Konuralp EFFECT OF THE CAVITY ANGLE ON FLOW STRUCTURES IN AN ANNULAR WEDGE ...

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TITLE OF THE MANUSCRIPT
EFFECTOFTHECAVITYANGLEONFLOWSTRUCTURESINANANNULAR
WEDGECAVITY
AUTHORS
HALISBILGIL,ZARIFEDOLEK
Abstract. KonuralpJournalofMathematics(KJM)isafully-refereedelec-
tronic journal. (KJM) is devoted to the high quality publication of current
rAeBsSeTaRrAchCTd.eTvheliosppmapeenrtasnianlytzheestahlel2fi-eDldSstookfeMsflaotwheinmaantnicuslaarnwdedagpepcliacvaittiieosnwsiitnhdalifl-
ifnerteenrdticsacvipitlyinaanrgyleas.reIanso.rTdehretojoaunranlaylzeisthpeubflloiwshsetdrutcwtuicreesa,tyheeatrw.odimensionalbi-
harmonicequationissolvedanalytically.Theflowisgovernedbytwophysicalcontrol
parameters:thecavityangleαandtheratiooftheupperandlowerlidspeeds(S=U1).
ByvaryingαforeachS,theeffectofcavityangleonthestreamlinepatternsandthUe2ir
bifurcationsareinvestigated.
1. This is a numbered first-level section head
This is an example of a numbered first-level heading.
1.1. This is a numbered second-level section head. This is an example of a
1. INTRODUCTION
numbered second-level heading.
Stokes flow generated within different shaped cavities is encountered in several
This is an unnumbered second-level section head. This is an example of an
manufacturingprocessesandengineeringapplications. Thelistofsomeoftheseap-
unnumbered second-level heading.
plicationscanbefoundinthereferences[6,12,14,17].Flowwithinthecavitieshasalso
b1.e1e.n1.aTfohciussisatatennutmiobnerfoedrctohimrdp-uletvaetliosnecatliflonuihdedaydn.aTmhiicssitsuadnieesxsaimncpeleitoisfaacnoummmbeorendly
uthsierdd-alsevaeblehnecahdminagr.kproblem.
Therearemanyworksintheliteratureoncavityflowsrelatedtoeddystructureand
Lemma 1.1. Example of the lemma
theirbifurcations.Gürcanetal.[8]analyzedthegenerationofeddiesinarectangular
cavity.Theyshowedeffectsofcavityaspectratioandspeedratioofthemovinglidson
thestreamlinetopologyandtheflowbifurcations. Flowbifurcationandeddygener-
(1.1) f(x)=0
ationforsteady,viscousflowinanL-shapedcavity,withthelidsmovinginopposite
dDireeficntiiotnios,nha1s.1b.eeTnhiinsvisesatnigaetxeadmbpylXeDoefliacedoeg˘filuniatinodn.Aydın[2]. ArunandSatheesh[1]
analyzedtheeffectsofaspectratioandReynoldsnumberonflowstructuresinarect-
Remark 1.1. This is an example of a remark element.
angularcavity.
ExMamosptloeft1h.e1.thTesheisstiusdaineseixnalmiteprleatoufreanreelaxtaemdptolecealveimtyeflnot.wareconcernedwiththe
squareorrectangularcavityflows, althoughthecavitiesmaybenon-rectangularin
Theorem 1.1. This is an example of a theorem.
applications. Gürcan&Bilgil[9],andGürcanetal. [10]investigatedbifurcationsand
eTdhdeyogreenmes1is.2m(eXchXaXniTsmhesoorfemSt)o.keTshflisowisiannaexseacmtoprleiaolfcaavtihtye.orEermtürwkitahnadpDaurersnuthne[t3-]
sicoalvlendot2e-Dinsttehaedhyeaadnidngd.rivenskewedcavityflowofanincompressiblefluidnumeri-
callyforskewanglesrangingbetween15oand165o.ErtürkandGökçöl[4]studied2-D,
Date:January1,2013and,inrevisedform,February2,2013.
2000 Mathematics Subject Classification. XXX.
Date:April1,2015and,inrevisedform,September14,2015.
Key words and phrases. xxx.
2010MathematicsSubjectClassification. 35Q35,00A69.
Theauthorissupportedby...
Keywordsandphrases. Cavity,Bifurcation,Eddy,FlowStructure,StagnationPoint,CavityAngle.
1
1
2 HALISBILGIL,ZARIFEDOLEK
steadyandincompressibleflowinsideatriangulardrivencavity. Asequenceofflow
structuresisillustratedbyGaskelletal. [7]forStokesflowinacylindricalcavity. Flow
structuresindifferentshapedcavitiesinvestigatedbyOzalpetal. [15]. Theyshowed
effectsofcavityshapeonflowstructurewithinthecavityindetail.
Ascanbeseenfromtheliteraturesurveygivenaboveandreferencestherein,most
ofthestudiesoncavityflowsareperformedoncavityaspectratioandspeedratioof
movinglids. Thereisaneedtoinvestigatetheeffectofcavityangleonflowstructure
inannularwedgecavities.Thisisaimofthisstudy.
2. MATHEMATICALFORMULATION
Weconsideredatwo-dimensionalcreepingflowinanannularwedgecavity r ≤
1
r ≤ r ,−α≤θ ≤α(Fig. 1). Thesidewalls, r = r , r = r arefixed. Theboundaries
2 1 2
θ =αandθ =−αaretwomovinglids,whichtranslatewithspeedsU andU inthe
1 2
radialdirectionrespectively.Theequationforthestreamfunctiongoverningthetwo-
FIGURE1. Geometryandboundaryconditionsfortheliddrivencavity
dimensionalsteadyflowofaviscousfluidis
(2.1) ∇2∇2ψ(r,θ)=0,
where∇2standsfortheLaplaceoperator
(cid:18) ∂2 1 ∂ 1 ∂2 (cid:19)2
(2.2) + +
∂r2 r ∂r r2∂θ2
inpolarcoordinates.
EFFECTOFTHECAVITYANGLEONFLOWSTRUCTURESINANANNULARWEDGECAVITY 3
Thederivativesofψgivethevelocitycomponents:
1∂ψ ∂ψ
(2.3) ur =−r ∂θ , uθ = ∂r ,
whereur anduθ aretheradialandazimuthalcomponentsofvelocity,respectively.
Thestreamfunctionisconstant(takentobezero)ontheboundaries
(2.4) ψ(r ,θ)=ψ(r ,θ)=0 , ψ(r,±α)=0 .
1 2
Theratiooftheradiiofthecylindersandspeedratioofthemovinglidsareourtwo
controlparameterswhicharedefinedby:
r U
(2.5) A= 2 , S= 1 .
r U
1 2
Intheplanepolarcoordinatesystem(r,θ),theotherboundaryconditionsare
(2.6) u (r,α)=S , u (r,−α)=1,
r r
andonthesidewalls:
(2.7) uθ(r1,θ)=uθ(r2,θ)=0,
wherewefixedU =S andU =1.
1 2
2.1. Eigenfunctionsolution. Thegeneralsolutionforthestreamfunctioncanbewrit-
ten[13]inseparableformas
∞
(2.8) ψ(r,θ)=(cid:88)[E sin(λ θ)+F cos(λ θ)]φ(n)(r),
n n n n 1
−∞
where
(2.9) φ(n)(r,λ )=a rλn+b r−λn+c r2−λn+d r2+λn ,
1 n n n n n
andλ arecomplexeigenvaluesgivenby
n
(2.10) sin(λ(cid:98)n)=±βλ(cid:98)n ,
where
1 1 1
(2.11) λ(cid:98)n=(i logA)λn and β=2logA(A−A).
ThesecomplexeigenvaluesarefoundviaaNewtoniterationprocedureasdescribed
byRobbins&Smith[165],Fettis[5]andKhuri[13]; andvaluesofthecorresponding
eigenvaluesλ aregiveninTable2.1.
n
Thecoefficientsa ,b ,c andd havetobedeterminedfromthesidewallbound-
n n n n
aryconditions.ThesecoefficientsaregivenbyKhuri[13].
ThecoefficientsE andF inEq. (2.8)havetobedeterminedfromtheupperand
n n
thelowerboundaryconditionsin(2,4)and(2,6). Itisclearthatthecoefficientsand
theeigenvaluesdependsonS andA,respectively(seefordetails[55,99]).
4 HALISBILGIL,ZARIFEDOLEK
Table2.1:Thefirst30rootsofλ forr =1,r =4.
n 1 2
n λ n λ
n n
1 1.86054+2.99451i 11 3.75801+48.66728i
2 2.46101+7.70825i 12 3.82201+53.203498i
3 2.78163+12.30346i 13 3.88080+57.739178i
4 3.00270+16.86922i 14 3.93515+62.27442i
5 3.17170+21.42265i 15 3.98570+66.80932i
6 3.30856+25.96964i 16 4.03295+71.34393i
7 3.42358+30.51282i 17 4.07728+75.87830i
8 3.52277+35.05353i 18 4.11906+80.41247i
9 3.60998+39.59256i 19 4.15854+84.94646i
10 3.68778+44.13037i 20 4.19598+89.48031i
TheinfiniteseriestherebyobtainedareinpracticetruncatedafterN terms,i.e.the
loweranduppersummationlimitsarereplacedby−N andN,respectively. Thecon-
vergenceoftheinfiniteseriesin(2.9)arenecessarytodeterminethenumberN which
assuresthatthetruncatedseriesiscloseenoughtotheinfiniteseries.
3. RESULTS
Wefirstanalyzedflowstructuresandtheirbifurcationsinahalf-annularringcavity.
Thenflowstructuresandtheirbifurcationsinanannularwedgecavitywihtdifferent
cavityangleareinvestigated.Effectsofthecavityangleonflowtopologyarerevealed.
3.1. Flow structures in a half-annular ring. A half-annular ring cavity, α = π and
2
3.2≤ r ≤18,consistingoftwostationarysidewallsandbothlidsmovingisconsid-
ered; theboundaryconditionsandsolutionprocedureareasgiveninsection2. As
inGürcanandBilgil’swork[9],mechanismsforeddygenerationareexaminedviathe
emergenceandthecoalesceofcorner-eediesorside-eddiesastheaspectratioisde-
creased;initiallyforsingledrivencavity,S=1,andthenforsymmetricflowS=−1and
S=1. Foreachcase,theflowstructuresandeddygenesismechanismsareillustrated
indetailwithfigures.
3.1.1. BifurcationsforaSingleLid-DrivenSectorialCavity. Theaimofthissectionis
toconsiderStokesflowinasingleliddrivenannularcavitywiththreestationarywalls
(fromFig1.,S =0impliesthetoplidisstationary;sincethesolutionsobtainedare
independentofwhichlidisstationaryandwhichonemoves,toaidvisualsationthe
flowpatternspresentedareforthecaseofastationarybottomlid). Althoughthean-
alyticalsolutionofthisproblemwasobtainedbyKhuri[13],hewasnotinterestedin
themechanism(s)ofeddygenerations.
Inthiscase,theboundaryconditionsaredefinedasfollows:
(3.1) ψ(r ,θ)=ψ(r ,θ)=0 , ψ(r,±α)=0
1 2
andimposingtheno-slipconditiononallfourwallsgives,
(3.2) uθ(r1,θ)=uθ(r2,θ)=0
(3.3) u (r,α)=1 , u (r,−α)=0
r r
EFFECTOFTHECAVITYANGLEONFLOWSTRUCTURESINANANNULARWEDGECAVITY 5
Thevaluesofthestreamfunctionandtheradialvelocityontopandbottombound-
ariesaregiveninTable3.1. FordifferentvaluesofN intheinfiniteseriesin(2.8),the
radialvelocityontopandbottomboundariesaregiveninTable3.2.
Toinvestigatestreamlinebifurcationsandhenceamechanismofeddygeneration
inthiscavity,theaspectratiowasdecreased,startingfromA=18wheretheflowcon-
sistofasinglemaineddyandtwosmalleronesnearthebottomcorners(Fig.2a).AsA
decreasesitisobservedthatthecornereddiesgrowinsizerelativetothelargecentral
eddy,tomeeteachotheronthebottomwallatacriticalvalueofA=15.49,seeFig.2b.
Atthiscriticalaspectratioaneweddyisformedwithasaddlepointandaseparatrix
withstreamfunctionvalueψ=0. AsA isdecreasedfurtherthisneweddycontinues
togrow(seeFig. 2c-d),thecentersoftwosub-eddiesapproachthesaddlepointand
thecenterneartheleft-bottomofthecavity(sayleftcenter)coalesceswiththesaddle
pointtoformacuspbifurcationatA=12.05. Hencethesetwocriticalpointsdisap-
pearandonlythecenterneartheright-bottomsideofthecavityremains(sayright
center).Infact,atthiscriticalaspectratiothedevelopmentofthissecondeddyisnow
completeasshowninFig.2e,A=11.10.Inthisfiguresmallcornereddiescanbeseen
onceagaindevelopingineachofthebottomcorners.Theprocessofeddygeneration
continuesastheaspectratiodecreases(seeFig.2f,whereA=3.20).
ThesamemechanismofeddygenerationhassimilarlybeenreportedbyGurcan
[8,11]forthecaseofarectangularcavityandGurcan&Bilgil[9]forasectorialcavity
(forα=π).
4
Table3.1: Thevaluesofψandu onboundariesforr =1,r =4andN =30.
r 1 2
r ψ(cid:0)r,π;30(cid:1) ψ(cid:0)r,−π;30(cid:1) -1∂ψ(cid:0)r,π;30(cid:1) -1∂ψ(cid:0)r,−π;30(cid:1)
2 2 r ∂θ 2 r ∂θ 2
1.0 -9.6458E-17 -3.2719E-020 1.7708E-15 0
1.2 7.1537E-04 -7.8157E-13 0.8817E+01 -0.1332E-10
1.4 -1.1147E-03 3.0229E-12 0.1071E+01 0.3918E-10
1.6 -1.4256E-03 3.7842E-12 0.1122E+01 0.3750E-09
1.8 9.9134E-04 -3.8356E-12 0.9247E+00 -0.3135E-09
2.0 -1.8560E-03 4.8247E-12 0.1163E+01 0.5496E-09
2.2 -1.4157E-03 2.3535E-12 0.1123E+01 0.3588E-09
2.4 -1.9610E-03 3.1865E-12 0.1139E+01 0.3818E-09
2.6 -1.5174E-03 8.4531E-12 0.1047E+01 0.2580E-09
2.8 2.2270E-03 -3.4649E-12 0.9173E+00 -0.3142E-10
3.0 -1.7164E-04 -2.5217E-12 0.1036E+01 -0.4235E-09
3.2 1.6848E-03 2.0594E-11 0.9621E+00 0.5959E-09
3.4 -2.1931E-03 4.9270E-12 0.1170E+01 0.7111E-09
3.6 2.1211E-03 -1.3457E-11 0.9658E+00 -0.1104E-08
3.8 -4.6814E-05 2.9814E-11 0.1195E+01 -0.1758E-09
4.0 -3.9111E-16 2.0768E-19 0.6123E-15 0
Table3.2: Thevaluesofu onboundariesfordifferentNandr =1,r =4.
r 1 2
6 HALISBILGIL,ZARIFEDOLEK
r -1∂ψ(cid:0)r,π;15(cid:1) -1∂ψ(cid:0)r,−π;15(cid:1)
r ∂θ 2 r ∂θ 2
1.0 0 0
1.6 0.9127 -0.1003-07
2.2 1.0922 0.8708E-08
2.8 1.0043 0.5807E-08
3.4 0.9439 -0.1674E-7
4.0 0 0
r -1∂ψ(cid:0)r,π;30(cid:1) -1∂ψ(cid:0)r,−π;30(cid:1)
r ∂θ 2 r ∂θ 2
1.0 0 0
1.6 1.1227 0.3750E-09
2.2 1.1236 0.3588E-09
2.8 0.9173 -0.3142E-10
3.4 1.1707 0.7111E-09
4.0 0 0
r -1∂ψ(cid:0)r,π;60(cid:1) -1∂ψ(cid:0)r,−π;60(cid:1)
r ∂θ 2 r ∂θ 2
1.0 0 0
1.6 1.0484 -0.2170E-10
2.2 1.0698 -0.9644E-11
2.8 1.0628 -0.2684E-10
3.4 0.8381 0.3904E-10
4.0 0 0
r -1∂ψ(cid:0)r,π;90(cid:1) -1∂ψ(cid:0)r,−π;90(cid:1)
r ∂θ 2 r ∂θ 2
1.0 0 0
1.6 1.0015 0.3599E-11
2.2 1.0003 -0.4064E-12
2.8 0.9996 -0.1018E-10
3.4 0.9928 -0.4382E-11
4.0 0 0
3.1.2. CaseS=−1. Inthecaseoflidsmovinginoppositedirectionswithequalspeeds,
(i.e.S=−1),theflowstructureissymmetricalaboutθ =0forallvaluesofA.Forlarge
aspectratios,asingleeddyoccupiesthecavity,seeFig. 3aforA=180. Astheaspect
ratioisdecreasedfrom180therearefourmainstagesinthedevelopmentofthesecond
andthirdeddies. Inthefirststage,a’Pitchforkbifurcationappearsatacriticalvalue
ofA =161.4. Thus,twoadditionalstagnationpointsaregeneratedinthecavity,(see
1
Fig.3bwhereA=130).
AsAisdecreasedfurther,theseparatrixcontinuestogrowandthesecondcritical
aspectratio,A =4.14,isobtainedatwhichtwodegeneratecriticalpointsappearon
2
thetwostationarysidewallswheresideeddiesareabouttoemergeasAisdecreased
further,seeFig.3c-h.
In the third stage, at A = 3.56, the heteroclinic connections coalesce with each
3
otherattheinteriorsaddlepointtoproducefourheteroclinicconnectionsbetween
thesaddlepointandthefourseparationpointsonthesidewalls,asshowninFig. 3i
(whereA=3.5).
Atthiscriticalaspectratio,A ,therearenowtwocompleteeddieswithinthecavity
3
andbetweenthemathirdisabouttobecreated.AsAdecreases,thesub-eddycenter
lying left of the saddle on θ = 0 approach the saddle point on θ = 0 and coalesce,
EFFECTOFTHECAVITYANGLEONFLOWSTRUCTURESINANANNULARWEDGECAVITY 7
(a) (b) (c)
(d) (e) (f)
FIGURE2. EddygenerationwithdecreasingAandS fixedatS=0.
a)A =18.0,b)A =15.49,c)A =13.20,d)A =12.50,e)A =11.10,f)
A=3.20
disappearing at A = 3.29. This is a cusp (saddle-node) bifurcation. At this critical
4
aspectratiotheformationofathirdeddy,betweentheothertwo,iscompletedsothat
threeeddiesnowoccupythecavity(Fig.3j).
Thisisamechanismforeddygenerationinwhichoneeddybecomesthree. The
numberofcompleteeddiesincreasesfrom3to5and5to7etc.viasimilareddygenesis
mechanism(seeFig.3k-m).
8 HALISBILGIL,ZARIFEDOLEK
(a) (b) (c) (d)
(e) (f) (g) (h)
(i) (j) (k) (l) (m)
FIGURE3. Eddygenerationviatheappearanceofsides-eddieswith
decreasingAandS fixedatS=−1forα=π.
2
a)A=180,b)A=130,c)A=90,d)A=25,e)A=12,f)A=5,g)A=4,
h)A=3.6,i)A=3.5,j)A=3.2,k)A=2.4,l)A=2,m)A=1.4
EFFECTOFTHECAVITYANGLEONFLOWSTRUCTURESINANANNULARWEDGECAVITY 9
(a) (b) (c)
(d) (e) (f)
FIGURE4. EddygenerationwithdecreasingAandS fixedatS=1.
a)A=5,b)A=3.1,c)A=2.85,d)A=2.8,e)A=2.4,f)A=1.74
3.1.3. CaseS =1. Inthecaseoflidsmovinginthesameradialdirectionwithequal
speed,(i.e. S =1),theflowstructureissymmetricaboutθ =0forallvaluesofA. Itis
clearthattheperipheralvelocityiszeroonθ =0.
Forlargeaspectratios, onlytwosymmetriceddiesoccupythecavity; seeFig. 4a
forA =5. Astheaspectratioisdecreasedfrom5,therearethreemainstagesinthe
simultaneouslydevelopmentofthethirdandfourtheddies. Inthefirststage,asAis
decreasedthefirstcriticalaspectratio,A =3.076,isreachedatwhichtwodegenerate
1
criticalpointsappearoneachstationarysidewall;seeFig. 4bforA =3.10. Theside
eddiesapproacheachotherasA isfurtherdecreased,suchthat,whenA =2.831is
2
reached, they coalesce on θ = 0. Thence a separatrix with a saddle point and two
10 HALISBILGIL,ZARIFEDOLEK
centers(i.e. twosub-eddies)isseeninthecavity(seeFig.4c-d). AsA decreases,the
sub-eddycenterslyingtotheleftofeachofthesaddles, approachthesaddlepoint
andcoalesce,disappearingatA =2.664whichmeansthatthisisacusp(saddle–node)
3
bifurcation. Hencefourfullydevelopededdiesarenowvisibleinthecavity(seeFig.
4e-f).
Thisisamechanismthatconsistsfromthreestepsforeddygenerationfromtwo
completeeddiestofour. Firstly, sideeddiesbornoneachstationarysidewall. Sec-
ondly,thesesideeddiesapproachandcoalesceeachothertoproducetworeflected
separatricesenclosingtwosub-eddies. Inthelaststep,thecuspbifurcationareseen
ontheeachseparatrixandtherearenowfourcompleteeddieswithinthecavity. A
similarmechanisminasectorialcavityisgivenbyGürcanandBilgil[9].
3.2. Effectofthecavityangleonflowstructures. Itistheaimofthissectiontotrack
thevariousflowtransformationsarisinginthecavityasαisgraduallyincreasedfor
0<α<π,andtoexposethemechanismsbywhichneweddiesemergeanddevelop
withinasectorialcavity.
Thisworkis,whichtoourknowledge,thefirstsuchstudyintheliteratureinterms
ofeffectofcavityangleonflowstructuresandbifurcations.
ForS=0,A=3andS=1,A=2,thevariousflowtransformationsaretrackedasαis
increasedandhencethemeansisidentifiedbywhichneweddiesappearandbecome
fullydeveloped.
3.2.1. Case:S=0,A=3. Solutionofthisproblemisintroducedinabovesection. For
narrowcavityangle,asingleeddyoccupiesthecavity;seeFig.5forα=15,wherethe
flowconsistofasinglemaineddyandtwosmalleronesnearthebottomcorners.
9 0
3 ,0 1 2 0 6 0
2 ,8
2 ,6
2 ,4
2 ,2
2 ,0
1 ,8
1 ,6 1 5 0 3 0
1 ,4
1 ,2
1 ,0
0 ,8
0 ,6
0 ,4
0 ,2
00 ,,00 1 8 0 0
0 ,2
0 ,4
0 ,6
0 ,8
1 ,0
1 ,2
1 ,4
1 ,6 2 1 0 3 3 0
1 ,8
2 ,0
2 ,2
2 ,4
2 ,6
2 ,8
3 ,0 2 4 0 3 0 0
2 7 0
FIGURE5. ThecavitygeometryandflowstructureforS=0,A=3and
α=15.