STABILITY OF NON-ISOTHERMAL POISEUILLE FLOW IN VERTICAL ANNULUS FILLED WITH POROUS MEDIUM

Abstract

.Thenon-isothermalPoiseuilleflowinporousmediahasbeena subjectofintenseresearch foroverfourdecades.Thistypeofflowinpipe/channel/annulusisusedinmanyindustrial situationssuchasextractionofbio-fuel[51],packed-bed chemicalreactors[2],oilrecovery process[63],solid-matrixheatexchangersandcoolingofnuclearplants[54],etc.However, mostoftheavailablestudiesaredoneinverticalchannelor pipe.Theresultsofchannel orpipecannotbeusedtopredicttheflowmechanisminannular geometry.Therefore,to understandtheflowconfigurationinannulargeometryformed bytwoconcentriccylinders a stephasbeentakeninthepresentthesis.Bothlinearandnonlineartheorieshavebeen usedtoexaminethestabilitymechanismoftheflow.Theobjectiveofthisstudyisto investigatetheeffectofgapbetweenthetwoconcentriccylindersontheaboveflowfor differentpermeablemediumaswellasnon-isothermalresources.Theannulusisfilled withahomogeneousandisotropicporousmedium.Anexternal pressuregradientanda buoyancyforce(duetotemperaturedifference)drivethefullydevelopedwaterflowin theannularregion.Theinnerwalltemperatureoftheannulus increaseslinearlywiththe axialcoordinatefromanupstreamreferencetemperatureand theouterwallisadiabatic. Inthelimitoffullydevelopedflow,thissimulatesaconstant heatfluxconditiononthe innercylinder.NotethatdependingonthesignofRayleighnumber,thefullydeveloped flowmaybestablystratified(i.e.,thebuoyancyforceactsin thedirectionofforcedflow) orunstablystratified(i.e.,buoyancyforceinthenegative directionofforcedflow).The linearstabilityoftheaboveflowforbothstablystratifiedandunstablystratifiedcasesare analyzedinthisthesis.FollowingthepreviouseffortsofYao&Rogers[123],theweakly i ii nonlinearstabilityofnon-isothermalPoiseuilleflowinverticalannulusfilledwithporous mediumisdeveloped.Thepresentthesisiscompiledinsixchaptersandthechapterwise descriptionisgivenbelow. Chapter1isanintroductoryandcontainssomebasicdefinitions,preliminariesofthe flowinporousmedium,briefdescriptionofhydrodynamicstabilitytheory,workdoneby variousauthorsinthefieldoflinearandnonlinearstability analysisofPoiseuilleflow,and justificationregardingthemodel,whichhasbeenadoptedfor thisproblem. Chapter2addressesthebasicflowcharacteristicofthenon-isothermalPoiseuilleflow inverticalannulusfilledwithporousmedium.Bothstablystratifiedandunstablystratified situationsareconsideredforthisstudy.Thenon-Darcy-Brinkman-Forchheimermodelis used.Thegoverningequationsaresolvedanalyticallyfora specialcase:formdragequal tozeroandnumericallybyChebyshevspectralcollocationmethod.Alongwiththeother controllingparameters,aspecialattentionisgiventounderstandtheeffectofcurvature parameter(C) oftheannulusontheflowconfigurationaswellasheattransferrate.The numericalexperimentsshowthatreducingthevalueof C enhancesthemaximummagni- tudeofthevelocityalongwithheattransferrateinthesystem.Theimpactof C (C > 10) ontheflowprofileaswellasheattransferrateisnegligible. Furthermore,theanalysis showsthatthetendencyofappearanceofbackflow,pointofinflectionandflowseparation (incaseofunstablystratifiedflow)intheflowprofileishighly sensitiveto C. Apartfrom this,forasmallincreasein Ra, adrasticchange(upsidedown)intheflowprofilecanalso beseen.Theappearanceofflowseparationshiftedfromthevicinityoftheinnerwallto theouterwall.Hence,toshedmorelightonthisphenomenonandtofindtheappropriate non-isothermalparameterspaceasafunctionofgapbetween thetwoconcentriccylinders, inwhichtheflowwillremainasfullydeveloped,stabilityanalysisisneeded. Chapter3containsthelinearstabilityoftheabovePoiseuilleflowforstablystratified case.Foragivenannulus,thestabilityofthebasicflowiscontrolledbydifferentparameters suchasReynoldsnumber(Re),Rayleighnumber(Ra),Darcynumber(Da),Prandtlnumber (Pr),heatcapacityratio(s ),viscosityratio(L),porosity (e ), andmodifiedForchheimer iii number(F′).Sincecurvatureparameter(C) playsavitalroletodescribethesizeofthe annulus,thereforeimpactof C onthetransitionmechanismofbasicflowforrelatively highpermeablemediumisconsideredinthischapter.Toavoid numerousparametricstudy wehavefixedthevalueofsomeoftheparameterssuchas L, Pr, and s at1,7and1, respectively.Thedisturbancemomentumandenergyequationsarenumericallysolved byspectralcollocationmethod.Wehavealsoanalyzedtheenergybudgetspectrumat criticalpoint.Thelinearstabilityresultsshowthatincreasing C aswellasdecreasing Da stabilizesthebasicflow.However,beyond C = 10theimpactofcurvatureparameteron thestabilizationofthebasicflowisalmostnegligible.From theenergyanalysisatcritical levelitisobservedthatthethermal-buoyantinstabilityis theonlymodeofinstability. Furthermore,theanalysisoflinearstabilityshowsthatalthoughtheimpactofformdrag uptoathresholdvalueisnegligibleoninstabilitybutitscontributioninenergydissipation issignificant. InChapter4,wehaveinvestigatedthestabilityofstablystratifiednon-isothermalPoiseuille flowofwaterinverticalporous-mediumannulususingweakly nonlinearstabilitytheory, withparticularemphasisontheimpactofgapbetweenthetwo verticalaxisymmetriccylin- ders.Foracomparativestudy,wehaveconsideredthreedifferentvalues(10−3,0.6,10)of C forthreedifferentvalues(10−1, 10−2, 10−3) of Da. Theflowintheannulusisgoverned bythevolume-averagedformsoftheNaiver-Stokesandcontinuityequationsderivedby [117].Tocarryouttheweaklynonlinearanalysis,westarted byanalyzingtherangeof Ra, beyondthecriticalpoint,inwhichthegrowthratevarieslinearlyusingperturbationseries solutionapproach.Fromthisanalysisithasbeenfoundthat forhighpermeablemedium thelinearrelationshipbetweengrowthrateand Ra holdsgoodforverysmallneighborhood ofcritical(bifurcation)point,howeverforlowpermeable mediumitisrelativelylarge. Thisgivesanimpressionthatthenonlinearinteractionisnoteffectiveforlowpermeable medium,whichisalsosupportedbyfiniteamplitudeanalysis. Thefiniteamplitudeanaly- sispredictsboththesupercriticalaswellassubcriticalbifurcationatandinthevicinityof bifurcationpoint,whicharealsoinvestigatedbynonlinear energyspectrum.Theanalysis iv ofthenonlinearenergyspectrumforthedisturbancereveals thatincaseof Da = 10−2 or C = 10−3 aninstabilitythatissupercriticalforsomewavenumbermay besupercriticalor subcriticalatothernearbywavenumber.Theequilibriumamplitudeincreasesondecreas- ingthemediapermeabilityaswellasreducingthegapbetween innerandoutercylinders. Inthelimitingcase(i.e.,at C = 10)thefundamentaldisturbanceofstablystratifiednon- isothermalPoiseuilleflow(SSNPF)ofwaterinverticalchannelfilledwithporousmedium willhaveminimumamplitude.Theinfluenceofnonlinearinteractionofdifferentsuperim- posedwavesonsomephysicalaspects:heattransfer,frictioncoefficient,nonlinearenergy spectrum,andsteadysecondaryflowisalsoinvestigated.Investigationrelatedtoimpactof superimposedwavesonthepatternofsecondaryflow,basedon linearstabilitytheorygives animpressionthatcellsofflowpatternarejustshifted.This istheconsequenceofnegli- giblemodificationinthebuoyantproductionofdisturbance kineticenergyandsignificant modificationintherateoftheviscousdissipationofdisturbanceenergyfortheconsidered setofparameters. InChapter5,theinstabilitymechanismoftheaboveflowisanalyzedforunstablystrat- ifiedcase.Linearstabilityanalysispredictsfirstazimuthalmodeastheleaststablemode intheentirerangeof C for Da = 10−1 and10−3. For Da = 10−2 firstazimuthalmodeis alsotheleaststablemodeexceptfor0.02 ≤ C < 0.1 wherezeroazimuthalmodeisthe leaststablemode.However,for Da = 10−2 (exceptfor0.02 ≤C < 0.1)and10−3 theleast stablemodeat n = 1 isunderR-T(Rayleigh-Taylor)mode.Energyanalysisatcriticallevel showsthechangeinthecharacteristic:stabilizingtodestabilizing,ofdisturbedkineticen- ergyduetoshearfactor(Es) onchanging C for Da = 10−1 and10−2, whichisthecauseof changingtheshapeofsecondaryflowfromuni-cellulartobi-cellular.Moreover,depending onthemediapermeabilityaswellascurvatureparameterthreetypesofinstabilitynamely, thermal-buoyant,interactiveandRayleigh-Taylorareobserved.ThisRayleigh-Taylortype instabilityisindependentof Re andbecomestheleaststablemodein(| Rac |, Re)-planeon decreasing Da. C takesasignificantroleontheappearanceofRayleigh-Taylor instability. Althoughforstablystratifiedcasenorelationbetweentheappearanceofpointofinflection v andinstabilityoftheflowisobservedbutforunstablystratifiedwaterflow,theappear- anceofflowseparationisthesufficientconditionforinstability.Furthermore,toanalyze thenatureoftheRayleigh-Taylorinstabilityandthefinite amplitudebehaviorofunstable disturbancethatoccursbeyondthelinearstability,especiallywhenthepermeabilityofthe mediumisrelativelylowwehaveusedweaklynonlinearstabilitytheoryintermsoffinite amplitudeanalysis.OuranalysisonLandauconstantandamplitudeasafunctionof Ra revealstwoimportantfacts.First,forboth Da = 10−2 and10−3 dependingon C aswell as Ra, Rayleigh-Taylorinstabilityshiftsfromsupercriticalto subcritical(reverse)atand beyond Rac. Second,theamplitudeprofileexperiencesasuddenjumpwheneverthetype ofinstabilitychangesawayfromthecriticalpoint. Finally,Chapter6presentsthesummaryandconcludingremarksofthisthesisandthe possibledirectionsofthefuturescope.