A Coaxial Vortex Ring Model for Vortex Breakdown

0.9 MB·English

by  Denis Blackmore

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A Coaxial Vortex Ring Model for Vortex Breakdown
DenisBlackmore
DepartmentofMathematicalSciencesand
CenterforAppliedMathematicsandStatistics
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0 NewJerseyInstituteofTechnology
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Newark,NJ07102-1982
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[email protected]
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a
∗ ∗ ∗
J

5 MortenBrøns
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DepartmentofMathematics

] TechnicalUniversityofDenmark
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D DK-2800Kgs. Lyngby,Denmark
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h [email protected]
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a ∗ ∗ ∗
m
ArnaudGoullet
[

DepartmentofMathematicalSciences
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v NewJerseyInstituteofTechnology
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0 Newark,NJ07102-1982
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[email protected]
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8 ABSTRACT:Asimple-yetplausible-modelforB-typevortexbreakdownflowsispostulated;onethatisbasedonthe
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: immersionofapairofslendercoaxialvortexringsinaswirlingflowofanidealfluidrotatingaroundtheaxisofsymmetry
v
i oftherings. Itisshownthatthismodelexhibitsintheadvectionofpassivefluidparticles(kinematics)justaboutallof
X
thecharacteristicsthathavebeenobservedinwhatisnowasubstantialbodyofpublishedresearchonthephenomenonof
r
a
vortexbreakdown. Moreover,itis demonstratedhowtheverynatureofthe fluiddynamicsinaxisymmetricbreakdown
flows can be predicted and controlled by the choice of the initial ring configurations and their vortex strengths. The
dynamicintricaciesproducedbythetworing+swirlmodelareillustratedwithseveralnumericalexperiments.
Keywords:Vortexringdynamicsandkinematics,swirl,advection,Poincare´maps,Melnikovfunctions,chaos,Shilnikov
chaos
AMSSubjectClassification: 37J20,37J25,37J30,76B47,76F06,76F20
1
1 introduction
Vortexbreakdownisofficiallyfivedecadesoldthisyear:Althoughthereisevidencetosuggestthatthisphenomenonwas
observedseveralcenturiesagoinvariousatmosphericcontexts,thefirstofficiallyrecordeddescriptionappearstobethatin
theexperimentalpaperofPeckham&Atkinson[45].Thiswasfollowedharduponbythemorevortexbreakdownfocused
investigationsof Elle [20] and Lambourne& Bryer [37], and what appears to be the first real attempt at analyzingthe
phenomenoninSquire[56]. Andevenafterhalfacenturyofresearch,interestinvortexbreakdownhasnotdiminished;
infact,ithasprobablyincreasedsignificantlyoverthelastfifteenorsoyears.
Theearlyworkonvortexbreakdownquicklycapturedtheattentionofasubstantialgroupoftalentedfluidmechani-
cians attracted by the importance, and challenge of unravelingthe secrets of this intriguingand rather mysteriousphe-
nomenon.Withthisintenselevelofscrutiny,itdidnottakelongtorealizefromtheexperimentalevidencethattherewere
apparentlytwodistincttypesofvortexbreakdownconfigurations:anostensiblyaxisymmetricbubbletype(B-type)man-
ifestation; anda definitelyasymmetricspiraltype (S-type)structureoftenobservedasemanatingfromthetrailingedge
ofa B-type breakdownform. Moreover,inspiredby Squire’sattemptto formulatea theoreticalexplanationofthe phe-
nomenaintermsofvariationallybasedcriticalstatecriteria,manysimilarandquitedifferenttheoreticaldescriptionswere
propounded.Thethreemaintypesoftheoreticalcharacterizationsemployedcriticalstateconcepts,boundarylayersepa-
rationanalogies,andlongwavehydrodynamicinstabilityformulations,andavarietyofflowpropertiesandcharacteristic
combinationsofparameters,suchastheRossbynumber,wereidentifiedasplayingkeyrolesintheformationofvortex
breakdownconfigurations. Outstandingexamplesofthesecontributions,inadditiontoSquire’spioneeringanalysis,can
befoundintheworkofBenjamin[6],Ludwieg[40],Hall[27],Leibovich[38],Escudier&Keller[21],Trigub[58],Spall
etal.[54],Berger[7],Rusaketal.[46],Gelfgatetal.[23],andKrause[34,35].Theseandothertheoreticalinvestigations
havedoneagreatdealtoexpandourunderstandingofvortexbreakdown,especiallytheB-typevariety.Nevertheless,the
venerableenigmathatisvortexbreakdownhasyettoyieldtoacompleteanduniversallyacceptedexposition.
Soon after the quest for a theoreticalexplanationof vortex breakdownbegan, a host of efforts directed at identify-
ing vortex breakdown flows via computationaltechniques applied to both the Navier-Stokes and Euler equations were
undertaken. Early results in this vein were somewhatinconclusive,but improvementsin numericalmethodsand major
advances in computers have produced results that are quite compelling. Notable examples of this important aspect of
vortexbreakdownresearchcanbefoundinGrabowski&Berger[24],Krauseetal. [32],Hafezetal.[26],Spall&Gatski
[55], Breuer [15], Weimer [60], and Krause [33]. All of these numericalinvestigationshave benefitted from and been
complementedbyanumberofquitesophisticatedexperimentalstudiesofvortexbreakdownthatbeganinearnestaround
thesametime,withSarpkaya[48],Faler&Leibovich[22],andUchidaetal. [59]amongthebestexamplesoftheearlier
experimentallybasedinvestigations.
Holmes[29]wasinstrumentalinstartingarelativelynewtrendinresearchinfluidmechanicsingeneralandvortex
breakdown flows in particular: During the last twenty years, many investigations of a theoretical, computational, or
experimentalnaturehavebeenconductedfromtheperspectiveofmoderndynamicalsystemstheory. Terminologysuch
asstructuralstability,stableandunstablemanifolds,homoclinicandheteroclinicorbits,andstrangeattractorshavenow
becomethe linguafranca for a largesegmentof vortexbreakdowninvestigations. Prime examplesof this trend can be
found in the work of Blackmore [8], Blackmore & Knio [12], Brøns and his collaborators [16, 17, 18], Gelfgat et al.
[23],Serre&Bontoux[49],Sørensen&Christiansen[51],andSotiropoulosandhiscoinvestigators[52,53].Viewingthe
2
resultsofthisdynamicalsystemsapproach,itissafetosaythatithasalreadyprovedquiteusefulinapprehendingsomeof
themorecomplexfeaturesofvortexbreakdownflows,andhasgreatpotentialforcatalyzingfutureleapsofunderstanding
inthisarea.
Dynamicalsystemstheorycertainlyinspiredtheworkpresentedhereonourvortexbreakdownmodel,whichwasfirst
adumbratedin[12]. Buttheideaofusingapairofcircularringswasalsoinfluencedandguidedbytheexperimentaland
computationalliteratureandinteractionwithsomeoftheleadingpractitionersintheseareas. Ontheonehand,weknew
fromthedynamicalsystemsperspectivethat,amongotherthings,tworingsarecapableofgeneratingthekindsofchaotic
kinematics that appear to be quite common in vortex breakdown flows [4, 10, 11, 13, 30], but on the other hand, our
choicewasinformedbytheapparentubiquityoftwocoaxialvortexringsinnumericalandexperimentalinvestigations.
Naturally,ourdecisiontoconcentrateonanidealfluidbasedmodelwasmotivatedbyadesiretokeepthemodelsimple
andalso havethe consequentrich symplecticstructureat our disposal; the advantagesof whichare, we believe, amply
demonstratedinthesequel.
The development and analysis of our two coaxial vortex ring model for B-type vortex breakdown unfolds in this
paper as follows: In Section 2, we formulate the mathematical model for the motion of a pair of coaxial vortex rings
immersedinaswirlingidealfluidflowalongtheaxisofsymmetryoftheringsinthecontextofHamiltoniandynamics,
whereinweconsidersuchfeaturesasthecompleteintegrabilityoftheresultingsystem. Next,inSection3,wederivethe
Hamiltoniandynamicalequationsfor the motion inducedin passive fluid particles(kinematics)by the dynamicsof the
rings in the swirling flow. The associated kinematic equationsare formulatedin two ways: in terms of a passive third
ringwithzerovortexstrength;anddirectlyasa(time-dependent)Hamiltonianone-degree-of-freedomsystem. Itisnoted
thattheresultingkinematicsisnotcompletelyintegrable,soitiscapableofexhibitingchaoticflowregimes. Inaddition,
we introduce the approach to studying the dynamics via a Poincare´ map associated to successive intersections of the
streamlineswithafixedmeridianhalf-plane.Thentheusualaspectsofthedynamicalsystemfortheringmotion,including
classificationofstationarypoints,areanalyzedinSection4. ThisisfollowedinSection5withananalogousinvestigation
ofthekinematicequations,whichincorporatesabriefanalysisofhomoclinicandheteroclinicorbitsconnectingtheseveral
fixedpoints.
Sinceitwasfoundthatthekinematicsforthecaseoffixedringsistooregulartosubsumethekindsofexoticstreamline
patternsthatoneexpectsinvortexbreakdownonthebasisofseveralcarefulexperimentalandnumericalinvestigations,
weformulate,inSection6,themodelequationsintermsofsmall(quasiperiodicbutnotnecessarilyperiodic)oscillations
oftherings. Particularattentionispaidtoobtainingasymptoticexpansionsoftheheteroclinicorbitcomprisingtheouter
boundary of the bubble for the trace of the kinematics in a meridian half-plane. Then, in Section 7, we prove using
Melnikov’smethodthatsmalloscillationsin the ringsproducechaotic streamlineconfigurationsof the type observed-
butnotproved-inearlierstudiesofvortexbreakdownphenomena. ThisisfollowedbySection8inwhicha varietyof
numerical experimentsare run to illustrate the dependenceof the kinematics on various model parameters such as the
swirlstrengthandrelativevortexringstrengths. Thenatureoftransitionsto chaosasthesize oftheringoscillationsis
increased is given special emphasis. Finally, we conclude in Section 9 with a discussion of our model, its plausibility,
anditsefficacyinpredictingandcontrollingvortexbreakdownbehavior. Wealsoindicatesomenaturalfurtherresearch
directionssuggestedbytheresultsobtainedinourinvestigation.
3
2 The Two Ring Model
First,weconsiderthedynamicsoftwocoaxialvortexmovinginanideal(=inviscidandincompressible)fluidinitiallyat
restinR3. Theaxisofsymmetryofthevortexringsischosentobethex-axis,andtheirrespectivepositivestrengthsare
denotedasΓ andΓ . Owingtotheaxisymmetryofthemotionoftheserings,theirpositionsarecompletelydetermined
1 2
bytheirintersectionpointswithanymeridianhalf-planecontainingthex-axis. Inordertofixideas,wechoosethishalf-
plane,whichwedenotebyH,tobethexy-planewithy ≥0(y ≥0,z =0),sothatthemotionoftheringsischaracterized
bytheirrespectivepointsofintersection(x ,y ):=(x ,y ,0)and(x ,y ):=(x ,y ,0)withthedesignatedhalf-plane.
1 1 1 1 2 2 2 2
Definings := r2 := y2 +z2, whichreducesto y2 on H, it iswellknown[4, 10,11,13, 16, 17] thatthe equations
ofmotionoftherings-desingularizedtoeliminateinfiniteself-inducedvelocities-maybeexpressedasthe2-degree-of-
freedomHamiltoniansystem
π/2 cos2σdσ
κ s˙ =4κ κ r r (x −x ) ,
k k j k j k k j 3/2
Z0 ∆12
(1)
κ2 8r π/2 (r −r cos2σ)dσ
κ x˙ = k log k −γ +2κ κ r j k ,
k k (cid:18)2rk(cid:19)(cid:20) (cid:18) δ (cid:19) (cid:21) j k jZ0 ∆312/2
forj,k =1,2,withj 6=k,κ :=Γ /2π,
k k
∆ :=(r −r )2+(x −x )2+4r r sin2σ, (2)
12 1 2 1 2 1 2
0 < δ ≪ 1 is a very small positive number representing the common radius of the two rings in the desingularization
procedure,and
1 ∞
γ := 1+log2+ e−ξlogξdξ ∼=0.558, (3)
2
(cid:18) Z0 (cid:19)
whichis an artifactofassuming a Gaussian vorticitydistributionin the cores(i.e. in the tubesofradiusδ) of the rings
inthedesingularization.Itshouldberemarkedherethatdifferentdesingularizationapproachesproduceslightlydifferent
parametersanddynamicalequations,buttheoverallqualitativeaspectsofthemotionareunaffectedbysuchchoices,and
thedifferencesinthequantitativepropertiesarequitesmall.
ByintroducingthePoissonbracket
2
1 ∂f ∂g ∂f ∂g
{f,g}:= − , (4)
κ ∂x ∂s ∂s ∂x
k=1 k (cid:18) k k k k(cid:19)
X
whichisquiteoftenemployedinvortexdynamics,(1)canberecastinthemanifestlyHamiltonianform
s˙ =κ−1∂ H ={H ,s }, x˙ =−κ−1∂ H ={H ,x }, (1≤k ≤2) (5)
k k xk 0 0 k k k sk 0 0 k
wheretheHamiltonianfunctionforthesystemis
2 8r π/2 cos2σdσ
H :=− κ2r log k −(1+γ) +4κ κ r r
0 (k=1 k k(cid:20) (cid:18) δ (cid:19) (cid:21) 1 2 1 2Z0 ∆112/2 )
X
2
8r
=− κ2r log k −(1+γ) +2κ κ (r +r )[F(λ )−E(λ )] , (6)
k k δ 1 2 12+ 12− 12 12
(k=1 (cid:20) (cid:18) (cid:19) (cid:21) )
X
4
where
r −r
r2 :=(r ±r )2+(z −z )2, λ := 12+ 12−, (7)
12± 1 2 1 2 12 r +r
12+ 12−
andF andGare,respectively,thecompleteellipticintegralofthefirstandsecondkindgivenas
π/2 dσ π/2
F(λ):= andE(λ):= 1−λ2sin2σdσ. (8)
Z0 1−λ2sin2σ Z0
p
Thisuseofellipticintegralsintheequatpionsofmotionisclassicalandisoftenused(cf. [10,30,36]). Wenoteherethat
(5)-(6)is(real)analyticonthefollowingsubsetofR4,whichservesasthephasespaceofthesystem:
X := (s ,s ,x ,x )∈R4 :s ,s ≥0, (s ,x )6=(s ,x ) .
D 1 2 1 2 1 2 1 1 2 2
(cid:8) (cid:9)
TheautonomousHamiltoniansystem(5)-(6)hasthefollowingindependentintegralsininvolution(seee.g. [2,4,10,
12,43,44]):
2 2
H , G:= κ s = κ r2, (9)
0 k k k k
k=1 k=1
X X
so it is completely integrable in the sense of Liouville-Arnold (LA-integrable). We note here that a system of two or
morecoaxialrings-evenif oneringhasavortexstrengthofzero-doesnothaveanyadditionalindependentconstants
ofmotionininvolution,asprovedbyBagrets&Bagrets[4]. Consequently,thedynamicsofthreeormoreringsisaptto
includechaoticregimesinmostcases.
2.1 Fullmodel including swirl
Wenowimmersethetworingmodelinanambientswirlingflow. Tobemoreprecise,weshallassumethattheringsstart
andremaininaswirlingflowaboutthex-axis.Inordertoinsurethatthedynamicalequationsforthefullmodelincluding
thecoaxialvortexringpairandtheambientflowstillhaveaHamiltonianformulation,weshallchooseourmodelsforthe
ambientswirlingflow to beaxisymmetricpotentialflows. A convenientwayto representsuchswirling flowsis viathe
useofcylindricalcoordinateshavingthex-axis(ratherthantheusualz-axis)astheaxisofsymmetry. Thesecoordinates
naturallyaredefinedas
x=x, y =rcosθ, z =rsinθ, (10)
where the angle θ is measuredcounterclockwisein the yz-planestarting at zero along the positive y-axis. The general
expressionfortheambientvelocityintermsofthesecoordinatesis
v =(x˙ ,y˙ ,z˙ )=(u ,v ,w )= x˙ ,r˙ cosθ−z θ˙,r˙ sinθ+y θ˙ . (11)
a a a a a a a a a a a a
(cid:16) (cid:17)
Weassumefortheambientswirlingflowthatx˙ isaquadraticfunctionofs=r2oftheform
a
u (s)=x˙ (s)=−α 1+a s+a s2 , (12)
a a 1 2
(cid:0) (cid:1)
whereα>0anda anda arerealconstantstobechoseninthesequel. Topreserveaxisymmetry,weassumethatθ˙isa
1 2
polynomialfunctionofs=r2ofdegreetwo. Whence,weseethattheambientvelocityassumestheform
v =(x˙ ,y˙ ,z˙ )= x˙ ,r˙ cosθ−z θ˙,r˙ sinθ+y θ˙ = −α 1+a s+a s2 ,−z θ˙(s),y θ˙(s) , (13)
a a a a a a a a a 1 2 a a
(cid:16) (cid:17) (cid:16) (cid:0) (cid:1) (cid:17)
5
whereweshallonlyconsiderthefollowingpolynomialformsfortheangularvelocity
θ˙(s)=Ω 1+b s+b s2 , (14)
1 2
(cid:0) (cid:1)
wherethereisnolossofgeneralityinassumingthatΩ > 0,andb andb arerealconstantstobespecifiedinthecourse
1 2
ofourinvestigation.
Now we are in a position to formulate the full coaxial vortex ring pair + swirling flow dynamics. Once again, the
axisymmetryenablesustospecifythemotion,modulorotationaboutthex-axis,oftheringsintermsoftheirintersection
pointswith themeridianhalf-planeH. Observethatthe“moduloarotationaboutthex-axis”isnotreallya problemin
describingtheevolutionofpointsonaring. Toseethis, supposeweconsiderapointonaring,saythefirstring,thatis
initiallyat(x (0),s (0),θ (0)),andweknowx (t)ands (t). Thenwecanfindθ (t)from(14)andasimpleintegration
1 1 1 1 1 1
as
t
θ (t):=θ(s (t))=θ (0)+Ω 1+c s (τ)+c s2(τ) dτ, (15)
1 1 1 1 1 2 1
Z0
(cid:0) (cid:1)
andthereisananalogousformulaforθ (t):=θ(s (t)).
2 2
Addingtheambientswirlto(1)onlychangestheequationsofmotionbyatheadditionofasimplepolynomialfunction
alongthex-axes;namely,thedynamicalequationsbecome
π/2 cos2σdσ
κ s˙ =4κ κ r r (x −x ) ,
k k j k j k k j Z0 ∆312/2
κ2 8r π/2 (r −r cos2σ)dσ
κ x˙ =−κ α 1+a s +a s2 + k log k −γ +2κ κ r j k , (16)
k k k 1 k 2 k (cid:18)2rk(cid:19)(cid:20) (cid:18) δ (cid:19) (cid:21) j k jZ0 ∆312/2
(cid:0) (cid:1)
havingthefollowingHamiltoniananalogof(5)-(6)
s˙ =κ−1∂ H ={H,s }, x˙ =−κ−1∂ H ={H,x }, (1≤k ≤2) (17)
k k xk k k k sk k
withHamiltonianfunction
2 2
a a a a
H :=α κ s + 1s2 + 2s3 +H =α κ s + 1s2 + 2s3 −
k k 2 k 3 k 0 k k 2 k 3 k
kX=1 h i Xk=1 h i
2 8r π/2 cos2σdσ 2 a a
κ2r log k −(1+γ) +4κ κ r r =α κ s + 1s2 + 2s3 −
(kX=1 k k(cid:20) (cid:18) δ (cid:19) (cid:21) 1 2 1 2Z0 ∆112/2 ) Xk=1 kh k 2 k 3 ki
2
8r
κ2r log k −(1+γ) +2κ κ (r +r )[F (λ )−E(λ )] . (18)
k k δ 1 2 12+ 12− 12 12
(k=1 (cid:20) (cid:18) (cid:19) (cid:21) )
X
Observethat(17)-(18)alsoisLA-integrableinasmuchasithasthefollowingpairofindependentinvariantsininvolution:
2 2
H, G:= κ s = κ r2. (19)
k k k k
k=1 k=1
X X
Our primaryconcernin the sequel is the affectof the ring dynamics- as describedby (17)-(18)- on the motion of
passivefluidparticles,whichisoftendescribedasthefluidkinematics.
6
3 Dynamics of Model Kinematics
Thedynamicsofpassivefluidparticlesinducedbythemotionofthecoaxialvortexringpairiftheswirlingflowcharac-
terizedby (17)-(18)can be describedin two primaryways: Either indirectlyand autonomouslyin termsof a restricted
threering problemin whichwe treata passivefluid particle as belongingto a thirdring of zerostrength, or directly as
a time-dependentsystem of ordinarydifferentialequationsin which we first solve for the dynamicsof (16) in order to
describeitsaffectonapassivefluidparticle. Eachoftheseapproacheshasitsadvantages,andweshallfinditconvenient
todevelopbothoftheminwhatfollows.
3.1 Passivethird ring approach to thekinematics
In this approach, we actually consider a three ring problem with two of the rings as above with their given nonzero
strengths, Γ and Γ , and a third (advected) ring of strength Γ = 0. Once again, the axisymmetry implies that the
1 2
motionofthe ringsis completelydeterminedby thatoftheir respectiveintersectionpoints, (x .y ), (x ,y )and(x,y)
1 1 2 2
with the half-plane H. The equations of motion of these three points are analogous to (16), and can be written as(cf.
[4,10,11,13,16,17])
π/2 cos2σdσ
κ s˙ =4κ κ r r (x −x ) ,
k k j k j k k j Z0 ∆312/2
κ2 8r π/2 (r −r cos2σ)dσ
κ x˙ =−κ α 1+a s +a s2 + k log k −γ +2κ κ r j k ,
k k k 1 k 2 k (cid:18)2rk(cid:19)(cid:20) (cid:18) δ (cid:19) (cid:21) j k jZ0 ∆312/2
(cid:0) (cid:1)
2 π/2 cos2σdσ
s˙ =r κ r (x−x ) ,
k=1 k k k Z0 ∆3k/2
X
2 π/2 (r −rcos2σ)dσ
x˙ =−α 1+a s+a s2 +2 κ r k , (20)
1 2 k=1 k kZ0 ∆3k/2
(cid:0) (cid:1) X
where
∆ :=(r−r )2+(x−x )2+4rr sin2σ. (1≤k≤2) (21)
k k k k
This system can be recastin whatamountsto a kindof piecewise Hamiltonianform. In particular, we may rewrite
(20)as
s˙ =κ−1∂ H ={H,s }, x˙ =−κ−1∂ H ={H,x }, (1≤k ≤2)
k k xk k k k sk ∗ k
s˙ =∂ H ={H ,s} , x˙ =−∂ H ={H ,x} , (22)
x ∗ ∗ ∗ s ∗ ∗ ∗
where{·,·}isasin(4),H isasin(18),and
7
a a 2 π/2 cos2σdσ
H :=H +α s+ 1s2+ 2s3 −4r κ r =
∗ h 2 3 i kX=1 k kZ0 ∆1k/2
2 2
a a a a 8r
α s+ 1s2+ 2s3 + κ s + 1s2 + 2s3 − κ2r log k −(1+γ) +
2 3 k k 2 k 3 k k k δ
(h i Xk=1 h i) (kX=1 (cid:20) (cid:18) (cid:19) (cid:21)
π/2 cos2σdσ 2 π/2 cos2σdσ
4κ κ r r +4r κ r
1 2 1 2Z0 ∆112/2 k=1 k kZ0 ∆1k/2 )
X
2 2
a a a a 8r
=α s+ 1s2+ 2s3 + κ s + 1s2 + 2s3 − κ2r log k −(1+γ) +
2 3 k k 2 k 3 k k k δ
(h i Xk=1 h i) (Xk=1 (cid:20) (cid:18) (cid:19) (cid:21)
2
2κ κ (r +r )[F(λ )−E(λ )]+2 κ (r +r )[F (λ )−E(λ )] , (23)
1 2 12+ 12− 12 12 k k+ k− k k
)
k=1
X
where
r −r
r2 :=(r ±r)2+(z −z)2, λ := k+ k−, (1≤k≤2) (24)
k± k k k r +r
k+ k−
F andE areasin(8),andthenewPoissonbracketisactuallythestandardonedefinedas
∂f ∂g ∂f ∂g
{f,g} := − . (25)
∗ ∂x∂s ∂s ∂x
(cid:18) (cid:19)
Thesystem(22)-(23)isanalyticonthesix-dimensionalsymplecticspace
X := (s ,s ,x ,x ,s,x)∈R6 :s ,s ,s≥0,(s ,s ,x ,x )∈X , (s,x)6=(s ,x ), 1≤k ≤2 ,
DK 1 2 1 2 1 2 1 2 1 2 D k k
(cid:8) (cid:9)
andclearlysharesthefollowingtwoindependentconstantsofmotionwith(16):
2 2
H, G:= κ s = κ r2, (26)
k k k k
k=1 k=1
X X
butnoadditionalintegralsininvolutionasdemonstratedin[4]. Fromthiswe inferthatthemotionoftheadvectedring
canbecomechaotic,whichisconsistentwiththeapparentchaoticmotionfoundinvortexbubbles[16,15,17,23,39,49,
52,53].
3.2 Direct formulationofkinematics
Inthedirectmethod,weobtainatime-dependentsystemofdifferentialequationsforthemotionofageneralpassivefluid
particlebyassumingthatthedynamicsofthepairofcoaxialringshasalreadybeendetermined;i.e.,wehavethesolution
(s (t),x (t)),(s (t),x (t))of(17)-(18),whichmeansthatwealsoknowθ (t):=θ(s (t))andθ (t):=θ(s (t))owing
1 1 2 2 1 1 2 2
to(15).
WecouldstartwiththeBiot-SavartlawaswasdoneinBlackmore&Knio[10]andderivethesystemofdifferential
equationsgoverningthemotionofpointsx = (x,y,z) ∈ R3. However,we simplifythingsbytakingadvantageofthe
factthattherotationalmotionoftheparticleaboutthex-axisiscompletelydeterminedvia(14)-(16)bythemotionofthe
intersectionpointwithHoftheadvectedcircularringcontainingtheparticle.Notonlydoesthisreducethedimensionof
the(continuous)dynamicalsystemfromthreetotwo,butitalsohastheadvantageofhavingaHamiltonianformulation,
8
albeita time-dependentone. Takingthis approach,it is easy to see thatthe desired system is alreadyat hand- namely
fromthe last two equationsof (20)with the positionsof the two rings specified as functionsof time. In particular, the
equationsofmotionoftheintersectionpointsoftheadvectedringwithHare
2 π/2 cos2σdσ
s˙ =4r κ r (x−x (t)) ,
k=1 k k k Z0 ∆3k/2(t)
X
2 π/2 (r (t)−rcos2σ)dσ
x˙ =−α 1+a s+a s2 +2 κ r (t) k , (27)
1 2 k=1 k k Z0 ∆3k/2(t)
(cid:0) (cid:1) X
where,
∆ (t):=(r−r (t))2+(x−x (t))2+4rr (t)sin2σ. (1≤k ≤2) (28)
k k k k
Thesystem(27)is(time-dependent)Hamiltoniansinceitcanbeexpressedintheform
s˙ =∂ H(s,x,t), x˙ =−∂ H(s,x,t), (29)
x s
wheretheHamiltonianfunctionis
a a 2 π/2 cos2σdσ
H(s,x,t):=α s+ 1s2+ 3s3 −4r κ r (t)
(cid:16) 2 3 (cid:17) Xk=1 k k Z0 ∆1k/2(t)
2
a a
=α s+ 1s2+ 3s3 −2 κ (r (t)+r (t))[F (λ (t))−E(λ (t))], (30)
k k+ k− k k
2 3
(cid:16) (cid:17) Xk=1
andwehaveusedthenotationin(24)interpretedas(known)functionsoftime. Wenotethatthissystemisanalyticonthe
phasespace
X := (s,x,t)∈R3 :s≥0, (s,x)6=(s (t),x (t)), 1≤k ≤2 .
K k k
Observethatitisclearfromeither(cid:8)(20)or(27)thatifthepassiveparticlestartsonthex-axis((cid:9)s=0),itremainstherefor
alltime. Tobemoreprecise,thehypersurfaces = 0isaninvarianthypersurfaceofthesix-dimensionalphasespacefor
(20),anditisaninvariantlineinthetrajectoryspaceassociatedto(27).Fromafluiddynamicsperspective,thepointson
s = 0forwhichx˙ = 0areofparticularimportancesincetheycorrespondtostagnationpointsontheaxisofsymmetry,
whichonewouldexpecttodefinetheleadingandtrailingedge(points)ofaB-typevortexbreakdownstructure.Weshall
havemuchmoretosayaboutthisinthecourseofouranalysis,whichfollows.Anotherobservationthatwillproveuseful
fortheanalysis,isthatthehalf-planeplaysaverynaturalroleasatransversalforafairlyobviousPoincare´ map-amap
that conveysa great deal of useful dynamicalinformation. As a preview, we give a brief description of this map. Let
(s(t),x(t))betheuniquesolutionof(27)startingat(s ,x ) ∈ H,withs > 0. ThenthetrajectoryinR3 windsaround
0 0 0
thex-axisuntilatafirstt > 0,itintersectsHagain. ThisestablishesasmoothmappingofHintoitselfviatheformula
i
Π ((s ,x )):=(s(t ),x(t )). Thishandymappingshallbeinvestigatedfurtherinthesequel.
0 0 i i
4 Analysis of the Ring Dynamics
As for the ring dynamics, governedby (17)-(18),it was pointed outin the precedingsection that it is LA-integrable, a
property often referred to as integrable by quadratures. Therefore, at least in a theoretical sense, we can integrate the
dynamicalequations(17)-(18).Itappears,however,thatthisintegrationcannotgenerallybecarriedoutinapracticalway
inordertofindatractableclosedformsolutionfortheequationsofmotion,exceptperhapsinsomeveryspecialcases.
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4.1 Stationary points
We begin with an investigationof the fixed points of the system, which we consider in the form of (16). Solutions of
s˙ =s˙ =0areeasytocharacterize.Thisfollowsfromthereadilyverifiablefactthattheintegral
1 2
π/2 cos2σdσ
I := (31)
Z0 ∆312/2
ispositiveforallchoicesofr ,r > 0,x andx (exceptinalimitingcasewhereoneringcollapsestoapointwithits
1 2 1 2
coreradiusdecreasinginaspecialway). Thusanecessaryandsufficientconditionfors˙ =s˙ =0isthatx =x . This
1 2 1 2
commonvalueshallbedenotedasξ :=x =x .
1 2
Naturally,itremainstofindsolutionsof x˙ =x˙ =0,andtothisendweintroducesomeconvenientsimplifications.
1 2
Thereisnolossofgeneralityinassuming
1=κ ≤κ:=κ and0<r <r (32)
1 2 1 2
since this can be attained by a rescaling of the time variable t and reordering the variables, if necessary. In order to
simplifymatters,weshallassumein(12)thata =a =0,sotheequationsweneedtosolvetaketheform
1 2
1 8r π/2 (r −r cos2σ)dσ
1 2 1
0=x˙ =−α+ log −γ +2κr ,
1 2r1 (cid:20) (cid:18) δ (cid:19) (cid:21) 2Z0 ∆ˆ312/2
κ 8r π/2 (r −r cos2σ)dσ
0=x˙ =−α+ log 2 −γ +2r 1 2 , (33)
2 2r2 (cid:20) (cid:18) δ (cid:19) (cid:21) 1Z0 ∆ˆ312/2
where
∆˚ :=(r −r )2+4r r sin2σ. (34)
12 1 2 1 2
Itisconvenienttodisplaythefullsetofequationsdefiningthefixedpoints,whichowingto(16)andourassumptionscan
bewrittenas
π/2 cos2σdσ
s˙ =Φ (r ,r ,x ,x ;α,κ):=4κr r (x −x ) =0,
1 1 1 2 1 2 1 2 1 2 Z0 ∆312/2
π/2 cos2σdσ
s˙ =Φ (r ,r ,x ,x ;α,κ):=4r r (x −x ) =0,
2 2 1 2 1 2 1 2 2 1 Z0 ∆312/2
1 π/2 (r −r cos2σ)dσ
2 1
x˙ =Ψ (r ,r ,x ,x ;α,κ):=−α+ [log(χr )−γ]+2κr =0,
1 1 1 2 1 2 2r1 1 2Z0 ∆312/2
κ π/2 (r −r cos2σ)dσ
x˙ =Ψ (r ,r ,x ,x ;α,κ):=−α+ [log(χr )−γ]+2r 1 2 =0, (35)
2 2 1 2 1 2 2r2 2 1Z0 ∆312/2
whereχ≫1isamultipleofthecurvatureofcircularcross-sectionsoftheboundaryofthe(virtual)coreofeachcoaxial
vortexring,andisdefinedas
8
χ:= . (36)
δ
Wenotethat,strictlyspeaking,thefunctionsin(35)alsodependontheparameterχ;however,giventhe“virtualnature”
of the definition of χ, we choose to fix its value at 1,000 and suppress it as a coordinate of parameter space in the
sequel. Asthefirsttwoequationsofthissystemrequirethatx = x := ξ,thissolutionreducestothefollowingpairof
1 2
10