On the Motion of a Vortex Ring and a Vortex Pair : Have we had a centurial misunderstanding?: University of the Ryukyus Repository

Title

On the Motion of a Vortex Ring and a Vortex Pair : Have we

_{had a centurial misunderstanding?}

Author(s)

Nagai, Minoru; Ameku, Kazumasa

Citation

琉球大学工学部紀要(66): 1-5

Issue Date

2004-03

URL

http://hdl.handle.net/20.500.12000/1454

BulLFacultyofEngineering・Univ、ｏｆｔｈｅＲｙｕｋｙｕｓＮｏ､６６，２００４

ＯｎｔｈｅＭｏｔｉｏｎｏｆａＶＯｒｔｅｘＲｉｎｇａｎｄａＶＯrtexPair＊

－Havewehadacenturialmisunderstanding？－

MinoruNAGAI*＊andKazumasaAIVmKU*＊

FacultyofEngineering,UniversityoftheRyukyus

Nishihara,Okinawa903-O213,Japan

dFnagai＠tcc・u-ryukyu・acwjp

Abstract・Tbexaminethcengmeeringapplicationofanartificialvortexring,theinducedvelocity

fleldaroundavortexrmghasbeenstudiedwiththepotentialflowtheory・Asresults，sincethe

inducedvelocitybecomeszerofarapartfiPomthevortexring,ｔｈｅｒｉｎｇｄｏｅｓｎｏｔｍｏｖｅｂｙｉｔｓｏｗｎ

ｉｎｄｕcedflowfield・Inothcrwords,avortexringdoesnotnecessarilymoveatthespeedasusuallyit

mighthavebeenmisundcrstood・

InthispapeBthefIowfieldinducedbyavortexringandthensupemnposingvariousstrength

unifbrmflowsonthefieldarecaIculatedanddiscussed・Ｔｈｉｓｐａｐｅｒａｌｓｏｃｌｅａｒｓｔｈａｔｔｗｏｐａrallel

slraightvortexlinesncighbormgtoeachotherdonotnecessarilyrotateorgoparallelwiththe

velocities,whichmayhavebeenmisunderstoodfbrIongtime．

Keywords：PotentialflowthemybVbrtexring,VOrtexpair,Inducedvelocity,Vbrtexmotion．

､０２：lenglh丘omeachvortexlme R：mdiusofvorteming U：unifblmflowvelociw Ubn：criticalvelociW xbXz：Cartesiancoordinates XYIpositionofvortexpair zmD:cylindricalcoordinates

ZR：positionofvortco[ring

(GにeklctteTs） ｃ：mdiusofaciIcularvortexco祀 り：strcamfimction r：cirmlation 入：＝(好功)/(cZ?+功） 兀：mtioofthccimumfbImcc 1.ｎｖＴＲＯＤｕｃｎＯＮ ＯｎｅｏｆｌｈｅａｕｔｈｏＩｓｈaslccentlyinspiⅡ巴dmeusageofartificial

voltexnngfbrvariousfluidmachmeTydevices(1)．Artificial

vorteKringmayseTveasanewwmdfmbrventilation､Ｗｈｅｎｉｔ ｉｓｓｃｔａｔｍｅｏｕｔｌｅｔｏｆａｄｉｍｓｅＷorte9Kringmightimpmvethe di節sereHiciencyatconsiderHbにrate．

Befb”Uleapplicationoflheartificialvortexring，authors

邸ａｍｍｃｄｌｈｅｉｎｄｕｃＣｄｆｌｏｗｆｉｅｌｄｂｙａｖｏｒtcxringwithUle

potentialflowlheoryThen,apmblemofthenecessalyfbTeto

fixthevoltcxlingtothespacewasmmedouLHowlaIgefbICe

wouldbeneededto5xlheringartificiallymtatingamundits

circumfbにntialvortexline？InotherwoIdsbifmevortexringis

notHxedtospacebmenngshallmovemewayjustasany

conventionaltextbookhastaughtusfbrlongtime． 2.THEORYANDDISCUSSION 21CaseofastationaIyvortexring

Inafluiddynamicstextbook，motionsoftwopamUelstmight

vortexlinesarealsodcscribedasiflheyaｪ℃moved応Iativelyto

eachotheTatlhevelociwuniquelydecidedbytheir”lative

mcngthofvortices・ｍ１ｈｉｓｐａｐ露aumors，calculationsand

discussionshavefbcussedthesebasicprOblemsofvortexnngs andvoltexpalms．

PotentialflowfieldamundaspacefixedvortexringisObtamed

asmEquation(1)andmFigu泥1．

v'一訂露梺`｡

‐表oMpw-厚M

DL-Zハ川鼠-2沢…]'’

準鶉鋤ルーzﾊnＷ]',

(1) ＮＯＭＥＮＣＬＡｒＵＲＥ

ｄＺ２：minimum,maximumlenglhfomvoItexring

D：lengthhOmvortexnng E：secondcompleteeUipticintegral F：filstcomplctecllipticintegml

Figuにlshowsstr麺nlmesmlhecentmlplancmcludmgIhc

centlalflowaxis・Ａｓｓｅｅｎ節mEquationlandlhefigU鹿,the

qowvclocityfarapartmmlhevortexringbecomeszcm,ｓｏthat

*Received:June20,2003．PrCsentedatUle5lhJSMBKSME

FluidsEngineemlgConfbrenccOWagoya,2002-11）

*＊DeparhnemofMeChanicalSystemsEngineering

ＮＡＧＡＩ・ＡＭＥＫＵ：ＯｎｔｈｅＭｏｔｉｏｎｏｆａＶｏｒｔｅｘＲｉｎｇａｎｄａＶｏrtexPair ２ enti定Howfieldisatl己sttothevorte9[1ing・II1olherwolds山e vortexringisfixedtoUlespace，evidentlybecausethe calculationhasbeenpracticedabout`hspacefixedvort甑Iing'１ lhisspeedas‘℃ritical，,、Ｉｔｉｓｃｌｅａｴcdthatifthesuperlmposmg unifbmlflowislargerlhanUlecriticalspeedlheouternowcan gothmughthemsideofthevmtexring・Ｉｆｔｈｅｓｐｅｅｄｉｓｗｅｅｋ，

outerHowcannotgommUghmthelingandlhefluidaroundthe

２

。』，｝／

､(Iiiiii).~〆ｆＬ．

~上←／１

_；：

：！

｢~￣い、､；

■(轍■↑､<■

0．１５１（

vortexringcannotgooutmmUlenearfield ０．１ m．ｉ

ｄｏ５１

_{….…-,←.:．…} 。､.Ｉ

罫輿､＿

星 ２ 皇 １ ０ 0 ０ 麹〆 ／: 。‐。｡?$O････ゲル･･･い・・・ｏ・」・ ０．０５：

／i

０．１ Ｉ ２ ２－１o1 Z FiglStreamlineamundastationalyvortexling （R=1,1=1,Ｕ=O） ２ Ｚ －２‐１o2 Z Fig､2S口已amlinesaJoundaslowb/movingvortexring （R=1,1=1,0=0.125） ThemducedflowspeedatlhecenterofUlengulCiscalculatedas fbUowmgEquation(2).Wemaycallthisspeedaslhecritical speed,bBcauseUlisspeedmaybecomcthekcyspeedmlhecase Whendlevortexringismoving”IativelytotheSpace． （CriticalSpeed）（２）

。-孟一ひ“

Inthcnonnalizedflowfield(R=1,吟1）inFigul巳１，thecritical speediscalculatedas0.5. ２ ～ 0 Z2Casesofseveralmovingvortexlings FigurB2showsthesupenmposingofaunifbnnflowonthe voTtcxringinducingflowindicatedinFig1rＣｌ,andtheunifb[m flowstr己ngUlisonefburlhoflhecTiticalSpeedLe,0.125.WhiIe thefigul巳ishPacedonthecoordinatesfixedtomevorte,、ing,ｗｅ ｃａｎｓｅｅｔｈｅｆｉgur巳ａｓｄｌｅｆｌｏｗａｍｕｎｄａｓｌｏｗｌｙｍｏｖｍｇｖｏｒtex ringlBlativeitolhespace,Inlhiscase,theflowaccompanied wiIhthevort甑ringischcuIatmgonlyinsideofasphericalarea， OnehastonoticethatitisnotsocalledHiU，ssphericalvortex flowbbecausemtheHill，svortexthevorticesarcunifbnnly distributedinthesphericalarBa． Ｚ -２ ‐】 0 ２ Fig3StIEamlinesamundavortexring movmgwiUlcriticalspeed （R=1,1=1,ひ=o,5） Figu照４ｉｓｔｈｅｈｉｌｄｃａｓｅｏｆｌｈｅｆｌｏｗｗｈｅＴＣ１heunifbrmflow strBngUlistwiceofmecriticalspeed・Itisthecaseof値stmoving vortexTing応lativelytolheSpace,Asdescribed曲oveblheouter flowcangothmughlhemsideoflhevortexnng･Thefhmiliar flowflcldaroundlhecigalBttesmokelingmaycmcSpondｔｏ lhisfigu妃． Figurc3showssccondcaseofamovingvortexring,inwhich thesupenmposmgunifbrmHowstrmglhisexactlylhecritical・ nefigul巳isobservedbylhecooldinatesfixedtothevolte9mng lhatismovingatmecriticalspeedr巳lativelytolheSpaca EvidcnUybBowspeedatlheccnterofthefiguにｂｅｃｏｍｅｓｚｅｍ， sincetheunifbrmnowandthevortexinducingflowaIc cancelledeachotheratUlecenteTbmUliscase,theouterflowcan nevergothJDughorpenetmteUleinnerfieIdofdlevortexring ThesupenmposmgunifbrmｆｌｏｗｉｓＩｈｅｓａｍｅＳｐｅｅｄｂｕｔｈａｓ ｏｐｐｏｓｉに。i応ctiontothemducedveloci坪Solhat,ｗｅｍａｙｃａＩｌ ComparingFiguにｓ２，３ａｎｄ４，itiscleamedthatonlytherelative movlngspeedtolhecritical印eedWhichislhefimctionofthe ci1℃ulationandmeradiusoflhevorte9〔ｒing，decideslheflow fields． 「 …－０．８．１－ -０．６ ： －－．-０．４－－ －－０．８…↑ ←=q-P￣P￣ -－．－１．…＿ ￣ げ一十⑭--0?

￣貢～～－＝－－－

－寸ＯＬ_ 0｡Ⅱ｡ －－－．－０－.雷一・，‐‐

BulLFacultyofEngineering，Univ・ｏｆｔｈｅＲｙｕｋｙｕｓＮｏ､６６，２００４ ３

U－２☆-会一喝にiiucaIspeed）⑥

Ｚ ～ ２

｡！

､､、０

／０

~…７

× 州／ 」Ｐ ’〆 。〆 〆 ｉｆ，』 デ ’ノノ》 〆 β 坤ノグ ウ ヴ ゴ ヰ〆〃｝ １，ｉ１１．３４．１……４．．…‐．．『 ０ 0 -1 ２ －２－１o _ｚ２ Fig.４StlEamlinesaroundafiLstmovingvort甑【ing （R=1,1=1,吟1） -２－２-10l2 Z

Fig､５s甘CamlinesamundastationaIyvortｅｘｐａｉｒ

（X=1,-1,1=1,ひ=O）

PmfbssorBahchelorShowedUlesamedIccflowfieldsinduced

byvortcxringsinhistextbook(2),howevablheywe『edescnbed

asthepossibilitiesoftheflowfield・Heindicatedlhatthe

di碇冗nccsoftheflowpattemmnonglhesｅＨｇｕ”ｓ８ｒｅｏｎｌｙ

ｃｏｍｅｆｏｍＵｌｅＣｈａｍｃｔｅｒｏfvortexringsnotcomemm血eoutcr

nowsh巳､9.1．ＨｅａｌｓｏｔａｕｇｈｔＵｌａｔｍｅｏｕｔｅｒｎｕｉｄｃｏuldgo

lhroughthemsideofvolte9【ringonb/whmthemtioofstrengUl

ofvoIteo【E1amenttoringladiusissmallerU1anacriticalvalue、

TheOIcticaUybheanalyzcdselfinducedmovingspeedofavolteo【

Imgappmximateb/aslhefbUowingEquaUon9),whereemeans

lhesmallmdiusoflhe``tubevortices,，．

Ｕ一念鰯(÷ノ

（４）

Acmally,BatChelorandanyolherpmfbssorsdidnotdiscusslhe

瓢penmposmgunifbrmnowsonavorte9mingmducingnow

field,solhatbdleyseemnottoaccepMhemolionofvolt邸ring

wilharbimlyvelociWmhefluｉｄ５Ｕｅｄｓｐａｃｅａｔｌ巳sL

Furthem1ore,ithasbeenundcrBtoodfbrIongｔｉｍｅＵ】atdlespeed

ofselfmovingvorte9mngcouldnotcalculated，otherwisebc

infinite，becauseoflhemaUlematicalsingularibノofIhering

519ment 2.4Casesofsevelalmovingvorte0[pahs

Figul巳６showsnowneldamundaslowlymovingvortexpahblt

isaspecialcaseofUleflowwhendlcmperimposedunifbnn

flowspeedisonefburlhofdlecriticalSpeed・Inlhiscase,dle

nowaccompaniedwilhlhevorteo(pairiscilpulatingonMna

nearchcularcylindricalzone、OnemightcaUitachCUlar

cylindricalvort邸nowlikeasHill，svortexsphe妃．

２ × １ -１ 2.3Casesofastationalyvolteo【pa1r

Equanon(5)andFigure5describelheflowficIdinducedbya

vorteoKpa1rbapairofpamUel副mightvortexlinesSincetheflow

vclociWElr8parttheorigmbecomeszemandthevoItexnngis

fixcdtolhespaceblhevortexpajrdoesnotmovcldativeb/tothe

Spacc､ＩｎｄｌｉｓｃａＳｃ,lhecTiticalouterunifbrmflowisobminedas

mEquation(6)．

妙--釜'w笏汁圭如吃ノ

（５） ２ －２－１・０１

_Ｚ２

Fig.６s賑amlinesamundaslowlymovmgvorteo[ｐair

（X=1,-1,Jと1,Ｕ＝UbrWVノ

Figul己７showsUlecriticalflowfield8mundamovmgvorte9【

painTheouterunifbnnnowandlhevortexinducingnowis

cancelledeachotheratlhecentemfthefiguに.Ifd1eouterflow

stJmgUlorlhemovmgspeedofthevort甑pairisequalorundeT

1hecriticalspeed，theouternownevergoeslhmughlhefield

areabetweenlhevort甑Iines．

‐‐圭鰄(芸Ｉ

耽小mxハ(＞]'ﾉ,]'’

ＮＡＧＡＩ・ＡＭＥＫＵ：ＯｎｔｈｅＭｏｔｉｏｎｏｆａＶｏｒｔｅｘＲｉｎｇａｎｄａＶｏrtexPair ４

InJapanesetextbook,ProfbssorTatsuminfbrexampleshowcd

U】esamestrBamlinesofFigurB6whα】avortexpa1rmovesatdlc speedofonefburlhofUleauthorscoldcriticalspeed，because

lhatisIhemduccdspecdusuallyunderstood,Tntsumisaidthat

Ulefigur己isobsewedbythecoordinatefixedtｏｔｈｅｖorte9(par justlikeasPrandtltoldThatis甘ueinUlecaseofslowlymovmg

voItexpairs,howevaiiflhecaseofstationaIyvortexpairs,lhe

flowfieldbecomestothat8sinFigu定S1tmightbedeclamed

matthesetwoorfburfigul巳sareenti妃1ydiHbTmtflowfieldsto

eachothe凪 × １ ３.DISCUSSIONandCONCLUSSION Itisclea配dlhatanyvolte9（ringsoranyvortem（pahBdonot

necessaｴilymovebyIheirownmducmgflowfie1.s,andlhat,d1e

flowpattemisdecidedbylheldativestrengUlsofouterunifblm

nowandvortices，inotherwords，宛lativemovingspeedof

voTtexrings/Pahstod1ecriticalspecd･Sofanalmostall

textbooksoffluiddynamicshavedescribedasifvort｡(Iingsand

voltexpairsshouldmoveａｔａｆｉｘｅｄｓｐｅｅｄｂｙＵｌｃｉｒｏｗｎｓｅｌｆｏｒ

ｍumalmducingflowfields． -１ ･２ Ｚ 0 -１ -２ Ｚ

Fig.７Str巳amlinesarcundavortexpair

movingwidlcriticalspeed （X=1,-1,Jと1,ひ=〔ん減ノ

TIleaumo盃doacceptU1epossibnityofmovementsofavoltex

nngmdpahHowcv甑1heydonotagrBelheconceptU1ateveTy

vortexIingandpairShouldmoveato町onefixcdvelociWasso

farunderstood・TheTBmighthavebeenacrucialmi副ｍｄ幹

standingonfluiddynamicsduringacenturialage､Oneoflhe

reasonsofmisunderstandingmaycomemmasimpleconfhsmg

abouMherdativeObservationsystemsofnowamundamovmg

bodyForthepmof；ｉｔｉｓｅａｓｙｔｏｓａｙｌｈａｔｕｓｕａｌｌｙｗｅｃａｎ

ｕndeIstanddledi碇冗nceofflowfieldsbetweenastationaJy

double‘ｉ､e､，sourCbandsink，andamovingdoubletWhiCh

becomesaflowaroundamovingcimularcylindricalbody．

２ 一

一・・…・…・・・

－－１￣ × －０．８ －－０．６ －０．４ ＝-０．２ －０．１ －－０ －．０．１ １ 0

[jEL墓二三

-１

ThcotherにasonofmisunderstBndingmi8htcomc廿０ｍ⑪e

mathematicalsingulariticsofvoltexfilaments・Itmaybesaid

matsupenmposingofanyflowsjustonlyonapointof

singularityismcanmgless,becauselhesingulaJitydoesnotbe

affbctedbyUlesupenmposition・Solhat,calculatedwell-known

innnitivespeedofvortexlingfilamentandlheonefblmhof

criticalspeedinlhecaseofvmtexpairfilamentsmaybejudged

simultaneouslytobemeaningless． －２

ｚＺ

￣Ｚ￣１０１

Ｆｉｇ．８ｓtｪcamlinesamunda値stmovingvortexpa1r

α=11~1,1=1,Ｕ=ＺＵｉａ"ノ

Fi創施８showsanolherspecialcaseoflheflowwhenlheouter

unifblmnowhaslhestlCnglhtwiceUlecriticalSpeedlnUliS

case,theomerflowcangolhroughmemnerm巳abetwcentwo

vmtexlineswiUlthccenterSpeedjustsameasmecriticalsPeed

butoppositeflowdiにction.Ｏbviouslyb1hisisdlecaseofafilst

movingvolteKparimhefIuidfiUedspaceatlpst．

Aquestionmaybeleft,saybmamass-IikcprObIemofvortex

nngandvortexlines､niactualphysicalpmblems,tomadoesor

q/phoonsfbrexample,twovorticesmaybemtelactedbyeaCh

othenandusuallylheweekvortexshallbeinfluencedmuch

mol巳thanlhesmngvortices･HoweveEhowcouldwedccide

Uleｍａｓｓofvolte9(ringfi1amentorlinefilaments?Zemmassand

mfinitiveSpeedisthusUlemathematicalsingularity・AnyvoTte9［

doesnotbeaflbctedbyanysupeIpositionandmaymoveatany

arbimIyspeeddecidedbylheotherphysicalboundariesor

initialconditions・Ｔ１１ｅｆｌｏｗｐａｔｔｅｍｓｏｆＦｉｇｍｃ１ｔｏＦｉｇUｒＢ８ａI℃

concludedasUledi匪祀ntcasesofastationaIyand/ormovlng

vort甑nngandavoTtcxpainAnyvortexnngsandpairsshould

ProfbssorPrandtlseemedtohaveanotherunders位ｎｄｍｇ曲out

lheseHows・Ｉｎｈｉｓｔｅｘｔｂｏｏｋｐ),Prandtltoldlhatthedi歴Imce

betweenFigu応sandFigu虚６onbcameECmlhedi錠rcnceof

Obscwationsystemsandsｏｍａｔｔｗｏｎｏｗｓｗ巳rp1hesame・Ｈｃ

ｓｅｅｍｓｔｏｈａｖｃｓａｍｃｍｍｍｄｅｒｓｔｆｍdingwilhBatcheloEThey

mighthaveaconvictiomhatanyvortexringsandanyvoltc9K

pairsshouldmovebytheirowninducingflowfields．

BulLFacultyofEngineering，Ｕｎｉｖ・ｏｆｔｈｅＲｙｕｋｙｕｓＮｏ､６６，２００４ ５

notnecessariIymoveatthefixedspeedlhatmightbe

misunde応toodfbrlongtime． authorsattheirtimeofCaIifbmialnStituteofTbchnologybhas scntauthorsmirlypositivecommen侭.Hesaid`UnIessl己alfluid effectsarcincludedanddueconsideranonisgiventothefbICes lEquiredtomove(orkecpstationaIy)thevoltices,aUsoIutions a1℃possible､TT1elCqui1℃dfb1℃esmayormaynotbefbasibleto aPply，，． FinallybitmustbenoticedonceagainthaMheabovecalculations anddiscussionsarcrEstrictedtotheinviscidsteadypotentiaI flowtheoIics・IteDKcludesU1eefIbctofviscositiesofactualnuid， norUlegenemtion，st2bUityanddecayprocessesofaphysical voIt欧ｒｉｎｇｏｒｐａｍＡｓｌｈｅｍａｔｔｃｒｏｆｆ嵐ct，ＰIcfbssorBachelor discussedthestdDiliwofalammarvoTtexIingandstudiedthe vortexringgenemtioninthewakesbehindblufrbodiesatla【ge

Reynoldsnumberc).Then,heconcludedd1epossibilitiesofIhe

stablelammarvoItexringsasdescIibedmrBgammgFigul巳ｓ２，３ and40ntheotherhand,ourdiscussionsandthefindmgsaIBm thepuldymviscidmathematicalpmbIemsoflheidealnow fieldsbutwilhsinguIaJitiesofvoltexHlaments． PmfbssorA1iOgUt,atRochesterlnstimteofTbchnologyandthe

ChiefOIganizerofthe41hASR胆』JSMEJomtFluidEngineeTing

Confbrence,planedandcaIIedfbrcollabomtionswithaulhorson apanclconcemingthemovingvorticesmtheConfbrencc、 Howeventheplanbasbcenpostponedtolhcnextchanceofthe Confbrence． Autho応expresstheirsincereapp1℃ciationstoalldiscusselBof Iheseacademicfieldsofnuiddynamicsandhopetohaveanother chancctodiscussomheprObleminnearfnturc． SincethetopicsmightbeconcemedwidlacmciayccnturiaI misunderstandingonfluiddynamics,aulho応hopefhlly”quest soonerinspectionofthealticleandlhenrevisionsoftextbooks andtheothewortcxconnectingpapeTsonnuiddbmamics． REFERENCES [1]Nagai，Ｍ､，VbrtexRmg,JapanPatentAppliedNo2001‐ 125003,20014(InJapanesc)． [2]Bachelo囚Ｇ､Ｋ，AnlnhuductiontoFluidDynamics， CambridgeUniv,囲已ssb19674,525． [3]PrZmdtl，Ｌ､，Fi1hrerdu1℃hdieSUtmungslehne，Friedlb Vicwcg＆Sohn,1141,72. M]Tntsumi,正,FluidDynamics,Baifh-kan，1,82斗，１１９（In Japanese） [5]BachelonGK.,Aproposalconceminglaminarwakcsbehind blufrbodiesatlalgeReynoldsnumbeTblFluidMeCh、1956, 388-398． [6】SharifTICandLeonardA,VbrtexRings,AnnualRevicwsof FluidMechanics,24(1992),235-279．

[7]LimmandNickclsTB.,VbrtexRings,FluidVbrtjces,ed、

GrcenSL,1995,95-153. ACKNOWREDGEMENT ProfbssorM.InoueandPmfbssorEmeritusYSenoo,Kyushu University,kindlydiscussedwimauUlo鱈onlhesesomewhat confUsmgbutimportantagqﾕdaandgaveUlemmuchusefUl adviscs・OtherJapaneseprofbssoIsinJSMEandmheSoci幻of FluidMech8nicscouldnotundelsmdlheautholB，ｐｏｍtsof findmgsandgavelhemlhefiBirlyncgativecommcntsProfbssor Max､PlatzeqNavalPostgmduateSchool,hasslmwcdmuch inte配standadvisedaulhorsmol己pmdentinspcctiontolhe

findmgsinhoducinganarticleonVbrtexRings⑥mdabookof

Fluid1brticcsm・PlatzeralsoinmducedcommentsofPmfbssor

SalpkayaatlhesamesChool,ahhoughlhosewelBnotappmval

wilhaumors，ＦｉｎａＩｌＭＤｎＧ､Y8tes,acoUeagueofoneoflhe