NUMERICAL SIMULATION OF VORTEX SHEDDING IN OSCILLATORY FLOWS

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ABella.&wdl1DfDrmatiouCouIpaPy 3OONortbZ«bRo.i,. AmaAtbJr" MI"l~l146USA

3l3n61...700 8OOIS21~

NUMERICAL SIMULATION OF VORTEX SHEDDING IN OSCILLATORY FLOWS

BY

© V I K A S KRISHNA, B .TECH.

A thesis submitted. to the School of GraduateStudies in partial fulfillment of the requirements for th e degreeof

Master ofEngineering

Facultyof Engineering&;Applied Science Memorial University of Newfoundland

June,1995

St. Jobn 's Newfoundland Canada

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Abstract

\"iscous forcestha t act onabody movingina fluid may form. asignificant part of thetotalforceacti ngonthebody.The use of the linearpotential-ftowtheory doe; nottake intoco nsi de ra tiontheviscous effectsthatcauseflow separation.

skin-frictiondrag,and lift. Variousmethodshave beendevelo pedandusedto calculate theviscous forcesnumericallysince Rosenhead'sinitialcalculations.A reviewofthe earlierwork done inthe area ofvortexshedding and calculation of viscous Bows is presentedin chapter1. Some vortex methods . thatare commonly used,are described. One of these,the DiscreteVort ex Method,isdescribedin detailin chapter2.Thisisdemonstrated for a bluffbody with sharp corners using thefeaturesof the Clement'smodel.However.this method does not simulatethe effects ofvo rti citydiffusion in the flow.Moreover,thebodymustalso have sharp edges. which aretaken tobethe separation points.

Anothermethod.that doesnetimposesuchlimitations.is theVertex -In- Cell('\lC) '-(ethod.Thisisdev'eloped and lirstappliedtostud yflows pastacircular cylinderinorder to validate themethod with existingresultsinchapter3.Confor- maltransformationscoupledwiththismet hodenableustostudy flows pastbodies ofother cross-sect ional sha pes,Variousmappings arederived.anddevelopedto simulate flows past a varietyofshapes. Afinwasadd ed on tothe body conto ur to simulatethe effectofaskeg inthe case of boat -eectious,The force coefficients, CDandCM•were calculatedat differentKeulegan -Carpenternumbers and verified forsectionslikeacircle.a flat plate.anda square,in anoscillatory flow,withthe

results obtainedbyotherworkers.They werealso calcula tedfor a finned-circle.a sectionofa boatthatisrounded and one that has haed chines.The theoryand the method .the necessary modificationsto the \lCmet hod.andtbe simulation resultsare presented in chapter-t•.-\conclus ionof theworkdoneinthe differ ent chapt ersalon gwith some comments aregiven in cha p t er5.

I dedicate this thesistomy wonderfulparents.

Prof.C.BaIKrishna and ).,hs.Sushila Krishna

Acknowledgments

I have received a great amount of support from an equally large number of people.Of all.I would first like to thank and express my deep gratitude to Dr.

D.W.Bass.my supervisor. for his excellent guidance and support. Inparticular.

I would like to acknowledge his suggestions at the crucial stages of this computa- tional project. They showed me the way out of seemingly impossible situations.

[ would alsolike to thank the Associate Dean of the Facultyof Engineering and AppliedScience.Dr.J.J.Sharp, for hie continuedencouragement and support at various stages of the )VI.Eng. program.I also thankDr. Bass.Dr. Sharp. and the School of GraduateStudiesof MemorialUniversity forthe generous financial support. withoutwhich it would have been difficult to remain hereand study.

Finally, I thank the staff of the Center forComputerAided Engineering for theirassistance and providing excellent computing services to the students of this faculty.

iii

Contents

Abstract

Acknowledgements

Contents

List ofFigures

List of Symbols and Abbre viations

1 Introduction 1.1 A Review

1.2 CommonVortexMethode .. 1.2.1 ThePointVort exMet hod 1.2.2 The VortexBlobMethod.... 1.2.3 Thevortex -In-CellMethod 1.3 Summary

2 PointVorte x Method(Clements) 2.1 TheModel...

2.2 Derivati ons ..

iv

ii iii

iv

vii

xi

10 13 . . . ... .• 16

rs

IS 21

2.2.1 Com plex Po tenti al 2.2.2 Vortex Velocities 2.2.3 Vortex Stre ngths 2.2.4 VortexPositio ns. 2.3 Sample Calc ula tions

3 The Vortex. In- Cell Method 3.1 Introd uc tion . 3.2 Theory . ..

3.3 The Method.. . .. .. .. .. ... .. . . . .... .. .... 3.4 Pressures.For ce Coefficientsand Vorticity

3.4.1 Pressur es .

3.4.2 ForceCoefficients . 3.4.3 Vorticity . . . .. 3.5 SomeNumerical Results

4 Results usingCo nfor m al Mapping 4.1 Int rod uct ion. ..

4.2 Modificationto theVICmetho d._ -1.3 Circularcylinderin oscillatory!l.ow 4.4 Flatplatein cross oscill a tory!l.ow. . 4.5 Cylinde rwitha fin . 4.6 Boat Secti ons

4.6.1 Rounded sections . 4.6.2 Sectionswit hsharp edges(har d chines) 4.7 Discuss ion..

21 23 26 28 29

31 31 32 3. 39 39 .0

"'

"'

47 -IT .8 52 53 5i 59 61 68 74

5 Conclusi on Referenc es

77

81

Li st of F igures

1.1 Area\Veigh ti ngScheme.

2.1 The physi cal z-plane .

2.2 The tr ans form edA-plan e..

2.3 Thetwo flows inthe a and the>..-planes[30].

2.4 Derivationof vortex strengt h. 2.5 Positi on of the vorticesatt

=

5.

2.6 Position of the vorticesatt=10.

2.7 Rate of change of circulationatthecorners .

3.1 Timehistory of thedragcoefficien t atBe

=

1000.

3.2 Surface vorticity distribution.t=0.3.

3.3 Surface vorticitydistribution.t=0.6.

3.4 Surface vorticity distribution.t=0.9.

3.S Surface pressure distribution .t

=

0.3.

3.6 Surface pressuredistri bution,t=0.6.

20 20 20 27 29

30 30

3.7 Surface pressure distribution,t=0.9. 44

3.8 Radialvelocity onthe z-axis behind thecylinder,Re=3000. 45 3.9 Radialvelocity on the z-axis behind the cylinder,Re=9500.. ... 45

vii

·U Effectofthe truncationvalue ofJonCDandC.\(. .. ... .. 50 4.2 Meshin thephysical plane(circulaccylinder)... 53 4.3 CD.C.\f t:s,KC(circu1arcylinder).3

=

439 54 4A FittingofF(t )to Morison's equation (circular cy linder). 54 4.5 Transformation schemefor a circle to a Bat plate. 55

4.6 Meshinthe physical plane{fiat plate). 56

-ti CD.CM us. KC(Ba t plate).J=439 56 4.8 Fitting ofF(t)to Morison'sequation(f1atplate). . . .... . 57 4.9 Transformationscheme fora circleto a finned-circle. 57 4.10Meshin thephysicalplane(finned-circle).. 58 4.11CD,C.W us.KC(finned~circle),J

=

320. fin-lengthwflfi. 60 4.12 Fitt ingofF(t)to Morison'sequation(finned-circle)... 60 4.13 Streak plot fora finned-cylinderatRe

=

426,KC

=

1.33,..1t

=

0.04 61 4.14Meshin thephysicalplane(roundedsection.no skeg)... . 64 4.15CO,CM^1:5. KC{round ed section.no skeg),;3=400 66 4.16 FittingofF(t)to Morison's equation(roundedsection .no skeg). 66 4.17Meshinthe physical piane(roundedsection.with skeg}. 67 ..1.18 Co, CM us. KC (rounded sect ion.with ekeg},J

=

878.fin-length sefl.S 67 4.19 Fitt ing ofF(t)toMori son'sequation(roundedsection,wit hskeg). , 68 4.20 Kerman-T reffta,Theodorsen-Gerricktransformations for a square[34J.69 4.21 Meshinthe physical plane(square). .... . 70 4.22CO,CMus.KC (squa.re),fJ=200 71

·1.23 Fitting ofF{t )to Morison'sequa tion(sq uare). 72 4.24Meshin the physic alplane(squarewithafin). ... .... 72 4.25CD,CMus. KC (squ are witha fin),()

=

258,fin~length=0.78 73

viii

-1.26Fitting ofF(t)to Morison'sequetioatsquarewitha lin). T3 -1.27 :!I.fesh inthe phys icalplane(boatsecti onwithafin).. .• ••...• 7-1 -.1.28Co.C.l.lL·S. KC(bo a tsection with alin).J=-I39.lin-length=O.S2 75 -1.29 Fitt ingofF(t)to :!I.forison·sequationlboat sectionwithalin). 75

ix

List of Symbols and Abbreviations

:vorticity

:frequencyof oscillatory How : circul a tio n

:Dirac delta :positionvector :time

ii

u,

.:>.t

R, KC

;l

:unit normal :velocity vector

:vorticity distrib utionfunction : core of vortexelement

:outerradiusof computational do m ai n :kinematicviscosity

:numberof newlycreatedvortices :time step

:streamfunction :vort ex stre ngth :fluid densi ty :Reyn ol ds number : Keulegan -Carpenternumber :Stokesparameter :drag coefficient :inerti acoefficient

II'

:complexpotencial :complex velocity :pressure force :frictional force :Jacobian oftrans form a tion : characteristiclengt h :elonga tionfactor :physicalplane : comput a tional plane

xi

Chapter 1 Introduction

For the smooth and sound operation of marinev-essels. such asshi ps and barges.the wave-induced roll motionsmust be accuratelypredicted.Calculations of the wave- exciting forces andthe motions of vesselsin wavescan be made usingthe linear potential-flowtheory. This,however, neglect sviscous effectswhich causeflow separationand skin-friction drag.Also.forthe linear theory to apply.the motions of the vessel and the wave amplitu desareassumed to besm allin magnitude.

Furthermore. the motions cal cul a ted usingthe lineartheo ry are quiteaccuratefor thedegr ees of freedom that are heavily dampedbutnotfo rthose that arelightl y damped or forthose where the damping isdominatedby";SCOU5effects and in particular for those withlar ge amplitudes.

The main reason for this inaccuracywasthe unavailabilityof goodmethods for determining the damping forces due tothe viscous effects on the submerged body.

The relative motion between me focalfluid andthe submerged vessel boundary causes flows which occur due to the unsteady boundary layer.its localized sepa- rations,and the associated vortexsheddingespeciallynear large changesinthe curvature of the hull. Brown et al.[1] presented experimental evidence to show

that a majorpart of the viscous force actingon abarge with bluffright angle keel edges was due tovort ex shedding. Workon this phenomenon has been carried out by manyresear chers using discrete-vortex-methods.

Vort ex methodshavebeen very widely usedforsim ula ti ng flows pastbodies in ordertodet erminethe forces actingon them. Incompressible flows at bight >500) Reyn oldsnumber are charact erized byregionsofconcentrated vorticity.Thisvor- ticityin the flowis transportedthroughtheHow using the localvelocitywhich intuen is calculated fromthevort icity field. Poisson'sequation forthestream functionissolvedto compute the local velocitiesto movethe vorticity inthe fluid.

It becomesquite convenient to represent the flowin termsof regionsof vortic- ity. Vort ex methods are.therefore.employedtosimulate dewsby discretisin g the flow into regions ofvorti city and then trackingthis discretisationina Lagrangian reference frame.

1. 1 A R eview

Rosenhead[2Jwas thefirst oneto use a distribution of discrete vort ices torepresent a v'ort exshee t. His results were welcomedby otherresearchers andthemethod has been usedsince then till today.Themostimportan t part of Rosenhead's work wastbe numerical treatment of tbe Helmholtz inst a bility phenomenon forlonger durations. Helmholtz instabilityisthe motionofadjacent port ionsof fluid ofthe same density with different speeds. The vortex sheet was replaced by adistri but ion of point vortices along its trace whose paths were determined by a numerical step- by-stepmethod. At any instantof time.the line joining the positions ofthese vortices was assumed to be the approximate shape of the sheet atthattime.

Rosenhead showedtha tthe effectof the instability upona surfaceof someform wastoproduceconcent rati ons ofvo rt ici ty at equal intervals along thesurface andalsothat thesurfaceof discontinuity tendstoroll uparoundthese points of concentrationtogetherwit h an accompanying increaseinthe amplitude ofthe displacement.Healso attributed.the approximate err or. thatcrep t in at each time step.to thetruncationofthe Taylorseries forthe displacement oftheelemental vortices during each step.The met hodwasnotapplica ble to simulationsinvolving longer durations as the errors int rod ucedateach timestep couldgetlarge enoug h toca use numerical difficulties.

Thedowpatterns thatRcsenbe adpresen tedforHelm holtzinstabilitywere very realist ic. Asa resultof this.his method ofcalculation was never really doubted.After1959.cri ticism ofhisworkwasmade byBirkhoffandFisher[31, Ham aandBurke[4].and others.Itwas pointedoutthatdiscrete vorticesmoving under theirowninfluence inatwodimensionalBow will always tend to a random distri buti on.Birkhoffand Fisher assertedtha t anydecreasesin distances between pairs of vortices would resultinan increase forot her pairs. Theybasedthis assertionon the theorem tha t the energyof asys te mof vortexfilamentsin atwo- dimensionalBow isconst ant. Since.for a system of vortices of equalstrength.

the energyis given bytheprodu ct ofalltheinitialdistances.notwo vortexlines cancoalesce with ou t aninfini teincreaseinthe separ ationofsome other pair.

Thus, accordingtothem , the vorti city ina uniformarray ofpointvortices cannot becomeconcentrated.Theyrecalcula ted. Rosenhead'5results and foun dthat the mot ion of theelement alvortices becamevery complica ted. inregions wheresome concentra tiontookplace,making it impossibletoproducesmoo thvortexsheets. Abern athyandKrona uer[.;Jexp lainedthemainfeatur esofthevortexstreet

formationmechanism from twoinit ially parallel uniformvortexsheetsofopposi te sign. They found tha t no argumentbasedsolelyon a linearized theory ofdist ur- baaces can generate the asymmetry commonlyassociatedwiththe interac t ion of the two sheets.Theyalsouseda numericaldiscretisationof the ve rtex sheetsas Rosenheaddid.Theirnon-linearmethod. workedvery well for a short timeafterthe start of the flow but randomness appearedto creepintothe calculations oftheco- ordinates ofthe point vortices whichevent uallybecame50pronounced thatit was saidthat. "therewasno longersufficientevidencetosuggest theexistenceofve rtex sheets" .Theyconclu dedthatcloudsofvorticityhave-e netstrengthdiminished by the vort icityswept into thecloudby the opposi te vortex row" . Michalke[6J generalized the work of Rosenbeadandof Abern ath yand Kronauer byincluding tbe effect of thethickn ess ofashearlayer. Herepr esen tedthe constant surface distributionof vorticity represen tingtheinit ial lylinear velocity profileof theshear layerby a discretenumber ofsurfaces of discontinuity .simulatingeach bya finite numberofeleme ntalvorti ces.

Furtherwork was donetostudy vort exshed di ng from two-dimensional bodies.

).(odelingaHowaround a sharp edgerequirestha t thevorti cityisshed from it inordertopreventinfini t eveloci ty at theedge. Theshedvorti ces forma spiral whichcoversmor e and moreareaasvorticityisshed..Accordi ngtotbe law of conservationofenergy .this vort icitycannot remain concentrat edinto a pointand acorest ru ct ureisform ed.Anton [7] wasthe first onetomodel vortex shedding from asharp edgeusingasemi-infini te pla tein1939. He found that thetotal vorticityshed isprop ortionalto a one third poweroftime and tbat the distance from the edgeto a pointon thevortexsheetisproportionaltothetwotbirds powerof time.Tode terminetheshapeandthevorticitydistri bu t ionof thesheet.

he dividedthe vortexsheet into an innercore andanouterlooplinki ng itto the oute r edge.Theinner core was approximatedbya spiraloftheKaden[81 type.

Wedemeyer [9]recalculated Anton'sresults using a different method. Howev er.

neitherAnton's norwedemeyer'ssoluti onisexactbecauseof theover-simplified representationofthe coreregion. ButWedemeyer'sresult.sare morecomplete.

They provide useful informationonthe growth anddist ribution ofvortici tyand its total circulation.Anto n's solutionforthe startingflow near a semi-infiniteplate wasthengeneralized by Blendermaan[10).Heincorpora ted.convex comers of any angle moving with velocityequ al tothetimeraised to anypowerina fluidat rest.These resultswere thoug htof as usefulstarting solutionsinst udyingvortex shedding from morecomplica tedboundaries in moregeneralmotions.

Otherworks inthis area include thoseof Giesing (11),Clements[12J,and Gerrard[131.Giesing used non-linearrepresenta tions forstudyi ng two-dimensional aerofoils in unsteady motion.Ofimportanceishis discussionon the kinematics ofvert exshedding.He claimed that the Kutta condi t ioncould be approximated byhavingazero veloci tydifference atthe trailing edgestha t are non-cusped.He showed thattheerrorinthesurfac e velocity distributionoccurr edonlyinrhe imm edi a te neighbour hoodof the edge.

Clementsmodeled the startingflowinthe near wake of a bluff-based.body.He broughtforwardsome newfea t ures in the treatmentofshedding conditionswhen onlydiscreteelementalvortices areused.As the velocityatthe sheddingpoint couldoth erwi se be infinite,Clementslettheshed vort icesst arttravelingwith a velocitytha t is calcula teda small distance awayfrom the point ofshedding.

This avoidedthe need ofaKutta conditio nwhile maintaining agoo damount of accuracy.Heobtained some interestingresults forthe rate of shedding of..-ort icity

into the shear layer from twoseparation points.but these also are sub jec tto the kindofcriticism made oftheRosenhead model.

In1967.Gerrardnumericallycalculatedthe magnitudeandfrequencyof the lift forceon a circular cylinder.Heintr od uced apair of elementalvortices into the flow some distance downstreamat eachtime step to avoid thedifficult prob lem of the determ ina tion ofthe separationpoints ofthe flow around a body without sharp edges.He used the sheddingrate obtainedexperimentallyto determine the strengthofthe vortices that are introducedat every time step. Even thougb the vortex positions lookedqui te irregular.the values of the liftforce and the scale offormation regions werequite close to the experimentalvalues for a range of Reynolds numbers.

Ofinterest tousisalso the vortex separation tha t occurs fromslend erwings and bodies. Bound arylayer separationtakes placeat the sidesof aircraft wings and otherslenderbodies even at small angles of incidence.Coherent vortexsheets are formed and the studyofthese is of importancewith respectto the non-linear lift, the roll-up of shed. vorticity,etc., in the fieldofaerona utics;withrespect to thenon-linearforce and momentinship manoeuvrability, rolldampingon bilge keelsand fins, erc..inthe area of ship hydrodynamics;and otherareas in offshore engineering.

Roy andLegendr ewere the onestomakethe majorcontri butionsinunder- standingthe phenomeno nofleadin g edge separationfrom slenderwingsin theearly 1950's. However, themost~u'.. ce9Sfuland purelyanalytical model wasdeveloped byBrown and Michael[14Jin1954. Theirresults forthe vortexpositio ns,pressure distrib utio n,etc. agreedonlyqualit at ivelywiththoseobtainedfromexperiments.

However, their me thodhas been used in manyat temp ts toincorp ora tethe effects

of non-co nicalplanfonns.thickness .andtheunsteady motion forthecalcu lation of stabilityderi-..arives.Ot her majornumericalmethods include Smith'sitera t ion technique[151and themulti-vortexmodel ofSack.Lundbe rg andHanso n[161.Any mod els developedafterthese involvedcomparisonwit h these two techniques.both ofwhichwereinmoderatetogoodagreement withexperi ments. These techniques werealsoused by researcherstostu dyvortex shedding from slend erbodies alone andfrom acombinationof awing anda bod y.Smith's met hodisnoteasyto gen- eral ize foruou-conical geomet ries.whereas Sack's methodis.Sacks.lundberg and Hanson used a Roeeuheaddiscretisationintheir mode landobt ained thekind of irregularityof the vortexsheets thatis expectedinapplica t ions ofthe Rcsenhead method.

Shipscan beconsidered asslender bod ies .Ina norm alstrai ght aheadcourse, boun darylayer separation.which leads to the format ion of coherentshear la yers.

occurs formostnullshapes.Theve rtexshedding tha t occursis responsible for the non-linearforce and moment me t in ship manoeunability.Studyingvortex shedding.using sectionsof ships.helps ininvestigatingthese effectsand impmving thegeometr yofthe null to minimizetheadverse and unwanted forces.Henni g[1; 1 usedasingle pairof discretevort icesto representthenon-linear sideforces on slenderships with Lewisformcross-sections. Fuwa [18J refined Henni g's work byusing amult i-vort exmodel forthe bilge vortexsepar a t ion on a shipwhichis being obliquelytowed.Manywritershave descri bed the vortices producedatthe bilges nearthebowsof ships,Tatinclaux[19]gave an approxima tesemi-emp irical treatm entofthe corresponding increme ntin ship resistanceor vortex drag. In 1971,Soh andFink (20J modeled and studiedvertexsepar a tion at the bilge keels ofshi ps using potential flow.They presented.a discretevort ex approximationfor

thesheets usingtheRosenhead method. The}'found that theuseofcons tant timestepscaused an earlyonset ofbreakdownof thecalculations with crossing ver t exsheets. Improvements were madebyusing varyingtim estepstoensure that theshedd ing pointand the lasttwoelemental vorticestobeshed.form ed.an equi-spacedsetofpoints,Amore thoroughanalysis ledtotheconclus iontbatthe erroris leastwhen theelement al verticesare equi-spaced.for the entiresheetat every instant forwhichthemotioniscalculated.

1.2 Common Vortex M ethods

1.2.1 ThePoint Vort exMethod

Hosenhead wasthe firstone to attem pt flow simulati ons usingavortexmethodin 1931.He approximatedthemot ion of a two-dimensionalvort ex sheetbyfollow- iogthe moveme nt ofa setofpointvorticesintime. Thevortici ty field.;J.was representedby

v

~(f,<)~ ~r,J(i'-~(!))

n.n

where ais the Diracdelta functionin twodimensions.ri (t)isthe locationof the,704 vortexatanins tant oftimet.N isthe totalnumber of pointvorticesinthedow.

and

r

i,isthe circulationof theitAvortex. Fora regionR.the totalcircula tion is givenby

(1.2) Here

n

is thenormal toan elemental area.ds,of regicnR.and

w

iswk.To satisfy the inviscidvorticitytransport equation.

8w

at +

(a.V)w

= o .

or, interm s of the materialderivative.

Vw Vt =0.

(1.3)

11.-1) the velocity of avort ex mustbe given bythe v'alueof tbevelocityfieldat the positionof a vortex.Therefore.

~ (t) =it(ri .t)

^(1.5)

Thevelocity field.iI(i'.t)is inturn determinedfromthe solut ion ofthePoisson equa tion

(1.6) The velocityfield mus t satisfy th e boundary cond itionofzero flow acrossthe surface ofthe body.Therefo re.

(1.,) IT the two-dimensional flow field has no interiorboundariesand the fluid isat restat infinity,the solutionto equation 1.6 can be obtained from the Biot-Savart integral

T )-

_1.. {

(i-~)xL(~.t)d" ^{(1.8 )}

UT,t - 2trlc.

I f - ,:f

r

whereListhelengt h of thetwo dimensional vortexsheet.By substituting equa- tion1.1inthisformula,wecan obtainri asthesolution of the following systemof non-linearordinary differe ntialequations:

5 = _..!.. t

^{((ri - rj)x krj}

dt 211",=-1# . Iri Ti l:.! (1.9)

Rosenhead didallthe calculationsby hand. \Viththe advent of computers, more accurate resultswere at temptedas morepoint vertices could be used.How- ever,usinga larger numberof point vortices of lower strengths did not yield a convergent solution. In many simulations.the motions of thevortices became quite chaotic.Therefore.although thepoint vortex method solvesthe Euler equa- tions,a system of point vortices does not represent a vorticity fieldvery well.

Bet te r results withthis method were obtainedusing fewer vortices togetherwrich a diffusivetimeintegra tionscheme.

1.2.2 TheVortexBlob Method

One ofthe other methods for simulating two-dimensionalflows is the vortex blob method.In this method. vortices with finite cores,or vortex blobs, are used as the computationalelements. The equation fortherepresenta t ion of the vorticity field nowbecomes

(1.10) where",!;is thevorticity distribution withinthe vortex located at rj with the normalization

Ie

^{r,(i)di' =1}

whereCisthe core boundary.We assume that

(1.11)

(1.12)

wherefis a distribut ionfunction whose shapeis commonto allvortex elements andatisreferred toas the core ofthe vortex eleme nti.For example,if thevort icity

10

distributionisgiven bythe Gaussian distribution such that

11.13) then the distributionfunction f(fJisgiv'en by

Thisfunc tionsatisfiesthe viscous part ofthe vorticitytransportequation

~~

^=lIV2....

Thevelocityinducedby thevorticity field.equation1.10. isgiven by 11.14)

(1.15)

where9is givenby

g(y )=2,

j,'

j(z )zdz 11.1, )

The use of distributedvortex coresorvortex blobsyieldsmore realistic vort icity distributionsand bounded inducedvelocitiesfor allthe vortexelements.Asper equation 1.16. thevelocity of thevortexblobsisgivenby

~^=iI(ri,t)=

_...!- i:

^(ri-rj)x:rjg~r;

-

^{rjt/qj) (1.18)}

dt 271'j= l, ;#i Ir ;-ril

whichisactually thevelocity fieldatthe center of the blob.

In contrast tothepointvortex method.an errorinthe spatialaccuracy arises inthevort exblobmethodbeca usethe computational elements are assumed to retain theirshapesthro ughout the simulation,whereasa real fluid withvortic ity may suffer a considerableamount ofstrain.Hald and DelPrete(211studiedthe

11

convergence ofthis method tothesolut ion of theEulerequationandshowed that convergence occurred only fora limited time interval with an errortha t grew exponentiallyintim e.

The use ofthevort icitydiffusion term.1/'\12:..,••in thevorti city trans port equa- tion has two im port an t consequences.Theyace.first lythe vorticitycreat ionatthe boundaryand secondlythe diffusionofthevort icity in the flow field.The diffusion ofvorticity can besimulatedeither byallowingthe vortex cores toincrease in size or. as proposed by Chorin[22].addingaran dom walk to the position of the vortices ateach timeste p . However.Greengra d{24]poin tedout that inthecore spreading techni que.the vortici tyis correctlydiffusedbutincorrectly convected .Therefore.

this techniqueisinac cur a teandcom-erges toasyste m of equations differentfrom theNavier -Stckes equations.Chorin'srandomvortex method.on the otherhand.

is acorr ect approximation .The random walk st eplength used hereispropor tional

to(v~t)1/2.the idea being thatthe effectsofviscositybe reproducedcorrectl yin

asta tis t icalsense. Itwas pointedoutthat in thecase of flowspast bluffbodies.

wherethe boundary-layerseparationisanim port an t physicalphenomenon. the crea tio nofvortici tyat the body contouranditstrans port a t ion alongthebody surface mustbe modeledcorr ectl y. Manyworkersintroduce dcirculationat the separationpoints that weredeterminedusing boundary-layercalculations. exp e r- imentalinformation.etc.Theydetermined the rateof creationofvorticityfrom the kinematic condition

(1.19) whereU:t:are the upper andlower speeds on the sides of the separating shear layer and(IiI'/dt)is the rate of change ofcirculation in the shear layer.

12

Another approachtothe creation of vorticity is toint roduce vorticesat the boundarytosat isfy the no-slipcondi t ion at the surface. Chorin employed this approachand presented a schemethat convergedto the solutionof theNavier- Stokesequations .Thethr ee stepsinhis schemewere the creation ofvert ices at the boundary to maintain the no-slip condit ion atthesurface.moveall thevortices with localveloci ties to take- car eof the inviscid partof theequationsof motion.

and finallythe diffusionofthevort icity simulatedby achan gein thecoresize.a, or adding a randomwalk to the position ofthe vort ices .

In both the pointvorte x andthevort ex blob method.the computationalel- ements are movedwithvelocitiestha t are calculated using a Green 'sfunction solutiontothe Poisson equation.If:.Vsuchelements are usedto simulatethe flow, then thenum ber ofopera t ions required. to compu teall the velocit ies isoftheorder of"v"'l.Thisis quite costlyin termsofthe compu ta tionaltimetha t acomp ut er spends simulat ingaflow. However.there are waysroun d this problemandthe computa tional timecan be brought down by combining a numberof element sill a given region into arepr ese nt a tive computationalelement. This givesrise toa differentclass of methodsfor simulatingflows. calledtheVert ex-In-Cell methods.

Another way, in which the numberof computationaloperations can bereduced.is tha t employedby Yeung et al.[25].They used. thefast adaptive multipolealgo- rithm for particlesimulations ofCarrieret al.[26)whichreduces thecomputa tional efforttothe orderofN.

1.2.3 The Vortex-In-Cell Method

Inthis method, a mixed Eulerian-Lagrangianapproachisused.The Lagrangian treatmentof the vorticityisretained. and thePoissonequ a t ion forthevelocity

13

fieldissolvedin a fixedEulerianmesh..Ascompar ed tothe previousrwo met hods wheretheopera tio ncoun t was oftheorderof.\"2.whereX wasthenumberof comourat ionalelem entsused.the operationcount hereisoftheorderof.\IlogJ!

~..here ),1isthetot al numberofgrid pointsinthe mesh.An addit ional number of stepsof theoeder ofNtha t are requiredareforthe generat ionof the mesh..-alues forthevorti city fromthe Lagrangian representa tionandtheinterpolation of the v·elocit ies from themeshbackintothe Lagrangian points.The number ofmesh points..\J,depends ontheproblemtobe solved.

Everyelement. i.e. everyvortex.lies in a particular cell of themesh. Its circulat ionis dist rib utedontoeachofthe fourcorn ers( nodes ) ofthatcell according totheareaweigh tingscheme

Figur e1.1:Area WeightingScheme.

14

(1.20)

where

r

~<>rt~Z'is the circulationof the vortex and.-iT=

Et=1

.-LA:..-1"being the area of a region as shownin figure 1.1. The Poisson equationforthe stream function .

(1.21)

is then solved.Oncethe stream functionisknown atallof the mesh points,then thev-elocit iesat themesh points can be computed using the central difference method forexam p le.

Wi.j+ l - Wij_t Ui.j=- - 2- h --

I.I;J= _li"+( j~W;_I,;

(1.22)

11.23)

wherehis the meshspacing. Once again,the area weightingschemeisemployed to determine the veloci tyof a vortexthat iscontai nedin a cell of themesh as givenby

(1.24)

As can be seen from these formulae,the vorticity distributiononto the mesh and the interpolationofthe nodal velocitiesrequires a fixed numberof operations per vortex givingan operationcount ofthe orderofNper time step. Christiansen {27}

and Baker [28Jused the \fIC method and reported some interestingresults . Chris- tiansen, for example,showed. that the vorticesthat have the same circulation pre-

15

cess about each other.Withtime . they eithe r coalesceor they do notdepend in g on theirinitialseparation.

TheVIC met hodalsosuffers from numerical errors as do othermethods. In additiontotheerrors that comeupinthe ..rortex blobmethod.thedistribu tio n ofthevorticity onto the meshnodes and theinterpo lationof thenodalvelocities to obtai nvortex..-elocities have errorsassociated. with them.Errorsalsocreepin duetothe differencin g proceduresusedtosolve the Poissonequa t iontodete rmine themesh velocities.However.resultsofnum ericalexperiments intwodimensions have indi cated thattheseerrors do not seriouslyaffect;the large-scalefeat ures.

Ha ckney etaI.[29]proposedanim provement to the\lCme t hod.They suggested usingnine surrounding nodes to distrib utethe vo rticityonto and using theirnin e nodalval uesoftheveloc ity tointerpo la te forthe calculatio nofthevelocityofa vort ex.inst eadof usin gthe nearestsurro unding four nodesonly.Theysta tethat the mesherro rs are reduced by two ordersof magnitude by this technique.

Thismethodisfurtherexplain edinfull detailinchapter 3. Itisvalidatedusing existing results fortheflow pas tacircular cylinder. Numerical experim ents for bodiesotherthana circularcylinde rwerecarriedout using this method coupled wit h conform altransfo rmations.The resultsare presented in chapt er4.

1. 3 Summary

Forincompress ibleBows, knowledg eof the vorticity distributionallows us to de- terminetheveloci ty field.Vort icitymoveswit hthelocal velocityintheinvisc id motionof a fiuid. Vo rt ex methodspresent a numericalalgorithmfor flowsim u- la tion based. on thesefacts. They offeranumberof advantages overthe Eulerian

16

schemes.Butevery advantage has a drawback associa ted withit.The userof a particular meth odhas to makea.judicious choiceofamethod exploiting the advantages andemployin gmeanstocircumvent the disad vantagesof a particular method.Someof theseareworth mentioning onceagain.

Vort exmet hods requireonly asm allnumbe rofstorageloca tionsbecauserom- putationalelementsarerequiredonly inthe rotational partsofthe Bow. The associa teddisadvantage isthat the numberofopera tio ns pertime stepispro- portional tothesquare of thenum ber ofvorte xelements orcoordinatesin the discretisation.Thisleads to a rapid increasein thecomput a tion time as the num- berofcomput ationalelementsisincre ased.Thebou nd ary condi tionat infinity can besatisfiedvery accurately,whereas the no-slip condi tionat the body sur- face requires somecare.Inthe caseoftur bulentflowsandlaminar flows athigh Reynoldsnumbers.fine-scalestructures may develop in anintermittentman ne r throughouttheflowfield.Vortexmethodsallow such regionstodevelopbyalocal concent ration ofcompu t a tio nal elements.This however leadstothe requirement of remeshing which.in turn.givesriseto other nume rical difficul ti es andapproxi- mations.However,theadvan tages of thevortexm-~thodshave madethempopular forsimula ting flews.:\Ian y workershave proposed improvementstominim izetheir disadvantagesand they arestill being used andwor ked0'0.

17

Chapter 2

Point Vortex Method(Clements)

Inthisstudy,a modelfor the two dimensionalinviscid Bow nearthe wakeof a bluffbody isconst ru ct ed. Thismodel is used inst udiesof vortex sheddingfrom bodies with well definedseparation points(sbarp edges),such as keels.wings.and fins.The applica ti ons ofthe method include maacevring ofcrafts.roU-d.amping of ships.etc.Discret epoint vortic es areusedforthe representationof theconti nuo us vortexsheets ..-\Sthemodelisbasedon inviscid Bow.itdoes nottakecare ofthe possible diffusion ofvorticityinthe flow.Therefore.itcannotbeusedtostudy flowsthat aresignific an tly affectedbyviscous diffusionmechanisms(forexample flowsatlow Reynolds numbers).The bluff bodyusedhere has sharpcorners.and thus .fixed separat ionpoints.Such a shape is chosentoobvia te the need (orany assumptions of the position of the separation points.

2.1 The Model

The bluffbodyis aplan e-basedtwodimensionalbodywithright angles between the sidesand therear face.The flowisassumed to separate at thecome rsof

18

thebodyand remainattached. on thesidefaces. The shearlayers shed.fromthe comers.i.e..the separationpoints.are approxi ma ted.byarraysofline vortices.

Thevelocityofa par ticu larvortexin thefiowiscalculated.byaddi ngthe two dimensionalirrotationalpotentialflow around the body andthevelocity induced.

at thepositi onoftha tvo rt ex byalltheot hervortices present in the flow.To obtai n these velocities.a Schwartz-Christoffeltransform a tionisusedtoproj ec t theexterior regionof thebody. which is assumed.toextendtoinfinity ups tr eam.

into anup per halfplanewi th the boundary ofthebody mapped alongthereal axis.This transformation . givenby.

: =

-(2',/<I[ " n- '(.I)

+

.1(1-

,1')"'1

(2.1)

: =

(2',/<)( il n[i.l+(l-

,1' )'''1- ,1(1 - ,1')"'1

(2.2)

transformsthe rearcorners of thebody:

=

±isinto thepoints .A

=

1=1 as shown infigures2.1 and 2.2. The irrotational flowusedhastwo compo nents. They are showninfigure2.3. Thefirst componen tisa flow~;thvelocity(lto. O) far upstr eamand downstreamandthesecondhas a velocity(0.-pUo) at:=0and 1r1J-+0 asIii-+00.Thisisacirculatoryflowin the region ofthe base andis used tocreatetheinitialasymmetricdisturbancewhenebeflowis startedfrom rest.

HencepwastakentobeO.lsin (¥ )fort<3 andzero fort~3.The flowstarts impulsivelyfromrestand initiallydevelopssymmetrically. Theintroductionof asmal lasym me tricdisturbanceresultsinan asymmetric interac tionoftheshear layers which amplifiesinto steadyvortex-sheddingmot ion.

19

*

,,--~ 8 • £-00

__ n " :...;r

. 1-The physicala-plane.

Figure2..

-, ,

. . Thetrans formed >'-plane.

Figure 2.2.

. twoflows in thezand the A-planes[30].

Figure2.3:The

20

2.2 D erivations

2.2.1 ComplexPot ent ial

Let the complexpotentialduetothe firstflow in thetransformedplane be ofthe fermi-

12.3) Fromthe relationship

(2.4) anddifferentia tingequation 2.3with respectto A. weget

(2.5) and equation2.1gives

(2.6) Therefore.

(2.7)

Tofindthe constant Allwe makeuse of the fact thatatz=00thevelocityisCIa andparallel tothe x-axis.Now,as>.-+00, UI-+[la.andVI-+O. Theabove equation,thus.gives

(2.8)

21

and finallysubstituting the valueofAl obtain ed backint o equation2.3gives

12.9) Now.fe e thecom plex potentialduetotheseco nd flow in thetran sfo rmedplan e beofthe forme-

(2.10) Sothe conjugateofthe veloci tydueto;"'2can be writtenas

Atthe origin.U2=0andva=-Uop.Therefore,bysubstitutingthis condit ionin equa tion2,11.we getA2as

12.12) andso

(2.13) The poten tialdue to a pointvortex of str engthkjatpositionAjisgivenby

(2.14)

Tosatisfythe conditionof zero flow acrossthebody, we introduceimage point vortices relative to the q-a.xis.Theseim agevort icesin {<0 havest re ngt hs equal in magnitude butoppositein sign. tothose in {>O.Thepotentialdue to theseis given by

22

;..Ij('\)=

it;

^{log('\ -Xj) (2.15)}

whereXjis the conjugateof'\i'Summingupthepotentialsdue tothetwo flows and tha t dueto point vortices.the total complex potentialat a position,\ in the transformedplaneis givenby

~(.\)=~, (.\ ) +~,(.\)+

I > ,(A)

+L:~;(.\ ) (2.16)

, ,

i.e.,

2.2.2 Vortex Velocities

Since the complexpotential.w

=

(j)

+iw.

where¢istheveloci ty potent ialand ""

isthe stream function.is a confo rm alinvariantandthe flow field due to a vortex trans formsintothat dueto a vortexofequal strengthatthetransform ofthe vortex position,the velocityatany point in the phys icalplanecanbe obtained by transforming thepositions ofallthe vorticesinto the,\-plan e. determiningthe velocity atthe transformofthe requir ed pointand thenreturning this velocityto the physicalplan eusingthe rela t ionship

viz)- ivlz)

= Tz ⁼ ~~.\) £

^(2.18)

Differenti a ting equation 2.17withresp ectto ,\ andsu bst itutinginequation2.18.

weget theconjug a te ofthevelocity at a locationZ;as

23

12.19)

where>';isthetransformed positionofthelocation=,.

Routh'sla w isthen appliedinthecase ofa point thatis alsoavortexposit ion. Sincethecomplex potential is transformelinvariant.we have

ik ik

..;~J: )-2;/og(:- :tl

=;.;.\,(..\) -

~log(>.-Ad (2.20)

where;.;"J.and....,\,are the pot entialsat=1and.\1duetoallcausesexcep tthe vort ex ofstr engthkat=1'Therefor e wecanwri te

:..1%1(: )=...\I().)-

~l09(~ = ~11 1

Expandi ng).==Ie:)inaTaylorseriesabout':1.

12.21}

A-A,=(,-=,)1'(, )

+ ~(, - = ,)'!" (=,)

+0(,-

' ,J'

(2.23) Substit ut ing thisinto eq uafio u2.2 1. we get

24

and

~

_dz^..=dw."_{dJ.. d}

~

^_

~ [\1

^0('"+0(,-

'oJ ]

^(2.25)

z 2:r f'{:d+0(=-=d Then as;;...=1and .A-+

.x"

12.26)

This givesthe expressionfortheconjugate velocityata vortexposition=1in thephys icalplanewitha modificationduetoRouth'scorrection. Csing(dlL' l d>") obtained from equation2.1i.this canbe written for a vertex position:,ingeneral

~I ..

From equation2.1.we have

Therefore.

and so

r (z ) izA 2/,(z )

=

8,(1_ A' )3/2

25

(2.28)

(2.29)

(2.301

(2.3 1)

Ncn-dimensionalisingwith respecttosanderasuch that

kj

= [.~~s: :;' =

;:u'

= ~:I:' = t;:;.! = s~o

we get

~I

^{dz' ".}=u~-it..·: ^{1 [., .}

f<', _{,, ~ 1}

-(l->'~)lr:! ^I•-Ip- 16(1-An

+ T

8Ai -Xi

- L

i"Ft

~-.-1

8>..; -

-1

A) ^(2.32)

2.2.3 Vortex Stren gths

As canbeseen from equation2.32,z

=

±is .i.e.A

=

=fl,are singu lari ties ofthe trans form at ion whichca us e infinitevelocity atthese points. Therefore.in-order to calcula te the velocityof a newly crea ted.vort ex .the velocityis calcul a ted. at thepoint :;'

=

±(l

+

eliand has a strengthIi:'equalto ElU~26t,where

l/ ;

isthe velocityat the point c'=±(1+e )i,6tis thetime step andthesum m a tio n is carried.

over all the time st eps sincethe last vortexwasintrod uced.The parameter,e.is a verysmal lnum ber usedto approximatethe boundarylayerthickn ess.However the vorticesare stillint ro d uced at::!=±i.the actualseparationpoints.Thisis becauseifthe fiow were a realviscous one.thevelocityat thesepar a tion points would be zero to satisfy theno-slipconditionandtheveloci ty withwhich the newly createdvortex leavesthe separationpointwould bethe velocity

U ;

attheouter edgeofthe boundary layer. The vortexstrengthscan be easily derived.wit hthe help offigur e 2.4.

The rate ofchange ofcirculationatpointBis given by 26

F' - --- - E(O.1+') dy

G ~x

- -""':;'- --1qO.-I)

r - - - - -

H(O.- I - ' )

Figure2.4:Derivation orvortexstrength

(2.33)

whereCisthe contourBEFG .ApplyingGreen'stheoremandtakinglJt:/8rto beverysmallintheboundary layer.weget

(2.34) (2.35) (2.36) (2.37)

Ifthe vortices are shed at intervals or

o t.

then the:strength ofanascentvortex that comes off point 8is

27

(2.38) Sim ilarly.thestrengthofavertexcomin goff' pointCisgivenby

C2.39)

2.2.4 VortexPositions

Theentire simulationiscarried outin the transformedplane.The non-dimensional ..-eloc ities in this plane are found using thefollowingchain-rule

C2.<0)

Once theveloci tiesarefoun d atallthevortex positions.the vorticesaremo v-ed accordingto therelationships

(

^{d~' )}

,'Ct+6')

="(')

+R.

df

(')6t

(

^d~')

«('+ 6t)=(Ct)+lm

df

^(t)6.

(2.<1)

(2.42)

During thetime developmen tofthe system,somevortices were found to ap- proach too close to tberear faceof the body and this caused them to havevery high velocitiesalongthebodyas a result of the closeness to the singleimage vorticesin the lower halfA- plac e.Toavoid this. absorptio nof tbe vortices thatapproached closertban0.058 to therearface was carried out by removing those vortices from the calculations.Otherfeatures of the model thatwrereusedbyClements[?]were notfoundtobenecessary to be incorporated due to the muchfaste rcomput ers

28

available today . Thesefeat ures includ eclustering ofthevortices into an equivalent single vortexofstrengthequal to thesum ofthe individual vortexstrengthsand positron as thepositionofthe centerof vorticityofthe clusterand also the shed- ding of thevorticesat integral multiples of timestepsins tead of sheddingvort ices ateverytimestep.Atime stepof6t

=

0.2was used forallcalculatio ns.

2.3 Sample Calculations

Acomputer program uathe C programminglangu age) waswrittentosimula te vertexsheddingfromthe bluffbodydescribedearlier.Shownin figures2.5and 2.6 are thepositions of the vorticesattimest=5 andt

=

10..Alsoshownisthe plot of therateof change ofcirculat ionatthe cornersof thebase ofthe bodyin figure2.7.

r.s

l ~ :"",,,,: : ^...

0..3 * ...

st» 0 ...: ,...

~:: e-CJ) -

-1.5 I

fromupp~cornl!t' ,.

ttomloWl!t' comer G

-2L_---L_ _'--_--'-_---'_ _---'-_----l

-r

Figure 2.5:Posit ionofthevortices att

=

5.

29

1.5 fromuppercomer.-.

I fromlo....tercorner 0

1

~~ .

^{..•• • '• •}

0..5

**~*~: ..\ _{~ o o " ~~} .. ..

yis 0

:*_*:*-::"';:'o" ~,, ;-o"~ - - -

a,

-0.5 ,,"" 0 " ""

-11---

"~""o,,:o ",, : :

^{0" 00}

·1.5

_,'-_--J._ _...L._ _- ' -_ ---'_ _- - ' - _- - - ' -I

Figure2.6:Position ofthe vortices att

=

10.

1.4r--t-r-- , - ---,- - r- , --r-r- -r-'- , - - - , - - - - , 1.2

atuppercomer- atlowerccenere - e - e -

o'---'-_L.-..J....-l._-'---l._- ' -- ' -_ L...J

0 10 1 5 2 0 2 5 30 3 5 -10 4 5 5 0

t

Figur e2.7:Rateofchangeofcircula tion at thecorners.

30

Chapter 3

The Vortex-In-Cell Method

3.1 Intro duction

The methoddescribed inthis chapte risthe Vortex- In -Ce lltvlC)met hod de velo ped byChristi~sen[27].Thismet hod has. since its int roductionby Christiansen.been continuously improved.to makeitmore accurate and econom icalwith respect to computer tim e.In fact,this met hoditselfisanimprovedversio noftheRandom Vor tex11ethod( RV)"I)ofCha rin[221.Oneoftheadvan tagesthat the\rlC method has overtheRv:'lmethod is thatthevelocity fieldclose to thebody is bet t er represent ed.This is because amesh,thatiscoincide n twit hthesurfaceof the body.

is usedwhich enab lesthesurfaceboundarycondi t io ns to be satisfied more precisely.

Theotheradvant ageis thatthe computationtimeis reducedtoO{Jllo gM

+

.:V) from;.y'lper timestep,wherelit!is thenumberof grid points andNis the number of computatio nalelements. Also, to simulatethell.ow moreaccurately,alargenumb er of point vorti ces isrequir ed.This can be efficientlyhandled by the vortex-in-ce ll method.To further redu ce thecomputationaltime, an absorp tion procedure is used.Inthisproced ure, thevortices thatcrossthe body conto urdue to therandom

31

walk are remo vedfromthe calculations.New particles.fewerinnum ber thanthose enteringthebody.are introducedintothe flow.Theyhave the circulationrequired tocom pensa te for that lost duetothe removal ofthe vortices thatenter the surface.

Thisvortex:method is stable.easy to set up.and accurate in predictingthedetailed and highlytransie nt f1.owstru ct ures which occurin the flow around a bluffbody.

3 .2 Theo ry

In this chapter,the theory for simulatingf1.ows around thesection of a circ ular cylinderisexplained. It can be furtherapplied to other cross-sectionshapes,like thatof a flatplate.a cylin derwith afin. and a boat-sectionwith a skeg using conformaltransform a tions.This is explainedin thenextchapter .

The Navier-Stokesequationsand thecontinuityequation governthe flow of a viscous.incompressibleand Newtonianfluid past a circ ularcylinder. These equationsin non-dimension alform. are.-

and

V.iI = O

(3. 1)

(3.2)

whereiIis theflow velocity,Pis the pressure,Reisthe Reynoldsnumber,andt istime.Thesequan ti tiesare obtainedbynon -di m ensio nalis ing withres pec t to the cylinde r radius(a),theunifortaflow speed(U ),and the density of the flui d(pl.

32

whereu",P".and t· are thedimensionalvelocity.press ur e.and timeres pectively.

However.tonumericall y simulatethe flow. the Nav-ier-Stokesequationscan be writtenintheform ofthe Poiss on equationforthestream function.e.

/3.3)

and thetransportequationforthecompone ntof vorticitynormaltothe two- dime nsionalplan eof flow.;.,.;.

The vorti cityisgivenby

oJ

= 1 '7

x

UJ k

(3.4)

(3.5)

The f1.ow is simulat edin apolar coordinat esyste m(r.O)in which the cylinde r surfaceisdefinedbyr=1. The boundaryconditio nsforequation3.3.imm edia tely afterthestart of the f1.ow(t>0), are

'"

an d 8tP

a;:

^-sinO

forr $ l (3.6)

(3. 7)

Thebound ary conditionsforequa tion3.4are

and

ii.iJ

=

⁰

w -+ 0

33

forr

s

1 as r e-ecc

(3.8) (3.9)

The boundary conditionsgivenby equations 3.6 and 3.8 ensurethe velocitywithin thecylindersurfaceis zeroand thecond it ions givenby equations3.7and 3.9 imply that thevort icity andthe perturbation tha titca uses inthe flow.atlarge distances fromthecylinde rsurfac e. are small .Theveloci tyi1can becomput ed fromthe finitedifference form ofthe equation

u=

~?:!!._{}?:!!._r80

ar

^{(3. 10)}

where rand {} are theunit vectorsin the radial andthe azimuthaldirecti ons.

respecti....ely.

3.3 The M ethod

The vorticitydistributioninthe flow. givenby.

,,(x ,y)

= to

^{r,e(x -xde (y-yd (3. 11)}

is representedby N point vortices. Here riisthe circulat ion of theill>vortex.(Xi, Yi) its position.and 6 isthe Dirac delta function.The computationaldomai n is defined by am x n(mistypically100 or 200 andnis129)exponential mesh defined over an annular region,1<r<ro,together with a modified polarcoordin a tesyst em (r', 9)(see below) in which the meshsizeisuniform.In this syst em . the Poisson equation for the stream function .equation 3.3. canbe written as

a'~ (,) a'", b(,)a~

,

802+ar iJr''/.+ r a;:;=-wr

34

(3. 12)

a(o-')

(rTr)'

b(r' ) (dr')_{rJ; +r dr},,pr'

2

an d B(e" ...-1)+1 (3.13)

Thisequationissolvedusingthe finitedifference methodanda.fastFourie r transformforthe azimut hal9-direction. A FORTRA:'J 77packagecalledFISH- PACK(Version 3.1,October1980),developedby John Adams.Pa ulSwarz tra uber and RolandSweet.The:'-lat ionalCen terforAtmospheric Research.BoulderCol- orado.US.A.was used. Both-...-r2andTilare expandedasFourier series in 8 and substi t ut ed intothe finitediffere nceanalogu e of equat ion3.12.This gives a set of tridiagonalsimult an eousequa tions foreach harmonicamplitudeof1Pwhich are solvedbytheGauss elimination method.The coordi natesystem(r,8)issuch that

r' = j 8 = i!i.8

O$j-cm

0::;;i<n 2rr/ (n- 1)

The coefficients.-landBin equation3.13are foundby solvingthe simultaneous equa tions obtainedbytakingthe valueof the oute r radiusTOtobeTfor

r

⁼m-1 in equation3.13andtheradi al mesh spacingat the cylindersurface.(r r'=1 - 1).

tobeequal to~.Here.IIis thekinematicviscosityof thell.uid and~tisthe timestep.Itisimportant tha troischosensuch thatthe boundarycondit ions.

givenbyequations3.7and3.9.hold . Avalue ofro

=

100 wasusedforallthe

simulations.

•-\.n area-weight ing scheme.illustratedinsection 1.3.3.isusedto distribu t e the.circula tioncarried.byeach vortex onto thefourcomernodes of thecell in whichthatparticular vortexiscontai ned.Having do nethat.ifrei,i)isthe total circul ation associated with thenode(i.j).theval ueof:.: atthatnod eisgivenby

.,(i.j)

=

^r{i.j)

~I

^(3.1')

r~8dr""""1

To modelthe action ofviscosityata solidboundary,newvort icesarecreated alongr=1at eachtimeste p withcirculationsuchthattheboundarycondition onthe tangential com po nentofvelocity.equation3.8.issatisfied.The additi onal circul ation.r(i,0),tha tmustbe int rodu ced at every nod e onthe surface.isgiven by

[(i.O)=

~8"'(i. 0)5;;1 - ..

^{- r(i.O) (3.15)}

~ow.since:fiJ

=

0alongr

=

^I.the finite difference form ofequatio n3.12reduces to

",(i.O)=- .(Oll,, (i. l )- 2" (i.0)+

"r;.

-111-

~b(OIl"(i. l l

^{-"(i.-I )) (3.16)}

Sett ing thestreamfunc tio n tobezeroinsideandon thesurfa ce ofthecylinde r.

we get

w(i.O)=-[.(0)

+ ~b(O)I"'(i. l )

^(3.17)

and['(i.O)isthecircula tiondistributed onto the mesh.alongthebody conto ur.

fromtheold vortices.Sincethevo rt icesaregeneratedalleverthe body conto ur.

36

we donot needto determinetheposition of the separation points.Theadditional circulationisshared equallyamong the newly createdvort ices. nu in number.at eachnode.An impulsive startca uses a large amount of circulationto becreated at each of thesurface nodes atthe first time-step.Therefore.toreduce the circulation carried by the initial vortices.a total of twentynewvorti ces were created atthe first time stepand threeat everysubsequenttime step.

All the vortices.those that are already presentin the flow togetherwiththose tha t have just been introducedonthe bodysurface. arethen moved usingthe OperatorSplittingTecluli.q ue(OST)of Chorin[23]. Equation 3.4 is splitinto a non-linear Eulerconvecti onequation

[%710

=-(uS').- and a linear diffusion equation

(3.1 8)

(3.19) Equation3.18issolvedbyconvecting the vortices.Theyare moved. with velocities givenby the finitedifferenceform of equation 3.10 using the nodal valuesof the stream functionofthe cellcontainingthe vortex. A second orderRunge-Kutta methodis applied.Thepositionof a vortexafterdispla cem entdueto convection from posit ionrjis given by

(3.20) where

d~ u(rj.t)At

d-;

u(rj+d~,t+~t)At 37

.-\second solution of thePoisson equationisrequired forthecalc ulati on ofd~.

For thesolutio n ofequa t ion 3.19.the vorti ces are diffusedbyaddinga random walkto the positio nof thevortices.If~X,Ji//and~Ytii/Iare two randomnumb e rs selected from a normal distribution withazero meanand a variance of2v 6t . then the position of avort ex after diffusionfromthe positi on(ZI.Y I)isgi'venby

(3.2 1) Since the effect of diffusionissimulat ed by adding a random walktothe posit ion of the vort icesusing random numbers,variablessuch as pressureforces, vorticity, str eamfunc ti on,and velocityetc. all haveasmall randomcomponent associa ted with them.Thisrandom com po nentgrowswith

.j

^f)"t/Re.Inother words,itde- creases as the Reynolds number increases sincethe randomwalkbecomessm aller.

During the processof convection. someofthe vortices mayenterthecylinder surface. Thesevort ices are reflected back intothe flow aboutthecylindersurface.

Furthermore,as we deal withoscillatory flows,man y vorticestend toremainclose tothebodycontourdueto the reversal ofthe flow.Therefore. theremay be some vortices thatent er thebod yduetothe diffusion process,i.e.,the random walk.

These vortices ,which enter the body,are coalesced atthe nearest nodesofthe contour mesh. New vorticesarethenre-injectedintothe fluid at aradialdist an ce, chosen from the same distribution of random numbers as for the diffusion process, fromthe cylinder surface. Thisleads toa considerable reduct ioninthe numberof vort ices present inthe flow,ena bling longerflowsimulati onsin time.

38

3.4 Pre s sures , Fo r ce Coefficient s and Vorticity

3.4. 1 Pr essur es

Itisknowntha t thevelocity and accelera tionareequal tozero atr

=

1. Therefore. equation3.1 canbesimpli fied and writtenas

-

~VP+2Re-h~'2jj

^{=0 (3.22)}

~Vp =

2R,-IV' U

=

2R,-IIV(V.Uj - t'xVxUJ (3.23) Fromthecont inuity equation,we have V.ii.=O.We alsohave;,j

=

IVx

iiik.

Subs t it utin gthese in equation3.23,weget

(3.24)

Taki ng ascalae productofeachsideofthe aboveequation with

8

gives

(3.25)

Theright hand sideofequati on 3.25representsthefluxof circulation perunit length acrossthecylinder surface.modelledoverone timeincrementbythe creation ofcirculationr(i,O)atthe surfacenodes.Thisis givenbyequation3.15.Hence, atanyinstant oftime,thepressure dist ribu tionaround thesurfaceofthecylinder can be obtained byintegratingthe pressure increme ntsalongtheminorarcs, (i- 1{2)6.8<0<(i

+

1{2 )t:.O,wh.ich'",given by

t:.P,=2f(i,O){t:.t

39

(3.26)

3.4 .2 ForceCoefficients

Thedragand liftforces are representedbytheir respectiveforce coefficients.CD andCL.Thesearecalculated by integratingthe pressuredistributionobtai ned from equat ion 3.26 .They are given by

(3.27)

(3.28) and

<l0

'28 [

i ] C,= --

E

sin UM )

E

r(i.O)

atJ=O ••1

Alternat ively,the force coefficients canbefoun d from the general form ulafor the force acting on rigid bodies in incompressible flow given byQuartepe lle&c Na po lit ano[30J.Thisrequiresthe knowledge ofthe entirevort icity field.For our case of acylinder inflow ataReynolds number above 1000.theformul a simplifies to the followingint egralexpression for thedrag forceFD.

whereAisthearea and

FD

= L

^dA{iix .:i).V '1:

'1r=rfr²

(3.29)

(3.30)

Using equation3.11todiscret izethe vorticityfield.equation 3.29givesthefollow- ing expressionsforthedrag andthelift coefficients

and

CD

= i: r,

[u;sin (20;)~^{.,OOS(20,) ]}

;-1 •

40

(3. 31)

c,

^=_

i: r.

[t/.Sin(29i )~IL;COS(28.)]

.=1 •

(3.32) wherer;istbe position ofthe ithvertexandU;andt:;areits '..elocities parallel to tbeflowand at right anglestoit.respectively..-\comparison betweenthe time histories of the drag coefficient calculated.from the integrationof pressures. equa- tion3.27 and thatobtainedfromthe formula given by Quanepell e&:::iapolitano.

equation3.31.isshewninfigure3.1.

3.4.3 Vorticity

Since thearea-weightin g schemesmoothes thevort ici ty distri bution along thecylin- decsurface.thesurfacevort icity:';0(8)cannotbe taken tobethenodalvaluesof:J alongr=1.Thus.the surface vorticity isfound by making use ofthenodalvalues ...(i,1), byusing aTaylor series expansionof ....(r).The formulaforthe surface vo rt iciry distributionis

. . ReL(i,O)iT

I

t..'o(,.<l9)~;';(I,l)+~~,...~

3.5 Some Numerical Results

(3.33)

.-\computerprogram (intheCprogramming languagewith callstostand ardFOR - TR.A..~

n

subroutines)wasdeveloped incorporatin g the abovemethod . Some flow simul a tions werecarri ed out.Shown below ace theplot sof the time historyof the dragcoefficien t for a submergedcircular cylinderin asteadyflow(figure 3.1),sur - fac evort icity distributions atRe

=

lOOO{6gur es 3.2,3.3, and 3.4),surfac e pressure distrib ut ionsatRe=1000 (figures 3.5,3.6.and 3.7),andthe radialvelocity on

41

the symmetryaxisbehind thecylinder(figu.res 3.8and3.9).

iJlt~ol~_

~kN&pOlie- .

Smith" Stauby

,JHI1.c

.,

.,' - - _ - - - '_ _---'-_ _--'-_ _--L._ _..J o

Figure3.1:Time history ofthe dragcoefficient atRe

=

1000.

As canbeseen from figure 3.1.there exis tsa goodagreementbetweenthe time historiesofthedragcoefficientcalculated fromequa tions 3.27(integra tionof pressures)and 3.31(Qua.rtepelle&:Nepclitanc).However,a random componentis associated with theCDcalculatedfrom equat ion 3.27.Shown here arealsothere- sults of Smithk Stansby(311forcomparison.)'Iore numerical experiments showed thatthe agreement did Dot significantl ydependonthe Reynoldsaumberfgrea ter than 10(0).Itimprovedbydecreasing the time-s tep,.It.

Infigures 3.2 -3.4over leaf, the randomcompo nen t in the vort icity distribution isquitelarge.This can bereducedbyaveragi ng over a numberof simulations or increasingn". The randomcomponentthatis associatedwith the surfacepres~

suredist rib utio nsin..figures 3.5 - 3.7decreasedwith an incre aseinthe Reynolds numbersin cethe randomwalkbecame smaller.[twasalsofound that changing TOcaused insignificant differences.Soavalue ofTO=100waschosenand used for

42

allsubsequentsimulations.The curves in figures 3.8and 3.9 were found to be

Figure3.2:Surface vorticity distribution.t=0.3.

Figure 3.3:Surface vorticitydistribution.t

=

0.6.

Figur e 3.4:Surfacevorticitydistribu tion.t

=

0.9.

43

.

^·2

: [S2J

P .J

...

·So20.l(I60 80 100 120loW160 180 9

Figure 3.5:Surfacepressuredistribut ion.t

=

0.3.

. _., : [S2j "".... ^s....

^by0

P .J

...

.so20.w60 80 10012t1140 160 180 9

Figure3.6:Surface pressure distribution.t

=

0.6.

. :[S2J

^{·2 .}

P .J

...

-so20 40 60 80 100120 140160ISO 9

Figure 3.7:Surfacepress ure distribution ,t=0.9.

Figure 3.8:Radialvelocity on the x-axis behindthe cylinder,Re

=

3000.

-L

2.4 2.2

1.6 1.8

r- L.4 1.2

-1.5L_--'-_ --'-_ -'-_ ---" ' -_ -'-_ --'-_ ---'----l 1

Figure 3.9:Radialvelocity on thes-axisbehind the cylinder,Re

=

9500.

45

sensitivetothevalueofn...They improvedwithan increaseinn...Forthecurves shown in figures3.8 and 3.9.n;=5was used.

The results presentedin figures 3.1 -3.9arein very good agreementwiththose presented in the paperbySmith andStansby(311,some ofwhichare shown in figures3.1. 3.3 and 3.6forco m pari so n.Thisprovidedus with a good valida tion of oursim ula tion program.On the basis ofthis.simula tionsfor Sowspast other bodi es.of cross-sect io nshap esot herthantha tof acylinder.were carriedoutwith the use ofconfo rmaltransformations.Thisisprese nt ed inthe next chap ter.

46

Chapter 4

Results using Conformal Mapping

4.1 Int r o duction

Themethod for the numericalsimulat ionof flows past a circular cylinder was expl ai nedinthe previous chapter.To study flows past bodies that haveothercross- sectionalshapes,the physicalplane containingthebody sectionismap pedinto a plane cont aining a unitcircle, whichcorres pondstotheconto urof the physical body.Thisis done by the use of conformal transformations.Generally,for the geomet ry of some sections tha t we are interested in, thetransfo rm a t ion of the bod y contourontotheunitcircleisa two-stepprocedure.First.anysharp comers, that the bodymay have,areremovedbysuccessive applicationsof the Karman-Trefftz transformation.Thisisreferred.toas the CornerRemoving Procedure.The second part is to mapthe body, with thesharpcornersremoved . onto aunitcircle via theTheodorsen-Garricktransformation.All the calculationsare performedinthe transformedplanecontainingtheunit circle.This renderstheproblemas being

47