# Kinematics III

Mechanics

Nov 13, 2013 (4 years and 6 months ago)

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





foraKelvinHelmholtz
KinematicsIII–p.2/21
Thenatureofvorticity
Vorticity

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

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

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



 

 
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

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

whichis:



 








representsthelocalbalancebetweenshearstrainrateandvorticitymagnitude.
FromNavier-StokesequationsthePoissonequationforpressurefollows:

Apressureminimumcanbepresentwhen

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

 

KinematicsIII–p.10/21
VortexidentiﬁcationafterJeongandHussain

cabedecomposedintosymmetricandantisymmetricparts

 

 

Theantisymmetricpartisthevorticitytransportequationwhilethesymmetricpartis

 

Theoccurrenceofapressureminimuminaplanerequirestwopositiveeigenvaluesofthe
tensor

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.

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
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

 

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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