DETERMINING FLOW FIELD SINGULARITIES FROM DRIFTER TRAJECTORIES
by
© H almar Halide
A Thesissubmittedto the School of GraduateStudies in partial fulfillmentof therequirements
for thedegree of Master of Science
Departmentof Physic s MemorialUniversity of Newfoundland
February,1992
St.John's Newfound land
111+1
orNahonalCanadaLibr ary Bib liDlhCql.lcfl3lionalc dl,l CanadllCanadian Theses Service Serveedestnesescanacennes
The autho rhasgrantedan irrevoca blenon- exclusivelicence allowing theNationalLibrary ofCanadatoreproduce,loan,disbibuteorsell copies ofhis/h erthesisbyany means andin any Iorrnor format,making thisthesis available tointerestedperso ns.
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L'aut euraeccoroeunelicen ceirrevocableat non exclusiveoerme ttentitlaBiblio l hoq uo nauonare du Canadaderep roduire,prl.ile r.
distribuer au venerades co piesde sauiese de qtrelquemenlereet SOliSqcelqucforme Que ce soil pour mettredes exemotoeoode cettethese3fadisposition oospc ruonocu oteressees.
L'auteurconserveIap-oonetedndroito'outou- quiprotege sa these.Ni1:1these nl des corans subsrannersde ceue-cl no dciv entctrc lmprirnes auautreme ntreprooens sansson autorrseuon.
ISS!; 0-315 -73316 -0
Can ada
Abs tract
sin~lIla ri li('Silllllf'nowIicl.l.Okufoaml Ehlll'SII1" )1'r(1!17Ii)1II111Mulinari andKirwan(1975) <11'\'('101',..1iIn'p;n'Ssiollh.,.,h llitjll,'l h;,lhas 1""'H1ll";1l' I,Ul'
pointedoutthutthisrl'p;rl'1'1'iu ll ll'l'll11i'llIl'iliflludallU'lIl,ull,\'il1iUll"II1i1I.,·I", ca us,'ita1'Sll11WS 11panldip;mwithlilt'Howl'l'ul,r,·Iixrdto tin'n·l1l.rui,1
"r
lilt, drifter cluster.Kir\\'l111dIIf.,(1!Ii'\S)fo rl1lllla tl',1a""ll1li.... lulhis,[i],'lI1lll a hy inverting non-linearl'Ulllli OiIS uhlll.int'dI.)'()klll,,,(1!1711)(ur IIU,l iuli IIt'a r :1TIll'DKran- horizontal ,Iiwrg" llc,',v"rlicity,sln'l,'hill~;III'!sll,'.. rill.&!;tld..r·
maticurete .\\' 1'soh..'tllf'non-linear '''Illatiu ns"fl\ir wlll1,i"I.,(1!J»l)I"
ohtaiuDKPandth.,l)(lI!litiollami n'lorilyuf ..limYfi.,I,1sillj!,lIlitril)'
rn"l1
a!lingl" dr ifh'rtraj<'C~tory, Thissu lllliu lI (llI'lIn ofur l h",111,..1till'01\s..ln lion)is1l1'llh<"llIl\lieallylIIun,t'tJ1lri ""limnlhalprt'''' ·lIlt..l in Kirwillid «L, (198l:l)audeorreetspn-vlouslyIllu[d"dlSIal/!;l'lmlit:"rl'UTSilllIw1J1l1,li.~lw,1 litemturc. [thasLll.'<mI;llcrl'1'sfull y I.l·1'!.".11Isingartilil'i;,tly,!!;f'II,·ra!.",1dal,iI, Thcllw lluxlisfllllllul1ll' lI!.allylilllill·,lolul, lull wTl" jlliT.,rlll·u1.llml.IlKI':,.'"ti1I11~iuvariaut. [L"Isuhustill' IIndl'Sirahlerl~aiuretlla!.ii n-qnin-sfourth orrh-rLinn-,]I' rivat ivl'S or d'lta. to.IIt'WIllt't hod, theliS method, tha t lIllt'll rqr;ft'l<siullwithout ,\rlificia[[y.s d li n ~tht'Ilcwre-ntreto tilt'du s terruut roirl ill11l"1'.SI'lItp,1.11.hllllalso1wI'li11l1f rl'llll{ully il's il'rl llyapplicanou to lll,tifidally
t1rirt.I' r.sillthec:lu.sl.flritft' ]wingmovr-dhythesanu-unique sing ulari tyintill'
I\pplyilll-lallthn ..~Illl'thudsto tlm 'f' lll'ighhnll rillgdrifter tracksuu-asurcd onSuhh-Islll11<1Hank elearlyindicatedthelimitationsof allihrpt'methods.
'1'11('ft'gn'ssioll krhni(1'1I'of Okubo and Ebbesmcyer (1!176),till'08method, 'railf,tl'1ll'l'1l1ts(' i lH'nowcentre-WiLSnot atthe-clustercentroid pos iti on ,The 0[":11Id,lltltlgivl'sambiglllJlls",'silitsillthatitcannotrlistinguish bctwe...-n solidhudyrutatiunabout
, I
pointalit!italuhlhatoscilla tes.Thelackora sin/1,lt,w\'11ddilwclHow('l'u t n-for ellthreedriftertrajectorieswassufficient tut-usnrt-t.lIl'liSmethodgavemcaninglcs»DI(P lhaibedlerg«iutcnnit tcnt IInd, lla lio ns, Never-theless, given lra.jt'('.loril'5near11wcll-dcfluedflowfield singnl'lrily.II't'"fillbl'assuredthaiboth the liSand 01<method canbe!lSi',]10ohtnintIll'Pt>,~itiOIl,velocityandDI\ P
fir
llll'singularit y.])cpcndingupon till'Sl']lflriltiollS!'all'softIll'drifters,theliSmethodceu bemuchlessor1J1on~scnsilin·t. o nojst'th'lll t11l'OK Ilwt.h"d.
iii
Acknowledgements
IutnAr"dlyill,l,·],t,·d 10tln-full"wiliKp,'upl,·wh"Il~~isll'rluu-tliro ugllthis sllldy.
I.lIr, II.(:.SilI"I'·I"S"ll.my slIp,'n'isul",,,,,l lillt'dtlu-IHISi,'i,ll'itlOrlilt' liSIlwl hUtI.Ilr.Sill" I,'rSllllalsoill1.I"(J(lul'l·,1 nu- tof.lu-llSI·fll!tll'SS u(
Illlrl"rs l,illlflillJl;uf1",\\·II! pillilph'n'ofl'virll'llI" Int«itIIlI'illlinp;rlll l'id .UfI·.
_.IkA.EoIL,...-.Ill".1\.L,mb,Dr..1.1l,'lllig.Hr.N.BidlandOr.~1.
~111 1" 1'1l 1l".pr(l\' id,~1sonu-inl,'rpn'llll.iollSand 1'll1lsLru,:lh'"sllAAt'stiollSill onh-rt"lIlak,' till'tll1'~iseh'un-rnndmoreinfonnativr-.
:I.Pro l.Ilr. A.Il. Kirwan .lr. gave hisr-ritica lreview011till'liSmetho d
<111,1:<ll~~,-st. ,'dInt.nn-sunlk-s t\'gan linp;tld.~methodawl]Jrovi,!,·t!hu-
P,-rsIH·,·n ""111111lu'II'",1iUll'l'prr'1. SOHlI' oflllf'n'silits ill terms oftl",
"ulls,'I"\'a liuliofpou-ntialvortlelty,
-J.Albn<:Ililidiup;ils:<istl',1JIll' withthe wmplltin ,!l" syste m,S,Kioroglou
iv
aIKd>l'il.(,11\'1Jp; IIt·ll<"ll',·,ll1l<" 1<1 1'rt'l'a rt· this llwsis.Ila.la! I\ ,1'"11'1,,
!1]'Olllk.)(J,·('rili,!!;f"r "ssisl'l ll1",'iu.!!;illllC'rillJ!;illl'I ,·,Iilill/'. Ilal' l.
;"i.TIle'(;"\" '· 1'11111l'1I1of]11,I.III<'s i" l,ru\"i.I'·11;1.t',,'l lC'rllllssl· I ",I;lr.~"il 'illl,1
I;.~I.\'filmily hytlwirnlillillllWls,'l wo ll n lj.\,'1Iil' 1I111"111"11I/'.,'1 fnlslr" I, ·'!
,111.1hun'd.
Contents
I Introdu ction
2 Culcul a tln g the Propertie sof Flow PiejdSing ularitiesfrom f.heTrajectory ofa SingleParticle:TheOK Method
:~ Tes t.ing OurOK Solutions 31
:1.1 l't ll~'.'il.rl't_l"hill~. ;1:J
:I.:! 1'111'1'Shl'ilrjll~. ,10
:1,:1 1'111""H,lta l iollwithTranslationVelocity ,16 :1.-1 TIll'ElfC'rt
o r
Halulu lll Noiseon lIw01\ Method. 55-I ANew Regr ess ionMethod forCnlculat ingDI(Pfrom a Cbs-
t.er of Drifters: the HS Method 62
vi
5 Tes tingandCcmpa eing TechniquesforObtnin iug DI\PIr-om DrirterTraj ect o r-ies
.'i, 1 Arl,ifi.-ially(;t·lH'rilll,.[ Dilla G.:! TIll'Elrl,,·tofN"i ~,·unIll<'I[SIlll' t h" r!
71
.'1.:1 OI'(';1l1il'lI al il
'i,:I,1 TIll'01\1IH'IIHu[ll '~ 11 l t~
ii,:l.:! TIll'liS111<' I II,uln'!illh~, .",,:1,:1 TIll'OFm"tl")I[rosults .
..IUi
" " " " .1[1 ii,:I..l Dill;,sil11ul ,di ull llsinp;1>1\ 1'u!>l;lilll'llrrulIl l l lt'OI\ " III
6 Sum m a ry 7 Bib liog ra phy
8 AP PEN DIX A
9 AP PENDIX B 10 APPENDIXC 11 APPENDIXD
vii
12K
IIjO
List of Figures
1.1 'l'ho disllla n 'l1ll'lIlofadlls[C'rof;J-d r i fh'r.~ina stetion a tyan- li<'yrlollirerldy.
2.1 (~lit.~~iJi";l li"lIor s;n.e;II]'lrili t·sillthr-pnranu-terspan 'furllw liuI'I!r I'l"IIIt'il )'lidd.Adaptl'.1[rumOkllhu(Hli O).
.1.1 TIll' 1>1\1'l"/lku lall'c1,llsin!( Ol( [ahn vc]iLn(II( IRWt\N[below]
l:j
:l7 sulutions, from trujcr-turieswithpurest retching defonnution.. :}.'i
;1.2 Tilt'pllrlid ,"s lrolj,'rlury,l1JtIthecalculnu-d positionoftlll1
tlow("'nln'fill't.lu-pure stret ching case11singOIC :1.:1TIU'pat Lil'll"Strnjr-ctoryandther-alc nle tcd positionofthe
nul\'"1'111n-furtill'pun's1.rddlingraseusingKIRWAN.... 38
viii
withpure~trt'tdli u~,TIlt' IIppl'r/hl\\'I'r1'101.showsvulucs01,·
tnlnedIl~inl!;tIlt'
o
[, /I, I\{WAN~()l ll l. i ull .:J..'i Tilt, DI\P("~ lr li bl. l'l l,ll~i ll gOK(.l!U)\'p)'1l11 II\ IHW/\N(Iwl" w) solut ions, fromtr ajl'(·toril':i withplln'shear rh-fm-mal.iun.
:l.Ii TIJI'01\ solution for partlrk-tl'llj"doryandposition1,[tIll'
[lowl'{'ll tn'fro m11lmjt,.'L!Jl",\'wit hpure~hl '1ll"i ll:!;.
:I.; 'l'hoI\l RWANsolut ionfIJr parti d,'lrajl'dnry<In.1position"f I:!
theHow rentret"ilkll!'lh ·d Iroru11tl'iljt't'1.orywithpu n 'sllt'ar illp,.·I:l :I.KSwirlanti trenslationn!Vt'l(ldlil',~raklllllLt'llInuu 1.I'ajl'.l" ri.,s
withpurl'shcuring,ll~illg01,(above) 1111.11,IHWAN[lu-low]
solutions.
:l.!l Themnuericullycalculatedtll'Pfor1"11'1'm1.alj,.11withtrans - lationcasell~jllgOKandKIB.WAN.
:1.10TileOK solution for partick- l.mjl'I·l.ol'yaw l pos iliolfoflilt' flow n-ntro nllcillatl 'lifroll) atraji'd-orylhat.l'u l,i l1,I'Sar""II.1 thetrallslatiugflowrcutr«.
:1.11 TIU'1\II/WAN :;ol llt i"ufo l'pilrtirl/'traject oryandpositio lJ or1.I1l'II,,\\,""lit!"!'l';,klllll11'dIrumn trnjoctoryth atro1.Hh' s ilrlllll,d~I Il't.nllls l iltill~How ceut n-.
:\. 11 Swirltun]trnnslu tion"d ud l;I':;ealculnto-lr!'Olllatmjl'l~tory tlwl,r()I, lltf '.~,lrullndittrauslu tiugnowcentre.The to p plotis
sn
rrum01\:solutionandtil('IOWNplotis from Kirwansolution. iiI
;\. l:lTIll'Il I\1'f:ilk lllill.c', l lIs; ngllw O I\method withSINratios(If ri,:1(lup), !'i:l(lIli,ld lc')uud.l:JO{hot.tom],
:\. 1,1TIll'1'1I1'1.irl("sI,rajf'd llryallliculcnla tedposit io nsoftill'llow rr-ntn-Il:<ingtil{'OKmothodwithSINrntiosor.).:3(top),
;1:\(l11;d (111') il1l11 :1:lO[bottom].
"i.I 'I'lli'muuorirully{' ilk llhlk llDl(P ofpU1"I- rotat ion withtrans- lut.ioun lSC'\IsingtIll' liSandth,-DE,,
."1.'2'l 'r<1 j,'doril'~flfthre-epartides withpure rotat ion,l1mHta trans- li11.;U11;Howf'('n1.1'I',TIll' flowf(,l1 t 1"l~trajec t oryhas beenrnlcu- la1.c "lusin!!;llll'liSmet hod .
!'i.:\ TIll'uuuu-rirallynl1,'u l,lll ·dvelocitiesferthl! p1l1"l'rotat;on lI'ill lLruuslaf.irm{·itS I'usinglilt'liSamitilt'OE,
76
78
5.'1 TIll' ilKI'rakul"L,'dIlsin,c; llll'liS111<,1.11,,,1withSI Nrntios"f 5,:1[top],!j:l(lHiddh,) and 1'j:1tI (huUulll).
centrellsingthe US11lt'thudwithSI Nnll,iusofrl.:\(h 'p ).
f):1(Illiddh')al1,1.'}:IO(huLLorn)..
!i.ti TIU' 1)1\:1'culculutcd usiugtlu- liSmdhod withSINn,lj"s"f
generatedusinghdo1"l'.
S,I
s.s
TIll'DI\:Pn,ktdal..·dllsiu/!;lIlt"OK ull'l,hod witll11 lOri,of,11'1 filterhavinga r-ut-oll' fn-qll"lll"yor~rydes/hulll·..'i.!J TileOI\ Pe;lk ul" ll'dusill,c; Lll('01\ 111l'l.lIo,1 withitlUll,unl"1 filterhavingOlcut -ellFn-qm-ncyuf~(:yd.'s/h"Uf.
.'1.10Thefrequency response
or
tile' IOlhurdr-rIUW-P iISSlill,'1'witll"nit-orr
rn'qll.·lll:)'urf.l)'d. '1i/h"'lr..Ii.LlTIll'dngill,!!;dft>dOillil"filu-r.dpositionsliSa n-xnlt
"r
11l'1'- Ionniug lllt' JUtI,01'11,·1'low-pusswithilr-ut-..H Frr-qur-ru-yur f.cyd "4 hOIIl".
xi
!1I
HI
!/.''j
.'l.I1 'I'll" DJ<1'n,slllt inl; fromlilt'Ol( unnlyais uu tlwtrnjer-toryof ',]r.fi'.
.'i.I:lTJ )( ~partidl"s trnjev-uu'y,1I1dnnwcentroIU HlltingIromtill' Ol(nnnlysisonthetrajectory of'dr.u'.
rl.I' 1Thetrnnslatlon vdudti,'sresultingfro m 1.1](' Ol(analysison till'trujcctory of 'dr.G'. .
rl.l .'i'I'he swirlvl'1ul"i til~rt'su ltillgfromthe01\analysis on the tfiljl"'luryof ' d l'.lj'.
!lij
98
fJ9
.100 rl.lli'1'11('!Jositiullurtill'driflt'r f(,liltiw totheHow centre rcsultiug
frumllw 0 1\analysison lht'll'ajl'dOfYIJC'tlr.6' 10 1 .'i.IiThopositions orth(~turec flowcentres resulting from tlw01\
iluill.vsi.s 1/11thotrajectory of'dr.! ",'dr.z",'drti'. .,lOll ii.lli'I'll<'st.l't'trh;ll~undslll'ilrin,l!;delonnation rate calculatedusing
till'liS<Im.!)"sis011thethreedrifters. .,108 .'i.I!1'1'111'vorticityan ddivergencecalculatedusing theHS ana lysis
.. . . . ... . 109 :i.:mTIll'Pill'tit-I.,S' trajectoriesand nowcentreposit ion resulting
{r(JIIItill'I-ISanalysis..
xli
" .110
5.11Till' UKI'Tl'!illlt illl:fromlilt"OEi1llalp,i"until'"dlls k r..
r
.Iriflr rs.
duslr-r of.lriflt'l1I.
ritrlll'll'rsur'dr.li'.
5.1.1'l'hr-1)1(1'n'sll llil\~from llll'01,;ll1al.l'.~is011till'"i' lllll;II"d trejrc tcey,
:;.2.1Tllt'UK!'rl"llullill,l1;rm lllll wIIS ;t1II.lysi""1llIu'lIl1,..·"illlll-
6,I TILl'lilr r-amlillt':lfromlI:m'lilrr lr hilll!:t"i,st·(IIPII" r);111<1llll....•
fromPIl,..'SIWMill,i;[lower].••,
xiii
.II~
.11:\
.l lli
.lli
.11,'\
.I:I~J
List of Tables
:l.l T;thlllil1.l'dkilll '11Wtif'propertiesobtainedusing01\antiKill- WAl\' and thl' TRUE valuesfortl](>purl' rotationwithtrnus-
laLitJll nISI'. 52
35 :1.:1 AVI'tagl~valuesofUI\I' obtainedfromnoisydata. Theaver-
agiup; was done over106points.The signaltonoise ratioof lIlt' rawdatawas fl,:I,!'i:J,5;10fortile'threecasesconsidered. 60
!i.l Kim·tIlil1.k para me terscalculatedbythethreedifferent rneth- lJ(1~forlIle'("itS('ofpure rotat ionwithtransl atingnowcentre. 79
!i.1 Awragl'valuesof l)KP,analysedIlsingtheHS method,ob- l.;lil ll' .1fromnoisydata. The averaging was110l1eover106 puiuts.'1'11(·Si/!:llll1ttllloisl'1'1I1iooflilt'rawdata Wf\.9 5.:l,53, .'i:lllfOI"Hlt' theooras(~considered.
xlv
S.:I Corrrlal itlilt·lJ("lIiri,·llls ..
r
tl... kiJwlllitli'"I",r,llIwl.'T1I1...11\·... ·11ImiMi
ur
tlrirLrTSfrum tht' OK iHlltlYl'is. • •.•.• .IlII SA Co rrel at ionl'udlkil' lllll uf Uwkin,·mal i.· lIam llU'l...." 10... \\"...·11simulated va llll'll," HlLlu-valur-s 1·"kllla ll..1 Inuulilt'Illn ..·
I1ldlLlJ(ls:OI\.IIS.0E. • •• ,1:.11.1
Chapter 1
Iutr od uction
TIlt""I,jl'l·ti\'1·ufthisthl>sisisto<lIIillysclIowIil.'ldllI,y~l1Jclyiur;tlriflt'Ttra- j ...·l.uric'S.Alrllj ,'C,lory illIl IMll1 uf11CIITrt>nLfuilowf' T,allobjl"Cltha t roilowli til<'lVal l'TllMW"IllC'l1lll.\Vr Irt' esp ecially iULl'fl'Sw<1in del.ermini ngDiITl'tI'Il- tial l\ illt'm"t.irPropertie s (DKP ) ofthefluidnowinthevicinityofvdocity :>in,;ula r ili•.,;,the-IILal.;ollarypuin!.:-(relativeto!10me-spllli allyuniformtrans- 1;ll~lIl )illllK'nowIii'll!.IIItwo-di meuslonalllcw,t!lNl.'proper tiesca llcause IIUII,..1I11;.lslIrf;u'c'ttlI){'illrn·"S(·,J/dl~rf'aS{'l.Iin area,rot ated,atrc t.chcdor shean ..1(S,IIIt';I'r , [!J!'i:lji\luli nari and Kirwan,]975; Sanderson,I!JS4).
'l'lu-renre 1I('\'c'Tal rt'HIIUIlSwhy weanalysetlrifl t'Tlrajt-dor;es. Thedis- trihntiu ll"I1ll00tt-rial,I'g,biola,can lK'rnoredir~('llyrela t e dtoLa g rangian
moasun-mcnts uf velocitythan'~ IlI(' ria llllll'nSllr,'I1Il'll ls .Ihi n l' l" Iraj"rt "ri"s rnn coverur01HIrnll~1'of smh-s Indllllill!!:100r!!:l',s\ ',dl'I1,,\\'s;lIl1ll1lt'Susrnh'..,I·
,Ii,'s(PouiainundNlilcr,[!}(om)withhiKl1SI,;,tiOlIt'I'slllllli'll l,SpaliOlIr<,s"I'll1'"1 ofLagrangian terhuiqucsis bct.tr-rthan lIluslEllll'riilll1,'r!LIJiqll"swithtill' posSlhl,'exr-r-pfiouof saldlll,'nhSt'f\'i\ti'JIIs.Spilt.i,d Tl'SOllllj,,"ofL,.c,ran.<:,iilll observationsisil,I"([llah' fur n,mpOlrisl1l1with("I,J.y-n'suh ·illll; 1l1l1lwriralllwd· elsfKirwauetol.,19!1O).
TIlt' DI,('llrt'ofilll"l't 'sLfornI'llrl"lyo[WilSOUs,Flol\"lil,ld ['r" IlI'rlil's"f drifll'rrlutn, sm-hliSDI\I',,' 111lIH' assl l1l l l a ll ~ 1inlo",ldy-n'1'lIh-IIIll.III,,,lloJ..,,fur
!JlIl'l'0St'Sof dilta illt."l"I ,"lat.innj ,'xLrap"l alitlllI1l1d11"1\'!i"I<II'l'l'di l"llou.1)1\I' oftlu-11m"fid e!provide a ,firedlinkIIIdyuumienl fun"'s.1,',,1'r-x.nupk-, b,\' usingilgroupof drift.,'rs,I{''i'tl(Illil )"1I11"1I[;ll<"j"1\', 'rl; " I1I"<'frolllI"!HIIIW 'SIII
pro\'idt'dalinkItl,lyllllllli"ill1ll''',",',''S''SIIItln-/\ [;,skOl lir-nr-n-nt, S,mt!,'r"tlil (1!J8i )in("1"I"I'(1dyunmies viailvortirity"quOItiollundlIlIill.l'."bofIn ,ln'rJ!.tr;, [cctorics.Bower(lfll':l H)"l(lLlllillt'e!till',IY llillllit"~"f!;\rl!."·illilplil'Hli'1IIl''1llil' 'I~
byt'stililatingpotenrinlvurfir.ityFromtIll' lraj'·''1.uri,'s"f111\ FOS lluMsill LIlt'GulfStrorun.'Phi''''f1ualsfollowaui"' I'yl' II01 Isurfa,""til"Ilrli,I,I'a<lllall amiNiilcr(l ImO)ct"plu)'t·dAli ensdrlf'u-rs, sitl.cdlill,·lrac·k",1drift.illl!. l >lif'y s,
Okul",(UI7:'';jh..sS]lUWlIl,hillHI';I' ranpluyan impurtautroll'indeter- lIlillill,l';1.Iw.listri lou1.il/ll Ofl11 il1l'I'i.d illtIW OCl" IIISllC!t iIS i llliltTll lll ll li ltiollllf
?''' ' ' l' l i UlktUIiilll'ilfidldnHvs at tlw ol'('1Hlsllrfacl'(O Il'I'II,I!/IW).
Okll " '1 (1!/7U)illv(~li~al('dLhl'triljl'rlOrips
(,r
pMLi rh'.~tl1II' to.~ itJgll]ilr jl.r sl.rtlc l llrPsilll./ll' lluwHI-f.] with variousllKP,llr-particularlylookedillho w Ilmv li..lll SillAlllilrili r-s;tlrl','kdtliSlll'rs ioll ofIioatahh-Ilar tirl,' sdue1,01111'hu- 1"111"1'.Thb workill1.l'rprd l'dobservationsofn~\·.'rsalandsllppn'ssiollllfLIlt',,1,sl' r\'a1.iul!sof,lrift, 'rduste-rs.Tilebaskun-t.lunlwasdr-wllll' ,·(1hyMolinari all.1Kirwan (1!lirl) iU1I1Okub«andEhht'snwy('r (l976},Okubocdfll"(1!J7{i)
"XI,(' lll lt'dUll'1.1'f'lIl1i111l('toculcnlatc Lngrsuginudeformationsand eddydiffu- si"il it·s,kitu-tuutic\·a rinhlt,.stlwtl'lI11Sr-a horizontalsJlwad ing ofthedl1sto~n;.
"ftill'rl'si,lnlll motion.Kirwan alit!Chaug(1979)considered thedr~Lof l,iil sill ,l';filiI'tosillllp lillp;Irequcncy. SiIlull'I'SUlid.ul.(1988)showedhow pn-vlunsim'l's t.ignLlIfshadcalculated the1I111nlwrufdegrees
o r
freedomill- l'llnl,..ll~'wln-ullll·.\'l'a lt-nl;\ll'd1'1Itly-di lrllsivitit·sfrom residualmotioncausedKil"\\,.111(1!ISS)pointedoutJ,iil~pr<>I.I"I1l~with 1.hl'n'j.\r,.~~iull1111,,1.'11';11'<1- di~lIlillUIC' OEn1<'1.I",,1ofOkuln••1IId E]'I>1':<1II",\'I'I"(1!17li),L,'lll ~nlll:;id"1 11 r1l1slpr ofdrirlf'l"spml",dd,'dillil:;I.ali o lla ryIll1l.kyd uni<'l"ld yi1~:<111 '11'11ill Ii~llrt'1.1.IIpfl',Lltt'initialand luterpositionsoftln-llriflwsan' d"lIuil'd bythelhlSlwd<IIIlIsulid drl'll's,fl'~Il('din ·ly, II Hlld11 an'1.1",l',,~iljllllsuf lilt' 11011'routn- andilllrifL,'rwilhn'Slwd LutIll'dust"r's l'I'IIII·oid.'I'll"hia"
prohh-milll, hi ~Ililrt klllal'"a:<pisdll" 1.0till'f,wl. Ihil1.1.lu'dll~I,l'r'sn'lIl.r"jd docs nutfuilll"idl'withtil!'[low C·I'lI l.rE'ofthc'pcHy.Fill"i, smidl ·arc'ildll~tPI (I~«.:lil,theOE model 1I'Illl111shewI,hil ltlu-eddyIII0I',.swil.li''('ft.i1i ll veloc-itythat equals IIILht~valueof 111<' DI':I'11I11Itil'li,',1hyII''I'h i~isIlu l trm-sill('"weart't[,'aliIlKwltha uon-moviug,',hly,]lllrl.llt'I",llll','~ l.ilu; d,' .1 Dh':P lI:;inj.\ this modelwouldabu !Jt"' UIlW Ioi"s..d.ElIlM.I!.ill.l!.1.I11't'111:<1.< '1' Iln'alIyitHTI' llsil1gIlwill1I0t:;"IVl'Hit'prohh-mJ,l~'lIIl",'IIIl'rlllsL"I'IlIH.\'
«ncounn-ranutln-rhins f.wLol"pwdul't,c1byyd11Irj.\t'I'.:;n,l,'slwar "nusstlll' lluw.Kirwan(HISS)dcmunstr uu-d11,;11,tlll'idltlVt'l"I'~rl,:;:;j"lIl1'dllli'IIU'S.l!.;IVl' flllltlal1wtll,ullyunrclinhh-.-:;l i I I Hl l t':<oftilt' 1l1':1'of[lowli.,I,1:;illWililfili,'s,
Figure 1.1:The displacement ofa cluster of J·driftersin a stationary anti- cycloniceddy.
Kirwa ndal. ,(1984, 1988,1990) found an alternative methodforcalcu- tating DKPfrom driftertrajectories.Theyassumed that drifterswere neara flowfieldsingularity and then invertedthesolutions of Okubo(1970)to cb- tainDKP from thedriftertrajectory.This method(subsequentlycalled the OKmet hod)is an elegantwayof obtainingflowfieldsingularity properties from onedrifter tra jectory. However,previous tests,madeby Sandersonand Goulding (person al communication)on simulate d data,indicated thatthe solutionsreported by Kirwanet ai., (1988)bad errors.Thisisnot surprising in view ofthe algebraic complexityof the inversionof the very non-linear
oquatious"fOkllh" (Will),
Kirwand uf.,(I!lSS)HI1I1p'i,',~uum-"onr l sl'":XIU','Ssi 'lIISfill'1111 's"I'll i " tls.
Havill,!!;I1HlIl t'tlll'SI'('IJrrI'I' t i " lls,tlu- 01,llH'tllo<lisIn;ll'd 1,.1"(1l1i11,ni n.!!;~" 11 orateddatu. By.ulll in ,!!;raudum motion sWI'\,\"<,111.,1"how01\,'slilll at"sIJf dilrl'l"I'l1 t ialkill"l11i1li(~jJnlllI'rt il'suftil"1I001'li"ldlIIi,l\11 1. I",,,If,',,1<',[ I,,"IIII'll suronu-nt('ITOI'.'I'll"01\ 111O'1h"dn-quin-s,[""'l"IlliWlli"lI"fhi~11"nJ, ,~I,ill!"
'!<'ri\'ill i\'l's"ftlu-eh-Hu-rtnlj"d,lJr,'" l>irr','n'lllilltiullofl'xl" 'rilllt'nl" Il,\'"I.
tninrdtlrirt, ,!"lri1j" ,·t"ri,,~is <Illinlll' l"I'l1l.l)'llllis,"prun 'ss.TlolI.~'·(llIsid",',,1.I,' smou lh ingisrequln-d.
t\tu'wl'(',!!;n'S~io"'lH'tl.odis,["1',,1"1)(,,1fut' ('all'lllal.in .l\111\ 1' Fnnn,1,'ift"1
rlllsll'rs,TIJi..;lH'W l"l'p;n'ssioll1.I'1'lIl1il lll"<l1)!'S'lo tsulfl'rr"l1IlI tll"aSSll1l1l' li"lI lIwLtill'o-utn-oftl,,'flow la-hlsillAlllarilJis"uiud <!,'utwlur1,1n-dlls 1I'I
rvntrcid.lus l ",,,lilIlSI~Ul11' tJrlI1l'Il,sSll 1l11,liolls"fUll'()I\11\1'111", 1,via.11,;11 llll' velocity oft,lll~Howceutn-is:;1";lIly, torlos,'tln-sl al.isl irillllun ly si s.WI' :;I1IJSI't[111'lItl y!'l,rl'r thisIIl'W1"l'~rI'ssi"n1.t·,.hllilPW axtill'liSIIwl,h"d,'lln:u,wr- romperotIl<'01\,liS1I11dOEIlIdlulllsilluuuualysisu]""UlII111Il,'r~" I Il'I';II,~1 andn-uluceuniedata.
\\·i1111.II.nmsillt' fllll' [tIlltlwing'1Ill·s titHI.Silll'I'l1l111n'~11Imuyconsis tofrdtlil's ill.;,11s, ·~,lt·s,dolw"r1rirl.l·rslhiLl11I1'lwarh}'me-asureLhcsallWf10lVfield sillglliitrit{!Dil l'illlalys is will pro.,.idt)alliUk rrs tillganSWI'Tillth!'ras t'of l.hWl'drifu-rI.rajl'rllJripsOilSall]t,IslandHank,ScolilU\SIlt']r.
Chapter 2
Calculating the Prop erties of Flow Field Singularities from the Traj ectory of a Single Particle: The OK Method
GClIl1mlly;~purt.ic lu'svelo citywillIll' nlllstilullydlilll~itlp;,'NilIlI<oVC,.;1111'1,·1' theinfluenceof aIlc wliehl.llcwevor,Wl~I'XJ)l~tthatlht'n 'rnil!,liI.Iwpuillts lnthonowfieldwhen"rolutlve to someuuifurmtrunslarlon,tilt'\'u rlllwlll'lI ls
"r
pilltid,''sy"J(),:itywncd dvanishsiul1lllmWolls ly(r..'li nors ky,]!Jfi2;Oku bo , 1!171J).Slid,I",illt.swill1,1',""llt·,1stntionuryorslugulur points in1111~follo wing work.W,'will Cllllsid"rL1liltll lC's,~stationary pointsd" rilwLlu-posltlonofh.11,,11''"'·lIlf(·(tlmtIllilyI)l'trau.•lilting ).NcnrtlH~How centre we'ISS'III1('tlwl til('ve-l"dty"0111h,'I'xpalllll'llas aTayl orseriesintilt'dis\.all((~fromtheIlow ...·lIlr.·.'I'hus, slrflifi l·lIl.ly dOSI'to OwlIowcent re thefirst.orderterms illthe 'I'il,vlureXl'm lsi,,"donunntr-andtlu- flowfield skinematic parameters can be d,'snil lt'd minglirwar vr-locif.ygrlu !i,'nl s.Flownearsiugulnriticsra ilhave a vnrh-tyorkilwllwtirprupNt il'SSIIthal particletrajectories maybesta b le01' 'llIsla!>I.,I.usmulllll'rh lrllilJio llsaboutl1wsing l1lal"ilYposit io n,Wt"willhe n'lIn'n]('dwithtwo-dhm-nsional velorityflelda,wh ichis ronslstentwithlIw '"lllls1.ra inls011till'l1\uti tlll
or
most ronunoulyIIs(',1 tY)ll'S of ucuanlcdriftingFi,l!; Ill"l':,U showsLilt'ronuuoutypesofsiugulurity (Ok u bo, 11l70) : (I) p"illts
or
llh"'I'p;t'n fl ' (eollvl'rg l'tll"l')where 1\11infinite numberofstrea m lines lIwd;(~)line-sof,liw rgl' llrl '(nJl1 Wl'geIH~l")fromwhich all infin itenumberof slrt'HIl11im·sdivl' r~I '(1"01l\'t'f,I!;l')asymptotically ; (3)neu t ralor saddlepoint s when-a nlllpll'o r
Slrt'ilI111i1ll'Smeetandth eot hersconv e rge anddive rgeaSYllIplulkall)';(·1)vurl..l·xI...inl ..1.,," 1which''In·..llllill....f"rlll .·lIil""-s.
s...tII,·
r-xnmpleaof sing••lar itit'SranIN'f"ul1.1illlIl"oo-au.I.im·..[1',,"n'r~"IIl'".'<111 Iw"hsr n ...l parallclI.. 1111'"'1<,>11 ns " n.,.1111ufa h"ri;:ulIl,,1nuw ItI\varol II...
l'O<ISl lill1'(Bjrrknl'l'lrInl.,1!1I1;Nr lllllillU'illl dl'il·rslIlI.I!Hili),TI...voru-x
lH7liHid lilnls,," II(II.,1!17X).
a.~a funct ion uf SPil('t·1111'11munytenus withhi~hurol"rSpillinl,INi Villi\"'"
wouklbo rt"lluirt"ll.lutill'\'irilli l}'uf till' siup;llhnily.h,, \\"'\'t'r,tl...lI" w lid,1 ('aube approxlmatedin u-rms uf line-ar \"I'I'N'il)'~r;"li"lIb,
Asralill~ilr.r;lIl11rllt011tb«Nil\'it'rStukt'S("lillilliulis l'ohuws 111,.1tilt',Iirf,'r·
t'lI<'1'I>rtw('('uv,·IOt'il)'1'111. poinl; au,1 a v,'hN'i1)'ill;1:«"l..,tlOll'"iul a,Iislall'"
faway fromiscales IJrupur liotlallu(I/'I(Srlll'rlJ(l'rilllOll.uVl'juy,I!),'I!J).Thu-, vt'IOf'iIYp;rculi"nts will!'of·..II'itS
r
11",I\ ilwili(llIST,)I,i'Snllll pil,..1"I~·rval i"n s tha1 an' cousistont with"('I,,.-ily p;ri\lli"nlsst'il ljll~liSr-~/"furitwi,I,·r;1I1~1' of oceanic scales. A "uns,'lllll'un'ofthiss"alillp;istll;ILilSr - .
UtII!' v"ludly gradien ts tend lolnlilli1y,Ultlmut clymok-cul..rvis,'usi1.y ,ldirwsiLSII1i1I11'sl possible sealo (ov('rwhichvl'lol"ily 1"1\11va ry lSlIl'lSlitlllially, Sll ...lueityp;T,,,li·10
"lI1.srill 1101.I",nmwluflnlu-intill'oce-an.Antvelocitygradi"lltsdo hcconn- Im'W'1" nsfIw('ollll'sslIml!"I'.Fllrtlll'I'I110 rt',nn-asun-monts<In' seldomnbleto rt'Sl,lv,'llUII'<' 1,1.111Ia uarrcwllaJl,1of allLlu-possih l,'st-ah-sof occanir-motion, TillIsadrift'~I'lmj ' 'i:tor y,forr-xamph-,is a smoothed versionof therealnow.
N, 'v('rUIl'I,'ss, vdndtygrmlil'ntsdue totll1' smallerscale .'dlli"sresolvedwill bl'.l!;n ';,l,'rthanthose duo totill'large rscaloeddios.Wemightexpect,there- furl', 1.0lindIU" ali;wdI"l'giolls ill tll(' flowfi(']dthathavestronggradien ts.For lllos1.lyhi.~luri ndwasons[Okubo,1970)IVI' railthesopointsnowsinguhu'itks '1I1l1"Xpt'c1.that it is SI'llsibl,'todescribellll' Ilowfield uoar singulari tie sas 'luylur s,'ri,'sexpausiunin 1.1'rmsof distanceIrom the singularity.
As,';,rrhfortIlt'frequencyof occurrenc eami properties of oo-anlc flow li" ldsill,l!;lllariti,~istherefore important for two reasons, First,nowfield Sil1,l!;lllal'il,i,'san'likelyl.ol11'OIlrch-arcstmathcmnti-alwindowinto the highly Illlll-liuml' tlymtmirillprocesseslhlltoftencontrolfluidflow. Second,till!
llllrizlllltal distribution
o r
partir.lf'Sorfloatabt escanbe greatlyinfluencedby llrul"'I"1.it'~tlr
any nearbyllowfieldsingularit ies.Third,thcco ncep tofflow lit·llIsjll~l1lal'i l,il'sranbe'usedto interpret nowllddpatternresultingeither frum11l1'solutionsofNitvic'r-Stok, 's1:11,]ccutiuuity !''lUilliollScloseto the sill,l!;lllaril.il·,~orthtl.~l'ohl.ainl',1(rumoxperiuu-uts.ClassiryillgsingularitieslJ
based011ll1n,lsolutions"flilt'NII\'il'r· Stukt'llllll,1nllll,illllitye"flllHli" lIli11<Is n nt101ll·1>t:101l·(Osw;Iolibch,1!I!iS;Hcu.I!l.""".1!1!i!J;ill ,,1I"'rr)'all,lFairli,'.
m-arlIo \\'lit·ld sin,r;1l1;1orili t-swith kuowuHI'I'.W,-tlll'llfull..wKirwou,I fI/., (/!lSI'I)to soh'(·tllt'iuverso pruM"lIl,nann-lyt..filltll )!,\'"r..llowli,·I,1 sillglllal'ilygi Vf>11 11pllI"lir lt,lrajl'elol"YillLlu-virillil.l'of till'sill,ll; III;lril.y.
FollowingOkllllO(1!170)till'lilll'aTwltll'it),li,·ltI ll'· ltrII11,,1\'li,·I,1SiIlAU' laritycan1>1.'t:"xprrssnlIlSili glItt:·Taylor:>I'rit'Sl'X,"IIISiullliS
~ =fl·.r +//!/
~ = t:·~ + (r!l
1:!.11 I:!.:t)
jlt:"l!,It:'l1t
o r
linw(Lr .•"Oustau ls ), (t1l,1ran.ly lin' I"NitiulIsrc·l" ti....·l"tl...singlllitril)"pililit.Noll'l!J,llWI:arl'l1SSl lI lli tlp;Ilhuve:111lLt 1111'sjn~IlIHrily,Iews
12
totheOK!'fl,h,canddas:
fl'=~
b'
= 6;e
C'= ~
d'=
Ii;.
wheretheOKPate:
st,·etching deforma ti oll"ale shearing deformat ion l'ate vorticit y
diver gence
1I=~-~
c=~-~ (2.3)
(2.4)
TheseOKP areoftenmorereadilyrelatedto our usualformulat ionsof dy- naruicalprecesses than thevelocitygradients .The stretchingdeforma tion rate isa measure ofchange ofshape by differentrates of stretchingalon g the xandydirectionswithoutchangeof areaor orientation,Posit iveadescribes astretching inthex directionandshrinking in theydirection ,The shearing deformatio n rate can be relatedto thechangeofshapeand/ ordistort ion of fluidelements as a resultthevelocityvarying ina.directionperpendicularto thevelocity. Thevorticity implies therateof changeillorienta tion without change in area orshape,so itrepresentstherotational nature ofthe singu-
13
lnrlty. TIll'horizoutul.1i\'.'r~t'I l<" ·is11Ill,':os n]"('uftIll'fr.u-f.ionnlral.'of:on-a innt'iISI'(Sawi,'r,l!l."i."i;Okubo, I!J7lJ;Molinar iandKirwan,1!l7r\;Kirwan, 1975),Theabo veDI\I' art'uotind.~ p,' n. I"1I1llfIh.'t'",m lillill..sy sl," 1ll alill,I principalroordineu- systemC1Llllu-III'lim,.1soth.u.llll'slll'arln~.hof..rr1\'lli"n
\'llllisill'sand tilt'Ilowist·ha l'llcl.'r i7,l,dl,yII,,·,.f,miltil<'mi"1l1ati,,"uftill' prin cipalmws(Batd wlor, Ill!i7:S,"lt'i"r,I!JFi:l).
'I'll.'I)], \,<11'\'relau-dto \'ilriolls1',\'Ifl'S
or
sill~lIlarit.ystrllrtllrt';oS show nin Iigu rc2.1.TIle'dillin 'or
<lX,'S.'MIh,'l"t'lilktllotill',.]lill',,,'l(,l'lSli,'1'..,,11'result ingIrcmIlHllliplllllt ill/!;(2,1)and(:!.:!)aswilllu- .lisl' llss"c!m-xt..SI'I1<'lHal,l' tra j ('rturi('s ncar the11011" ("('nll'\,ilfl'drawn uut.lll'Fi,l';lI rt·,TIlt' 1'11111'""r
III<' trajec to ry Ill'p l'ndsuponthert'h.tiv(' illl!'orta l\l'"of sln't.,'hill/!;-slll'ilrilll:,,1<,.formation aurl vorticity[plottedontlll'lJl'llillilll'),m, llllt'.liwr/!;(' II<'I· (I,I" I,l,', 1 Oiltheubsisca].
~ , ,
I, ,'~
/
Figure2.1: Classificationof singularitiesin the paramete rspaceforthe linea r velocityfield.AdaptedfromOkubo(1970).
Defining the differentia loperatorD ( )
= 1;( )
we can rewrite(2.1) and (2.2)as(D-aO)x- bOy=
a
(2.5)(2.6)
Elimin atingxbymultipl ying (2.5)by c' andoperat ing on (2.6) with(D-aO) andad dingthe result ingequationsgives
[D'-(00+,")D+a","-b"d"J y
=
O. (2.7)15
The characteristicequation of(~.i)
hasthe followingroots.
Therootshave been expressedin termsofthe DKPfl,b,C.rlhyusing(~.:I l . Providing thattherootsI'llrlaredistinct, i.e.a1
+
Ii-c1#-
0, till'P;l' IINlIlsolution foryis
(2.!1)
where(:'1andC2arearbitraryconsta nts. Wewill considertill !CIlS~rr=I'~
laterinthischapter.Substitnti ng the solution fory as given by ('l.!I)into (2.6) we obtain the followingsolutionfor;c.
Theinit ialparti cleposition relati vetothesingularityis:
x(t=O)= Xo y(t
=
0)=
Yo.16
Sd till~I=lJIOld~lll,~til ll 1.illl;(2. 1l) nUll(2.12)luto(2.10)lind(2.!J)n-aper-
t·~Xu- }ij (" 'l-fr )
"1-"2
Yo("1-d~)-c~Xo
SIlI,s t i Lllti ll ~(2. 1:1)and(:U,I)inlo(2.9), wege l
(U l) (2.1' 1)
{(f,Xo+ t:Xo-fl }~d'1 + YO}p'~11+{-(IIXn
+
cXo-nYo)q+
}O} C'~1 2~{(~
-(I )Yu+
(/I+
c)XO}C·,1+~{( ~ +
n)Yo-(/1+c)XO}c....1 (XI1(1J+c)+
}IJ(I'_IIHc'l l;,! YO(JJ+
Il)-Xo{b+
c)}cT~r} (2.15)llll f lli ll lrs1. itll lill~(2.1:1)<Hili(2.14)into (2.10) gives
,r(I)
wlu-re
{XII
+
(Ii Xo+
Wo -d-;')'/lcT,1+
{Xo-(IiX o+
bYo-cYO)q) CT21}.,
~{( ~ +
n)Xo+
(Il-c)}'O}CT11+ ~{(~
-11)Xo -(/I_c)YO}CT21 (XII(,)+
II)+
Yo(1l-I~)}e', l-{Xo(/!-]1)+
Yo(b -ene',I}(2.16 )2/,
(2. 17)
Notelha tWI'IrI1V(~expressed(2. 15),(2.16) and (2.17)inter msoftheDK?
rathor-thanvelocitygr adk-nts.
17
The \'11111('O{1'2tOgl·thl'I'wlthtill'di\"'1").!;I'U'·",/ \\·,'n ·l1s<,t1l.y()kll!'"(1!l7U) torli~~sirytilt'nownl,l,l,~illp;lIlilril)"as:ilL\\'ar d/ulIl,\\';1I'l1lm,la l..~acltll.·.ill' wlll'd/uutwanispiral,vortexaudHm-soffum·,'r,!.!.,' I...r-.Oklllo..(1~Ij'U),·all...1 l' till'lIillgula rilypiln mll'l,·...
Kirwand ,d"(I!lHH)illl"t'rl,'dtill'al", \·,'soluti onint,llI'iri1111'1lI1" I..
oht uinthekluemattcprOIW1"li.,,-;"friu,l!;sfrom1.rilj,·cl ori,·s
"r
drift,'l's intill' Gulfo r
M,'xir o,'1'111')" dil'ilh',1IllI'I'd odl.yli,'l,1iutuhilIlSI"li,,"ill,,1swirl rompom-nts.Thiskindoftn-atuu-ntrunI",rOllu,1ill ;,ll11osl'll<'ri,'slll,li,'s,Taylors-ricsexpansionIWilr t.Ill'liiuAlllaritypoilll..II..n-,Kirwan,1'1/"(I!1,'iK) 01110 \\'('<1rho HowIicklsingulutityl.e.Llw rillg"' ·U1.1'..1.0umvr-.
lJ .,. +u.
11." +" .
(:UK) (1, I !JJ
TIH~suh~,'ripts'I'aml,-;(11'liutl''tl"<llls l"l,io ll'and'swirl' ,n's]wr!.iv,'l y.TIll' tl'illlsialiol\component istIll'trenslatioualvl'llll·i1.yoftill'sill~lI lilriLyp"sll,ioll illtlu-vdo!'ityfr-ld.Theswirlvelocity ofa )I"r ti,-ll'lWill'till'llow'·\'Il1.rt' describesrot a tionandothermu1.iu nrel" Liv\'tothetrHtlsla1.ilil!;n-ntn-,,rIIll' flow singu larity, The(li lrl~r{"lti;\1kilWllla tiepurtum-tersruu, l1wrdun' ,Ill'
I,
{d
+
a)J'/'2+
(b - c),II/'1.v. = (b+4r/2+(rl - rt)U/'1..
SIlI,slil.llli llJ.\('l.'2CJ)int o('2,11'1) ;t11l1('J.. 21)iutc(2. 19 )gi\'t'~
('2.:W) (2.21)
IIdf)
"kCI)
Un
+
(Ih+
Ild:r.k/'2+
(bk-cdYk/ 'J.lin
+
(h+
ck):r k!'l.+
(Ilk - flk)Yk/2 (2.22) (2.2:1)Wllt'l'l'tilt' slll.sni ptkillllic1l. lt'sva luescalculatedill rho timeinte rva lIk ::;;
1~'Hl. '1'111't'Xllt('ssiu lIsfor,r kandIlkoh t a illt>llfrom (2.15) lLUd(2. I(j)have tfn-Ionn:
.rd' ) {[X'dIll'
+
lid+
)'d bk-Ck)jChl;(l-fl;l1-[.\'k(l/k-11d
+
Yk(bk -ck)le{r,. (I-tl;l)}/ 'J.11k (2.24),lfk(I) {[Xdh
+
cd+
Yd l1k-lId Je(r,I;(t- ll;})+
[-X k(flk+
Ck)+
Yk(Pk+
ak)]ehlfl- t.)]}/'J.Pk. (2.25).'\."anti\'knrr- tlwrocrdiuatcsofthefluidparcelrelativeto thesingularity pointat. I=
'k.
TIlt'positiunofthepart iclerelat ivetoth esingularityforan-til"tctnl\',·Im'itil's;11.lillil'sIk:51:5'HI'TIn-\'nrinl,[,'slid /).I'dl).
'''k(I),lIdl)allvarywith1 uvr-rtilt'illll'n ';I!h::::I:51.'+1-'l'his \'"ri,,(j"l1
rhurcalltakr-nttlIll'constant overtill'int er val/~:5I :::1"+1 ,1111 1 u ll l~h they11111)' cll'lUgl',
IJr
l'mlfSI',Frumilll,I'n'a llointer val.III;1I111lh,'I'w"rds, lhis assumptionrI'lluin 's tltalf'.,..
1"1', II,b,I',.farl'shjwlydlilllp,ill).{ o,,"rIIn-lilll.·iutervnllu-twccufi xes.
Su bstituting(2,:H)all,l(2.2.'i)iutuC!. 22)!IlIII(2.2:1)lonrl"\'alll"l illp, llwl1l at/=Il" withtlu~subs criptkSUPPrl'sM,d,v,iw s:
U1'+ {(d +ri)[X'1
+
X"-Xu+
.\'/1+nIJ- .'1-
I'U,-1')1+ (IJ-c)[X(b+ d -X(I,+r)+1'1' - \',/+"'1+l'/I)}n "\hr
+
{(b+c)[X/1+
X/I -Xrl+
X/I+nit-I')-l"{"-")1
+((I- (l)[X (b+I:)-X(lJ +r)+Y,,·- l'n
+
1"(,+ )'"111-1,,,
Aft('I'r,lIll'dlillglindn'llrr;lIlgi ligtenus,tilt'a!fm'l'("lll nl illll ,~n-durr-to :llh'+(11 +rl)X
+
(b-r,)I' :lliT+
(b+(:)X+(11-/1 )\','"
(2,2Ii) (:.1.,:.1.7)
X, V,It,b,
r "
rlall,l<lilly two('(llJiLliolls(2.26)and (2.27)presentlyaVilillloll' tilsulw fortln-ru,WI'1I1~'111\lIut ]w r sixequutious.Thes eaddilioflillequations",UlIll'ulllailll'(11Iydilft'f(~lItill tin,l{(2.22)and (2.2:))withrespectLothue.
Tilkiu){tlwiiI's!,deriVit!.iVl~withnspccitotillle of (:.1..22)nud (2.2:1)and 'ISSlJlHin.e;il .~r,l'fOfl 'thatfl,b,c,tl,llr,liT1Ln~cons tant gives:
(d
+
fl)X'+
(b -el y' 2 (b+
(')x'+ ttl-
Il)Y'2 .
Null',in~('lIl'falwu run not l'xpcdfl,b,c,d, UT,liTtoheindependent of tiuu-.Hul WI'i' SSl lllWthatlIw
um«
sta leforthem tothilngl~is long compa rl,(1 I."1.111~till'Hull's,'a]e for;1:,y to chnnge.Th iscondition mus t besilt islie<:1 for ,r,IIslIlli"il'lllly small providing<I,b,r.,dnre1I0 tall identicallyaero. Whether"I'n"I,.I',IIun-sllflir il' lIt1ysma llfor,Lgivendata issomethingtha t musthe
dwc'k,,,1furliSI'''rl of 1Il<'data analysis.HC' rl1lind subsequently , the prime dC'l1uksdilfl'l"l'ulialiullwithn~p('d10 tlme. Substituting(2.24)atul(2.25) ililurlu-iI!lm'\. <" llllltiunll andevaluatingthem att
=
tkgives(/J- c)h { X( b + c)
+
Vp- Va} +J'1{-X(b+ c)+Yf/+Ya }}}/4]1 21orthen~lIhslit lllillgIorauud,I)ll~il1p;(;!,~,1)HIIII(~,~;j ),,111.1,'vil luiol.illl!.1.11<' resultlng equutlcnsall=I k,c;iw s
(If
+
tl)/'+
(1.-ely"2
(b +c);c" + (II - f1)!/'
.,
(d
- 1I)H U, (II+
c)+
Yl'-Y nJ+
22
til<'<I1mVI~"IIUllli"IlS1."':OIllt'
Apl,r,l'ill~tlll'saul!'procodun- torthethirdderi vativewithrC5[wc ltotinu-
-I-(b-f'IHP.'(iI+c)+ Y/I-Ya}
+
t·~ {-X(b+f~)+)'/1-I-Yrl}]}/ '11J (il-I-dr~'+(II-II)Y'"
:l,fl
+
(i""~- -,- -
and
1i,'IP
+
:lli\- - , - - ,
\\'"get
(~.:tll
The above eight equation s,from (2.26) through (2.3 3),areequations(A3) to(AJO) of KirwanefIll., (1988).
Putti ng
r =
(l1+
b1- (:1+
,P,we canrew rite equa tio ns(2.28),(2.29), (:l.ao),(:.!.:ll ),(2.:12),and(2.aa)as2X da.+2Ydb-2Y dc 411.'-X l' (2.2s')
-2 Y da
+
2Xdb+
2X dc 4v'-Yf (2.29')X('l rll
+
rJa+
Y(2d1+
r)b -Y(2(fl+
f)c 811."+
Xd(2d1-31') (2.30' ) -y(Ul+
f)a+
X(2d1+
r)b+
X(2d2+
flc 8,"+
Yd(2d'-3r) (2.31' ) 41'Xda+
4rydb--.tryde 1611.'"+
X(4tt _4dl~_r
1) (2.32') -4fYdll+
4fXdb+
41'Xdc 16vN'+
Y(4tr -4d2r _
1'2) (2.33'). WI'.nowsolve the equati onsanalytically. Using (2.28')toeliminateY from(2.:10' )givesX(2d'-r)' ~ 4(2d'
+
r)u'-
16du". (2.34) Using (2.29' )to eliminateXfrom (2.31')givesY(2d' -r)' ~ 4 (2d'H)u -16du". (2.35) Using (2.28')to eliminat eYfrom (2.32') gives
X(2cf -1')2
=
8fu'-16 11."'.25
(2.36)
y(~"1_r)'"
=
srI,'_Hi»".
Using(2.:H)tndiminiltr~Xlrcm(~.;lfi )p;iVt's
andsimilarly(2.;l,'j)ranbeusedto r-lhuiuau-
r
Fnuu(2.: rn
Lop;i\'"IlsillA(2.:11'\) to!'liminat" FIrom(2.:1!I)givt·s
(~.;Ij)
tl
=
·U!/l!t l" --n0'1I'"il2tJ2 -I' = ,IlIJ~=rJ2-f1~- I?
+
r~ (:HI) wlll'1'l· M1illthcdctcnulueutof[ (,,+ ,,)/'
AI
=
(b+c)/'i
:,W
(2.'1'2)
E'plaliflll(j!A I)iudicatcs UI<l1APcan1)('substituted for2
..r -
rinboth"'llIat i<Jl1s(j!.aH)<llId(1.:19).Eliminatingd fromther('lIulling equation,gives
M~
=1:/:" = :'t:~' .
(2.4'l)Tlmsw,'JIilVl's"lv,~1fo r "IIsin~oqnation (2AO) amihave uuothc-r"filia t io n (2A:!)rd ilLing rho rourelulugunknownDI( P to daIH!10the velocity rleelvn- tivI'lI.NOlI', W"Il l"l'ill,1positionlo solvetheequations {or the reutalning mrillhll'.,X,V,fl,11uudc.Note,WI' callwlvl>thb problemwithoutdcflu iug JII,IlIlt !lu SIJill orde r to 1lJ"<'S"I'WjJarallelswithKirwaurl al.,(1988).
Substituting(2.41)into(2.:)(1)givf's
(2.44) Slibslitlll,inp;(VII) into(2,:17) givf'!!
\V"still1m\/('Ulnl('varia blesII,h,cto 50 1\'1'for. Elimina t ingaFrom (2.2S ') ;11U1('l.'l!)')togivelinexpressio n(orhin tenusof cas
11= 8(l( Xv'
+
Yu' ) -4X Y ,/r- 'ltil(X2-Y2)C4fP(X 2tY2) .
27
(2.46)
Sll[,~titll ti l1gtIll' ubovo vallie:;of(Iilild,.into(2,-11 ),l!;i\"<'s;111I'Clllati"nlIla!
isquudmtlcinI:.Tosim plifytheequationant]suh'c'fur1',II'"1I.~t'nUIII'" I"r 111gl'hrasort \\'lIrl'[i.e.MACS't"MA ),11.turns OlltI.lliL ttill'l'!illa1.io nhas1'\'. peatedroots. \\11,tlu-usnhstitull'till'solutionfor ('lnto(~ " W )alld(2.H7) to solveforfIandI,. Osing
r
=,,/.1£1 - <IiIll•(2"[(1),(2,.1-1).(2 ..J!".),(2..l :1) we-substitu toforl',Il,X,V,Af1in\.I1t'n'sllltiuAI'qlla l,j"usI.""lttiliu1.1...fol1owillg so111tiOllSilltenusoftilt'tim" .11·ril'i1ti\'t,:,,,r\"'l,,,-it,\'_
b =
It'V''' -211"1)"
+
I/"l"U' U" - I L "tI'
'''' "" "" " "
11 I) -vV -uU +1III
'''' "" "', ""
vV -IIV+11II - U II
(1.·1X) (2A !I ) (V 'ill)
0111'solutionsforX,V,n,II,c urralldilfel-"lIlIrmulllus.·ofKir willlrl.,d., (1988).Onlyill1ILl' case
ur
rldoesthoOl(so lu l.iollgiw tIll'SM !l I'l'xl'r,'s:;io li asKirwanrlfll.,(I!JS8).Toslim Ill',weprpsI~lllallsolutionsfor tl1l'1>1(1'iUldpflsi1.iullSn-]..tivl'
t,uI.lw tlowl'l'lItr" obLailll,,1IISill1l;theOKIlwt llnd a!!below ,/ 1)'"-21/·1)"+u"'v'
u'v "-tt"V'
,m "I' ,,,' ""
!II) - I)I) - uII
+
1111,," ,," ,'" ""
vv - I)I) +11Il- U t i
U'V" - l t V '
d=
x
III! - V I I
~ (rll- 2 M ')u'- jj'"
M~
(Ill-2Ml)v' - v'"
M' .
From(1. 26)and(2.27)weM'{'th a t.the swirlvelocity ,ldlJ H't!by(2. 18)and
It,
= ~(n +d)X + ~(b-c)Y
II,
= ~{b+ c)X + ~(d -ft)Y
1/1'
=
! I - I t ,Vr = v- v•.
1' ·52)
1'·5:3) (2.54) TIlt'illlu\'{'solutions wet!' ohl,lilwd 011thonssurup tinnslhi~t lll
+
b2_r.1oF O.l.c.thedHl[('I·tt'ri~Lkrootsinoquatlon (2.8)arediffcn-ut:Ldusnowuse19
continuitytoshew thatlin'ahu\'p~lJl11tiom;i11~uhuhlfurtlu-l11ullip l.·,"to"l
para meterssothatit
=
f1,h=',+1,,',= f',J::::
II,tlu-wlm'ilSf..l,II"·""'1\l'SIt+1!J/ 2
,,+t.1'/'1., (~.r,(i )
us to calculate' fur
u.i"i:
nsiugl'qllitl.io1l.';('1. Aii,2,·I!I,;,l)j(l).1'\,,11'1111.1.he-n-u'p '"-2U"tl"+u"' t,'
- n'u" U"Il' +O( t ) ('1...,7)
,m "" "" ,,"
IJ V -1/1' +11Tl- , I U
U'IJ"
_II",,'
+0 (1). ('1...'i!J)Ifwelet ( ...0intotill'ahot'I'1'llllatiIIllS,WI'lind thatI.hl·~' ·"qll'l l iuIlS will reduceto (2.'1::1, 2.,m,ViO).Similarly, tllt'SI'l1'Sullsom:Ills""I,tllill".!by perturbing(I andc.From tIlis, WI' i\l'llun'thattlu-01,suluticusstill110101
rOTboththeunrcpcatod antimultipleroot1'<l SI'S.
:m
Chapter 3
Testing Our OK Solutions
111tilt'pn'vi(llI~rha ptl'fII'I~sho wrsl howaslugleIJilrticlt' traje ctorywouldbe 1I~1 '(1toc'akllla( l':thepositionX,Yorthepa rt icletelative lo thenowcentre
lI~ilJp;(2.+1)and(2.'15),tbetr-anslationvelocityUrandVrof thellowcentre
IIsing(2Ji:l) and(2.54)(11\11 theDKPa.b,c,rloftheHo wcentre using (2.48), (2A!J),(2.50) awl(2.'10),n-spe-ctiwly. We willnowtestthese solut ionsusing
;lrtili('iiII1Y ,l!;t' IH'fULt'(! datil tha thaskn o wnnowproperties.Severa lartificially
~"ll(, l" at('(l llat asdscoveringl~aSt~of pu rr stretchingand )lure shearingwith-
"Illtrnns!ill.iollvulccl ty andpurerot atioll with1I111fo r111trans lation velocity
an-\lSI "1.W(·l'onsll'lwLIlu-shuuletedtrajectory of a.particlefrom itsinitial
pu"it.ion n-latiw(0tIlt'llow('t!U lr l'Xo• 1'G; the tr<lllslution velocity
or
then ow
centro{IT,Vr;1I11l1tilt'[)I\ P( I ,b,r,«l:oflIlt'tlow lk-kl.'l'lu- 01, ltwll1.. d is the nlI!wt lton1lrllli ll,I'LlI\'kinl'tllilli. 'par.mn-Ie-rsHI' t!J I'IWilrlili,'iilll,r).:;,' u- erl\tnltrajl'doril's.Kirwan's sohukms(I\ irwill.,.,,/1.,1!11i1\)will],1'IISt-,] lH calru la tckluematicpal";\t1ld~rs[rumtilt' sauu- dillaSC'!,l<,FrotunowUII\\'0'
willrefer(aUu" solution,IIiVt'tlhy'~lIIlHl. i l>tlS(A :~) ,(M),(AI~)t(1\1:1),(1\]-1), (AI5),(AHi), and (A I 7) of Kirwanrt.ttl.,(1!)8H)astilt']\i r W;ltlS"l , tl.iu li.
This so lutionisim'ort'ect,huthas!J1'('U Wlt'tIhy otlu-rs{lpI\' isaudKirwuu, IDS7;Lewlsrf(d."1989; JudC'Strt,/1.,IlIH!l;I\il' wiltl (/III.,1!J!lO) L"r;d,'HI;l1.,' [)I( P ofdriftt"r dilL". Theytlw rt,rlJrt,lllt'ri1.somesLudyI"illl"'l' l' ft' LI.Iws<' drift('I' il.llalysC's.
TheOK metbcd l'!'ql1irl'stukinghiK!I unlt'r .1.'!'iWl1.iwsoftill'!,ilrlidl' tmj('clo ry,lIig h order .ll'rh"tti\'.'sart'lJm Lllhl"ttl1",1.1111lt"ISIll"l ' IlI I' III ,t'rrors andtrajcctcrylluctua tlonsCUllSt'l[bylilHillIsenh,I~!dit's.WI"will,I.1wt't'Cul"l', analyselh{~robustnessorthe0[\mdllmlbyllppl)'ill ,l.!;itLII1111ilrtili,'i ally gt'liera t o..'t.!trajPdorytowl.lcha knownamountofrnudum IlOis(' has1"'I'U
3.1 P ur e Stretch ing
Wt ~1I0WI:£msit]l'rlIwCIL~11ofMeddywithpu re strctchiegand notra nslation v<'!odly.Tl ll ~ str(~illl1li .wsforSlirha flowaresketche din Appendix13. Inlhb ('ilSf~,lIw ahsnluL(' pos ition(i "v)ofthepa r tirll\isthesame asits position rdat.i\lt~totill'lIo w centrenrulisobtnined from (2.15)and(2. ]6) with;t,Y
Tocalculatethetime ,lcrivativf>s1l11111Cr ica l1)'.we use a forwarddiffercnc- iu,e;sd lt' n wfurthl'rir~ltilt!'!'points(Mathews,1987)
uU) v(ll
-:U(I.)+4i:(t+~ )-; (1.+26.)
2"-
-:JY(I.)
+
19(t+
6)-ii(t+
:M)2"- .
(:J.l) (3.2) n'lI[c'rc'ddifferencing atthemiddlepoints
n(ll -.r(1+26.)+S.i(t +~;~8 ;t(l -D.) +X(I-26.) (3.:3) 11(1) -ii(I+ 28) +8 :lj(l +~~ ~8Y(t-6)+Y (l. - 26.), (3A )
111111aiuu-kwarddilfl'n'nd llgat thelastthr ee points It(l) ;1itt)-,1i{l- .6)+i:(i-2d)
2.6.
:1jj(I ) -'1fi(t- A)+!i(t-26.)
2 A .
:j:l
(a.5) (3.6)
Here,tI.de note's tIL"time il1k r\'all wl wrt' ll two nJlIS< 'c'uli n 'l'"il1l~illtill"tiuu-
&,ries.Throughout thl'tl ll~istho abovet1111111' rit' al<!i lfl'l"I'IWillp;sdll'lII.,swill Ill'used to obtainnpIotill'fourthurdcrtluu- ,It'ri"i,tiws"I'posit.ions. 'n'sls wert'alsodone using lime dl'ri\'alivt'suhliti llC'daualyurallyfrom(:.!.I ;, )allli (l .IG),but thesearcnot reported lu-re.Tllt's l' t{'s1.llP;i<"", of"()IIl·s l,. th"saul!' result s;us obtainedusingnuuu-rjr-ally,'lI1t'u)atl'drhnod,'rin ll il'l'l'.
We sotthe slrl'lrhillgd,J urlll "tiollrat"atII""0.1, 111l,1til<' utht-r1)[,1'
atII ""c=rI=0, Lot. tIll' initialpositionwithrc'spl','l 10till'11,,\\'n'u1.rc'
[whichisalsotileabsoluteinitialposit ionillthisn"'~I~)Ill'Xu
=
Il.mnand Vo=
0..1.TIll'DKPand theposltkmha w aunitoflill/(,- I1I1I,llI lIil,,listalw,' , respectively,Thetime intervalhl·t\\'l~'nsnn"'ssivl'I'0sitioll ~is 1'110""11to h,' :!units.Sllhstit utiuglIwsl'values rorXo,Yu,«,ii,r.tllutu"llllali Ull s(::Uf,) and(2. 11;),we canohtuin<Ilime s,'ri"sorplls iliulIS.TIll' slim,',1~ lt i's"tb used to evaluu t«Kirwan'ssolutions(Ki f Willitot{II,.I!JS~).Fromnow 011,lIw numeri c allyrnlcul utedresults of th"r;,' llt'I'atp t!dat alIsillp;eJIII's"llll,illl!,'1.01,1,..OKproblemwillbelabelled'0 \,',Values obtained Fromllll',l\etl(' I'iLt!,ddill«
usingsol utionsgivenbyKirwanatIii"(l!mt))will lH ~1"lwlll'llas'l'IIlWAN'.
Figure3.1:TheDKPcalculated, usingOK (above)andKIRWAN(b elow) solutions,fromtrajectorieswithpurestretchin g deformation.
Figure 3. 1 showstime series ofthe DKPcalculatedfromthe simulated tra jectorywithpurestretchingmotion.Theupper diagram isobtainedby usingtheOKsolut io ns((2.40),(2.48) ,(2.49),(2.50)) andgivesthe exp ected valuesofallparameters,e.g,a== 0.1,b ==c==d == Oin llnits of time-1•These parametersare constant and in accordance withthoseusedtogeneratethe trajectory.Thelowerdiagram,ontheother ban d, isproducedby KIRWAN solutions.Itshowsthatonly the divergenceis calculatedcorrect ly.Theother
35
threeparamet e rsdono t«mform withtillIS"used101!:'~Itt'I'IItl'till'sinmlal.t'd traj ectories.
Notethi~linbothlilt'uppe-r 1I11111uwcr,liilgril llls,Wt',10lIulilll"lllll,'StIliI!' resultsatthebcginuiu g andattIll' "lidofIiir-t.inwsl'l'i,'ssill!'('ill'llIl1.ai l1s spur jousIeatu rcs, Thisis ruusedbyth,~numerimldilft'T!'III,iitti'lliI'l'tu"<SS thatisb-ingUSl't1.IIIligllrca.:tWI'show tIll'pusi1. iolLoftilt'pilrtid l' III Intervals8, Clearly,atsmalland largo tilllt'sI,h!'1l1l\"li"l l ' lI uJ\"I',~qnh-klyill 01\l~direction andslowlyinthe otherdi n·diuli.Th is1"11'Isl.u1I1l1lH'I'indl'rrurs al,smalllindla rge Hull'S. Wndouothavethispl'lJhll~rllwllt'lilllakillKlIS('of the analyticallycomp uu-dtim e derivativesuf('qllHLiuIlS(:t,l!))itlld(:t.lli).
Havingcalcula ted thelJl\P,wentklll llh~lhppllsitiulloftln-p1Ltlidt~1'l'11l- Livetolilt'no w centreusing(':.J.A~)am](:U:I), TIll'pusi tioll uflllO'I\lI(','""111,1'1' is calculatedby suhttactiuglht~jlDsiHOIIorlJilrlid,'\'l'11~l.iVt'IIILll<'How<"('l l!.n' fromtheahsolntc par'Lirjcpositiongeneratedllsi n,u; (2.1.1),\lItl(~"lfiJ,'1'11<' resultis shownin figure:1,2.IIIthisrigll n~,lllt~
.*,
sYllllmld"lIlJk,~Llw1'.11"- tide's trajectory.1I1t!the'0'symholco rrespuncls!.Uthl~flowt·"lltn~.WI'!ill,l thatthe0[<so lutiongivesthenUWcentrellllh.~oriJ!,iH(O,UJwll,~ r"iL"~holll,l be.0.16, 1 --;---;;;;;O--_ = -~
····~n icre
o
!
OJ.
. . . .
'."
.
<It.--flowce nt re nospositi on z direct io n
Figure 3.2:The particle'strajectoryandthe calculatedposit ion of the !low centrefor the pure stretchingcaseusing OK.
Theaboveprocedures are usedagainto locate theflow centrefromthe I<IRWAN solutions, i.e.subt racting(A.13)and (A.H)ofKirwan etaI., (1988) fromthe particleposition.Theresultispresentedin figure3.3.In thisfigure, the flow centre moves far from theorigin,whereasit shouldbe stationary at theorigin.
37
. 2
D.6.~ 0.4
~" 1-
";~L \""
eftd" " '"
pOl itio n%direelion
Figure3.3:Theparticle'strajectory and the calculated position of theflow centreforthepurestretching case usingKIRWAN.
"avin! obtainedtheDKPand thepositionrelativeto the flowcentre,we proceed to calculate theswirland flowcentrevelociti~.To dothis,wemake use ofequati ons(2.51),(2.52), (2.53)and(2.54).Theresultispresentedin figure3.4 inwhich weplot thetime series
or
transla.tion velocities,UTandVT, log"· :"~with the swirlvelocities,u.andv•.Thetopdiagram isthe result given bytheOKsolution.Here,wefind thatthe tra nslat ionvelocity(UT ,VT) is zero. Thisiscorrectsince oursimulatededdyhas110translat ionvelocity.38
The swirlvelocity describes theparticlevelocity relati ve to thevelocity of the flowcentre.A part icleplacedin thistype ofBow withpurestretchint;
ha"int; apositi vestretchingdeformati on rate,Q,willmovetowardthex-axis and away(romthey-exis(seefigure3.2).Asaconsequence,the eastward componentofvelocity,u.. increases. Ontheotherhand,the northward rompouentofvelocity,V" will decrease .These featuresareobtainedbythe OKsolutions.
'J
_ vrOK~ .
-- "~0.01
~ ..
:> •• •••••• ••• ••• •• • •• • •_ -
40120:;--- "':;--- -:;;. .- - " ':;--- 60 10 10 W
I: "2
io.o , ....
.s ...•. . • • • .
~ O /:,:,~,:,: :;,~"".'~.~'"
-4:1.01
;0 )0 «l 50 60 10 10 W
Figure3.4:Swirland translational velocities calculatedfromtrajec tor ies withpurestretching. Theupper/lower plotshowsvalues obtained usingthe OK/KIRWANsolution.
39
Thebottom diagram is uht "i w',lllsill,l!;l\lHWAN\s..,lllli"ns,IIsh"wstiS lha l till'crldylias trauslut ion \'I'lorith-s111'1!.nn-iurnnsistc-ut with111<'I':ll'ill1l
Intilt'ras,'of]1111'1',~ll't'lI'hill,l?;,1\'.'limlthntIIIl' 0]\s"lllli..,n n>l'l',',-l l,I' obtninsalltln-killt'1J)lIti., pnrunu-h-rs.()IItill'111,1ll'I'hand.]\111\\-''';'1:s,,11l1j,ms
~jv,'lm-crrer-t \'i1hwsruralltll(· kin"l1Iil1,i" l'i1r;lIl1,'I!'rso'x"" pidiv,'r.~" IIC'I"
3.2 P ure Sh earing
Equations(2.1.')lind(2.lIi)Hr"1111\\'uwd1" sillliliall'lraj"duri,o:;n-sIlHill.l!.
Irom a st iitio llllry "ddywithpun- sll"Hrill,l!;. Till' stn'illl1liw'S(" I's!wll;,lI"w ill'"sk"tdlt'dillApllt'lidix1l.lu l liis <'as,'till'1>1\1'\';IIII"S111111iniliilll",siti"l1 are:b
=
0,1,11=r=«[=0,XII=n.OH:I,}';I=
u.t.TlwtiltH'illl, 'ITal lcrwcon Iwn SIlC('('SSl\'"points ISdl"SI'u l "Ill' :! units TIll' n'silltill,l!;1raj.'d,,,ry WitS il1llllys,'t1 hy 11,,111till'01\ilJl(II\IHWA;'I: 1I1<'11o,,,ls Hlldt.11<'n'slillill~1)1\1' an-pl'('sl'II1.,'(1 illli,l!;IIl"l' :1.;1,·lIJ
Figure 3.5:The DKPcalculated,usingOK(above) andKIRWAN(below) solutions,from traje<:torieswithpureshear deformat ion.
The diagram a.tthe top of figure3.5isobtained by using theOKsolutions And~vestbecorrect values of all parameters,e.g.It=0.1,a
=
c=
d=
0 inuuiuoftime-Iwhich are are independentof time.Tbebottomdiagram showsDKPohtainedusingtheKIRWANsolut ions.It showsthat onlythe divergence is calculatedcorrectly.Theshearingbhas thecorrect magnitude, butthewrongsign.Theothertwo parametersare inconsistentwiththose usedtogeneratethe simula tedtrajectory.41
····I"niclt
t...-flowcen rre
po,ilion zdirectio n
Figure3.6:TheOK solutionforparticletrajectory andpositionof theflow cen trefrom a trajectorywithpure shearing.
Having obtainedthe DKP,we calcula.tethepositionortheparticlerelative to theflow centreusing(2.44)and(2.45). Asforthepreviouscase, the positionof the flowcentreis computed bysubt ract ing the positionof particle relative totheflow centre abovefrom thetrajectorygeuereted using (2.15) and(2,16).The positionsobtainedfromtheOKsolutions arepresentedin figure
3.e.
The OK solutionsshowthatthe flow centreliesaLthe origin(0,0)a.~itshould.
position~dir ectio n
Figure 3.7:The KIRWAN solu tion forparticletrajector yand posit ion of the Bowcentrecalculated from a trajectorywithpureshearing.
UsingKirwan's solutionforthepart iclepositionrela tive totheflowcentre posit ion, i.e.(AI3) and(AI4) of Kirwandal., (1988), and subtractingthem fromthe trajectorygeneratedusing(2.15) and (2.16)we obtainthe plotof flowcentre trajectory'0'in figure3.7.Theparticle trajectoryobtained from (2.15) and (2.l6)is plotted with"". Itshowsthatthe How centr emoves rapidly,whereasit should be st ationary.
43
Figure3.8:Swirlandtranslationalvelocitiescalculatedfromtrajedoril)s with pure shearing,using OK(above) and KIRWAN(below) solutions.
Having obtainedtheDKPand the position relat ive to the flow centre, we proceed further to calculating thetranslation and swirlvelocities. In this regard,we makeuse of (2.51),(2.52), (2.53)and (2.54).Theresult is presented in figure3.8in which weplotthetranslationvelocities,Urand Vr,together with theswirl velocities,u, andv,.The upperdiagramare resu ltsgivenby theOKsolutions.Here,wefind that there is00translation
wl,wilyilSshown LyLIlt'bothvalur-s of(ITandV1'lx-lngequaltoeero. Thisi.~ I'orrpc~t silln~our.~il1l11l;ll('t1I~c1dyhasnutranslatio n velocities. The Lrnjm-tury of;0parti<'1c'illthislYlw ofpureshearinglIow withI)posil ivl'
il1't'.~lIlt,!lul h l·omlltllll'l1t.S ofswirlvdol~ity(u,,t'.)enlarge.'I'hesof.'alll rl's iln'I'ropc'rlydl'SlTilwdlIyIIII'OK so lut ions.
TlIC'lowerdiilp;rlll11isulltaitll'dllsing KIRWAN'ssolutions.Itshowsthat lllC'l'lllly hilSLrunxlutiuuvelocitios which,Jr('notconsistentwithtill'values IISI'I[top;plll'rilt.t'tIll'ln tj..dory,\Vt'nlaos~thatth~ll.decreases whereas/J,
inr-n-ases.Tln'sc'valuesMI'iUfllusisll'ulwithtln-trajectoryhdng1111<1.lyzl'{l.
WI'hll\'c'lIOSll'dtire'0/\ <IudK/HWANsolnricusbyiUlillysiu,l!;trajectories with!,1Ir<' slu'<Ir illp;.TIll'01\.~Ol lllil>llS.r;-iVl'll '·um 'rt.r~prrSl'ntlltiollof tllC'now lip),!.011 thl' othl'rhand, KIHWANsolutionsfa iltu give corr ec testimates forallkhn-mutjr-pllrlllHC'l('rscxrcptthe div('rgcnre.
3 .3 Pure Ro t a t ion with Tran sl ation Velocity
TIll'artifh-laltra joctoryfortln-1'111"('of pun' rut.nfiuuis~t 'I U' l' ll l t 'dIIsill ).!;r.lu- followingexpressinus:
fh'f t :ll'/I,'( wl) V/'I+:l.•ill(wl )
Pnllingtile'itnp;1l11lT\·('IUI"ityw
=
:1.1-[xW-1,if.,.=
I'~I'=uu! illl "(3.7) and(:),8), we ,11;1'1all'1ljl~:loryfo r Jilin'rotationulunu.IIl.rall slillill /l,How centro.TheLinn-intervallx-t we -nt.WOsll,',','ssi l'l'points isdll'SI'1ittlI...,'i units. TheOK aml KIR\VANsolutionsill"<'usedto:lllil ly Sl'till'J!.1'lw ril1,·,1 trajertory.Note thil l,('(11 111110111"(:1.7)and(:I.Ii)ronhlr-quallywe-ll,I''';''r i lu'a [lowfk-ldthut movr-sasits\" ',,1 Sotll i11ullpoiutsuu l.I",s]Il',·tJillV<'III<'saUII"
(10111.dis!,la",·,I)l"ilT Ularlra j P("t"ri,·s1,('la t !,, !'1,0it1I1lifllrllllrilli slalj,, "{IT,Vr.
Alliucr ti nl oscillutiouSl1P"l'l'tlSl ,d011uuilortntrI1lls!..1louistlUI'"xIlill p l,'"I"
lhis lYJl"
o r
ruction.!;:
:I
x e, '<
"
0, '"
..
x '<" . ,
0
~.
2 3
lime (ed dyperiods) w
- '
2 3
time (eddyperiods)
Figure3.9:Thepumerieallyulculat ed DKP forpure rotat ionwithtransla- tion case usier;OKandKIRWAN.
Figure3.9 showsthe time seriesofcalculated DKPforpure rotati on and translation.The upperdiagamisobtainedby using theOKsolutions.It showsasingle lineatc=6.28X10-2and threeoverlappinglinessbowing
tl=b
=
d=
O.Thesevaluesareidentical tothoseusedto generatethe trajec tory.Thelower diagramproducedbyKIRWAN showsthat onlythedivergenc e, d,iscalculat ed correctly.Thisfinding is simila rto those of thepure st retc hing
47
andputt'sh~aringshownearlier,Thevorticity,c.is only onefourthoftilt'
truevalue but itis constant.The stretc hingbAndshearcoscillate atrwk e the rotationalfreq uencyof theeddy.
_ !WIde
1'0,iHo n•Jiredion
Figure3.10:TheOKsolut ioDforparticletrajectoryudpositionof theflow centre calculated.from a.trajectorythat rotatesaround thetranslatingBow centre.
HavingobtainedtheaboveDKP,wecalculate the posrtiouof a particle relat ive totheBowcentreusing(2.44)and (2.45).Thepositionof theRow cent reis obtained bysubtractingthe positionof particlerelative to thenow
48
,,1t1.ililll'dIromtln-OK solutum arcshownill I1gnHl,1.10.IIIthis {Iiagl'mn, till'...'~yllllJ(J1,!l'IlUI.I'litilt' flow rcntn- trajr'doryandl1w '-'symbol ren-e-
'~IJl III ' 1st"lIll'pm'tit'll":;trajertory.Using the 01<solution,weflndthatthe
t.O,I!;"lu 'wl,'tIl{' silllillatl"[,lat a ,TIll'trenslarionvelocitywi llbe presented lak r ll !l,
trsill~Kirwuu's solutionfor partie.!l>posit ionrelativeto the flow centre, j.e-.(AI :!)alii! (Al.l)ofKirwan rIal.(1988) ,and subtrnctfngthem from till' positiu llof lJilrti d{'gl'lwrat{,d!I,~iJlg(:J.7) and(3.8) we obtain the plot of /lUI\" {'{'!lI,T!'Imjl'(:tlJry',f , 'ill figure :1.1J.The particletrajectoryobtained fruUl(:I,i)uud(:1.1l)is plottedwith '- ',Figure :J.l 1 showsthat the flow
"l'ntn'nw\,{'slIu rtlll'Hst \\'Hrtlwhilerot at ing. The radius of its circle is greater thanl.11l'uctunltrujcctcry.This featureshows1I1ill I<IHWA Ngives another
· ··· .IIow_
~ : ,,~ /~;§!f~;~~:~~<~~~:~~\
::" \O ,::t,: ~erd \':\i
; \:"?) JJ
-31.'30:,-1- , ;; -; ; - ---:;----;,;"----,, ,; : ' -- :, ,'; - - ; ' ; - - ;, po. i tion zdir eclio n
Figure3 .11:TbeKIRWANsolution forpart icletraj ec toryand positionofthe Row centre calculated froma trajecto ryth at rotates uoundthetranslatio&
£lowcentre.
Having foundtheDKPand thepositionrelat ivetothe flow centre, we proceed to calculatethe80w centreandswirlvelocities.le ordert.oget these, wemake use ofequations (2.51),(2.52),(2.53)and (2.54).Inligure3.12we plota timeseriesofthetranslationvelociti es,UTandVT,togetherwitbthe swirlvelocities, u.and u•.The topdiagramsbows resultsgivenbythe OK solutions.Here,we lind thattheparticlehas atra nslat ion velocitywithboth UTandVTequalto0.01(t helines areovcrlap pingo n theplot).Thisisexactly
:;0
