Kinematics III

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13 Νοε 2013 (πριν από 3 χρόνια και 7 μήνες)

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KinematicsIII
Brieflyaboutvorticitydynamics.Dererminationof
NULLvectorsandcriticalpointsinnumericalsolutions.
Classificaitonofcriticalpoints
KinematicsIII–p.1/21
EnstrophyofaKHbillow
Motivation:Thefigurebelowshowsenstrophy









foraKelvinHelmholtz
billowduringitstransitionalphasetoturbulence.SimulationdatamadebyDrJosephWerne,
ColoradoResearchAssociates.
KinematicsIII–p.2/21
Thenatureofvorticity
Vorticity

isavectorexpressinglocalrotationalbehavior.
Inturbulentflows,vorticityappearsasstronglyelongatedandcoherentstructures
whichwemaycallvortices.
Thelengthsofthevorticitystructuresarecomparablewiththeouterscalesof
turbulence.Thecross-sectioncorrespondstothemicroscaleofturbulence.
Vorticitystructuresrevealthestructureofturbulentdissipation.
Thevorticityfieldissolenoidal(ithasnopoles)




.
Itispassivelyadvectedbythefluidmotionwithoutcreatinganycounter-forces.
Intheidealnon-dissipativecase,thevorticityfieldlinesare“frozen”intothefluid.
Vorticityfieldlinesmayreconnect.Reconnectionproducesabruptchangesinvorticity
topology.
Vortexreconnectionhappenswhere



andinareaswithhighstrainingand
dissipation.
Duetointermittentnatureofvorticity,itiseasytovisualize.
KinematicsIII–p.3/21
Brieflyaboutvorticitydynamics
IncompressibleviscousflowsaredescribedbyNavierStokesequations:



 













(1)
Where

isanexternalforce.
Exercise:
Vorticityisgivenas




.Apply


totheaboveequationwith



toshowthat





 













 







 

 

(2)
showthatthiscanbewritten

 
 





 

  





(3)
where



israteofstraintensor.
KinematicsIII–p.4/21
Sourcesandsinksofvorticity
Accordingtoequation(3),withoutanyexternalforces,vorticitycannotbecreatedoutof
nothing











 
 
Ifthereisinitiallyvorticity






,stretchingwillincreaseitsmagnitude.Physicallythisisa
consequenceofconservationofangularmomentum,




 




 








Theoff-diagonalcomponentsofthestraintensor








representsourcesorsinksofvorticityduetotiltingandtwistingofthevortexstructures.
KinematicsIII–p.5/21
Frozeninfieldlines
Foranidealfluid



,thevorticityfieldlinesare“frozen”intothefluid.Wehave


 

 




Therateofchangeofthevorticityfluxthroughamovingcontourseethethefigurebelow
becomes

 


































KinematicsIII–p.6/21
VorticityandBiot-Savart’slaw
Forincompressiblefluids

 




withtherelation




,impliestheexistenceofa
vectorpotential

suchthat




.

canbechosensothat




.
Exercise:Showthat






.
UseoftheGreen’sfunctionof


,intheabsenceofboundaries,yields




















 



 




















ThisisBiot-Savart’slaw.Foranincompressibleflow,

and

equivalentlyexpressthe
motion.Itisinterestingtonoticethat




,where

denoteconvolution.Theoperator

isgivenby








 


















 
Formoreaboutvortexdynamicsseeforexample[2].
KinematicsIII–p.7/21
Motionoftwocounterrotatingvortices
Giventwocounterrotatingvortices






separatedbythevector

.
AccordingtoBiot-Savart’slaw:
Vortex-1will“induce”avelocityonvortex-2inthedirection




.
Vortex-2will“induce”avelocityonvortex-1inthedirection







.







KinematicsIII–p.8/21
Ontheidentificationofa“vortex”
Whydowewanttoidentifyavortexcoreorasimplycalledavortex?Thatisbecausethe
stateandtopologyoffluidmotioniscloselyrelatedtothetopologyofvorticity.Vorticityis
easytovisualize.
InthediscussionoftheidentificationofavortexwepresentamethodintroducedbyJeong
andHussain[5].
Itisnoteasytogiveaprecisedefinitionofavortex.
Avortexisrelatedtorotationalmotion,itiscoherentand“tube”like.
Thepresenceofavortex





,notnecessarilytheotherway.
Avelocityshearhasvorticitybutdoesnotformavortex.Velocityshearsarerelatedto
“vortexsheets”.
Theswirlingmotionofavortexisrelatedtoalocalpressureminima,butnot
necessarilytheotherway.
Thepresenceofclosedstreamlinesmayindicatethepresenceofavortex,butnot
necessarilytheotherway.(NotGalilean-invariant).
Huntetal.[4]definedaneddyastheregionwithapositivesecondinvariant

,ofthe
velocitygradientmatrix


.
KinematicsIII–p.9/21
VortexidentificationasinHuntetal.
Huntetal.’susedthesecondinvariant

whichis:
















 




















representsthelocalbalancebetweenshearstrainrateandvorticitymagnitude.
FromNavier-StokesequationsthePoissonequationforpressurefollows:







Apressureminimumcanbepresentwhen



Thiscriteriaisaccordingto[5]notnecessaryvalidclosetoaboundary.Thenanew
definitionisrequired.InformationonlocalpressureextremaiscontainedintheHessian
 




ofpressurewhichissymmetric.TakingthegradientoftheNavier-Stokesequations,
wefindtheaccelerationgradient












 






KinematicsIII–p.10/21
VortexidentificationafterJeongandHussain




cabedecomposedintosymmetricandantisymmetricparts










 





















 















Theantisymmetricpartisthevorticitytransportequationwhilethesymmetricpartis




 































Theoccurrenceofapressureminimuminaplanerequirestwopositiveeigenvaluesofthe
tensor




.Neglectingtheeffectofunsteadyirrotationalstrainingandviscositythisisthe
sameastheoccurrenceofamaximumof





duetovorticalmotion.Thevortexcoreor
vortexisdefinesasaconnectedregionwithtwonegativeeigenvaluesof





.The
eigenvaluesarerealand








,then




withinthevortexcore.



 
identifiesavortexcore
KinematicsIII–p.11/21
Comparizonof

 
and



Q

SimulationdataprovidedbyJoeWere,CORA,
postprocessingandrendering,FFI.Thereisaperfect
similaritybetweenQand
 
.IntheQand
 
panels,vortex
sheetsareabsentwhiletheyarevisibleinthe


panel.
KinematicsIII–p.12/21
Determinationof

FollowingGreene[3]:Locatingthree-dimensionalrootsbyabisectionmethod,utilizing
topologicaldegreetheory.Theapplicationofthisisindeterminationofzero-vectorswithin
numericalsolutionswhereavectorfieldisgivenatdiscretepoints
 















 















.Forsimplicityweassumethatthe
coordinatesareCartesian.Thevolume





 






















iscalleda
voxel.Itisarectangularparallelepiped.Weassumethatalinearinterpolantcanexpressthe
vectorswithineachvoxel.
If



at








 



,thenintheneighborhoodthefieldcanbewritten:
 



 


  

 








 


Thetopologicaldegree,

,ofthefieldintheparticularvolumeunderconsiderationisgiven
by


nulls
sign




KinematicsIII–p.13/21
Topologicaldegree
Wetreatonlythecaseswherenullsareisolated,noneoftheeigenvaluesof


vanish.
Thusthedeterminantisnonvanishing.
Thetopologicaldegreeisstronglynonvanishing,nullscanappearinagivenvolumeonlyby
crossingtheboundary,orbyproductionofpairswithoppositesignsofthedeterminant.
Thetopologicaldegreeyieldsthedifferencebetweenthenumberofnullsofpositiveand
negativedegree.Itdoesnotprovideacountofthenullpoints.
Thetopologicaldegreeisadditive:


Evaluationoftopologicaldegree
Consideravoxel,itiseasytosubdivideitintosubvoxels.Thetopologicaldegreeofthefield
insideavoxelisevaluatedtodeterminewhetherthefieldvanishesinside.Awayof
evaluatingthetopologicaldegreeisasfollows:
Thefieldisevaluatedattheeightcornersofthevoxelinphysicalspace.
Eachofthesixsidesofthevoxelisdividedintotwotriangles.
Foreachtriangle,thevectorsineachtrianglevertex








aredrawninaspace
thatwecall

space.










spanasolidangle

in

spaceshownbelow.
Physical space

x

y

z

1

2

3
 space
KinematicsIII–p.15/21
Evaluationoftopologicaldegreecontd
Thevectorsfromall12trianglesspanadodecahedron
Iforiginof

spaceisinsidethedodecahedron,then




andthereisoneormore
nullsinsidethevoxel.
Letusdenotethethreevectors










,let


betheanglebetween


and


,then















Thesolidanglespannedbythethreevectorsfromtriangle

becomes



  

 





 










  










 










  










KinematicsIII–p.16/21
Evaluationoftopologicaldegreecontd
Thesignof

istakentobethesameasthesignof








,then











isaninteger.If




thereisatleastonezeroinsidethevoxel.
Howtoevaluatethepositionofthezerowithinthevoxel?
Greene[3]suggestsalinearinterpolationutilizingthefourcornersofatetrahedron.This
takesintoaccountonlyhalfoftheavailabledata.Wesuggesttousealleightvectorsofthe
voxelvertices.
KinematicsIII–p.17/21
EvaluationofthepositionoftheNULL
Themostgeneralwayofwritingalinearinterpolantwithinthevoxelforcomponent

is


 

 









 































Newtoniterationisthenperformed.Guess


asthepositionofthezero.Thecomponents
oftheJacobianis













 



thenanimprovedpositionofthezero,


,isobtainedaccordingto










 


where




 












 
andsoontoconvergence.
KinematicsIII–p.18/21
Implementationandusage
Noteveryzerosarepickedupwiththismethod.
Theremaybeuptothreezeroswithinavoxel.
Incasetherearetwozeros














Thenthevoxelmaybesubdivideduntilthesubvoxelcontainsasinglezero.
Exercise1:
Giventhefollowingvectorfieldon





 
























Computethefunctions


andtheJacobian

KinematicsIII–p.19/21
Classificationofthezeros
ClassificationofcriticalpointsaretreatedamongothersbyChongandPerry[1].Ashort
summaryoftheirworkfollows.
Theeigenvaluesof


arecomputedclosetothezero.Thatleadstothefollowingequation



 



since

isa


matrixtheeigenvaluesaresolutionsof













Wediscussherea2-dimensionalcaseforsimplicity,thenwecanhave
Ifbotheigenvaluesareimaginary

theneighboringfieldlineslieonclosedloops
surroundingtheNULL,theyarecalledOpoints.
Iftheeigenvaluesarereal,oneispositiveandtheotherisnegative

X-points
KinematicsIII–p.20/21
Exercises
Exercise1:
Giventhevectorfield













Usetopologicaldegreetheorytodetermineifthereareanyzeroswithin

 




 


.
Computethevectorcomponentsateachcornerandshowhowtogetbacktothe
vectorexpressiongivenabove.
Findzerosanalytically.
Classifythezeros.
Exercise2:
In2D,givevectorsateachcornerofasquare,showthatitispossibletohaveuptotwozeros
withinthesquare.
KinematicsIII–p.21/21
References
[1] M.S.Chong and A.E.Perry.A general classification of
three-dimensional flow fields.Phys.Fluids,2(5):765–777,
1990.
[2] Alexandre J.Chorin.Vorticity and Turbulence.Applied
MAthematical Sciences 103.Springer-Verlag,New York
Berlin Heidelberg,1994.ISBN 0-387-94197-5.
[3] J.M.Greene.Locating three-dimensional roots by a bisec-
tion method.J.Comput.Phys.,98:194–198,1992.
[4] J.C.R.Hunt,A.A.Wray,and P.Moin.Eddies,stream,
and convergence zones in turbulent flows.Technical report,
Center for Turbulence Research Report CTR-S88,1998.p.
193.
[5] J.Jeong and F.Hussain.On the identification of a vortex.
J.Fluid Mech,285:69–94,1995.
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