POLITECNICODITORINO
CorsodiLaureainIngegneriaAerospaziale
TesidiLaurea
Studiodiunflussoturbolentoisotropo
mediantelateoriadellereticomplesse
Studyofanisotropicturbulentflowwiththecomplex
networktheory
Relatore: Ing.StefaniaScarsoglio
Luglio2016
Candidato:
DavidePrino
Sommario
Ifenomeniturbolentisonolargamentediffusiinnaturaeneicontestitecnologici,essinonsonouna
prerogativadelsolocampoaerospaziale,mariguardanoipiùsvariatiramidell’ingegneria.Perfare
qualcheesempiobastipensarealflussod’acquacheescedalnostrorubinettodicasa,allecorrenti
atmosfericheeoceanicheoalflussod’ariacheattraversalepalediunaturbina.
La prima esperienza scientifica sulla turbolenza risale alla fine del diaciannovesimo secolo e fu
condottadaOsborneReynolds.Essaconsistevanell’osservazionevisivadeimotidiunfilettofluido
all’internodiuntubo,ciòèrealizzabileutilizzandountubotrasparentecollegatoadunserbatoioe
servendosi di un colorante per distinguere un certo filetto fluido dal resto del liquido. Reynolds
osservò che i fenomeni turbolenti si presentavano alle alte velocità, infatti la turbolenza è
caratterizzatadaunaltonumerodiReynolds.Apartiredaquestopionieristicocontributosonostati
fatti grandissimi passi in avanti nello studio della materia, attualmente si usano sia simulazioni
sperimentalichenumeriche.
Datalanaturacaoticadelfenomenol’approccionaturaleperlostudiodiquestamateriaèditipo
statistico, tuttavia questi tipi di studio possono rivelarsi estremamente complessi. Nel seguente
lavorosipresentaunapprocciodifferente:utilizzandoilvisibilityalgorithmsitrasformaunaserie
temporale (in questo caso l’evoluzione temporale dell’energia cinetica) in un grafo.
Successivamente,attraversolostudiodialcunecaratteristichetopologichediquestografo,sicerca
diarrivareadunainterpretazionefisicadelfenomeno.Questolavorosibasasull’ipotesichealcune
caratteristichefisichedelsistemasitrasmettonoalgrafo.Ilgrandevantaggiodiunostudiodiquesto
tipoèchesipossonoricavarealcuneinformazionisullafisicadelproblemautilizzandounalgoritmo
disempliceimplementazione.
Nel capitolo 1 si descrive in generale il fenomeno della turbolenza, descrivendone le principali
caratteristicheedandoparticolareimportanzaallaturbolenzaomogeneaeisotropa.
Nelcapitolo2siriportaunaintroduzionesullateoriadeigrafi,sidannoleprincipalidefinizionie
notazioneesidescrivonogliindicicheverrannopoiutilizzatinellatrattazione.
Nelcapitolo3siparladelvisibilityalgorithmesifornisceunapossibileimplementazioneconMatlab,
cercandodisottolineareipassaggicritici(ilcodicecompletoèriportatoinappendiceA).
Nelcapitolo4sidescriveildatabasedalqualesiattingonoidatiperilseguentestudio,sitrattadel
John Hopkins Turbulence Databases. Esso è costituito di più parti, ma viene descritta più nel
dettaglioesclusivamentelapartecheconcerneunasimulazonenumericadiunflussoturbolento
omogeneoedisotropo.Sifornisconoinfinelecoordinatedeipuntichesarannosuccessivamente
presiinconsiderazione.
Nel capitolo 5 si studiano le caratteristiche dei grafi ricavati dai due punti descritti nel capitolo
precedente, infine si comparano i risultati ottenuti per i due diversi punti e si cerca di dare una
interpretazionefisicaaidatiottenuti.
Contents
INTRODUCTION....................................................................................................................... 1
CHAPTER1:TurbulentFlows………………………………………………………………………………………….. 2
1.1 Characteristics……………………………………………………………………………………………………………. 2
1.2 Kolmogorov’shypotheses…………………………………………………………………………………………….3
1.3 Homogeneousandisotropicturbulence……………………………………………………………………… 4
CHAPTER2:GraphTheory…………………………………………………………………………………............... 5
2.1 Definitionsandnotations……………………………………………………………………………………………. 6
2.2 Metricsandindices…………………………………………………………………………………………………….. 7
CHAPTER3:Visibilityalgorithm…………………………………………………………………………………….. 9
3.1 GeneralProperties………………….………………………………………………………………………………….. 9
3.2 Propertiesandimplementation………………………………………………………………………………….. 10
CHAPTER4:JHTDBdatabase………………………………………………………………………………………….. 12
4.1 Descriptionofthedatabase………………………………………………………………………………………… 12
4.2 ObtainingData……………………………………………………………………………………………………………. 14
CHAPTER5:Networkanalysisandphysicalinterpretation……………………………………. 16
5.1 HDCnetworkanalysis…………………………………………………………………………………………………. 16
5.2 LDCnetworkanalysis……………………………………………………………………………………………….…. 20
5.3 Comparisonandphysicalinterpretation…………………………………………………………………….. 23
CONCLUSIONS…………………………………………………………………………………………………………………... 25
APPENDIXA:Implementationofvisibilityalgorithm………………………………………..…….. 26
BIBLIOGRAPHY……………………………………………………………………………………………………………..……. 29
Introduction
Turbulentphenomenaarelargelydiffusedbothinnatureandtechnologicalcontext,theyarenota
prerogative of the aerospace area, but they concern so many branches of engineering. Some
examplescanbe:waterflowthatgoesoutfromawatertap,atmosphericandoceaniccurrentsand
airflowthroughthebladesofaturbine.
The first scientific experience on turbulence was made by Osborne Reynolds in the nineteenth
century.Usingglasspipesconnectedtoareservoir,heobservedtheflowpatterninthepipesat
variousspeedsofwaterbyinjectingathinsteamofdyeinthemainstream.Hefoundoutthatat
lowvelocitiesthedyefilamenttravelledstraightandparallelstothewallsofthetube,insteadat
high velocities it diffuses over the whole depth and loses its identity. The former is the laminar
condition,whereasthelatteristheturbulentcondition.Infact,turbulenceischaracterisedbyhigh
Reynolds number. Since this pioneering contribution, considerable developments were done in
understandingthenatureofturbulence,nowadaysphysicalandnumericalsimulationarelargely
usedinthestudyofthissubject.
Due to the chaotic nature of the phenomenon, a statistical approach is the natural method for
studying turbulence, but these studies can be extremely difficult. In this work another kind of
approachispresented:usingthevisibilityalgorithmatimeseries(inthiscasethetemporalevolution
ofthekineticenergy)istransformedinagraph,then,studyingsometopologicalfeaturesofthe
graph,aphysicalinterpretationofthephenomenonisgiven.Thisworkisbasedonthehypothesis
thatsomephysicalcharacteristicsofthesystemaretransferredtothegraph.Thegreatadvantage
ofthiskindofstudyissimplicityofvisibilityalgorithm.
Chapter 1 is an introduction on turbulence: its main properties are described and particular
importanceisgiventotheisotropicandhomogeneousturbulence.
Chapter2dealswiththegraphtheory:atfirstthemaindefinitionsandnotationsaregiven,then
theindicesusefulfortheforwardanalysisaredescribed.
In chapter 3 the visibility algorithm is defined and a possible implementation with Matlab is
presented,givingparticularemphasistothecrucialpoints(completeMatlabcodeisreportedin
appendixA).
Inchapter4theJohnHopkinsTurbulenceDatabasesaredescribed,theyarecomposedbymany
parts, but only the numerical simulation of an isotropic and homogeneous turbulent flow is
describedindetail.Thenthecoordinatesofthetwopointssubsequentlystudiedaregiven.
Inchapter5thecharacteristicsofthetwographsobtainedfromthetwopointsdescribedinthe
previous chapter are studied. Eventually obtained results are compared and a physical
interpretationisgiventothesedata.
1
Chapter1
TurbulentFlows
WhentheReynoldsnumberexceedsacertainlimit,disturbancesareamplified.Whenamplification
islarge,disturbancesbreakdownintochaoticmotion.Repeatedbreakdownofdisturbancesleads
tocompletelychaoticsustainedmotion,commonlyknownasturbulence.
It is difficult to give a precise definition of turbulence. Hinze defined turbulence as follows:
“Turbulentfluidmotionisanirregularconditionofflowinwhichvariousquantitiesshowarandom
variationwithtimeandspacecoordinatessothatstatisticallydistinctaveragescanbediscerned”[2].
Thelateralmovementoffluidparticlesinthecaseoflaminarflowisduetomoleculardiffusion,so
itisverysmall.Instead,inturbulentflows,thismovementcanbecausedalsobythepresenceofan
eddy,whichisalargegroupoffluidparticlescharacterisedbyhighvalueofvorticity.Duringitslife
aneddycanchangeitsshape,stretchandrotateorbrakeintotwoormoreeddies.
Ifoneweretofocusattentionatapointinflowfield,passageofsmallandlargeeddiesthroughthis
point would induce velocity fluctuations of small magnitude and large frequency, and large
magnitudeandsmallfrequency.
TheturbulentflowisdescribedbytheNavier-Stokesequations,butthedeterministicsolutionof
thisproblemisveryhardtofindduetothepresenceofnonlinearity,soitisusuallystudiedwith
statistical and computational methods or, like in the present work, with the complex networks
theory.
1.1Characteristics
Turbulenceisachaoticphenomenon,whichhassomeimportantcharacteristics:
• The presence of fluctuations both spatial and temporal, which make the flow threedimensionalandnotsteady;
• Three-dimensionalbehavioureveniftheinitialconditionsaretwo-dimensional.Transition
fromlaminartoturbulentflowstartswithsmalltwo-dimensionalvibrations,thatgrowand
becomevortices,whichwarpandmoveduetomutualinduction;
• The non linear motion become more important for high Reynolds number, in fact the
convectiveterm,whichisnonlinear,assumesmorerelevanceandtheviscoustermisnot
able to dull the fluctuations. Due to this characteristic a small perturbation in the initial
conditionscaninvolveabigoneinthesolution,whichbecomebiggerandbiggerwhiletime
isgoingon.
• Thepresenceofthree-dimensionalfluctuationsofvorticity;
• Thisphenomenonhasawiderangeofscales.Thebiggesteddieshavethedimensionofthe
overallturbulentregionandtheycontainthemajorpartofthekineticsenergy,whilethe
dimension of the smallest ones depends on the Reynolds number, the higher is Re the
2
smaller are the eddies, in fact the ratio between the two scales has the same order of
magnitudeoftheReynoldsnumber.Thankstothevortexstretchingthereisatransferof
energyfromthebiggerscalestothesmallerscales,untilthegradientofvelocitybecomesso
high(becauseoftheconservationoftheangularmomentum)thattheviscousstressescause
thedissipationofkineticsenergy;
• Thediffusivityisverystrongbecausefluctuationscausearapidmixingofmass,momentum
andheat,higherthaninthelaminarcase,inwhichthediffusivityisonlyduetomolecular
agitation.
• Thedissipationisstrongtoo,becauseoftheviscousstressesinthesmalleddies.
Tosumupturbulenceisachaoticflowcharacterizedbythree-dimensionalvorticity,nonlinearity
andhighvaluesofdiffusivityanddissipation.
1.2Kolmogorov’shypotheses
Kolmogorovformulatedthreehypotheses:
1. First Kolmogorov’s hypothesis (local isotropy): for high enough Reynolds number the
turbulentmovementofthesmallscales(l<<l0,wherel0isthelengthscaleoflargesteddies)
arestatisticallyisotropic;
2. SecondKolmogorov’shypothesis(firsthypothesisofsimilarity):forhighenoughReynolds
numberthestatisticalphysicalquantitiesofsmallscales(l<<lEI)haveauniversalformdefined
byviscosity(ν)andenergytradedbetweenthebigandthesmallscales(ε);
3. ThirdKolmogorov’shypothesis(secondhypothesisofsimilarity):forhighenoughReynolds
numberthestatisticalphysicalquantitiesofthescalelwithlDI<l<lEIhaveauniversalform
definedonlybyε.
Figure1.1:TherepresentationoftherangesoftheKolmogorov'shypotheses
Generally,thelargesteddiesareanisotropic,buttheylosethisfeatureduringthetradeofenergy
betweenthebigandthesmallscales,thisphenomenonhappensintheinertialrange(lDI<<l<<lEI).
They lose also information about their geometry, so the motion of the small scales is universal,
whichmeansthatitissimilarineveryflowsthathavehighRe.Ifνandεareknown,dimensional
3
scale, scale of velocity and temporal scale (Kolmogorov’s scales) can be defined, so that the
dimensionlessquantitiescanbebuilt.Thesequantitiesarestatisticallythesameforeveryturbulent
flowwithhighRe.Peculiarityofinertialrangeareintermediateeddies,thatarebigenoughnotto
beinfluencedbyviscosity,butsmallenoughtobeintherangeofuniversality.
1.3homogeneousandisotropicturbulence
Thehomogeneousandisotropicturbulenceisaflowinwhichtherearenointeractionswithsolid
objectsandthereisnoameanflowofvelocityimposedfromtheoutside.
Aturbulentflowishomogeneousifthevelocityfielddoesn’tchangeduetoatranslationofthe
referencesystemanditisisotropicifitdoesn’tchangeduetoarotationofthereferencesystem.
Itisphysicallyrealizedlettingafluidpass,withaconstantflow,throughagrid.IfReynoldsnumber
is high enough, beyond the grid the flow becomes turbulent and it can evolve in a free space,
generating the homogeneous and isotropic turbulence. Feeding the turbulence for a long time
requiresasourceofenergybecauseofthestrongdissipationoftheturbulence.Theproductionof
energyisconcentratedinthebigscalesandthedissipationinthesmallones.Inthiscasethesource
isrepresentedbytheflowthroughthegrid.
4
Chapter2
Graphtheory
Many real systems from very different fields, for example chemical systems, social interacting
species,theWorldWideWeb,arecomposedbyalargenumberofhighlyinterconnectedunits.They
canberepresentedbyagraph,wheretheunitsaredrawnaspoints(nodes)andtheinteractions
betweenthemasedges.Theanalysisofmanydifferentnetworkshasproducedanimportantresult:
realnetworkshaveaseriesofunifyingprinciplesandstatisticalpropertiesincommon,forexample
thedegreedistributioncanexhibitsapowerlaw,theyhaverelativelyshortpathbetweenanytwo
nodes(small-worldproperty)andthepresenceofalargenumberofshortcyclesorspecificmotifs.
Thepresentworkisbasedonthehypothesisthatsomephysicalbehavioursarereflectedinthe
topologyofthenetwork,sotheanalysisofthegraphcanbeausefulalternativetoamoreclassical
statisticalstudyforknowingmoreaboutturbulence.
Figure2.1:aneuralnetworkmadeof100neurons
5
Figura2.2:Anexampleofsocialnetwork.
2.1Definitionsandnotations
An unweighted (i.e. all the links have the same values, there is no priority in correlations) and
undirected(i.e.thelinksdon’thavedirection)networkcanberegardedasagraphG=(N,E),where
NisasetofnodesandEasetofedges.
Thegraphcan’tcontainloops,i.e.linksfromanodetoitself,ormultipleedges,i.e.morethanone
connectionbetweentwonodes.
Twonodesjoinedbyalinkareaddressedasadjacentorneighbouring.Animportantconceptisthe
reachabilityoftwodifferentnodes,awalkfromnodeitonodejisasequenceofadjacentnodes
thatbeginswithiandendswithj.Itslengthisthenumberofedgesinthesequence.Apathisawalk
inwhichnonodeisvisitedmorethanonce,theshortestpathbetweentwonodesisanimportant
characteristicofthenetwork.
Agraphis“connected”ifthereisapathforeachpairofnodes.
AsubgraphG’=(N’,E’)ofGisagraphsuchthatN’iscontainedinNandE’inE.IfG’containsallthe
edgesofGthatjointwonodesinN’,G’iscalled“subgraphinducedbyN’”.Thesubgraphofthe
neighboursofagivennodei(Gi)isthesubgraphinducedbythesetofthenodesadjacenttoi(Ni).
Itcanbeusedamatricialrepresentationofagraph,itconsistsinaNxNmatrix(theadjacency
matrix)whoseentryaijassumesthevalue1ifthenodesiandjarejoined,elseaij=0.Forundirected
graph it is a symmetrical matrix, and all the terms on the diagonal, aii, are zero, because of the
absenceofloops.
6
AnotherimportantmatrixforthecharacterizationofthenetworkismatrixD,inwhichtheentrydij
isthelengthoftheshortestpathfromnodeitonodej.Itissymmetricalforundirectedgraph.
2.2Metricsandindices[4]
Thedegreekiofanodeiisthenumberofedgesincidentwiththenode.
!" = %∈' $"% (2.1)
Theaveragedegreeofanetworkis:
!" =
(
'
" !"
=
(
'
"% $"% (2.2)
ThedegreedistributionP(k)istheprobabilitythatanodechosenuniformlyatrandomhasdegree
k,soitisthefractionofnodesinthegraphhavingdegreek.
Theshortestpathisthelengthoftheminimumwalkbetweentwonodes,withoutvisitingthesame
nodemorethanonce.Itplaysanimportantroleinthetransportofinformationandcommunication
withinanetwork,sincethetransferofinformationisfasterifnodesareconnectedwithshortpath.
Thediameteristhemaximumvalueofshortestpaths:
) = max -"% (2.3)
"%
Theaverageshortestpath(orcharacteristicpathlength)istheaveragenumberofedgesalongthe
shortestpathforallpossiblenodesinthenetwork:
-"% =
(
"/% -"% ' '.(
(2.4)
Thisdefinitiondivergesiftherearedisconnectedpointsinthenetwork.Thiscausesanissue,anda
way to avoid the divergence is to introduce the efficiency E, which is an indicator of the traffic
capacityofanetwork.
0=
(
' '.(
(
"/% 1
23
(2.5)
Thebetweennesscentralitybiisameasureoftherelevanceofanode,infactthecommunication
betweentwonon-adjacentnodesdependsonthepathsconnectingthem,sotheimportanceofa
givennodecanbeobtainedcountingthenumberofshortestpathgoingthroughitandcalculating
bi(themorethepathpassingthroughthenode,themoreimportanceisgiventothenodeitself).
4" =
536 "
%/7 5
36
(2.6)
7
wherenjkisthenumberofshortestpathsconnectingjandk,whilenjk(i)isthenumberofshortest
pathconnectingjandkandpassingthroughi.
The clustering coefficient (or transitivity) measures the probability that two nodes, which are
neighbourofagivennodei,arelinkedtogether.Thismeansthepresenceoftriangles.
8" =
9:2
72 72 .(
=
3,< ;23 ;3< ;<2
72 72 .(
(2.7)
WhereeiisthenumberofedgesinGi(thesubgraphinducedbythenodei)and!" !" − 1 /2isthe
maximumnumberofedgesthatGicanhave.aijandamiverifythatnodesjandmarebothneighbours
ofi,thenajm=1ifjandmarejoinedbyanedge.
Itcanbeusefultocalculatetheaverageclusteringcoefficient.
B=
(
'
' 8" (2.8)
The modularity of a network is a measure of the structure of a complex network for detecting
communities.Ahighvalueofmodularityindicatesastrongdivisionintogroups.Nodesbelongingto
the same community have a lot of connections between them, this implies a faster rate of
transmissionofinformation.
8
Chapter3
Visibilityalgorithm
3.1Generalproperties
Thevisibilityalgorithmisasimplecomputationalmethodtoconverttimeseriesintographs.Every
pointofthetimeseriesbecomesanodeoftheassociatedgraphandtwonodes(ta,ya)and(tb,yb)
areconnectedifanyotherdata(tc,yc)betweenthemfulfilthefollowingpropriety:
CD < CF + C; − CF
HI .HJ
HI .HK
(3.1)
(C; − CF )/(MF − M; )isthegradientofthestraightlinethatjoinsthetwodataaandb.Soaandbare
linkediftherearenotanydatabetweenthemwithavalueabovethisline,(thevisibilitylinedoes
notintersectanyintermediatedataheight).
Infigure3.1isrepresentedthetimeseriesy=sin(t),wheretisavectorwithtenvaluesbetween0
and2π,withthesamegapbetweentwoadjacentvalues.Onthisfiguretheedgesthatfulfilthe
visibilityalgorithmarealsodrawn.Theedgesontherightofthefourthnodeareofdifferentcolours,
inordertocreatelessconfusion.Then,infigure3.2,isreportedthegraphoftheseriesmadewith
Gephi. This is an open-source and free software for visualization and exploration of all kinds of
graphsandnetworks.Itcanbedownloadedatthelinkbelow1.
Figure3.1:thetimeseriesy=sin(t),withlinksdrawninordertofulfilthevisibilityalgorithm.
1
https://gephi.org
9
Figure3.2:thegraphoftheseriesy=sin(t),madewithGephi.
3.2Propertiesandimplementation
InthisworktheimplementationofthevisibilityalgorithmtobuildadjacencymatrixAisdonewith
Matlab,(macroisreportedinappendixA).Thisprogramalsocalculatessomeimportantindices:
averagedegree,degreedistribution,averagepathlength,diameterandclusteringcoefficient.
Tobuildtheadjacencymatrixisimportanttoconsidersomepropertiesofthevisibilityalgorithm,
thatmakethecalculationfaster.
Firstofall,thegraphobtainedwiththismethodisundirected,sotheadjacencymatrixissymmetric,
itcanbebuiltonlythetriangularmatrix(inthiscasethesuperiortriangularmatrixisbuilt).Then
therearenoloops,sothevaluesonthediagonalareallzeroes.Thegraphisalwaysconnected,in
facteachnodeseesatleastitsnearestneighbours,leftandright,thismeansthatallthetermsof
theadjacencymatrixontheleftorontherightofthediagonalare1.Forthesereasonsthe“for
cycle”thatsoundsoutthelineofthematrixstartsfromi=1andendswithi=(N-1),whereNisthe
numberofnodesandthesizeofthematrix,infactAisinitializedtozeroandtheonlytermofthe
lastlinethatneedstobecalculatedistheoneonthediagonal,whichis0.Insteadthe“forloop”
thatsoundsoutthecolumnsstartswithj=i+2,thetermsonthediagonalanditsnearesttermsare
knownapriori.
Toverifyifapairofnodesarejoinedtheintroductionofanauxiliaryvariablecisneeded,atfirstit
assumesthevalue1,thatmeansthatthetwonodesaresupposedtobeconnected,thenathird
“forcycle”controlsifthereisnovisibilitybetweenthenodes,ifsocisputequaltozeroandthe
10
cycleisinterrupted.Thereasonofthechoiceofinitializingcto1isthefactthattwonodescanbe
supposedtobeneighboursunlessthecontraryinproved,butisnottruetheviceversa.
Attheendofalloftheseiterationstheadjacencymatrixiscalculated.
ThenextstepofthemacroistobuildmatrixDthatcontainsthelengthofthepathbetweenallpairs
ofnodes.Thiscanbedonereferringtothefactthattheentry(i,j)ofmatrixAkisequaltothenumber
ofwalksoflengthkfromnodeitonodej.InitiallyDisposedequaltoA,infactforeverypairof
nodesjoinedbyanedgethelengthofthepathis1,thena“forloop”fromk=2tok=(N-1),whichis
themaximumlengthpossibleforapath,analyzesonebyonethematricesAktofindalltheentries
thatfulfilthefollowingproperties:
• )",% = 0,ifitisnotzeroitmeansthatapathbetweenthenodesiandjwasalreadyfound;
•
O7",% ≠ 0,ifthisistruethereisatleastawalkoflengthkbetweenthenodesiandj;
ifboththepropertiesarerespecteditcanbesaidthat)",% = !.
Itisimportanttoproceedincreasingk,andnotdecreasingit,becauseitmustbefoundthelength
ofthepath,whichistheshortestwalkthatjoinsapairofnodes.
Anotherissueoftheprogramistocalculatetheaveragepathlengthandthediameterofthegraph,
whicharerespectivelythemeanandthemaximumvalueoftermsofmatrixD.
ThedegreeofthenodeiiscalculatedsummingalltheentriesofalineoracolumnofmatrixA,the
averagedegreeissimplythemeanvalueofthedegreeofallthenodes.
The maximum degree is also calculated because it is useful for the calculation of the degree
distribution,whichisavectorwhosenumberofcomponentsisequaltothisvalue.Thefirst“for
cycle”definesthevalueiofthedegree,whilethesecondonesoundsupthevectorthatcontains
thedegreeofallthenodestofindthenumberofnodeswhichhasdegreeequaltoi.
The local clustering coefficient ci is calculated using the equation (2.7). It is important to pay
attentiontothefactthatifthedegreeofthenodeiis1,cidiverges,butifanodehasonlyone
neighbouritsciis0,thisiswhyMatlabcodeusesan“if”commandtocontrolifthedegreeisequal
to1.
Eventuallyallthegraphicsareplotted:timeseries,degreedistribution,degreeofeachnodeand
local clustering coefficient. The degree distribution is managed using functions “histogram” and
“ksdensity”,thatevaluatethedensityestimateatrispectively20and50pointscoveringtherange
ofthedata.
11
Chapter4
JHTDBdatabase
4.1Descriptionofthedatabase
ThedatausedinthepresentworkaretakenfromtheJohnsHopkinsTurbulenceDatabases(JHTDB).
It contains four kind of turbulence fields: a homogeneous and isotropic turbulence, a forced
magneto-hydro-dynamicturbulence(MHD),achannelflowandahomogeneousbuoyancydriven
turbulence.Alltheseflowsareobtainedfromadirectnumericsimulation(DNS);thismeansthat
theNavier-Stokesequationsaresolvednumericallywithouttheuseofaturbulencemodel.Allthe
temporalandspatialscalesaresolved,fromtheintegralscalestothedissipativescales,starting
fromappropriateinitialconditionsandboundaryconditions.Thismethodisthesimplestapproach
totheturbulenceproblem,butithassomelimitations.DuetothecomplexityoftheNavier-Stokes
equationsitcannotbeusedtosolveeverykindofflow.Thenitneedsveryhighcalculationpower,
making the simulation very expensive, but, in the last years, thanks to the development of
technologywhichleadstoproducingpowerfulcomputers,itislargelyusedinresearch.
From JHTDB database the velocity components, the pressure and the magnetic field can be
downloaded.
Figure4.1:arepresentationofvorticityoftheforcedturbulentflow,takefromtheJHTDBwebsite
12
Thedomainofthefieldisacube2πx2πx2πandthevelocityorpressurevaluesarereportedona
spatialgridwith10243nodes.Theviscosityisν=0.000185.ThetimestepisΔt=0.0002,butthe
storingtimeintervalis0.002,finallythesimulationtimeintervalist∈[0,2048],obtainingthe1024
timestepsavailable,thetotaltimeofthesimulationisaboutonelargeeddyturnovertime.
Themaincharacteristicsoftheisotropicfieldarereportedinthetablebelow:
2Q
Distancebetweennodes
-=
= 6.142×10.V 1024
Totalkineticenergy
0HWH =
7
Dissipation
7
_=
Kolmogorovlengthscale
Kolmogorovtimescale
Integralscale
Largeeddyturnovertime
15]`9
= 0.118
a
`_
= 433
]
bcd =
]V
f=
a
ij =
k=
= 0.695
]! 9 X ∙ X∗ = 0.0928
Taylormicro-scale
Taylor-scaleReynoldsnumber
1
X ∙ X∗
2
l
2`9
]
a
( g
= 0.00287
( 9
= 0.0446
0 !
-! = 1.376
!
mn = k
`′
= 2.02
Table4.1:characteristicoftheturbulentflow
To obtain data from the database an authorization token is needed, obtainable asking the
permission to the database administrator, this is useful for knowing the number of users of the
differentservicesavailable.
Onthewebsitethereisasimpleinterfacefromwhichtheusercanchoseoneofthefourkindof
flowavailable,thefield(pressureorvelocity),thestartingcoordinates(temporalandspatial)and
thesizeofthecutout.
FilesdownloadedfromthedatabasearesavedinHierarchicalDataFormat(extension.h5).
13
Figure4.2:theinterfaceofdata-cutoutservice
4.2Obtainingdata
Inthisworkthetimeseriesoftwopointsofahomogeneousandisotropicturbulenceareanalyzed.
From the database are taken all the 1024 time values of the velocity, then the kinetic energy is
calculated:
(
p = (`9 + r 9 + s 9 )
9
(4.1)
whereu,v,warethethreecomponentsofthevelocityvector.
Thechoiceofthetwopointsisbasedonanotherwork2,inwhichacutoutoftheisotropicturbulence
of the JHTDB database is analyzed with the complex network theory, using a procedure that
convertsthespatio-temporaldatainaspatialnetwork.Inparticular,thisworkfocusonthestudy
oftwopointsofthenetwork:anodewithhighdegreecentralityvalue(HDC)andanodewithlow
2
“Nuoveosservazionisullacaratterizzazionespazialediflussiturbolentiattraversolateoriadelle
reticomplesse”byGiovanniIacobello,magistralthesis.
14
degreecentralityvalue(LDC).Thecoordinatesare(385,401,508)and(372,387,510)respectively
forHDCandLDCnode.Highvaluesofkiindicateregionswithsimilarinstantaneousvorticity,this
meansthatthereareturbulentpatternscoherentlymovingovertheacquiredtimescaleTL.The
domainofthisstudyisaspherewithradiusequaltoTaylormicro-scale,thatisthecharacteristic
dimensionofthesmallestdynamicallysignificanteddiesoftheflow.Itistheintermediatelength
scaleatwhichfluidviscositysignificantlyaffectsthedynamicofturbulenteddiesoftheflow.Length
scaleswhicharelargerthanTaylormicro-scale,theintegralrange,arenotstronglyinfluencedby
viscosity,whilebelowtheTaylormicro-scalethereisthedissipationrange.
15
Chapter5
Networkanalysisandphysicalinterpretation
In this chapter the two networks built with the visibility algorithm applied to time series of the
kineticenergyoftheHDCandtheLDCnodesareanalyzed,usingtheindicesdescribedinchapter2.
Distinctfeaturesofatimeseriescanbemappedontonetworkswithdistincttopologicalfeatures,
sotheresultingnetworksinheritcharacteristicsofthetimeseriesandthisseriescanbeinvestigated
fromacomplexnetworkperspective.Thankstothisassumptionaphysicalinterpretationcanbe
giventotheresultsobtained.
5.1HDCnetworkanalysis
StartingfromtheHDCnode,belowisreportedthegraphicofthetimeseries.
Figure5.1:thegraphicoftheHDCenergytimeseries
Thetimeseriesisconvertedinagraphwiththevisibilityalgorithm,thenthegraphisdrawnwith
Gephi(figure5.2).Nodeschangetheircolourfromwhitetoredduetotheirdegree(redmeanshigh
degree).
16
Figure5.2:graphofthetimeseriesofHDCpoint
Theaveragedegreeofthegraphis<k>=159.176
Degreedistributionisreportedinfigure5.3.Itisevidentthatthemajorityofnodeshaveadegree
minorthan250,whilethereareonlyafewnodeswithveryhighdegree(morethan600).
Figure5.3:degreedistributionP(k)ofHDCpoint
17
Infigure5.4isrepresentedthedegreeassociatedtoeverynodeofthegraph.Itisimportantto
rememberthatthevisibilityalgorithmconvertseverypointofthetimeseriesintoanodeandthere
isagapof2betweentwoconsecutivetimesstored.Thenumberassociatedtoanodeisthevalue
oftimerelatedtothenodeitself(attime2correspondsnode2,attime4thenode4,etcetera...).
Figure5.4:agraphicwhereisreportedthevalueofthedegreeforeverynodeofthegraph
Comparingthisgraphicwithfigure5.1itcanbeseenthatthepeaksaresituatedatthesamepoints.
Thisisnotsurprising,consideringthefactthatapeakofthetimeserieshasalargevisibilityonthe
othernodesandcoversthenodesnexttoit,infactafterapeakonthisgraphicthereisastrong
decreaseofdegree.
The diameter and the average shortest path are respectively D = 7 and <d> = 2.602, it is not
necessarytointroducetheefficiency,duetothefactthatthegraphdoesnothavedisconnected
points,so<d>isnotdivergent.
Theaverageclusteringcoefficientis<c>=0.642andthelocalclusteringcoefficientsarereportedin
figure5.5,theyhaveveryvariablevalues.Thereisanalternationofregionswithhighvaluesofci
andregionswithlowvalues.Alowvaluemeansthatthenodesaremoreindependentonefrom
eachother,whilethepresenceofalargenumberoftrianglesinvolveshavingastrongtransitivity.
Thestrongvariabilityofthisindexprobablyinvolvesastrongdivisioninclusterofthegraph.
18
Figure5.5:agraphicwhereisreportedthelocalclusteringcoefficientforeverynodeofthegraph
Inordertohaveabetterknowledgeofthedivisionincommunitiesofthegraph,themodularityis
studiedwithGephi:Q=0.321
There are 9 communities. In the graphic below is reported how many nodes each community
contains.
Figure5.6:agraphicofthesizeofeverycommunityofthegraph
19
5.2LDCnetworkanalysis
Thetimeseriesandthecorrelatedgrapharereportedbelow.
Figure5.7:thegraphicoftheLDCenergytimeseries
Figure5.8:graphofthetimeseriesofLDCpoint
20
Theaveragedegreeofthegraphis<k>=277.16
Infigure5.9isreportedthedegreedistribution.Therearealotofnodeswithavalueofdegreeless
than200.Thereisalsoalargegroupofnodeswithdegreenear300.Veryfewnodeshaveavery
highdegree(morethan800).
Figure5.9:degreedistributionP(k)ofLDCpoint
Thenextgraphicassociatesateverynodeitsowndegree.Considerationsonthisfigurearethesame
oftheonesmadeinthepreviouschapter.
Figure5.10:agraphicwhereisreportedthevalueofthedegreeforeverynodeofthegraph
21
ThediameterandtheaverageshortestpatharerespectivelyD=7and<d>=1.875
Theaverageclusteringcoefficientis<c>=0.687andthelocalclusteringcoefficientsarereportedin
figure5.11,thegraphichasquiteacontinuousforminthefirstpart,then,inthelastpart,thevalues
aremorevariable.
Figure5.11:agraphicwhereisreportedthelocalclusteringcoefficientforeverynodeofthegraph
ThemodularityisQ=0.26
There are 5 communities, in the graphic below is reported how many nodes each community
contains.
Figure5.12:agraphicofthesizeofeverycommunityofthegraph
22
5.3Comparisonandphysicalinterpretation
Inordertogiveaphysicalinterpretationtotheresultsdiscussedintheprevioustwoparagraphs,it
isimportanttocomparethedataobtainedforthetwopoints.Inthetablebelowarereportedthe
maintopologicalfeaturesofthetwonetworks.
HDC
LDC
Averagedegree
159.176
277.16
Diameter
7
7
Averagepathlength
2.602
1.875
Clusteringcoefficient
0.642
0.687
Modularity
0.321
0.26
Numberofcommunities
9
5
TheaveragedegreeofHDCpointislower,thismeansthattherearefewerconnectionsinitsgraph.
Aprobablecauseofthisbehaviourmaybethepresenceofmoreperturbationsinthemotionofthe
fluidparticles,whichinvolvesinrapidfluctuationsofthekineticenergy.Thisdecreasesthevisibility
between nodes. Perturbations may be caused by the continuous formation and breakdown of
vorticalstructures.
Both the graphs have some nodes with a very high degree, as seen in chapter 5.1 these nodes
correspondtopeaksinthetimeseries,whichhaveahighvalueofkineticenergy.Atthesepoints
thereisastronginfluenceoffluidparticleswithalargevalueofkineticenergyonparticleswith
lowervalue.
Thediameterisequalforthetwopoints;itdoesnotgiveinformationaboutthedifferencebetween
HDCandLDCpoint.Duetotheconceptualsimilarityofthediameterandtheaveragepathlength
(theyarebothrelatedtothepathlengthbetweennodes)alsothesecondindexisnottakeninto
account.
TheclusteringcoefficientofHDCpointislower,soitsgraphhasasmallernumberoftrianglesand
thenodesaremoreindependent.Alsothisfactcanbeexplainedwiththepresenceofvortexes:
perturbationsmaketheclosureoftrianglesharder.
Asimilarconclusioncanbeobtainedstudyingthemodularity,whichishigherfortheHDCpoint.A
highvalueofQmeansastrongdivisionincommunities,infacttheHDCgraphhasmoregroupswith
smallerpopulation.Thepresenceofvortexescausesoftenperturbationsofthesystemdynamics
andasaresultsthesuccessivestatesloseconnectivity.
Inthefollowingfigurearereportedthegraphicsoflocalclusteringcoefficientsforthetwopoints:
23
HDC
LDC
Figure5.13:ontheleftthegraphicofthelocalclusteringcoefficientoftheHDCpoint,ontheright
thesamegraphicoftheLDCpoint
Theformofthelastpartofthegraphic(characterizedbythepresenceofalotofpeaks)calculated
intheLDCpointissimilartotheformoftheothergraphic.Probablyduringthelasttimesofthe
simulationvortexesarecomparingintheLDCpoint.Thishypothesisissupportedbythepresence
of many peaks in the graphic of the degree in this step of time, that can be interpreted like
perturbations.
Inconclusion,theHDCpointischaracterizedbythepresenceofmanyfluctuationsofitsenergy,
maybecausedbyacontinuousformationandbreakdownofvorticalstructures.Thismeansalow
connectivityofthenetwork.WhileLDCpointhaslowerenergy,butmoreconstant,whichincreases
connectivity.
24
Conclusions
In the present work two points of a forced isotropic turbulent field have been studied with the
complexnetworktools.Atfirstthevisibilityalgorithmhasbeenappliedinordertotransformthe
timeseriesofkineticenergyintoagraph,thenitstopologyhasbeenanalyzed.Theaimofthestudy
was to find out correlations between the physical characteristics of the turbulent field and the
indicesusedforstudyingthetopologyofthenetwork.
Vorticalstructurespromoteinfastfluctuationsofthekineticenergy,andthisinturndecreasesthe
visibilitybetweenthenodesofthegraph.
The results obtained suggest that, to find out a point which has been much influenced by the
presenceofvorticalstructuresduringthetimeofthesimulation,wehavetosearchforpointwith
precisecharacteristics:
• lowaveragedegreevalue;
• lowclusteringcoefficientvalue;
• highmodularity;
• thepresenceofmanycommunities.
Thediameterandtheaveragepathlengtharenotmuchusefulforthisaim.
Eventually, the local clustering coefficient, due to its feature to be “local”, can give information
aboutthetemporalevolutionofthephenomenon,forexampleitcansuggeststhebirthofnew
vortexesatacertaintime.
Thus some physical behaviours can be analysed using the visibility algorithm and the complex
networkstheory.Thismethodcanbeusefulbecauseofitssimplicity,sincewecangetinformation
aboutthesystemwithoutusingthestatisticalapproach.
25
AppendixA
Implementationofvisibilityalgorithm
diametro=0;
path_medio=0;
C=0;
grado=zeros(1,N);
grado_medio=0;
%% MATRICE ADIACENZA
A=zeros(N);
for i=1:(N-1)
A(i,i+1)=1;
for j=(i+2):N
c=1;
for m=(i+1):(j-1)
if X(m)>=X(i)+(X(j)-X(i))*((m-i)/(j-i))
c=0;
break;
end
end
A(i,j)=c;
end
end
for i=2:N
for j=1:(i-1)
A(i,j)=A(j,i);
end
end
%% MATRICE DELLE LUNGHEZZE
D=A;
for k=2:(N-1)
B=A^k;
for i=1:(N-1)
for j=(i+2):N
if D(i,j)==0 && B(i,j)>0
D(i,j)=k;
end
end
end
end
for i=2:N
for j=1:(i-1)
D(i,j)=D(j,i);
end
end
%% DIAMETRO E PATH MEDIO
somma=0;
for i=1:N
for j=1:N
somma=somma+D(i,j);
if D(i,j)>diametro
diametro=D(i,j);
end
26
end
end
path_medio=somma/(N*(N-1));
%% GRADO DEI NODI
K=zeros(1,N);
for i=1:N
somma=0;
for j=1:N
somma=somma+A(i,j);
end
K(i)=somma;
end
grado(1,:)=K;
somma=0;
grado_max=0;
for i=1:N
somma=somma+K(i);
if K(i)>grado_max
grado_max=K(i);
end
end
grado_medio=somma/N;
%% DEGREE DISTRIBUTION
P=zeros(1,grado_max);
for i=1:grado_max
for j=1:N
if K(j)==i
P(i)=P(i)+1;
end
end
end
%% CLUSTERING COEFFICENT
c=zeros(1,N);
for i=1:N
somma=0;
for j=1:N
for m=1:N
somma=somma+A(i,j)*A(j,m)*A(m,i);
end
end
if K(i)==1
c(i)=0;
else
c(i)=somma/(K(i)*(K(i)-1));
end
end
somma=0;
for i=1:N
somma=somma+c(i);
end
C=somma/N;
%% GRAFICI
% serie di tempo
time=0:2:2046;
27
figure
plot(time,X)
xlabel('time')
ylabel('kinetic energy')
title('TIME SERIES')
% degree distribution
degree=1:1:grado_max;
figure
plot(degree,P);
xlabel('degree (k)')
ylabel('number of nodes')
title('degree distribution')
grado_min=K(1);
for i=2:N
if K(i)<grado_min
grado_min=K(i);
end
end
k_vettore=linspace(1,grado_max,50);
[g]=ksdensity(K,k_vettore);
figure
plot(k_vettore,g,'r')
xlabel('degree (k)')
ylabel('density')
title('DEGREE DISTRIBUTION')
hold on
pdf_k=histogram(K,20,'Normalization','pdf');
% grado dei nodi
figure
plot(time,K)
xlabel('nodes')
ylabel('degree')
% local clustering coefficient
figure
plot(time,c)
xlabel('nodes')
ylabel('c')
28
Bibliography
[1] RenzoArina,2015,Fondamentidiaerodinamica,Levrotto&Bella.
[2] R.J.Garde,1994,TurbulentFlow,JohnWiley&Sons.
[3] L.Lacasa,B.Luque,F.Ballesteros,J.Luque,J.C.Nuno,2008,Fromtimeseriestocomplex
networks:Thevisibilitygraph,ProceedingoftheNationalAcademyofSciences,vol.105no.
13,4972-4975.
[4] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D. U. Hwang, 2006, Complex networks:
Structureanddynamics,Physicsreports,424,175-308.
[5] A. K. Charakopoulos, T. E. Karakasidis, P. N. Papanicolaou, A. Liakopoulos, 2014, The
application of complex network time series analysis in turbulent heated jets, Chaos: An
InterdisciplinaryJournalofNonlinearScience,24,024408.
[6] Giovanni Iacobello, 2016, Nuove osservazioni sulla caratterizzazione spaziale di flussi
turbolentiattraversolateoriadellereticomplesse,magistralthesis,PolitecnicodiTorino.
29