A universal three-dimensional instability of the wakes of two-dimensional bluff bodies

J.FluidMech.(2016),vol.792,pp.50–66.c󰀄CambridgeUniversityPress2016doi:10.1017/jfm.2016.9450

Auniversalthree-dimensionalinstabilityofthe

wakesoftwo-dimensionalbluffbodiesAnirudhRao1,MarkC.Thompson1,†andKerryHourigan1

1DepartmentofMechanicalandAerospaceEngineering,FluidsLaboratoryforAeronauticalandIndustrialResearch,FLAIR,17CollegeWalk,MonashUniversity,Clayton,Victoria3800,Australia

(Received23June2015;revised12November2015;accepted1February2016;firstpublishedonline29February2016)

Linearstabilityanalysisofawiderangeoftwo-dimensionalandaxisymmetric

bluff-bodywakesshowsthatthefirstthree-dimensionalmodetobecameunstable

isalwaysmodeE.Fromthestudiespresentedinthispaper,itisspeculatedtobe

theuniversalprimary3Dinstability,irrespectiveoftheflowconfiguration.However,

sinceitisatransitionfromasteadytwo-dimensionalflow,whetherthismodecanbe

observedinpracticedoesdependonthenatureoftheflowset-up.Forexample,the

modeEtransitionofacircularcylinderwakeoccursataReynoldsnumberofRe󰀇96,

whichisconsiderablyhigherthanthesteadytounsteadyHopfbifurcationatRe󰀇46

leadingtoBénard–von-Kármánshedding.Ontheotherhand,iftheabsoluteinstability

responsibleforthelattertransitionissuppressed,byrotatingthecylinderormoving

ittowardsawall,thenmodeEmaybecomethefirsttransitionofthesteadyflow.

Awell-knownexampleisflowoverabackward-facingstep,wherethisinstabilityis

thefirstglobalinstabilitytobemanifestedontheotherwisetwo-dimensionalsteady

flow.Manyotherexamplesareconsideredinthispaper.Exploringthisfurther,a

structuralstabilityanalysis(Pralitsetal.J.FluidMech.,vol.730,2013,pp.5–18)

wasconductedforthesubsetofflowspastarotatingcylinderastherotationratewas

varied.Forthenon-rotatingorslowlyrotatingcase,thisindicatedthatthegrowthrate

oftheinstabilitymodewassensitivetoforcingbetweentherecirculationlobes,while

fortherapidlyrotatingcase,itconfirmedsensitivitynearthecylinderandtowards

thehyperbolicpoint.Forthenon-rotatingcase,theperturbation,adjointandstructural

stabilityfields,togetherwiththewavelengthselection,showsomesimilaritieswith

thoseofaCrowinstabilityofacounter-rotatingvortexpair,atleastwithinthe

recirculationzones.Ontheotherhand,atmuchhigherrotationrates,Pralitsetal.

(J.FluidMech.,vol.730,2013,pp.5–18)havesuggestedthathyperbolicinstability

mayplayarole.However,bothinstabilitieslieonthesamecontinuoussolution

branchinReynoldsnumber/rotation-rateparameterspace.Theresultssuggestthat

thisparticularflowtransitionatleast,andprobablyothers,mayhaveanumberof

differentphysicalmechanismssupportingtheirdevelopment.

Keywords:instability,parametricinstability,wakes

†Emailaddressforcorrespondence:

mark.thompson@monash.eduModeEinstability51

1.Introduction

Recentnumericalstudies(Pralits,Giannetti&Brandt2013;Raoetal.2013a,b,

2015a;Navrose&Mittal2015)ofrotatingcircularcylinderwakesshowthe

appearanceofanewthree-dimensionalinstabilitymodethatdevelopsonthesteady

two-dimensionalwake.Thismode,modeE(namedinthealphabeticorderofthe

modesobservedbyRaoetal.(2013a)),wasinitiallyobservedforα󰀁2andRe󰀁200,

whereαisthenon-dimensionalisedrotationrateofthecylinder(surfacetofree-stream

speed)andReistheReynoldsnumberbasedonthecylinderdiameter.Theonsetof

modeEoccursatlowerReynoldsnumbersastherotationrateisincreased.Thismode

canalsobeobservedforlowerrotationratesofα󰀂2,iftwo-dimensionalperiodic

shedding(i.e.,Bénard–von-Kármán(BvK)vortexshedding)isartificiallysuppressed.

Raoetal.(2015a)speculatedthatmodeEisessentiallythesamethree-dimensional

modeasobservedforrotatingcylindersplacedclosetoawall(Stewartetal.2010;

Raoetal.2011,2013c).

Toinvestigatefurthertheoccurrenceandthenatureofthistransition,andindeed

howwidespreaditis,linearstabilityanalysisisperformedforarangeofbluff-body

geometriesandflowset-ups,wherethewakehasbeenartificiallystabilisedtobe

steadyandtwo-dimensional.Giventhesesteadybaseflows,itisshownthatmodeEis

thefirstthree-dimensionalmodetobecomeunstableinbluff-bodyflowsandappears

tobea‘universal’modethatisobservedirrespectiveoftheconfigurationofthebluff

bodyunderconsideration.

Theremainderofthisstudyisorganisedasfollows:§2dealswiththenumerical

methodemployedinouranalysis,followedbytheresultsin§3;in§4theresults

areexaminedintermsofstructuralstability,firstintroducedbyGiannetti&Luchini

(2007)andLuchini,Giannetti&Pralits(2008),anddiscussedandinterpretedina

widercontext,exploringtheroleofgenericphysicalmechanismsintriggeringthis

instability.Finally,§5providesfurtherinterpretationsandconclusions.

2.Numericalmethod

Toobtainthetime-dependentflowsandstabilitymodes,theincompressibleNavier–

Stokes(NS)equationsaresolvedintwo-dimensionalCartesianoraxisymmetric

geometriesusingaspectral-elementformulation.Thecomputationaldomainconsists

ofseveralhundredquadrilateralmacroelements,withhigherconcentrationinthe