortices impinging on the downstream cylinder are similar to the first two regimes and the third regime is not
connectedtotheshearlayersoftheupstreamcylinderasillus- observedinthepresentstudyandthepossiblereasonswillbe
(cid:12)
trated by the spanwise vorticity (ω = ∂u /∂x ∂u /∂x ) provided in Sec. IV C. We also notice that the present A /D
z 2 1 1 2 y
contours in Figs. 23(a) and 23(b). In contrast, at S /D = 5, in the second regime shows a plateau instead of increasing
x
the completely detached vortices from the upstream cylin- monotonically with V as shown in the experimental results
r
der in Fig. 23(c) cause the WIV of the downstream cylin- of Hover and Triantafyllou13 and Assi et al.14,15 This dis-
derwithlarge-amplitudevibrationbeingmaintaineduntilthe crepancy can be attributed to the different treatments of Re
maximum V considered in the present study. Assi et al.15 betweenthepresentstudyandtheexperiments.Inthepresent
r
FIG.7. VariationofmodalamplitudeswithVr:(a)singleflexiblecylinder,(b)upstreamcylinder,and(c)downstreamcylinder.Bluecircles—singleflexible
cylinder;redsquares—tandem,Sx/D=2.5;blackdiamonds—tandem,Sx/D=3.5;pinktriangles—tandem,Sx/D=5.