Numerical Investigation of Particulate Gravity Currents

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Numerical Investigation of Particulate Gravity Currents
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1 online resource (240 p.)
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english
Creator:
Shringarpure, Mrugesh Surendra
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University of Florida
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Gainesville, Fla.
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering, Mechanical and Aerospace Engineering
Committee Chair:
BALACHANDAR,SIVARAMAKRISHNAN
Committee Co-Chair:
MEI,RENWEI
Committee Members:
UKEILEY,LAWRENCE S
VALLE-LEVINSON,ARNOLDO
CANTERO SPOSETTI,MARIANO IGNACIO

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Subjects / Keywords:
capacity -- carry -- currents -- geohphysical -- gravity -- suppression -- turbdity -- turbulence
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
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Mechanical Engineering thesis, Ph.D.
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government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Abstract:
Gravity or density currents are horizontal flows driven by hydrostatic pressure gradients caused due to density differences under the influence of gravity. In other words, gravity currents are caused when fluids with different density interact in an unconstrained environment. This involves a current spreading into the column of ambient fluid at the bottom (heavier than the ambient) or top (lighter than the ambient) or at some height from the bottom (the ambient is stratified and the current density lies between the stratification limits). Gravity currents are ubiquitous in nature and various industrial scenarios. This work considers gravity currents that are driven by active scalars like sediments, dust particles, powder snow etc, and initiated by continuous discharge of current. In particular I will consider submarine particulate gravity currents also known as turbidity currents. Three aspects of these currents are analyzed in detail. In this work it is shown that the front condition of gravity current initiated by constant discharge into the ambient can be substantially influenced by ambient flow direction. From theoretical considerations an expression for front condition based on the current depth ratio and ambient flow direction is derived. These results are validated using numerical simulations that were performed as a part of this work and experimental observations available in the literature. Currents initiated by variable inflow are also simulated and their front condition is compared with steady fronts. The interaction of turbulence and stratification caused by sediments inside turbidity currents is studied using a mathematical model that considers mono and bi disperse suspensions of sediments. Through numerical simulations,turbulence suppression mechanisms controlled by stratification are identified and characterized using the parameters that govern the flow. The key parameters are Reynolds number (measure of flow intensity), Richardson number (measure of stratification) and sediment particle settling velocity (measure of particle size). Parametric grouping Richardson number x Settling velocity is identified that quantifies turbulence suppression. A scaling relation for this parametric grouping with the flow intensity is also proposed. It is also shown that adding fine sediments into the suspension can increase the carrying capacity of turbidity currents.
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Mrugesh Surendra Shringarpure.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
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Adviser: BALACHANDAR,SIVARAMAKRISHNAN.
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Co-adviser: MEI,RENWEI.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-06-30

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NUMERICALINVESTIGATIONOFPARTICULATEGRAVITYCURRENTS By MRUGESHSURENDRASHRINGARPURE ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2013

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c r 2013MrugeshSurendraShringarpure 2

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Tomyparents,SurendraandUmaShringarpure 3

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ACKNOWLEDGMENTS IwouldliketoexpressmydeepestgratitudetomyadvisorsPr ofessorS.BalachandarandDr.MarianoI.Cantero.Withouttheirguidance andsupportIwouldhave perhapsneveraccomplishedthiswork.ProfessorBalachand ar'sroleasamentorand anadvisorhasbeenimmense.Myinteractionswithhimhaveal waysgivenmeafresh perspectiveandenrichedmewithdeepunderstandingofthes ubject.Hehasalways beenencouraging,kindandpatienttowardsme.Igreatlyadm irehispassionforscience andithasbeenandwillremainaconstantsourceofinspirati onforme. IcannotemphasizetheimportanceofDr.Cantero'sroleasan advisorenough. EarlyoninmyPh.D.program,whenProfessorBalachandarwas thechairmanof departmentofMechanicalandAerospaceengineeringandhad hishandsfullwiththe administrativerole,Dr.Canterolledtheroleofamentora ndanadviserextremelywell. EventhoughhewasthousandsofmilesawayinArgentinaitdid notlimitourinteraction; wehadinnumerablechatsandvideocallsthathelpedmetogra pplewithandtake controloftheresearch.Iamgratefultohimforbeingsuchag reatadviserandavery goodfriend. Iamalsogratefultomyproposalandnaldefensecommittee: Professors LawrenceUkeiley,RenweiMeiandArnoldoValle-Levinsonfo rtheirvaluablecomments, correctionsandsuggestiontomywork.Iwouldalsoliketoth ankalltheProfessorsat MechanicalandAerospacedepartmentunderwhomIhavetaken manycoursesand learnedsomuch. Iwouldliketothankallmyofcematesduringalltheseyears :Dr.HyungooLee,Dr. GordonTaub,Dr.MinaMankbadi,Dr.ManojParmar,Dr.Yue(St anley)Ling,Francesco Kuljevan,NadimNzgheib,Subramanian(Subbu)Annamalai,G eorgesAkiki,Roberto Diaz,CharlesCook,ChirstopherNealandGoranMarjanovicf ormakingresearchand lifeingeneralsomuchfun. 4

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ManythankstoallmyfriendsatuniversityofFloridaforthe soccergames,potlucks, gatornights,etc.YoumademystayatUniversityofFloridaf ulloffondmemoriesthatI willcherishallmylife.Aspecialthankstomyanc eeAvantiforherlove,patienceand encouragement.IwouldalsoliketothankmyparentsSurenda andUmawhohave givenmesuchagoodlife.Ialsowanttothankmywonderfulsis terMrunmayeeforher love.AbigthankyoutoallmyrelativesandfriendsbackinIn dia. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ..................................4 LISTOFTABLES ......................................9 LISTOFFIGURES .....................................10 ABSTRACT .........................................14 CHAPTER 1INTRODUCTION ...................................16 1.1Motivation ....................................18 1.2ScopeandObjectivesofthisDissertation ..................24 1.3StructureoftheDissertation ..........................25 2FRONTCONDITIONSOFHIGH-REGRAVITYCURRENTSPRODUCEDBY CONSTANTANDTIME-DEPENDENTINFLUX ..................29 2.1Background ...................................29 2.2ASimpleAnalysisofReturnFlowEffect ...................33 2.3NumericalApproach ..............................36 2.3.1ProblemFormulation ..........................36 2.3.2BoundaryConditionsandNumericalScheme ............39 2.4Results .....................................42 2.4.1ConstantInow( =1 ) ........................42 2.4.2VariableInowCurrents( 6 =1 ) ....................47 2.5Deductions ...................................53 3DYNAMICSOFCOMPLETETURBULENCESUPPRESSIONINTURBIDITY CURRENTSDRIVENBYMONODISPERSESUSPENSIONSOFSEDIMENTS 71 3.1Background ...................................71 3.2ProblemFormulation ..............................74 3.2.1MeanFlowEquations .........................76 3.2.2DimensionlessEquations .......................78 3.3NumericalMethod ...............................80 3.4LaminarSolution ................................80 3.5Results .....................................82 3.5.1MeanValuesandTurbulentFluxesinStratiedFlows ........83 3.5.2IsolatedEffectsofNon-UniformStreamwiseForcing ........86 3.5.3TurbulentKineticEnergyBalance ...................88 3.5.4EnergySpectra .............................92 3.5.5ReynoldsStressBalance .......................93 3.5.6ReynoldsStressEvents ........................95 6

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3.5.7Near-BedCoherentStructuresandConcentrationFluc tuations ..98 3.5.8MechanisticViewofTurbulenceSuppression ............101 3.6Deductions ...................................102 4TOWARDSAUNIVERSALCRITERIAFORTURBULENCESUPPRESSION INDILUTETURBIDITYCURRENTSWITHNON-COHESIVESEDIMENTS ..121 4.1Background ...................................121 4.2Methods .....................................123 4.3EvidenceofTotalTurbulenceSuppression ..................124 4.4TheoreticalConsiderations ..........................126 4.5ReynoldsNumberScaling ...........................127 4.6Deductions ...................................129 5ANALYSISOFTURBULENCESUPPRESSIONINTURBIDITYCURRENTS DRIVENBYBI-DISPERSESUSPENSIONSOFSEDIMENTS .........134 5.1Background ...................................134 5.2ProblemFormulation ..............................137 5.3DimensionlessEquations ...........................139 5.4LaminarSolution ................................144 5.5Results .....................................146 5.5.1CompleteTurbulenceSuppressionCriteriaforBidisp reseSuspensions .................................149 5.5.2TurbulenceStatisticsofFlowswithNearCriticalTur bulenceDamping ....................................151 5.5.3TurbulenceStatisticsofFlowswithSimilarTurbulen ceDamping ..155 5.6CarryingCapacityofaTurbidityCurrent ...................156 5.6.1ExperimentalEvidence .........................157 5.7Deductions ...................................160 6HYBRIDOPENMP-MPIPSEUDO-SPECTRALCODE:VALIDATION&OPT IMIZATION .......................................180 6.1ModifyingGoverningEquationsandRelatedParameters ..........184 6.2Validation ....................................185 6.2.1RecoverLaminarSolution .......................185 6.3FullyTurbulentSolution ............................186 6.4Optimization ..................................188 6.4.1OptimizeMatrixMultiplicationOperation ...............188 6.4.2OptimizeFourierTransformOperation ................190 6.4.3OptimizeCommunication .......................191 6.5TimingtheNewOptimizedHybridCode ...................195 7CONCLUSIONSANDFUTUREWORK ......................215 7.1Conclusions ...................................215 7

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7.2FutureWork ...................................220 APPENDIX AANOTHERDERIVATIONOF Fr r ..........................223 BEVALUATIONOFDEPTHRATIOFOR =1 CASE ...............225 CMATHEMATICALMODELOFTURBIDITYCURRENTSDRIVENBYPOLYDISPERSESUSPENSIONOFSEDIMENTS ...................226 C.1DimensionlessEquations:SelectionofAppropriateSca les ........228 C.2CompleteTurbulenceSuppressionCriteria .................230 REFERENCES .......................................233 BIOGRAPHICALSKETCH ................................240 8

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LISTOFTABLES Table page 2-1Listofsimulationswithconstantinow .......................56 2-2Listofsimulationswithvariableinowcurrents ..................57 3-1Listofsimulationsofturbiditycurrentsdrivenbymono -dispersesuspension ..105 3-2Turbulentkineticenergybudget ...........................105 3-3Reynoldsstressbudget ...............................105 4-1Listofdirectnumericalsimulations(DNS)ofturbidity currents ..........131 4-2Listofexperimentalandeldobservations .....................131 5-1Listofsimulationsofturbiditycurrentsdrivenbybi-d ispersesuspension ....164 5-2Turbulentkineticenergybudgetofallthenear-critica lcases ...........165 5-3Turbulentkineticenergybudgetofsub-criticalcases ...............165 6-1Summaryoftransposeoperationsintheoldandthenewopt imizedversionof thecode ........................................197 6-2Comparisonoftheexecutiontimeofvarioustasksintheo ldandnewversion ofthecode ......................................197 9

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LISTOFFIGURES Figure page 1-1Schematicrepresentationofagravitycurrent. ...................28 2-1Aschematicofbottomgravitycurrentowingintoanambi entcolumn. .....55 2-2Controlvolumearoundthefrontofthecurrentfordiffer entoutowconditions .55 2-3Schematicrepresentationofthecomputationaldomain .............57 2-4Timeevolutionofthecurrentfordifferent Fr in values ...............58 2-5Frontvelocity( ~ u N )andfrontheight( ~ h N )basedontheboxassumptionfor = 1 cases ........................................59 2-6 f ( Fr in )= h N = h N ( Fr in =1.00) prolethatcollapsestheeffectof Fr in .......59 2-7Prolesof Fr N asafunctionoftimefor =1 cases ................60 2-8EffectofoutowboundarylocationonthefrontFroudeva luefordifferentvaluesof ........................................60 2-9Comparisonoftheactualvolumeofcurrentinthesimulat iondomainwiththe exactvalueforvariousinowconditions ......................61 2-10Densitycontoursof Fr in =1.00 and =1.25 simulationatdifferentinstances .62 2-11Densitycontoursof Fr in =1.00 and =0.90 simulationatdifferentinstances .63 2-12Prolesoffrontvelocity( ~ u N )andfrontheight( ~ h N )for =1.25 .........64 2-13Prolesoffrontvelocity( ~ u N )andfrontheight( ~ h N )for =0.9 ..........65 2-14Prolesofscaledfrontvelocity u andscaledfrontheight h for =1.25 ...66 2-15Prolesofscaledfrontvelocity u andscaledfrontheight h for =0.90 ...67 2-16ProlesoffrontFroudenumber( Fr N )for =1.25 and =0.90 .........68 2-17Comparisonof Fr N ofcertaincasesof =1.25 with Fr B and Fr NR .......69 2-18Comparisonof Fr N ofcertaincasesof =0.90 with Fr B and Fr NR .......70 3-1Schematicrepresentationofthemodelofaturbiditycur rent ...........106 3-2 ~ c and ~ u prolesofvariouscases ..........................106 3-3Meanvelocityprolesofvariouscasesinwallunits ...............107 3-4Variationin ~ u t ~ c b and ~ c rms b withthesettlingvelocity ...............107 10

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3-5Prolesof ~ w 0 ~ c 0 = ~ V z ~ c and ~ u 0 ~ w 0 forvariouscases ................108 3-6Prolesofrmsvelocityuctuations .........................109 3-7 ~ c and ~ u prolesofcaseswherestraticationeffectsaresuppresse d ......110 3-8 ~ w 0 ~ c 0 = ~ V z ~ c and ~ u rms prolesofcaseswherestraticationeffectsaresuppresse d 111 3-9TKEshearproductionandTKEdissipationproles ................112 3-10Prolesofsediment-inducedTKEdampingandtheratioo fsediment-induced TKEdampingtoTKEproduction ..........................112 3-11Energyspectraat z + 12 z + 40 and z + 150 ................113 3-12ProlesofReynoldsstressproductionandsediment-in ducedReynoldsstress damping ........................................114 3-13ProlesofweightedprobabilitydensityfunctionofQ2 eventsandprobability densityfunctionofQ2eventsatbed-normallocationof z + 18 .........115 3-14Contoursof ~ u 0 ~ w 0 and ~ c 0 forcase1at z + 12 ..................116 3-15Contoursof ~ u 0 ~ w 0 and ~ c 0 forcase5at z + 12 ..................117 3-16Iso-surfacesof ci forcase0 ............................118 3-17Iso-surfacesof ci forcase5 ............................119 3-18Iso-surfacesof ci forcase6 ............................120 4-1Meanconcentration( ~ c )prolesofvariouscases .................131 4-2Bed-normalprolesofTKE( k = u 2 )andTKEproduction( P H = u 3 ) ........132 4-3Evolutionofbed-normal-integratedturbulentkinetic energyproductionand dissipationwithReynoldsnumber .........................132 4-4Evolutionof K c with Re ...............................133 5-1 ~ c c + r f and ~ u prolesofcasesfromsetB .....................165 5-2 ~ u 0 ~ w 0 and ~ w 0 ~ c 0 c prolesofcasesfromsetB ...................166 5-3 ~ u rms ~ w rms and ~ v rms prolesofcasesfromsetB ..................167 5-4 ~ c c + r f and ~ u prolesofnear-criticalcasesfromeachset ............168 5-5 ~ u rms ~ w rms and ~ v rms prolesofcriticalcasesfromeachset ............169 5-6 ~ u 0 ~ w 0 and ~ w 0 ~ c 0 c = ~ V cz ~ c c prolesofnear-criticalcasesfromeachset ......170 5-7 ~ V cz ~ c c and ~ w 0 ~ c 0 prolesofcriticalcasesfromeachset .............171 11

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5-8 ~ c crms prolesofnear-criticalcasesfromeachset .................172 5-9TKEProductionandTKEDissipationprolesofnear-crit icalcasesfromeach set ...........................................173 5-10ProlesofTKEdampingofcriticalcasesfromeachset .............174 5-11 ~ c c + r f and ~ u prolesofsub-criticalcasesfromeachset .............175 5-12 ~ u rms ~ w rms and ~ v rms prolesofsub-criticalcasesfromeachset ..........176 5-13 ~ u 0 ~ w 0 and ~ w 0 ~ c 0 c prolesofsub-criticalcasesfromeachset ...........177 5-14TKEProductionandDissipationprolesofsub-critica lcasesfromeachset ..178 5-15ProlesofTKEdampingforsub-criticalcasesfromeach set ..........179 6-1BlockdiagramrepresentationofthehybridopenMP-MPIp seudospectralcode 198 6-2Blockdiagramrepresentationoftheconvertoperationt hattransformsdata betweenrealspaceandcomputationalspace ...................199 6-3Blockdiagramrepresentationoftheconvertoperationt hattransformsdata betweenenlargedrealspaceandcomputationalspace .............199 6-4Velocityproleoflaminarchannelowdrivenbyconstan tpressuregradient thatisobtainedfromthehybridcodesimulationandtheanal yticalsolution ..200 6-5Laminarvelocityandconcentrationproleforchannel owdrivenbymono dispersesuspensionobtainedfromhybridcodeandanalytic alsolution .....200 6-6Percentagerelativeerrorinthevelocityandconcentra tionforachannelow drivenbymono-dispersesuspensionofsedimentthatisobta inedfromhybrid codesimulation ....................................201 6-7 ~ u prolesofturbulentchannelowobtainedfromthehybridco deandtheold sharedmemorycode ................................201 6-8 ~ u rms ~ w rms and ~ v rms prolesofturbulentchannelowobtainedfromthehybrid codeandoldsharedmemorycode .........................202 6-9Prolesofpercentagerelativeerrorin ~ u rms ~ w rms and ~ v rms ofturbulentchannel owobtainedfromthehybridcode .........................202 6-10~ u 0 ~ w 0 prolesofturbulencechannelowobtainedfromhybridcode andold sharedmemorycode ................................203 6-11 ~ u and ~ c prolesofturbulentchannelowdrivenbymonodispersesus pensionofsedimentandsimulatedusingthenewhybridcodeando ldshared memorycode .....................................204 12

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6-12Percentagerelativeerrorin ~ u and ~ c foraturbulentchannelowdrivenbymono dispersesuspensionofsedimentandsimulatedusingthehyb ridcode .....204 6-13 ~ u rms and ~ w rms prolesofturbulentchannelowdrivenbymonodispersesus pensionofsedimentandsimulatedusingthehybridcodeando ldsharedmemorycode ........................................205 6-14Percentagerelativeerrorin ~ u rms and ~ w rms foraturbulentchannelowdriven bymonodispersesuspensionofsedimentandsimulatedusing thehybridcode 205 6-15 ~ v rms and ~ c rms prolesofturbulentchannelowdrivenbymonodispersesus pensionofsedimentandsimulatedusingthehybridcodeando ldsharedmemorycode ........................................206 6-16Percentagerelativeerrorin ~ v rms and ~ c rms ofturbulentchannelowdrivenby monodispersesuspensionofsedimentandsimulatedusingth ehybridcode .206 6-17 ~ u 0 ~ w 0 and ~ w 0 ~ c 0 prolesofturbulentchannelowdrivenbymonodisperse suspensionofsedimentandsimulatedusingthehybridcodea ndoldshared memorycode .....................................207 6-18Percentagerelativeerrorin ~ u 0 ~ w 0 and ~ w 0 ~ c 0 ofturbulentchannelowdriven bymonodispersesuspensionofsedimentandsimulatedusing thehybridcode 207 6-19Thesequenceofoperationsinvolvedincomputingthead vectiontermsinthe conservativeform. ..................................208 6-20Thesequenceofoperationsinvolvedincomputingthead vectiontermsinthe nonconservativeform. ................................208 6-21Descriptionofthesymbolsusedinalltheowcharts ...............209 6-22Anoptimizedmethodofcomputingtheadvectiontermsin theconservativeform 211 6-23Anoptimizedmethodofcomputingtheadvectiontermsin thenon-conservative form ..........................................213 6-24FlowchartshowingthesequenceofoperationsintheNCb lock .........214 A-1Thecurrentsystem(a)inthelaboratoryframe;(b)inafr amemovingwith u F = Ua (1 r ) totheright. ................................224 13

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AbstractofdissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy NUMERICALINVESTIGATIONOFPARTICULATEGRAVITYCURRENTS By MrugeshSurendraShringarpure December2013 Chair:Sivaramakrishnan,BalachandarMajor:MechanicalEngineering Gravityordensitycurrentsarehorizontalowsdrivenbyhy drostaticpressure gradientscausedduetodensitydifferencesundertheinue nceofgravity.Inother words,gravitycurrentsarecausedwhenuidswithdifferen tdensityinteractinan unconstrainedenvironment.Thisinvolvesacurrentspread ingintothecolumnof ambientuidatthebottom(heavierthantheambient)ortop( lighterthantheambient) oratsomeheightfromthebottom(theambientisstratiedan dthecurrentdensitylies betweenthestraticationlimits).Gravitycurrentsareub iquitousinnatureandvarious industrialscenarios.Thisworkconsidersgravitycurrent sthataredrivenbyactive scalarslikesediments,dustparticles,powdersnowetc,an dinitiatedbycontinuous dischargeofcurrent.InparticularIwillconsidersubmari neparticulategravitycurrents alsoknownasturbiditycurrents.Threeaspectsofthesecur rentsareanalyzedin detail.Inthisworkitisshownthatthefrontconditionofgr avitycurrentinitiatedby constantdischargeintotheambientcanbesubstantiallyin uencedbyambientow direction.Fromtheoreticalconsiderationsanexpression forfrontconditionbased onthecurrentdepthratioandambientowdirectionisderiv ed.Theseresultsare validatedusingnumericalsimulationsthatwereperformed asapartofthisworkand experimentalobservationsavailableintheliterature.Cu rrentsinitiatedbyvariable inowarealsosimulatedandtheirfrontconditioniscompar edwithsteadyfronts.The interactionofturbulenceandstraticationcausedbysusp endedsedimentsinturbidity 14

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currentsisstudiedusingamathematicalmodelthatconside rsmonoandbi-disperse suspensionofsediment.Throughnumericalsimulations,tu rbulencesuppression mechanismscontrolledbystraticationareidentiedandc haracterizedusingthe parametersthatgoverntheow.ThekeyparametersareReyno ldsnumber(measure ofowintensity),Richardsonnumber(measureofstratica tion)andsedimentparticle settlingvelocity(measureofparticlesize).Parametricg roupingRichardsonnumberx Settlingvelocityisidentiedthatquantiesturbulences uppression.Ascalingrelation forthisparametricgroupingwiththeowintensityisalsop roposed.Frombi-disperse suspensionsimulations,itisshownthataddingnesedimen tsintothesuspensionhelps thecurrentcarrylargesizesedimentparticles.Furthermo re,itleadstoincreaseinthe carryingcapacityofturbiditycurrents. 15

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CHAPTER1 INTRODUCTION Gravityordensitycurrentsarehorizontalowsthatarecau sedduetoexistence ofbulkdensitydifferencesinauidsystemundertheinuen ceofgravity.Toillustrate thisconsideradenseuidreleasedinacolumnoflightambie nt,thedensitydifference betweenthetwouids(undertheinuenceofgravity)genera teshorizontalpressure gradientsthatcausesthedenseuidtoowintheambientcol umnbydisplacingthe ambientuidoverit.Figure 1-1 showsschematicallythisprocessofspreading.Aprime exampleofthis,whichalmostallofushaveexperiencedisth egustofcoldairthatone feelsimmediatelyafterthedoortoanair-conditionedroom (orarefrigerator)isopened. Thisgustisnothingbutagravity/densitycurrenttriggere dbythedensitydifferencein theairinsideandoutsidetheroom(orrefrigerator).Other moreprominentexamples ofgravitycurrentsareduststorms,turbiditycurrents,de brisows,oilspills,estuaries releasingtheirwaterintotheocean,lahars,lavaows,sea breezefrontsetc(formore examplesseeAllen[ 3 ],Hopnger[ 39 ],Simpson[ 81 ]). Ingeneralgravitycurrentscanbebottom,toporintrusions .Whenthecurrent densityismorethantheambient,itwillowalongthebottom displacingtheambient overit.Similarly,whenthecurrentdensityislessthanthe ambient,itwillowontop oftheambientcolumn.Intrusionsareaspecialcasewhereth ecurrentowsatsome intermediateheightintheambientcolumn.Thisscenarioar iseswhentheambientisa stratiedcolumnandthecurrentdensityliesbetweenthetw oextremedensitiesinthe stratiedambient. Gravitycurrentscanalsobeclassiedasconservativeorno n-conservative.In conservativegravitycurrentsthebulkdensitydifference isconservedthroughoutits evolution.Forexample,whendifferentuidscomeincontac twitheachotherthe resultinggravitycurrentisconservativebecausethedens itydifferenceisduetotheir materialproperty.Furthermore,conservativegravitycur rentscanalsoresultfrom 16

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passivescalarslikesalinity,temperatureorchemicalspe ciesthatareonlyresponsible forimposingadensitydifference.Oilspills,dambreak,re leaseofcontaminantinto theambientsurrounding,etcareexamplesofconservativeg ravitycurrents.Notethat conservativegravitycurrentsaresometimesalsocalledth ermohalinedensitycurrents. Ontheotherhand,innon-conservativegravitycurrentsthe densitydifferencesare notconservedoveritsevolution.Considergravitycurrent slikeduststorms,debris ows,turbiditycurrents,pyroclasticowsetcthatareren dereddensebythedispersed phasedembeddedinthem.Thisdispersedphaseisfreelyexch angedwithhorizontal bedoverwhichthesecurrentsarespreadingthroughdeposit ionanderosion.Over time,thisfreeexchangecansubstantiallyalterthetotala mountofdispersedparticles carriedinsuspensionbythecurrent.Asaresultallthesecu rrentsareclassiedasnon conservativegravitycurrents.Itisapparentthatnon-con servativegravitycurrentswill existsaslongsastheyareabletoretainthedispersedphase responsibleforimposing thedensitydifference.Ingeneralthedispersedphaseisca lledanactivescalarasit iscloselycoupledwiththedynamicsoftheowandthismakes theanalysisofnonconservativegravitycurrentsmuchmorecomplexthantheir conservativecounterparts. Wecanalsoclassifygravitycurrentsintermsofthemethodo freleaseofcurrent intotheambient.Todescribethisclassication,letusrep resentthevolumeofcurrent releasedintotheambientas V = qt ,where t istime, q issomeconstantand isa nonnegativenumber. =0 representsxedvolumerelease,i.e.theentirevolumeof current( q )isreleasedintotheambientattime t =0 .When > 1 ,wecanseethat therateofreleaseofcurrent( V = qt 1 )intheambientincreaseswithtime.Such typeofcurrentsarecalledwaxingcurrents.When < 1 ,therateofreleaseofcurrent decreaseswithtimeandeventuallybecomeszeroas t !1 .andhencetheyarecalled waningcurrents.Mostofthegeophysicalowshaveacomplic atedmethodofrelease, butingeneraltheycanbemodeledasseveralsuccessiveinte rvalsofwaxing,waningor constantinowrate. 17

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Fromtheaboveitisclearthatgravitycurrentscanmanifest inavarietyofsituations,but,irrespectiveoftheirorigintheyexhibitsimil arowstructuresi.e.theyhave anintrudingfrontora”head”followedbya”body”anda”tail ”(onlyforxedvolume release).Figure 1-1 showsaschematicofagravitycurrent.Gravitycurrenthead forms theleadinginterfacebetweenthecurrentandtheambient.H eadisalocationofintense motionandmixing.Ifthevolumeofcurrentreleasedintothe ambientislargethen theheadisfollowedbyabodywhichcomprisesofcurrentwith nearlyuniformheight. Kelvin-Helmholtzbillowsareobservedattheinterfacebet weenthebodyofthecurrent andtheambientuid.Beyondthebodyisthetailofthecurren t(tailisabsentincaseof continuousreleasecurrents).Inthisregion,thecurrentd ensityissubstantiallyreduced byheavyentrainmentofambientuidanditformstheregionw herethecurrentdissipatesintotheambientuid.Iftheinitialvolumeofthecurr entissmallthentheheadis immediatelyfollowedbythetailofthecurrent.Mostoftheg ravitycurrentsobservedin natureandinindustrialapplicationsarestrongenoughtoh aveintensevortexshedding andhighmixingalongitsinterfaces.Invariablythecurren tisintheturbulentregime andhasowfeatureswithwiderangeoflengthandtimescales .Tosummarize,gravity currentsareextremelyturbulentmaterialfrontsthatareq uitecommoninnaturaland industrialsituations. 1.1Motivation Spreadingofoneuidintoanotherisafundamentalprocessi nmanygeophysicalandindustrialsituations.Evensmalldensitydifferen ces(lessthan 1% orusually calledastheBoussinesqlimit)cantriggerlarge,extremel yenergeticgravitycurrents thatmakesthemanimportanttransportmechanismofmassand energy.Following exampleswillhighlightthedisruptiveanddestructiveabi lityofgravitycurrents.During volcaniceruptions,largequantitiesofashmaybedumpedin totheatmosphere.This ashmakesthesurroundingairheavyandspreadasanintrusio ncoveringlargedistances.Thisspreadingcanhaveacatastrophicimpactonthe weatherandtheeconomy 18

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ofthesurroundingregion.RecentvolcaniceruptioninIcel and(April-May2010)dumped largequantitiesofashintotheatmospherethatspreadover largeareasofnorthern Europeandbroughtaboutacompleteshutdownofcommerciala irtrafcoverwestern andnorthernEurope.Similarly,gravitycurrentstriggere dbymanmadedisasterslikeoil spillsarealsoknowntocausetremendousdamagetothemarin eecosystem.Thedeep waterhorizonoilspillinthegulfofMexicohadahugeenviro nmentalandeconomic impact.Othernaturalphenomenonlikeearthquakes,tsunam iscansometimestrigger extremelyenergeticunderwaterparticulategravitycurre ntsthatcandestroysubmarine communicationcables[ 50 ],damageoilpipelinesandotherequipmentsonthesea oor[ 44 46 73 ].Manymoreexamplesofextremelyenergeticgravitycurren tsthatare responsiblefortransportinglargequantitiesofmaterial overhugedistancescanbe foundintheliterature(seeHuppert[ 43 ]formoreexamples).Thereforeunderstanding thistransportprocessbecomesextremelyimportantnotonl yfromascienticpointof viewbutalsofromthepointofviewofsafetytohumansandoth ermanmadestructures fromitsdestructivepower. Itisobviousfromtheaboveexamplesthatmostofthesecurre ntscanbeclassied ascurrentsproducedbycontinuous(variableorconstant)r eleaseintotheambient. Although,theseowshavebeenextensivelystudiedoverpas t30to40years,gravity currentsgeneratedduetoreleaseofxedvolumehasreceive dfarmoreattention. Fixedvolumereleasecurrentsareconceptuallysimpleandi tiseasytosetuplaboratoryexperimentstoanalyzethem.Someofrealworldproblem slikesuddenreleaseof contaminantinthesurrounding,powdersnowavalanchestri ggeredbyfailuresinaccumulatedsnow,debrisowtriggeredbycollapsingbuilding, etc.,canbeconveniently modeledasxedvolumereleases(alsocalledaslockexchang e).Overthepastseveral years,withhelpofexperiments[ 41 42 74 78 ],variousmathematicalmodelsandtools likeboxmodel[ 42 ],shallowwaterequations[ 33 34 85 86 ]etcandmorerecentlywith helpofhighdelitydirectnumericalsimulations[ 16 22 35 36 ],spreadingofplanar 19

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andaxisymmeticcurrentscausedbysuddenreleaseofcurren thavebeencharacterized ingreatdetail.Furthermore,withtheadventofpowerfulco mputers,directnumerical simulationsofhighReynoldsnumber,turbulentcurrentsha verevealedingreatdetailthe roleofturbulentstructuresintheevolutionofthecurrent [ 16 22 69 ]andthegrowthof variousinstabilitieslikelobeandcleft[ 35 ].Ontheotherhand,gravitycurrentsinitiated bycontinuousreleasehavereceivedfarlessattention.One oftherstexperiments oncontinuousinowcurrentwereperformedbyDidden&Maxwo rthy[ 27 ],Huppert [ 41 ],Maxworthy[ 55 ].Grundy&Rottman[ 34 ]investigatedplanarandaxisymmetric currentsproducedbyvariableinowandpresentedselfsimi larsolutionstotheshallow waterequations.Gratton&Vigo[ 33 ]revisitedtheshallowwaterequationsforinertial gravitycurrentsproducedbyvariableinowandpresenteds olutionforwiderangeof densityratios(ratioofdensityofcurrenttodensityofamb ient).Boxmodelshavealso beenproposedtoanalyzevariableinowgravitycurrents(d etaileddescriptionofbox modelforplanarandaxisymmetricowscanbefoundinUngari sh[ 87 ]).Severalofthe abovemodelsandmathematicalformulationsinvolvesevere simplicationslikeinterfacialfrictionbetweenthecurrentandambientisneglected, frictionbetweenthecurrent atthebottomsurfaceisneglected,motioninsidethecurren tisassumedtobeuniform andtheambientcolumnheightisassumedtobeinnity(large ascomparedtocurrent height,alsocalledasdeepambientassumption),etc.Furth ermore,thepredictivecapabilityofthesemodelsdependonusinganappropriatefrontc onditionforthecurrent(the frontconditionforthecurrentcanbedenedasafrontFroud enumber Fr N = u N = p g 0 h N where u N isthefrontvelocity, h N isthefrontheight, g 0 =( c a ) = a isthereduced gravity, c isthedensityofthecurrentand a isthedensityoftheambient).Oneofthe earliestanalysisofthefrontconditionwasdonebyvonK arm an[ 45 ]andlaterbyBenjamin[ 9 ].TheanalysisbyBenjamin[ 9 ]considersasteadyfrontmovingintoanambient column.Fromcontrolvolumeanalysisaboutthefrontofthec urrentanexpressionfor Fr N wasobtainedintermsofthedepthratio a = h N = H ( h N istheheightofthecurrent 20

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frontand H istheheightoftheambientcolumn).Sincethefrontofacont inuousrelease currentcanbeunsteady,thisclassical Fr N formulationcannotbeapplieddirectly.A rigorousevaluationoffrontconditionforvariableinowc urrentsisneededandremains anopenquestion.Furthermore,anyproposedanalyticaldes criptionwillhavelimitations thatneedtoassessedthroughdetailedexperimentsandnume ricalsimulations. Clearly,analyzingcontinuousinowgravitycurrentspres entsaformidablechallengewhichisfurthercompoundedwhenthecurrentsarenonconservativeinnature. Althoughtheunderlyingmechanismsofmanyofthenon-conse rvative(orparticulate) gravitycurrentsaresimilar,thenatureofthedispersedph aseanditsinteractionswith theambientcanchangesubstantiallyfromonescenariotoan other.Forexample,the inertiaofdispersedphaseparticleshasastronginuenceo nthedynamicsoftheow. Dispersedparticleswithsmallornegligibleinertiatendt ofollowtheuidandtheresultinggravitycurrentmaybequitesimilartoitsconserva tivecounterpart.Ontheother hand,dispersedparticleswithnon-negligibleinertiawil lhaveastrongcouplingwith thedynamicsoftheow.Owingtoitsnon-negligibleinertia suchparticleswillexhibit fascinatingbehaviorlikepreferentialaccumulationandt urbophoreticmigrationetc.Furthermore,inagravitationaleldtheseparticleswilltend tosettledownorrainoutofthe current.Basedontheinertiaofthedispersedparticlessom eofthesebehaviormaytend tobeprominentascomparedtoothers.Consideringhewidega mutofnon-conservative gravitycurrentscenarios,thisthesiswillrestrictthesc opetosubmarineparticulate gravitycurrents(alsoknowsasturbiditycurrents). Tubiditycurrents: Turbiditycurrentsaresubmarineparticulategravitycurr entsthat arecharacterizedasowswithdilutemassconcentrationof sediments(Boussinesqlimit applies),consistingofnesiltandsandparticles(sizeca nvaryfrom 10 m to 200 m ), highReynoldsnumberowsandcanlastforminutestodays[ 68 ].Theycanbetriggered byvariousmechanismslikesubmarinevolcaniceruptions,e arthquakes,unstable accumulationofsedimentsalongtheslopingoceanoor,wav esupportedsediment 21

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gravityows,dumpingofsedimentrichwaterattherivermou th,etc.Onceinitiated therunoutdistanceofthecurrentdependsonthebalancebet weenthesediments suppliedtotheowandthecapacityofthecurrenttoretaint heminsuspension.Since turbulentmixingisthesolemechanismbywhichthesediment sarekeptinsuspension, understandingtheinteractionofturbulenceandthesedime ntparticlesiscriticalfor predictingtheevolutionoftheow. Turbiditycurrentsareextremelyenergeticandcanerodema ssiveamountsof sedimentstocarveoutvariousfeaturesontheoceanoor.So meofthesubmarine canyonsarebelievedtobecarvedoutbyprolongederosionca usedbyturbiditycurrents [ 29 ].Turbiditycurrentsarealsoknowntoattainhighspeedand cantravelforseveral hundredkilometers[ 48 ].Thesecharacteristicswerehighlightedfromtheanalysi sof 1929Grandbanksslopefailure.Oceanographersanalyzedth edepositsassociatedwith theslideandproposedthataseismiceventproducedaturbid itycurrentthattravelled rapidlyatthespeedofabout 10 m = s andbrokemanytransatlanticcommunication cables.Theestimatesforthespeedwerereasonablyaccurat ebecausetheexacttimes whenthetransatlanticcommunicationsbrokewasknownandt hisinformationwasused tobackcalculatethecurrentspeed[ 50 ].Owingtothesepropertiesturbiditycurrents areoneofthemostimportantmechanismsofsedimenttranspo rtinthesubmarine world.Infactturbiditycurrentsplayacrucialroleintran sportingorganicmatterfrom thecontinentalshelf,deepintotheoceanwheretheyaretra nsformedintohydrocarbon reservoirsovergeologicaltimescales.Thishasbeenoneof theprimemotivationsfor petroleumgeologisttostudythetransportpropertiesoftu rbiditycurrentsandtodevelop modelstointerpretsedimentarystructuresofsandstonere servoirswiththedeposition patternsofturbiditycurrents(seeBouma[ 12 ]).Ontheotherhandresearchershave proposedconditionsunderwhichtheturbiditycurrentisen ergeticenoughtocause neterosionofsedimentsfromtheoor.Suchcurrentsarecal ledselfaccelerating currents.Theconditions(calledignitionconditionsinth eliterature)underwhichthis 22

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stateisachievedbyturbiditycurrentswereproposedbyBag nold[ 8 ],Fukushima etal. [ 29 ],Parker[ 64 ],Parker etal. [ 66 ].AndmorerecentlyexperimentsbyPantin &Franklin[ 62 ]andSequeiros etal. [ 77 ]wereabletoproducesuchcurrentsinthe laboratory.Insummaryturbiditycurrentscanbethoughtto existin3statesormodes, namely,depositional(thereisnetlossofsedimentsfromth ecurrenttotheoceanoor), erosionalorselfaccelerating(thereisnetentrainmentof sedimentsfromtheocean oor)andbypassorautosuspensionmode(thereiszeronetu xofsedimentsfromthe currenttotheoceanoor). Typicallyanyanalysisofturbiditycurrentswillhaveoneo rmoreofthefollowing objectives:1.Topredicttheextentoferosionofsedimentsfromtheocea noororthesediment carryingcapacityofthecurrent. 2.Topredictthefeaturesofturbiditedepositslikethegra in-size,distributionof grain-sizeandtheextentofdepositionetc.Similarlytopr edictbedmorphology. 3.Topredicttheextentofowandthespeedofthecurrent.Th ishelpsinestimating thedangersoflayingdowncommunicationcablesorpipescar ryingoilincertain regionsintheocean. 4.Heightofthecurrentandthespatialevolutionoftheow.Inthepastfewdecadesmanynumericalmodelsbasedonshallo wwaterequations (equationsbasedonboundarylayerapproximations)havebe enproposedtoanalyze turbiditycurrents.Inmostofthemodels,sedimentsareass umedaspassivescalar andhencehaveuniformconcentrationinthecurrent.Someof thenewermodelshave proposedtouseturbulentmixingclosuresinsidethecurren tandalongtheinterfaceto modelverticallyvaryingsedimentconcentration.Ingener alsuchmodelsareassembled byusingvariousapproximationsorempiricalrelations(i) forsedimententrainmentform thebedlikeGarcia&Parker[ 30 ],(ii)forturbulentmixinginsidethecurrentandalong theinterfaceliketheoneproposedinParker etal. [ 67 ]or ortheturbulencemixing modelproposedbyMellor&Yamada[ 57 ]and(iii)forthefontconditiontheyusethe 23

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empiricalformulationgivenbyHuppert&Simpson[ 42 ]andthensolveshallowwater equations.Therearemanyfundamentalquestionsthatneeda ddressingintheabove modelstoimproveitsoverallpredictivecapability.Someo fthequestionsareasfollows 1.Howdoesthefrontconditionbehaveforcontinuousinowg ravitycurrents? 2.Whatisthedependenceoffrontconditionofthecurrenton theambientow direction? 3.Howdoessedimentstraticationinsidethecurrentaffec tturbulence? 4.Whatarethemechanismsthatmaketurbiditycurrentsdepo sitional? 5.Whatistheroleofpoly-dispersityinturbiditycurrents andwhatisitsinuenceon themechanismsthatmaketurbiditycurrentsdepositional? 6.Givenagravitycurrentofcertainintensity,canwedene itssedimentcarrying capacity? Therefore,welldesignedexperimentsandhighdelitysimu lationplayacrucialrolein probingsuchaspectsofturbiditycurrentsandtoimproveth eoverallpredictiveabilityof themodels. 1.2ScopeandObjectivesofthisDissertation Nonconservativegravitycurrentsinitiatedbycontinuous inowofcurrentexhibit fascinatingphysics.Therearemanywaysofapproachingthi scomplexproblem, butbroadlywecanclassifythemintothree,(i)Analytical, (ii)Experimentaland(iii) Numerical.Analyticalanalysiswillinvolvemathematical lydescribingtheowand formulatingitscomplexityintosimplermathematicalmode ls.Laboratoryexperiments andeldobservationshelpusinvisualizingtherealow,va lidateanalyticalmodels anddevelopempiricalrelationships.Numericalapproachc ansometimesbeagreat alternativetoexperiments.Itcanprovidegreatexibilit yinexploringthecompeting mechanismsintheowinisolationandthereforerevealimpo rtantinsightsthathelpin improvingtheanalyticalmodels.Irrespectiveoftheprosa ndconsofeachapproach, allofthemhavetobeemployedincomplimentarymannerinord ertodescribethe dynamicsofsuchcomplexowsaccurately. 24

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Therecentadvancementsincomputershaveallowedustoperf ormhighdelity simulationsatincreasinglyhigherReynoldsnumbers.This providesuswithanextremelyusefultooltostudytheseowsandderivedetailedi nformationofthenest structuresassociatedwiththeow.Thisabilityisfundame ntalindevelopingahierarchal approachtowardsmodelingtheseows.Nowwecangatherinfo rmationfromthenest structuresintheowandlinkthemtothemacroscopicfeatur esthatareeasilyobserved inexperimentsandeldobservations.Thisgreatlyboostso urabilitytobuildmacroscale modelsthatarebasedonsoundphysicsandincorporatethede tailedinformationofthe inuenceofmicro-scalestructuresontheow. Theobjectivesofthepresentworkistoaddresssomeofthefu ndamentalquestions presentedintheprevioussection.Morepreciselytheobjec tivesare(i)todevelopsimpliedmathematicalmodelsofturbiditycurrentsandcarry outhighdelitysimulations soastoexploretheinteractionofturbulenceandsediments tratication,(ii)toidentify mechanismsthatareresponsibleforsuppressingturbulenc einturbiditycurrents,(iii) toquantifytheturbulencesuppressionmechanismsinterms oftheparametersofthe owandtodevelopscalingrelationsthatcanextrapolateth eseresultstoeldobservationsand(iv)toexploretheinuenceofbi-dispersityon theturbulencesuppression mechanismanddenesedimentcarrycapacityofturbiditycu rrents. 1.3StructureoftheDissertation Thisthesiscontains7chapters:chapter1istheintroducti on,chapters2to6 documentanddescribethendingsofthisworkandchapter7p resentstheconclusions. Notethatchapters2to6areselfcontainedwiththeirownint roduction,literaturereview andashortconclusion. Chapter 2 dealswiththepropagationofaBoussinesqgravitycurrentw hichis createdbyaninowatthebottomofahorizontalrectangular container.Inparticular, attentionisfocusedonthefrontconditionofthecurrentwh ichischaracterizedbythe Froudenumber Fr N = u N = ( g 0 h N ) 1 = 2 .Followingaproceduresimilartotheonegivenin 25

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Benjamin[ 9 ],itisshownthatthefrontconditionissignicantlyinue ncedbytheposition oftheoutowboundary(i.e.,bythedirectionofowoftheam bientuiddisplaced bythecurrent).AnexpressionforthefrontFroudenumber( Fr N )asafunctionofthe depthratio a = h N = H ,whichtakesintoaccountthedirectionoftheambientow,i s presented.Twolimitingcasesareconsidered:no-returnam bientow(whentheoutow andinowareattheoppositeendwalls)andfull-returnambi entow(inowandoutow onthesameendwall).Navier-Stokessimulationsofconstan tinowcurrentsinnoreturncongurationwerecarriedout.Unsteadygravitycur rentsarealsosimulated byimplementingtimedependentinowboundaryconditions. Twosetsofsimulations werecarriedoutthatrepresentacceleratinganddecelerat ingcurrentsbyimplementing waxing(increasingintime)andwaning(decreasingintime) inowrates.Thendingsof thischapterhavebeenpublishedinShringarpure etal. [ 80 ]. Chapter 3 dealswithturbiditycurrentsdrivenbymono-dispersesusp ensionof sedimentsandthemainobjectiveistounderstandthedynami csofturbiditycurrentsin bypassmode.Inturbiditycurrentsthesettlingtendencyof sedimentsiscounteredby owturbulencewhichspendsenergytokeeptheminsuspensio n.Thisinteractionleads toadownwardincreasingconcentrationofsuspendedsedime nts(stablestratication) intheow.Thisstablestraticationcansubstantiallyred uceturbulenceandpossibly extinguishit[ 15 19 ].InthischapterweexpandthesimpliedmodelbyCantero etal. [ 15 ]andputforthapropositionthatexplainsthemechanismofc ompleteturbulence suppressionduetosuspendedsediments.Thendingsofthis chapterhavebeen publishedinShringarpure etal. [ 79 ]. Chapter 4 exploresthephenomenonwhereturbulenceinaturbiditycur rentis abruptlyextinguishedowingtoincreasedsuspendedsedime ntstratication.Three parametersareidentiedthatcontroltheowdynamics:Rey noldsnumber( Re ), Richardsonnumber( Ri )andsedimentsettlingvelocity( ~ V z ).Fromtheturbulentkinetic energyequation,theconditionforcompleteturbulencesup pressioncanbeexpressed 26

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asacriticalvaluefor Ri ~ V z .Thischapteralsoexploresthescalingrelationforthecri tical valueof Ri ~ V z withtheowReynoldsnumber( Re ).Thendingsofthischapterhave beenpublishedinCantero etal. [ 23 ]. Chapter 5 extendsthemono-dispersemodelofturbiditycurrentstoin cludebidispersesuspensions.Threesetsofsimulationsarecarrie doutsuchthatineachset thesettlingvelocityofthecoarsesedimentsisxedandthe proportionofcoarseto nesedimentsisvaried.Ineachsetwestartwithasuspensio ncontainingalotofne sediments( r f 1 ),whichkeepsthestraticationeffectstominimum.Insubs equent cases(suspensions),thequantityofcoarsesedimentsisin creasedattheexpenseof nesediments.Thisensuresthatthestraticationeffecti ntheowincreaseswhile keepingthesedimentloadconstant.Theresultsofthesesim ulationsarepresented inthischapter.Furthermore,theturbulencesuppressionc riteriaproposedinChapter 3 and 4 isextendedtobi-dispersesuspensions.Thischapteralsop roposesthatthe turbulencesuppressioncriteriacanalsobeinterpretedas asedimentcarryingcapacity ofthecurrent.Thisparadigmistestedbycomparingitwithe xperimentalresults.Using thescalingrelationsproposedinChapter 4 ,asimplemethodisproposedthatcan computesedimentcarryingcapacityathigherReynoldsnumb ers. Chapter 6 describesthehybridopenMP-MPIpseudospectralcodethatw as developedandoptimizedbyourresearchgroup.Thistoolena blesustoperformhigh Reynoldsnumbersimulationsonmassivesupercomputers.Th ischapterdescribesthe outlineofthealgorithmandvariousoptimizationfeatures thatwereincorporatedtomake itanefcientandascalablecode. 27

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Figure1-1:Schematicrepresentationofagravitycurrent. 28

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CHAPTER2 FRONTCONDITIONSOFHIGH-REGRAVITYCURRENTSPRODUCEDBY CONSTANTANDTIME-DEPENDENTINFLUX:ANANALYTICALANDNUME RICAL STUDY 2.1Background Gravitycurrentsproducedbyinow(i.e.,byacontinuoussu pplyofdense/lightuid intoaless/moredenseambientuidforasignicanttime)pl ayamajorroleinvarious geophysical,environmentalandindustrialprocesses.For example,sustainedriver owsintooceansorindustrialwasteowsintolakesandpond scanoccuroverdays andweeks.Time-dependentinowdrivengravitycurrentsar eparticularlyrelevantin oceanographicapplications.Inproblemssuchasexchange owsthroughchannels andgatewaysthethicknessoftheambientuidlayerandthed irectionofowofthe displaceduidareknowntoplayanimportantrole[ 54 72 ].Thesecontinuouscurrents canbeclassiedintothreeregimes:therateofreleaseincr easeswithtime(waxing), remainsconstantordecreaseswithtime(waning).Theeluci dationandrigorous modelingofthesecontinuouscurrents,however,hasreceiv edlessattentionthanthe classicalxed-volumecurrentsproducedbylockreleaseco nguration.Thislackof informationisparticularlyimportantconcerningtheimpl icationsofthetheoryofthefront Froudenumber,whichisdenedas Fr N = u N = ( g 0 h N ) 1 = 2 ,where u N h N arethespeed andtheheightofthefront,and g 0 thereducedgravity.Hereweattempttoclosesome gapsinknowledgeconcerningtwo-dimensional,Boussinesq ,high-Reynoldsnumber, time-dependentinowsystems. Inatimedependentinowsystemitisconvenienttoconsider thevolumeofthe currenttovaryas V = qt ,where q and arepositiveconstants.Withthispower-law dependencyforthecurrentvolume, > 1 correspondstoincreasingrateofinow (waxingcurrent), =1 correspondstoconstantrateofinow,and 0 << 1 correspondstodecreasingrateofinow(waningcurrent).T hetheoreticalstudiesusing shallow-waterformulationshavebeenperformed[ 34 ].Themainvariablesarethe 29

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thicknessofthecurrent, h ,andthe z -averagedspeed, u ,asfunctionsof x and t .The pertinentcontinuityandmomentumequations(SeeUngarish [ 87 ])formahyperbolic system. Thesolution,ingeneral,requiresfurthereffortsandappr oximations.Letthesource beat x =0 ,andthefrontofthecurrentbeat x N ( t ) ;seeFigure 2-1 .Asthecurrentows inthe x direction,a”return”motionintheambientmaydevelopinth eregion 0 < x x N Thepresenceorabsenceofthismotionhasinterestingconse quences.Whenthe returnmotioncanbeneglected,thesystemadmitsself-simi larsolutions[ 4 33 34 ]. Theanalyticaladvantageoftheseresultsisoftenreducedb yincompatibilities(or complications)associatedwithrealisticboundarycondit ions.Forexample,itmaybe necessarytodividethecurrentintosubdomainsbyinternal jumps,andanon-constant Fr cannotbeimplemented.Amoreconvenientanalyticalresult isprovidedbythe box-modelapproximationwhichpostulatesthatthethickne ssofthecurrent, h ,isnot afunctionof x [ 27 41 42 87 ].Foraconstant Fr ,boththeself-similarandbox-model resultspredict x N = CFr 2 = 3 ( g 0 q ) 1 = 3 t ( +2) = 3 (2–1) where Fr isthefrontFroudenumberofthecurrentand C isadimensionlessconstantof theorderunity. Thischaptershallpresentresultsforvariousvaluesof ,butfocusattentionon theconstantrateofinow, =1 ,caseinparticular.Theimportanceofthiscase isthatitproduces,undersimpleoutletboundarycondition ,agravitycurrentwith time-independentspeedofpropagation u N andconstant-thickness, h = h N .This facilitatestheanalysisinseveralaspects,asfollows.In thiscasethereisactually nodifferencebetweentheself-similarsolutionandthebox -modelpostulatethatthe thicknessisindependentof x .For =1.0 ,thevalueof C in( 2–1 )canbecalculatedwith condence.However,eveninthissimplecasethespecicati onofthevalueof Fr isnot straightforward. 30

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Whentheoutowisatthesamestreamwiselocationastheino w( x =0 )boundary, thenastheconstantinowcurrentpropagatestotheright,t hereisareturnambient owdirectedtotheleftabovethecurrent.Inthiscaseweach ieveacloserealization ofBenjamin'sidealizedparallelowcurrent,basedonwhic htheclassicalevaluation offrontcondition(orfrontFroudenumber Fr )anddissipationhavebeenderived (seeBenjamin[ 9 ],Figure4andassociateddiscussion).Intheclassicalloc k-release problem,areturnowofthiskindisalwayspresentandthere lativemagnitudeofthe returnowtothemaincurrentis h = ( H h ) ,where H isthedepthoftheentirelayer ofuid.Intheotherextreme,wheretheinowisat x =0 andtheoutowislocated atthedownstreamendofthetank,therewillbenoreturnowi ntheambientabove thecurrent.Inbetween,therecanbesituationswheretheou towispartlyat x =0 andpartlydownstream,allowingforsomereturnow.Herewe willpresentasimple extensionoftheBenjamin-typeanalysisthataccountsfort hepresenceorabsenceof thereturnow.Theresultsshowastrongdependenceofthefr ontFroudenumberon themagnitudeofthereturnow. Currentswithinowhavebeenstudiedinthelaboratory(e.g .,Didden&Maxworthy [ 27 ],Hogg etal. [ 38 ];andalsoincylindricalsystems,e.g.,Britter[ 13 ]).However, theavailabledataarenotamenabletoeasyandconclusiveco mparisonswiththe analyticalresults.Inexperimentsthecurrentissubjecte dtofrictionatthebottomwall. Furthermore,thecurrent'sinterfacewiththeambientuid isnotsharpduetomixing andentrainment,andtheoutowboundaryconditionsarenot alwaysunderstrict control.Inmanysystemstheremaybenooutowandanopentop surfaceisused toaccommodatethevolumeoftheinux.Thediscrepancieswi ththetheorycanbe attributedtotheseeffects,butuncertaintiesremain. Numericalsimulationsareexpectedtoremedysomeofthesed eciencies.The presentnumericalsimulationswillemployfree-slipcondi tionsandwillemploystringent 31

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controloftheoutow.Moreover,thepositionofthefrontan dinterfacecanbedeterminedwithgoodaccuracy.Overall,thecomparisonswiththe analyticalpredictions providesharperconclusionsandcoversawiderrangeofpara meters. Thevalueof Fr N ininowproblemsisstillanopenquestion.Directmeasurem ents areratherinconclusive,because h N isnotaclear-cut,preciselymeasurableexperimentalvariable.ThetypicalbehaviorisseeninFigure7ofH ogg etal. [ 38 ],wherethe representativevalueisseenas Fr N =0.81 .Thisvalueof Fr alsoprovidesgoodagreementbetweenthemeasurementsofMaxworthy[ 55 ]andthebox-modelpredictionof u N Thereareindicationsthat Fr N increaseswiththedepthoftheambient.Anja&Huppert [ 4 ]suggesttheuseof 0.91 foradeepambient.However,theclassicalBenjaminvalue forthedeepambient Fr N issignicantlylargerat p 2 .DoesthismeanthatBenjamin's theorycannotbeappliedtoinowproblem,andifso,whatist hecorrectreplacement? Oristhediscrepancyaresultofunavoidableincompatibili tiesbetweentheidealized theoreticalowandtheexperimentalrealization(effects ofviscousfrictionandlocal mixing,say).Ourworkattemptstoanswerthesequestions. Inthecaseofasteadyinow( =1.0 )wecanemployaBenjamin-typeanalysis toacontrolvolumeattachedtothefrontofthecurrent.Weth usobtainajumpcondition applicabletoaquitegeneralinviscidcurrentgovernedbya hyperbolicsystemof equations.Usingthisapproachwecanderive Fr N andtheenergydissipation,as functionsofthedepthratio a = h N = H ,fortheproblemsunderinvestigation.Interestingly, wendthat,dependingontheoutowconditions, Fr N ( a ) islargerorequaltoBenjamin's resultforthesame a .Thedissipationresult,however,isnotaffected;thisals omeans thatacurrentcannotexceedthehalf-depththickness.Thet heoreticalpredictionis comparedwithnumericalsimulations.Interestingly,weob servetheanalyticalresultsto beequallyapplicableevenundernon-constant(waxingandw aning)inowconditions. Thestructureofthischapterisasfollows.Insection 2.2 the Fr N frontconditionsare derivedandsomeanalyticalresultsconcerningthepropaga tionarepresented.Section 32

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2.3 presentsthenumericalapproachandsection 2.4 presentsresultsandcomparisons withthetheory.Briefconcludingremarksintheformofdedu ctionsthatcanbedrawn fromthisworkarepresentedinsection 2.5 2.2ASimpleAnalysisofReturnFlowEffect Herethederivationof Fr N andenergydissipationformulasarebrieydescribed. Dimensional variablesareused.Thederivationisgeneralandisvalidfo rnonBoussinesqsystemsaswellandthusopensthewayforextensi onsofthistoother systems. InFigure 2-2 thevelocityoffrontinthelaboratoryframeisdenotedby U (= u N ) andtheheightofthefrontby h (= h N ) isthepressurereducedby + a gz andthe reducedgravity g 0 =( c = a 1) g ,where a and c aretheambientandthecurrent densities.Thevelocity u 0 inthemovingframe ( x 0 ) isgivenintermsofthevelocity u in thelabframe ( x ) as u 0 = u U .Subscripts r and l denotetheright(upstream)and left(downstream)sides.Thesubscript B willdenotetheclassicalvaluesderivedby Benjamin.Itisassumedthattheforemostbottompointofthe currentisastagnation pointoftheadjacentambientuid. Twosetupsareshown.InFigure 2-2 (a)theoutow(sink)andinow(source)are atthesamestreamwiseend, x =0 .Thecurrentpropagatesintoastagnantambient. ThiscorrespondstotheclassicalBenjaminproblem.Inthet wo-layerformulationweuse the z -averagedspeeds,andthe z -hydrostaticpressurebalance.Ontherightsideofthe controlvolumethespeed(totheleft)is U ,and r =0 .Ontheleftsidethespeed(tothe left)is U = (1 a ) intheambient h < z < H ,andzerointhecurrent.Thepressureonthe leftis l = 8><>: 1 2 a U 2 a g 0 z (0 z h ) 1 2 a U 2 a g 0 h ( h z H ). (2–2) 33

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Thebalancebetweentheow-force u 0 2 + integratedfrom z =0 to H ,ontheleftand rightsides,canbeexpressedas a U 1 a 2 ( H h )+ 1 2 a U 2 H Z h 0 a g 0 zdz Z H h a g 0 hdz = a U 2 H (2–3) Solvingfor U ,weobtain U ( g 0 h ) 1 = 2 = (2 a )(1 a ) 1+ a 1 = 2 = Fr B (2–4) whichisBenjamin'sclassicalprediction. ThesetupshowninFigure 2-2 (b)correspondstoamoregeneralformulationofthe problem.Hereitisassumedthatthemotionofthedisplaceda mbientuid(displaced bythemotionofthedensecurrentalongthebottomboundary) isnotrestrictedin anydirection.Inotherwords,theambientuidcanowoutth oughboththeleftand rightstreamwiseendsofthedomain.Torepresentsuchanout owconditionanew parameter r isintroducedthataccountsforthefractionalamountofamb ientuidthat exitsthroughtheleftboundary.Thereforewhen r isequalto 1 wehavethefull-return owconditionexactlythesameastheclassicalanalysisbyB enjaminand r =0 isthe ”no-return”owcondition,i.e.,ambientuidexitsonlyth roughtherightboundary.With ananalysissimilartotheoneusedbyBenjamin,wegetthefol lowingresultforthefront Froudenumberofthecurrent. U ( g 0 h ) 1 = 2 = 1 [ 1 (1 r ) a ) ] (2 a )(1 a ) 1+ a 1 = 2 = Fr r (2–5) Theaboverelationcanbere-writtenasamodiedBenjamin's Froudenumber: Fr r = ( r ) Fr B (2–6) where ( r ) isacorrectionfactorwhichcorrectstheclassicalBenjami n'sfrontFroude numberconditionaccordingtothenatureofthereturnow.T hecorrectionfactorhas 34

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thefollowingform ( r )= 1 [ 1 (1 r ) a ] (2–7) Byphysicalintuitionitisapparentthatthefrontmotionof thecurrentwillbegoverned bythehinderingeffectorthestagnationpressureimposedb ytheambientatthecurrent front( a U 2 (1 (1 r ) a ) 2 = 2 ).Thishinderingeffectdependsontheoutowlocation oftheambient.Inthefull-returnowcase( r =1 ,similartothecongurationusedby Benjamin)theambientisentirelymovingintheoppositedir ectiontothecurrenthence thehinderingeffectorthestagnationpressure( a U 2 = 2 )onthecurrentishighest.The extentofthishinderingeffectiscapturedbythecorrectio nfactor whichdecreasesas r increasesfrom 0 to 1 .Thismeansthatatagivendepthratio a thefrontFroudenumber willbethelowestfor r =1 andhighestfor r =0 .Foranyother r thefrontFroude numberwillliebetweenthetwoextremes.Thegeneralformre ducestotheclassic Benjamin'sFroudenumberrelationfor r =1 andfor r =0 weget U ( g 0 h ) 1 = 2 = 2 a 1 a 2 1 = 2 = Fr NR (2–8) Thesubscript NR denotesthe”no-return”featureoftheambientow.Interes tingly,this formofthe Fr functionisgiveninSimpson[ 81 ],p.176,butwithoutderivation,reference andcomparisons.Therealsoseemstobenosubsequentuseoft hisformulainlater works.Considertheenergyuxgivenbyintegrationof u 0 (+(1 = 2) u 0 2 ) from z =0 to H .Thedifferenceinuxbetweentheleftandtherightdenest herateofdissipation, D Forsetup(a)weobtain D = D B = 1 2 a g 0 3 = 2 h 5 = 2 Fr B 1 2 a 1 a 2 (2–9) whichistheclassicalrateofdissipationobtainedbyBenja min.Forsetup(b),aftersome algebra,weobtainthesameresult( 2–9 ).Themainconclusionisthat,foranyambient owconditions,theinowcurrentsenergeticallybehaveac cordingtotheclassical scenario:(a)Thethicknessisrestrictedto a 1 = 2 ;and(b)Theyaredissipative(except 35

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for a =1 = 2 ).Inthelock-releaseproblemthedissipatedenergycomesf romthepotential energyoftheinitiallylockeddenseuid.Intheinowcaset hesupplyofenergyismore subtle.Heredissipationmeansthattheworkdonebythefron tisnotfullyconvertedinto kineticenergyoftheambient, K a .Thedeeplimitof a 0 isilluminating.Thespeedof theambienttendsto 0 ,and D attainsamaximum(Anotherpossiblederivationofthe analytical Fr anddissipationresultsarebrieyoutlinedinAppendix A ). Inthisstudy,inadditiontotheBenjamin'sclassical r =1 system,attentionwillbe focusedonthe r =0 case,i.e.no-returnowcondition.Inthislimitthecorrec tionfactor is ( r =0)= 1 1 a (2–10) Overtheentirerangeofdepthratio(i.e., 0 < a 0.5 ),thevalueof Fr NR islarger thanthatof Fr B .Inthisrange Fr NR ( a ) isalmostconstantatabout 1.39 ,andshows only 1.7% variation.Themaximumvalueof p 2 isachievedat a =0 and 0.5 ,and inbetweenaminimumvalueof1.366isreachedat a =0.27 .Ontheotherhand, Fr B variesbyafactorof 2 overthisinterval.Thishasapracticalimplicationthatpr ediction modelsbasedonaconstantFr assumptionaremoreapplicableto”no-return”ow cases.Typicalexperiments(e.g.Maxworthy[ 55 ])employanopentopsurfaceandinthis scenariotheowisexpectedtobeclosetothe” NR ”type. 2.3NumericalApproach 2.3.1ProblemFormulation Inordertotesttheabovetheoryontheeffectofreturnowon thefrontFroude conditionofthecurrent,2DplanarBoussinesqbottomgravi tycurrentsaresimulated. Thesimulationsconsistsofarectangularboxshapedcomput ationaldomainthatis lledwithlightambientuid.Heavyuidisinjectedthroug hthelowerleftcornerof thecomputationaldomain,whichformsthebottomgravitycu rrent.Thelightambient uiddisplacedbytheheavyuidowsoutofthedownstreamen dofthedomain, i.e.,throughtheboundaryontheright.Whenthefrontofthe currentissufciently 36

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downstreamoftheinowboundary,thissetupcloselymimics theno-returnambientow conditiondescribedintheprevioussection(seeFigure 2-2 (b)forthe r =0 no-return conguration).Hence,thiscomputationalsetupiscalleda sno-returnconguration (seeFigure 2-3 ).Similarly,whentheoutowisontheupstreamendofthedom ain,i.e., outowandinowareonthesameside,suchasetupwillbeaful l-returnconguration. Experimentsoflockreleaseproblemarenaturallyoftheful l-returnowtype,wherethe voidcreatedbythereleaseofcurrentislledbythebackow ofambientuid.Thisis howevernotthecaseforanexperimentalsetupwithacontinu ousinow.Intheprevious sectiontheoreticalconsiderationsshowthatthedirectio ninwhichthedisplacedambient uidowsplaysasignicantroleindeterminingthefrontco nditionofthecurrent.Here thesimulationresultsofinowgravitycurrentswithno-re turnowarecomparedwiththe theoreticalpredictionsoftheprevioussectionandtheimp ortanceofincorporatingthe natureoftheambientowfeaturesintheanalysisofgravity currentsisexamined. Theinterestingaspectofthepresentsimulationistoconsi deratimedependent current,whichwaxesorwanesaccordingtothepower-law V = qt ,where V ( t ) isthevolumeofthecurrentperunitwidth.Accordingly,the rateofinowintothe computationaldomainisalsotimedependentandisgivenby qt ( 1) .Thistime dependentinowcanbeachievedindifferentways.Theinow height( h in )canbeheld xedandtheinowvelocity( u in )increasedordecreaseddependingonwaxing( > 1 ) orwaning( < 1 )current.Similarly,theinowvelocity( u in )canbeheldxedandthe inowheight( h in )increasedordecreasedovertime.Inanycase,thestateofi nowcan becharacterizedbytheinletFroudenumber,i.e. Fr in = u in = p g 0 h in .InowFroudenumber measurestherelativeimportanceoftheinowvelocitytoth evelocityscaleofthegravity current.Iftheinowvariationisonlythroughinowveloci tythentheinowFroude numberwillincrease(ordecrease)forawaxing(orawaning) current.Conversely,ifthe inowvariationisonlythroughinowheightthen Fr in willdecrease(orincrease)fora waxing(orawaning)current. 37

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Incontrasttothelock-releaseproblem,inthecaseofinjec tedgravitycurrent, closetotheinjectionport,theowmayexhibitajet-likebe havior.However,thejet-like behaviordecaysdownstreamandthussufcientlyawayfromt heinletthegravity-driven natureoftheowdominatesandthetheoryof x 2.2 willbeapplicable.Thiscross-over fromthejet-liketogravitycurrent-likebehaviorisakint othetransitionfromthejet-like toplume-likebehaviorinaninjectedplume,andthetransit ionpointwilldependon inletFroudenumber.ForlargevaluesofinletFroudenumber ,thejet-likebehaviorof theinowwillpersistfardownstreamfromtheinlet.Inthep resentcontext,especially forcasesinvolving 6 =1 ,iftheinletvelocityorheightaloneisallowedtovary,the n inletFroudenumbermaycontinuetoincreaseordecreaseove rtime.Asaresult,an unreasonablylargecomputationaldomainmayberequiredto capturethegravitycurrent behavior,sufcientlydownstreamofthejet-dominatedreg ion. Furthermore,atimedependentinowFroudenumberwillcomp licatetheinterpretationofresults.Hence,inthepresentsimulationsthein owvelocityandheightare changedsimultaneouslysuchthattheinowFroudenumberwi llbeheldxed.Inother words,forachosenconstant Fr in thetimedependentinowvelocityandheightaregiven as u in = qt ( 1) 1 = 3 g 0 Fr 2 in 1 = 3 and h in = qt ( 1) 2 = 3 g 0 Fr 2 in 1 = 3 (2–11) Forthepresentanalysiswedenethetimescaletobe t s andthelengthscaleto betheinowheightatthistime,i.e., h s = h in ( t s ) .Theirvaluesarechosensuchthatthe correspondingvelocityscalewillsatisfy u s = h s t s = p g 0 h s (2–12) Withsomealgebraitcanthenbeshownthatthesechosentimea ndlengthscalescan beexplicitlyexpressedintermoftheinowparametersas t s = q g 0 2 Fr in 1 = (4 ) and h s = 2 q 2 g 0 Fr 2 in 1 = (4 ) (2–13) 38

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Theresultingdimensionlessgoverningequationsare @ ~ u @ ~ t +~ u ~ r ~ u = ~ r ~ p + 1 Re ~ r 2 ~ u +~ e (2–14) ~ r ~ u =0, (2–15) @ ~ @ ~ t +~ u ~ r ~ = 1 ReSc ~ r 2 ~ (2–16) where e = f 0,0, 1 g istheunitvectorinthedirectionofgravity, ~ isthedimensionless densitygivenby ~ = a c a (2–17) Thedimensionlessparametersinthegoverningequationsar e Re = u s h s and Sc = (2–18) where isthekinematicviscosityoftheuidand isthediffusioncoefcientofthe agentresponsibleforthedensitydifference.Althoughthe generaltheorypresented in x 2.2 isvalidfornon-Boussinesqcurrents,theabovegoverninge quationsandthe simulationresultstobepresentedbelowarelimitedtotheB oussinesqmodel.This simplicationisusedbecauseitallowsforquickerconverg enceofthenumerical simulations.Moreover,innon-Boussinesqcurrentstheow isadditionallydependenton therelativemagnitudeofthecurrent( c )andtheambient( a )uiddensity.Therefore Boussinesqassumptionisemployedinordertosimplifythep roblem. 2.3.2BoundaryConditionsandNumericalScheme Inthesimulationstobereportedhereboththeinletvelocit yandheightarevaried suchthattheinletFroudenumberremainsaconstantoverthe entiresimulation.From thelengthandtimescaleswecandenethenon-dimensionali njectionparametertobe ~ q = qh ( 4) = 2 s g 0 = 2 = Fr in (2–19) andthusrelateittotheinletFroudenumber. 39

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Thetimeandlengthscales( t s and h s )arechosensuchthatthewaxingandwaning inowsimulationsstartatanon-dimensionaltime ~ t =1 ,withaninitialinletheightof ~ h in ( ~ t =1)=1 ,andaninitialinletvelocityof ~ u in ( ~ t =1)= Fr in .Notethatatthisinitialtime thereisalreadysomeinjecteduidofvolume V ( ~ t =1)= ~ q inthedomain. Thetimevariationoftheinletheightandvelocityaregiven by ~ h in ( ~ t )= ~ t 2( 1) = 3 and ~ u in ( ~ t )= Fr in ~ t ( 1) = 3 (2–20) Intheconstantinowcaseof =1 ,boththeinowvelocityandheightremainconstant overthedurationofthesimulation.Thesesteadyinowcond itionscanbeenforcedina straightforwardmanner. Inthecaseoftimedependentinowaspecialtreatmentisreq uiredatthestart ofthesimulation.Forexample,notethatincaseof < 1 boththeinowvelocityand heightbecomesingularas ~ t 0 .From ~ t =0 to ~ t =1 aconstantinowcondition isenforcedsothatat ~ t =1 thenetvolumeofcurrentinthedomainisexactlythe sameasthetheoreticalvalueforthegivensetofinowparam eters.Thismeansthatat ~ t =1 ,thevolumeofthecurrentinthedomainmustbe ~ V ( ~ t =1)= Fr in = .Duringthis startingphasetheinletheightisxedtobe ~ h in =1 andtheinletvelocityisxedtobe ~ u in = Fr in = .For ~ t > 1 inowconditionsarespeciedusingthechosenvariablein ow parametersforthesimulation(seeequation( 2–20 )).Asaresultweareinterestedinthe behaviorofthecurrentlongaftertheinitialstartingphas e(i.e.,for ~ t 1 ). Benjamin-typeanalysisneglectsthefrictionalforcesatt hebottomofthecurrentand alongthetopboundary.Suchanidealizedconditionishardt oachieveinexperiments. However,tofacilitatecomparisonwiththeory,inthesimul ationswesetthebottomand topboundariestobestress-free(orfree-slip).Thiswilll imittheviscouseffectstobe primarilyattheinternalinterfacebetweenthecurrentand theambientuid. TheoutowboundaryisgivenbyNeumannboundarycondition, i.e.,thenormal derivativeoftheowpropertiesontheverticalwall(thelo cationofoutow)iszero. 40

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Alltheboundariesexcludingtheinowandoutowlocations imposenopenetration condition.Mathematicallyalltheaboveconditionscanbew rittenas d ~ u d ~ z =0 on ~ z =0 and ~ z = ~ H (2–21) ~ v =0 on ~ z =0 and ~ z = ~ H (2–22) d ~ dz =0 on ~ z =0 and ~ z = ~ H (2–23) d () dx =0 at ~ x =30 andfrom ~ z =0 to ~ H (2–24) ~ u in = ~ q ~ t ( 1) = 3 at ~ x =0 andfrom ~ z =0 to ~h in (2–25) ~ =1 at ~ x =0 andfrom ~ z =0 to ~ h in (2–26) ~ =0 at ~ x =0 andfrom ~ z = ~h in to ~ H (2–27) Insummary,thisproblemhas5parameters Fr in Re Sc and H .Here,Schmidt number( Sc )issetto 1 asitwasobservedthattheresultsremaininsensitivetothe exact valueof Sc when Sc isof O (1) and Re islarge(seeBonometti&Balachandar[ 11 ]). Simulationsatvarious Re werecarriedoutandavalueof 1000 wasfoundtobesufcient tolimittheviscouseffectsandtohaveasharpinterfacebet weenthecurrentandthe ambient.Heightofthedomainistakentobeaparameterandva rioussimulationswere performedwithdifferentvaluesforthedimensionlessdoma inheighttoinvestigateits inuenceonthecurrent. Twodimensionalsimulationswereperformedbysolvingthet ime-dependent, incompressibleNavier-Stokesequations.Theschematicof thecomputationaldomain isasshowninFigure 2-3 ,whichisnottoscale.Theactualcomputationaldomainis30 dimensionlessunitsinlength.Ahighlyresolveduniformgr idsystemisused(typically, 41

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x = y = H = 300 ,whereHistheheightofthebox)andemploythenitevolumem ethod todiscretizethegoverningequations.Asecond-ordercent raldifferenceschemewas employedforallspatialderivativesandtheQUICKschemefo rtheconvectionterms. Thenumericalsimulationwasadvancedintimebyafractiona lstepmethodwithsecond orderAdams-BashforthschemefortheconvectiontermsandC rank-Nicholsonscheme forthediffusionterms.Forpressurecorrection,thepress urePoissonequationissolved andthedivergence-freevelocityiscomputedateachtimest ep.Thiscodehasbeen extensivelyvalidatedinthecontextofthermalplumes[ 84 ]andparticulateows[ 53 ]. 2.4Results 2.4.1ConstantInow( =1 ) Twoimportantobservationscanbemadefromtheanalysisof x 2.2 :a)forno-return conguration,thefrontFroudenumbermaybesignicantlyh igherthanthatofthe full-returncongurationandb)forno-returnconguratio n,thefrontFroudenumberis narrowlyboundedovertheentirerangeofdepthratio.Thiss ectionwilltesttheabove theoreticalpredictionswithhelpofconstantinowandloc kreleasesimulations. Table 2-1 showsthelistofallthesimulationscarriedoutforthisstu dy.Inallthese simulations,theoutowisxedonthedownstreamboundarya ndinowisonthe upstreamboundary.Gravitycurrentswithdifferentdepthr atioareobtainedbyappropriatelyspecifying Fr in and H .Inallthesimulations,thefrontFroudenumber( Fr N )is computedsothatcomparisonwiththetheoreticalpredictio nscanbemade.Thefront Froudenumberisdeterminedbytrackingthefrontofthecurr entandevaluatingitsvelocityandcharacteristicheight.However,preciseevalua tionoffrontspeedandheightis nottrivial.Thesourceoftheproblemlieswithdeningthef rontheightofthecurrentina robustwayandwithaccuratelyextractingthisinformation fromthesimulationdata.Providedtheproduct ReSc islargeandnumericaldiffusivityissmall,asharp,wellde ned frontpropagatesintotheambient.Thesharpfeaturemakesi teasytotrackthefrontand accuratelycalculatethefrontvelocity. 42

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ShowninFigure 2-4 aaretheNavier-Stokessimulatedcurrentshapesforthe simplestcaseofcontinuousinowgravitycurrentgivenby =1 and Fr in =1.0 (case1). Theseshapesarecalculatedbydeningthecurrentheightas ~ h (~ x )= Z ~ H 0 ~ (~ x ,~ y ) d ~ y (2–28) andwillbeusedforfurtherdiscussion. Figures 2-4 band 2-4 cshowthecurrentshapesfordifferentinletFroudenumbers atvarioustimesduringthesimulation.For Fr in < 1 thekineticenergyofthecurrent attheinletislessthanthepotentialenergy,becauseofwhi chthecurrentplunges downclosetotheinletandthenstartspropagatingdownstre amatitscharacteristic heightandvelocitywhicharedifferentfromthoseofthe Fr in =1 case.Sincetheheight ofthecurrentissmallerthantheinlet,correspondinglyth ecurrentvelocityishigher thantheinletvelocityandtogethertheycontributetoanin creaseinthefrontFroude numberabovetheinowFroudenumber.Notethattheheightof theuidlayer(domain height)isheldxedandasaresulttheheightofthecurrentc omparedtothelayerdepth decreaseswithdecreasinginletFroudenumber. Figure 2-5 showstheevolutionofthefrontvelocityforsomeofthecase slisted inTable 2-1 .Thefrontvelocityshowsuctuationsearlyon,butafterth isinitialphase itsettlesdowntonearconstantvalue.Thisconstantfrontv elocityisexpectedasthe currentisdrivenbyaconstantinowofheavyuid.Computin gthefrontvelocity,gives anestimatefortheinertialforceassociatedwiththecurre nt.However,anestimatefor thedrivingorbuoyancyforceisalsoneededtodenethefron tconditionofthecurrent. Thefrontofthecurrentischaracterizedbyintensemixinga ndnon-uniformity( g 0 changeslocally).Henceitisinherentlydifculttoprecis elyestimatethebuoyancy effect.Typicallybuoyancyforceisestimatedbydetermini nganequivalentheightof thefrontwhichwillrepresenttheexcesspotentialavailab letothecurrent( g 0 isheld xed).Estimatingthisfrontheightisnotstraight-forwar dasthereareseveralpossible 43

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denitions.Thesimplestdenitionissuggestedbytheboxm odelwhichutilizesthe factthattheamountofcurrentinthedomainisalwaysconser ved.Inthisdenition,the shapeofthegravitycurrentisassumedtoberectangularata lltimes.Thelengthofthe boxisdenedbythefrontlocationandtheheightoftheboxca nthenbetakenasthe frontheight.Therefore,theevolutionofthegravitycurre ntcanbeexpressedinterms oftheevolutionoflengthandheightoftherectangularbox. AsseeninFigure 2-4 ,in thepresentsimulations,thefrontofthecurrentremainssh arpandcanbeaccurately trackedtoobtainthefrontvelocity.Fromthefrontlocatio nandtheamountofheavyuid injectedintothebox,theheightofthecurrentcanbeestima ted.Previousexperiments haveusedsimilarmethodstodenethefrontheightofthecur rent[ 78 ].Thefrontheight calculatedusingthisdenitionisindicatedby'(I)'inTab le 2-1 Analternatemeasureofcurrentheightcanbedenedbyavera gingonlyovera smallregionnearthefrontofthecurrent.Clearly,forthis methoditisnecessaryto denetheextentoftheregionnearthefronttobeusedforave raging.Physically,the frontregionofthecurrentischaracterizedbystrongmixin gandnon-uniformvelocity, whilethebodyischaracterizedbynearlyuniformvelocity. Therefore,welookforthe regionnearthefrontofthecurrentwherethe ~ z -averagedstreamwisevelocityhasa maximum.Thislocationcorrespondstotheregionwherethea mbientuiddisplaced bythefrontsplashesbackontothecurrentcausinglocalcon striction,whichleadsto increasedstreamwisevelocity.Thislocationisapproxima telywherethecurrentcanbe dividedintoahead(orthefront)andthebody.Nowthefronth eight h N isgivenbythe averageheightofthecurrentbetweenthelocationofmaximu m ~ z -averagedstreamwise velocityandthefrontofthecurrent.Thisdenitionisrefe rredtoas'(II)'inTable 2-1 Boththesedenitionsarerobustandgivesimilarvaluesfor thefrontheight.The boxshapeassumptionissimpleranditspredictionsresulti nasmoothevolutionof thefrontheight.Incontrast,thepredictionbasedonlocal averagingisinuencedby Kelvin-Helmholtzinstabilitiesoftheinterfacenearthef ront,whoseeffectontheheightof 44

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thecurrentcanbeobservedinFigure 2-4 .Asaresult,frontheightestimatedfromlocal averagesuffersmildtime-dependentoscillation. Thedifferencebetweenthetwoestimates(IandII)arisesfr omthestreamwise extentofcurrentheightbeingaveraged.Averagingoverthe entirelengthofthecurrent leadstotheboxestimate.Itisobservedthattheboxestimat eoffrontheightisslightly higherthanthatfromlocalaveraging.Thisimpliesthatthe computedfrontFroude numberwillhavesomescatterbasedonthefrontheightdeni tion.Duetothesimplicity oftheboxassumption,henceforth,thediscussionwillbeli mitedtotheconservative evaluationoffrontheightusingtheboxshapeapproximatio n. Figure 2-5 showsthetimeevolutionofthefrontheightforalltheconst antinow cases.Itisclearfromtheprolesthatthefrontheightrema insnearlyindependentof time.Ascanbeexpected,thefrontheightisastrongfunctio nof Fr in .Forcaseswith Fr in < 1 ,thecurrentplungesdownstreamoftheinletandtheresulti ngfrontheightis lowerthantheinowheightof ~h in =1 .Figure 2-6 showstheplotoftheconstantfront heightforallthesteadyinowcasesasafunctionof Fr in .Althoughthereisaweak dependenceondepthratio,thefrontheightisdominantlyde terminedbytheinow Froudenumber.Asimplecurvetthroughthedataisobtained anddenotedby f ( Fr in ) Figure 2-7 showsthetimeevolutionoffrontFroudenumberforalltheco nstant inowcases( =1 )withdomainheight H =5 (i.e.cases1,2and3),whichdifferonly bythevaluesselectedfor Fr in .ThefrontFroudenumberprolesarequiteclosebutdo notfullycollapseontopofeachother.Thisdifferenceisdu etothedependenceof Fr N onthedepthratio( a = ~ h N = ~ H ).InFigure 2-7 thedepthratiochangesindirectlydueto thedependenceofcurrentheight( h N )on Fr in .Asadirectevaluationoftheinuenceof depthratio,incases1and5wemaintain Fr in =1.0 butdecreasethedomainheight andtherebyincreasethedepthratio.Itisobservedthatthe frontFroudenumber( Fr N ) changesfrom1.241(forcase1)to1.259(forcase5)whenthed omainheightchanges from 5 to 2.5 ,respectively(seeTable 2-1 forsimilarcomparisons). 45

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Comparisonwiththeory. Figure 2-8 presentsthetheoreticalresultsof x 2onthe frontFroudenumberofno-returncurrent ( Fr NR = Fr r ( r =0)) andfull-returncurrent ( Fr B = Fr r ( r =1)) asafunctionofthedepthratio a = h N = H .Alsopresentedinthe gureareresultscorrespondingtothepresentconstantin ow( =1 )simulations.At smalldepthratio a ,orfordeepambient,i.e. H !1 ,both Fr NR and Fr B reachavalue of p 2 .Thedifferencebetween Fr B and Fr NR increasesawayfromthedeepambient limit.Thedeviationreachesitsmaximumat a =0.5 .Observethatthepresentconstant inow( =1 )simulationresultslieabovetheclassicalcurveof Fr B ,indicatingthat theBenjamin'sfrontFroudenumberformulaunder-estimate sthefrontvelocityofthe no-returngravitycurrent. AlthoughthefrontFroudenumberofno-returncurrentspred ictedby Fr NR doesnot matchexactlywiththepresentsimulationresults,itcaptu resthequalitativebehavior ofthesimulationresultsmuchbetterthanthepredictionsb y Fr B .Forexample,over awiderangeofdepthratios, Fr N obtainedfromthesimulationslieinaclosebandof 1.25 0.05 ,whichissimilartothe Fr NR prolethatitremainsboundedwithinanarrow rangeof 1.37 0.04 .Theexistenceofaweakminimuminthe Fr NR proleatadepth ratioofabout 0.27 ,compareswellwiththelocationofminimuminthecomputedf ront velocityat a 0.25 .Thediscrepancybetweenthesimulationresultsandthenoreturn owtheorycanbeattributedtotwomainfactors:(a)themixi ngandviscousforces presentalongtheinterfacebetweenthecurrentandtheambi entthatisnotaccountedin thetheoryand(b)errorintheestimateofthefrontheightof thecurrent. Asdiscussedbefore,dependingonthedenitionoffronthei ghtemployedaslight scatterisobservedinitscomputedvalue.Thisdirectlylea dstodifferencesinthefront Froudenumbercomputedfromthesimulation.Thismeansthat Fr N computedusingthe boxdenitionwillbesmallerthanthelocalaveragingdeni tion. Inlockreleaseconguration,interfacialmixingandaddit ionaldissipativeprocesses havebeenconsideredtoberesponsiblefortheconstantfron tvelocityintheslumping 46

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phasetobesignicantlylowerthanthetheoreticalpredict ionby Fr B .Also,recent simulationresultsofbothplanarandcylindricallockrele asecongurationsshowed noticeableReynoldsnumbereffect[ 16 22 ].AstheReynoldsnumberofthecurrentis increasedfrom 895 to 8950 ,thefrontvelocityintheslumpingregimeincreasedby 10 to 15% Inthisstudysimulationsoflockreleasecongurationwith nostressbottomwalland at Re =1000 arealsoperformed.Differentsimulationswerecarriedout bychangingthe lockheight,whichresultedincurrentsofdifferentdepthr atio.ThefrontFroudenumber oftheselockreleasesimulationsarealsoshowninFigure 2-8 asopencircles.The frontFroudenumberoflockreleasesimulationscontinueto decreasewithincreasing depthratio. Fr N decreasesfromabout 0.99 at a =0.08 (deepambient)toabout 0.78 at a =0.4 .Theseresultswereobservedtobecomparabletothepreviou slockrelease experimentsbyHuppert&Simpson[ 42 ]andShin etal. [ 78 ].Thefactthattheno-return owresultsarequalitativelysimilarto Fr NR andthelock-release(full-returnow)results aresimilarto Fr B showsthatthedirectionofambientowplaysasignicantro leinthe frontconditionofthecurrent.2.4.2VariableInowCurrents( 6 =1 ) Thissectionpresentstheresultsofvariableinowgravity currentsimulationsand discussessomeoftheirfeatures.Variableinowgravitycu rrentscanbeclassiedinto: (a)waninginow( < 1 )wheretheinowrateofthecurrentprogressivelydecrease s withtimeandeventuallybecomes 0 as ~ t !1 and(b)waxinginows( > 1 )wherethe inowrateofthecurrentprogressivelyincreaseswithtime .Withtimedependentinow thecurrentdoesnotreachasteadystate. Table 2-2 listsallthevariableinowsimulationscarriedoutinthis study.Values of arechosensothattheyrepresentboththewaxingandwaningi nows.Inallthe simulationstheoutowboundaryisxedatthedownstreamen d.Inowisonthelower halfoftheupstreamboundaryandtheinowconditionisgive nby( 2–20 ).First,the 47

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accuracyoftheimplementationofinowconditionisvalida tedbycomparingtheamount ofinjecteduidinthedomainwiththeexactamountgivenby ~ V =~ q ~ t .Thiscomparison isdoneatalltimestillthefrontreachestheendofthedomai n.Figures 2-9 a, 2-9 bshows thiscomparisonforthecasescorrespondingto =1.25 and =0.90 respectively. Thereisgoodagreementwiththetheoreticalcurvefor =1.25 .For =0.90 atlarge timesthereislessthan 2% error.Simulationsofwaninginow( < 1 )haveingeneral proventobeachallenge.Thereasonforthediscrepancyisth atthecurrentbecomes progressivelythinandagridthatadequatelyresolvesthe owoverthelongdurationof thecurrentishardertoachieve. Figures 2-10 and 2-11 showthedensitycontoursatvariousinstancesduringthe simulationofcasesA1andB1.Bycomparingtheseframesitap pearsthatthegravity currentdevelopsaselfsimilarshapeasitpropagatesthrou ghtheambientcolumn. Somemixingofthecurrentisalsoseenalongtheinterface.A sthecurrentpropagates, thedisplacedambientuidfallsbackontopofthecurrentcr eatinganeck.Theheight ofthecurrentsufcientlyawayfromtheneckremainsunifor m.Theshapeofthecurrent providessomesupportforusingtheboxshapeapproximation torepresentthefront heightofthecurrent. Withvariableinow,itisexpectedthatthefrontpropagati onspeed( ~ u N )willbehave similartotheinow( ~ u in ).AsseeninFigures 2-12 aand 2-13 a,for =1.25 thefront propagationspeedincreaseswithtimeandfor =0.90 thefrontpropagationspeed decreaseswithtime.Inallthecasesthereisaninitialosci llatorybehavior,buteventually thefrontvelocity( ~ u N )settlesdowntoasmoothevolution.Thefrontvelocityford ifferent Fr in casesaresimilarandappeartodifferonlybyascalefactor. Figure 2-12 band 2-13 bshowthetimeevolutionofthefrontheight( ~ h N )forvariouscasescorrespondingto =1.25 and =0.90 respectively.Similartothefrontvelocity,frontheighti ncreases (ordecreases)withtimeincaseofwaxing(orwaning)inow. Aswithfrontvelocity, 48

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theeffectofinletFroudenumber( Fr in )istoprimarilyrescalethetimeevolutionoffront height. HereweproposethedependenceoffrontheightoninowFroud enumbertobethe sameasthatforconstantinowcurrents.Thus,thefronthei ghtofthevariableinow currentscanbeexpressedinthefollowingseparableform ~ h N ( ~ t Fr in )= f ( Fr in ) h ( ~ t ), (2–29) wherethescalefactor f ( Fr in ) ,asperourassumption,isthesameasintheconstant inowcurrentshowninFigure 2-6 .Notethattheaboveexpressionweaklydepends onthedepthratiothrough f andalsothedependenceon issuppressed.Thetime evolutionofthescaledfrontheight, h ,isshowninFigures 2-14 band 2-15 b.Theresults forthedifferentinowFroudenumberscollapsequitewell, validatingtheapplicability of f ( Fr in ) for 6 =1 .Power-lawcurvetsforthecollapsed h ( ~ t ) havebeenobtained forboth =1.25 and =0.9 andpresentedinFigures 2-14 band 2-15 b.The correspondingexpressionforfrontvelocitycanthenbewri ttenas ~ u N = Fr NR p f ( Fr in ) u ( ~ t ), (2–30) wherethefrontFroudenumberdenitionisusedtoobtainthe appropriatescalefactor. Herethedependenceondepthratioisboththrough Fr NR and f ,andboththesedependenciesareexpectedtobeweak.Thetimeevolutionofthesca ledfrontvelocity, u isshowninFigures 2-14 aand 2-15 a.AgaintheresultsforthedifferentinowFroude numberscollapsequitewell. ThefrontFroudenumberofthecurrentisevaluatedfromthes imulationresultsof frontvelocityandfrontheightas Fr N =~ u N = p ~ g 0 ~ h N .Figures 2-16 aand 2-16 bshows thefrontFroudenumberprolesforallthecasescorrespond ingto =1.25 and =0.90 respectively.Atearlytimes,duringthetransientphase,i nertiaandpotential energyofthecurrentarenotinbalance.Thistransientphas eischaracterizedby 49

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oscillatorybehaviorofthe Fr N prole.Atlatertimestheinertiaandbuoyancyofthe currentapproachclosebalanceandaccordingly Fr N tendstowardatimeindependent prole.Thisscenarioprevailsuntiltheviscouseffectsre mainsmall.Intimeasthe currentpropagates,theviscousforceswillslowlygrowtoe ventuallychangethefront dynamicsofthecurrent.Theextentoftheviscousforcedepe ndsonthelengthand inowinertiaofthecurrent(i.e.,on Fr in ).Forwaxinginows,wherethefrontspeedis increasing,viscousforcesarerealizedthroughthedragim posedbytheambientalong thelengthofthecurrent.Eventually,whenthecurrenthast ravelledsufcientlengththe viscousforcesbecomesignicantcomparedtothecurrentin ertialeadingtoachange inthefrontdynamics.Inwaninginows,theeffectofviscou sforcesismoresignicant. Here,apartfromthedragforceexertedoverthelengthofthe current,theinertiaof thecurrentisdecreasingduetodeceleratingfrontvelocit y.Thedecelerationleadsto adecreaseinthefrontReynoldsnumber( u N h N = )implyingrelativeincreaseinthe importanceofviscouseffects.Asaconsequence,viscousef fectswillbecomesignicant soonerforwaninginowsandforsmallerinowinertia(i.e. ,forsmaller Fr in ).Thiseffect canbeobservedfor Fr in =0.5 and =0.90 case.Inthiscaseatlatertimes,the currentfrontvelocitydropstoadimensionlessvalueofabo ut 0.75 andthecurrentheight hasreducedto 0.5 .CorrespondinglythefrontReynoldsnumberhasdroppedto 400 Becauseoftheincreaseintheviscouseffects,thefrontFro udenumberafterreaching steadyvalueslowlystartstodrop.Thisbehaviorisexpecte devenforhigher Fr in cases butthedomainisnotlongenoughtowitnessthis. Comparisonwiththeory. TheBenjamin-Liketheoreticalanalysisof x 2.2 isbased onsteady-stateassumption.Nevertheless,inFigures 2-17 and 2-18 wecomparethe timeevolutionoffrontFroudenumberforvariousvariablei nowsimulationswiththe theoreticalresultsofno-returnowandfull-returnowcu rrentsof x 2.Unlikeaconstant inowcurrent,thefrontspeed u N andfrontheight h N ofavariableinowcurrentevolves 50

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intime.Thereforethedepthratioandcorrespondingly Fr B and Fr NR arefunctionsof time. Inthewaxinginows(seeFigures 2-17 )asthefrontheight ~ h N graduallyincreases withtime,thecurrentdepthratio( a )willalsobeanincreasingfunctionoftime.From Figures 2-8 itcanbeclearlyseenthat Fr NR istightlyboundbetweenthevalues 1.37 0.04 ,while Fr B isastronglydecreasingfunctionofthedepthratio.Furthe rmore,from theexpressionsof Fr B and Fr NR wealsoknowthatfordeepambientorverysmalldepth ratio,both Fr B and Fr NR convergetothesamevalue( p 2 ).Asthecurrentdepthratio increases,thetwoexpressionsincreasinglydeviatefrome achother.Alloftheabove characteristicsareobservedthrougheachoftheframesofF igures 2-17 .Consider frame(a)whichpresentscaseA3,i.e. ~ H =3 and Fr in =0.75 .Theseinputparameters leadtodepthratiothatrangesfrom 0.31 to 0.367 (refertoFigure 2-12 bforprolesof ~ h N ).Henceinframe(a),the Fr B and Fr NR prolesarefarapart.Similarlyinframe(c) whichpresentscaseA5,i.e. ~ H =5 and Fr in =0.25 ,thecurrentdepthratioranges between 0.099 and 0.11 ,whichcorrespondstoadeeperambientorsmallerdepthrati o. Hence, Fr B and Fr NR prolesaremuchcloserthaninframe(a). Nowobservethesolidlineineachframewhichcorrespondsto thefrontFroude number( Fr N )calculatedfromthesimulationdata.Inframe(a)the Fr N curveliesjust belowthe Fr NR curve.Inframe(b) Fr N curveliesbetween Fr B and Fr NR curvessuch thatitisslightlycloserto Fr NR than Fr B curve.Whileinframe(c) Fr N curveinitiallystarts below Fr B butthenitcrossesoverandmovesslightlyaboveit.Thisbeh aviorof Fr N proleisingoodagreementwiththe Fr N ofconstantinowsimulations(seeFigure 2-8 whichalsoincludesdataforwaxinginowsimulations).Tom akethiscomparisonshown inFigure 2-8 ,instantaneousvaluesof Fr N anddepthratio a forallthe5casesbelonging to =1.25 wereselected.Theseinstantaneousvalueswerethenplotte dalongwith thesimulationresultsof =1 Fr B and Fr NR curves.Thedatapointsfor =1.25 correspondto Fr N anddepthratio( a )takenat ~ t =20 andtheirrangeislistedinTable 51

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2-2 .Itmustbementionedherethatthetrendshownbythedatapoi ntsisindependent ofthespecicchoiceof ~ t .Thismeansthatfrontconditionofsteadyfrontsandunstea dy frontsofwaxinginowarecompatiblewhentheirdepthratio andambientowdirection aresimilar.Thisalsoconrmstheexpectationthatthetheo reticalanalysisdevelopedin x 2.2 canbeappliedtothefrontsofwaxinginowcurrents.Furthe rmore,theagreement betweenthefrontconditionof =1.0 and =1.25 simulationssuggeststhatambient owdirectionwillplayasimilarroleinthefrontcondition ofwaxinginowcurrents. Theresultsofwaninginowscanalsobeexplainedinasimila rmanner.Waning inowsleadtounsteadycurrentswhosefrontheightdecreas eswithtime.Sucha currentischaracterizedbydepthratiowhichisdecreasing intime.Figure 2-18 shows thecomparisonof Fr N for =0.90 caseswithcorresponding Fr B and Fr NR predictions. Frames(a)and(b)refertocasesB1andB3respectively.Inth esetwoframestrends whicharesimilarto =1.25 casesareobserved.Currentswithhigherdepthratio have Fr N muchhigherthan Fr B (seeframe(a)inFigure 2-18 ).Evenatsmallcurrent depthratio Fr B cannotcapturethetrendshownby Fr N (seeframe(b)inFigure 2-18 ). Furthermore,trendsin Fr N asafunctionofdepthratioareobservedtobesimilarto =1.0 simulation.Again,toobservethissimilarity,instantane ousvaluesof Fr N and depthratioforallthecasesof =0.90 areplottedinFigure 2-8 .Theseinstantaneous valuesaretakenat ~ t =17.5 andtheirrangeistabulatedinTable 2-2 .Itshouldbenoted thatthetrendobservedfor =0.90 isindependentoftheprecisevalueof ~ t selected. Thedatapointscorrespondingto =0.90 lieclosetoandshowsimilartrendasthe =1.0 cases.Thisimpliesthatfrontconditionofwaninginowsre maincompatiblewith thefrontconditionofsteadycurrentsaslongastheirdepth ratiosaresimilar.Ingeneral, fromFigure 2-8 itcanbeconcludedthatthefrontconditionofconstantino w,waxing inowandwaninginowcurrentsissimilaraslongasthecurr entdepthiscomparable andtheambientowdirectionisthesame.Somescatterinthe numericalresultsare observedFigure 2-8 .Thisbehaviorisexpectedasthefrontconditionisanasymp totic 52

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idealization.Inarealisticsystem,deviationsfromthesi mplicationareboundtooccur duetoviscouseffects,discretizationerrors,localoscil lations,uncertaintiesconnectedto thedenitionof ~ h N ,etc. 2.5Deductions Classicallyalltheanalysisofgravitycurrentfrontswasr estrictedtoambientowing intheoppositedirectiontothecurrent(full-returnowco nguration)andtheclassical Benjamin'sFroudenumberexpressionwasusedforsuchtypeo finviscid,steadygravity currentfronts.Here,throughtheoreticalconsiderations andsimulations,itishighlighted thattheambientowdirectioncangreatlyinuencethefron tconditionofacurrent.In x 2.2 amostgeneralfrontFroudenumberexpressionforagenerica mbientowdirection isgiven,i.e.thedirectionofambientowisparameterized by r whichspeciesthe extentofreturnowofambientinagivensetup.Toillustrat ethevalidityofthistheory anextremecaseofno-returnambientow( r =0.0 )wasconsideredandvariouscases ofcontinuousinow =1 currentsweresimulated.ThefrontFroudenumberobtained fromthesesimulationswascomparedwiththepredictionsof Fr NR Fr B andlockrelease simulations(lockreleaserepresentsgravitycurrentswit hfull-returnowofambient). Thiscomparisonconrmsthepredictionthattheambientow directioncansignicantly inuencethefrontconditionofthecurrent. Variableinowsimulationswerealsocarriedoutwiththesa meno-returnow conguration(outowandinowareontheoppositesidesoft hedomain). =1.25 and =0.90 valueswereselectedwhichrepresentwaxingandwaningino ws respectively.Frontconditionofthesevariableinowsimu lationswerecomparedwith thefrontconditionofconstantinowsimulationsandtheor eticalfrontFroudenumber expressions Fr B and Fr NR .Suchacomparisonrevealedthattheoreticalexpressions developedforsteady,inviscidgravitycurrentfrontscanb eappliedtoinviscid,unsteady frontsasjumpconditionsacrossthedensitydiscontinuity (frontofthecurrent).Itwas alsofoundthatthefrontconditionofvariableinow(unste ady)gravitycurrentsare 53

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comparabletothefrontconditionofcontinuousinow(stea dy)currentsforsimilardepth ratios.Furthermore,thefrontconditionsremaindependen tonthevalueof andthe dependencewasobservedtobenon-uniformovertherangeofd epthratios.Fordeep ambient,theeffectof onthefrontconditionisweakascomparedtoshallowambient Simulationshavebeenusefulinvalidatingthetheory,butt heapplicationofthe theorytorealandlaboratorygravitycurrentsshouldbedon ewithsufcientcare.Ina realisticsettinggravitycurrentsaresubjecttodragfrom thebottomboundarywhich cansignicantlyalterthefrontconditionofthecurrent.W ehaveperformedadditional simulationsofconstantinow( =1 )gravitycurrentswithno-slipbottomboundaryand noticed 15 20% reductionin Fr N ofthecurrent.Moreexperimentsandsimulationsare requiredtoquantifythiseffectsothatusefulempiricalre lationscanbedeveloped. 54

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Figure2-1:Aschematicofbottomgravitycurrentowingint oanambientcolumn. Figure2-2:Controlvolumeaboutthefront.(a)Outowonlef t(onsameendwallas inowsource);(b)Whenoutowisnotlimitedtoanyparticul ardirection.Theamountof outowonleftisnowparameterizedby r 55

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Table2-1:Listofsimulations.Inallthesimulationstheou towboundaryconditionisonthedownstreamendwhilethein owisontheupstreamendofthedomain.Depthratio a (I)isbasedontheboxshapeassumptiontoestimate h N and a (II) isbasedonthe h N estimateusingtheaveragecurrentheightnearthefront. a ( )isthepredictedvalueobtainedbysolving ( B–4 )giveninAppendix B case Fr in ~ H ~ L ~ u N ~ h N (I) ~ h N (II) a (I) a (II) a ( ) Fr N (I) Fr N (II) 11.01.005301.1640.8790.8630.1760.1730.1621.2411.25321.00.755301.0540.7340.7070.1470.1420.1471.2301.25331.00.505300.9230.5630.5450.1130.1090.1281.2301.25041.00.355300.8290.4510.4530.0900.0910.1141.2341.23251.01.002.5301.1680.8600.7600.3440.3040.3251.2591.3 40 61.00.752.5301.0430.7270.6420.2910.2570.2951.2231.3 02 71.00.502.5300.9020.5660.5130.2260.2050.2581.2001.2 59 81.01.002301.1930.8400.6850.4200.3420.4031.3011.44191.00.756301.0530.7370.7150.1230.1190.1221.2271.246 56

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Table2-2:Listofsimulationswithvariableinowcurrents .Duetoitsunsteadynature depthratioofthecurrent a isafunctionoftime.Followingtablegivestherangeof a for eachcase. case Fr in ~ H ~ La (range) Fr N A11.251.005300.231-0.2591.283A21.250.755400.195-0.2291.255A31.250.753400.314-0.3691.321A41.250.505300.153-0.1751.231A51.250.255300.099-0.1141.232B10.901.002.5300.326-0.3101.199B20.900.752.5300.273-0.2491.187B30.900.502.5300.213-0.1911.181 Figure2-3:Schematicrepresentationofthecomputational domain.Thissetupcorrespondstono-returnconguration,i.e. r =0 inFigure 2-2 (b). 57

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x h 5 10 15 20 25 30 0 2 4 6 t=15 t=20 t=25 case1~~ ~ ~~ (a) x h 5 10 15 20 25 30 0 2 4 6 t=15 t=20 t=25 case2 ~ ~ ~ ~~(b) x h 5 10 15 20 25 30 0 2 4 6 t=20 t=25 t=15 ~ ~ ~~~ case3 (c) Figure2-4:Timeevolutionofthecurrentforsomeofthecase slistedinTable 2-1 .(a) case1,(b)case2and(c)case3. 58

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t h N ,u N 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 1.2 1.4 h N case1(I) u N case1 h N case2(I) u N case2 h N case3(I) u N case3 ~~ ~~ ~ ~ ~~ ~ Figure2-5:Frontvelocity( ~ u N )andfrontheight( ~ h N )basedontheboxassumptionfor someofthecaseslistedinTable 2-1 Fr in f(Fr in ) 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 cases19curvefit f(Fr in )= h N h N (Fr in =1.0) (b) f(Fr in )=1.002Fr in 0.6147 Figure2-6: f ( Fr in )= h N = h N ( Fr in =1.00) proleobtainedfromcurvetthroughvarious datapointsthatrepresentallthesimulationslistedinTab le 2-1 59

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t Fr N 5 10 15 20 25 0.6 0.8 1 1.2 1.4 1.6 case1case2case3 ~ Figure2-7:ProlesoffrontFroude( Fr N )asafunctionoftimeforcertaincaseslistedin Table 2-1 a Fr 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Fr B Fr NR a =0 a =0.90 a =1 a =1.25 Figure2-8:EffectofoutowboundarylocationonthefrontF roudevaluefordifferent valuesof .Dashedline( )representsthecurveofBenjamin'sFroudenumberexpression.Solidline(—)representsthecurveofno-returnF roudenumberexpression.' symbolrepresentsthefrontFroudenumberofthelockreleas e( =0 )simulations.' symbolrepresentsthefrontFroudenumberof =0.90 simulations.' 'symbolrepresentsfrontFroudenumberevaluatedusingtheBoxmodelas sumptionappliedtothe constantinow( =1 )simulations.' 'symbolsrepresentsthefrontFroudenumberof =1.25 simulations. 60

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t V 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 40 45 A1A2A4A5Exact a =1.25~~ (a) t V 0 5 10 15 20 25 30 35 0 5 10 15 20 25 B1B2B3Exact a =0.90~~ (b) Figure2-9:Comparisonoftheactualvolumeofcurrentinthe simulationdomainwiththe exactvalueforvariouscasesbelongingto(a) =1.25 and(b) =0.90 61

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Figure2-10:Densitycontoursof Fr in =1.00 and =1.25 simulationatdifferentinstances.(a) ~ t =5,(b) ~ t =10,(c) ~ t =15and(d) ~ t =20 62

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Figure2-11:Densitycontoursof Fr in =1.00 and =0.90 simulationatdifferentinstances.(a) ~ t =5,(b) ~ t =10,(c) ~ t =15and(d) ~ t =22.5 63

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t u N 0 5 10 15 20 25 30 35 0.25 0.5 0.75 1 1.25 1.5 A1A2A3A4A5 a =1.25 (a) ~~ t h N 5 10 15 20 25 30 35 0.4 0.6 0.8 1 1.2 1.4 A1A2A3A4A5 a =1.25 (b) ~~ Figure2-12:Prolesof(a)frontvelocity( ~ u N )and(b)frontheight( ~ h N )fordifferent Fr in casesof =1.25 asafunctionoftime. 64

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t u N 0 5 10 15 20 25 30 35 0.5 0.75 1 1.25 1.5 1.75 B1B2B3 a =0.90 (a) ~~ t h N 5 10 15 20 25 30 35 0.4 0.6 0.8 1 B1B2B3 a =0.90 ~~(b) Figure2-13:Prolesof(a)frontpropagationvelocity( ~ u N )and(b)frontheight( ~ h N )for different Fr in casesof =0.90 asafunctionoftime. 65

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t y u 0 5 10 15 20 25 30 35 0.75 1 1.25 1.5 1.75 A1A2A3A4A5 a =1.25 (a) ~ t y h 5 10 15 20 25 30 35 0.6 0.8 1 1.2 1.4 1.6 f(Fr in )=1.000(A1) f(Fr in )=0.842(A2) f(Fr in )=0.842(A3) f(Fr in )=0.654(A4) f(Fr in )=0.415(A5) curvefit a =1.25 (b) y h =h N /f(Fr in )=0.852t 0.137 ~ ~ ~ Figure2-14:Prolesof(a)scaledfrontvelocity u and(b)scaledfrontheight h for different Fr in casesof =1.25 asafunctionoftime. 66

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t y u 0 5 10 15 20 25 30 35 0.5 0.75 1 1.25 1.5 1.75 B1B2B3 a =0.90 (a) ~ t y h 5 10 15 20 25 30 35 0.4 0.6 0.8 1 f(Fr in )=1.000(B1) f(Fr in )=0.836(B2) f(Fr in )=0.646(B3) curvefit a =0.90 y h =h N /f(Fr in )=0.946t 0.0696 ~ ~ ~ (b) Figure2-15:Prolesof(a)scaledfrontvelocity u and(b)scaledfrontheight h for different Fr in casesof =0.90 asafunctionoftime. 67

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t Fr N 5 10 15 20 25 30 35 0.4 0.6 0.8 1 1.2 1.4 1.6 A1A2A3A4A5 (a) a =1.25 ~ t Fr N 5 10 15 20 25 30 35 0.4 0.6 0.8 1 1.2 1.4 1.6 B1B2B3 a =0.90 (b) ~ Figure2-16:Prolesof(a)frontFroudenumber( Fr N )fordifferent Fr in casesof =1.25 asafunctionoftime.(b)frontFroudenumber( Fr N )fordifferent Fr in casesof =0.90 as afunctionoftime. 68

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t Fr 5 10 15 20 25 30 35 0.8 1 1.2 1.4 1.6 1.8 A3Fr B Fr NR (a) ~ t Fr 5 10 15 20 25 30 35 0.8 1 1.2 1.4 1.6 1.8 A2Fr B Fr NR (b) ~ t Fr 5 10 15 20 25 30 35 0.8 1 1.2 1.4 1.6 1.8 A5Fr B Fr NR (c) ~ Figure2-17:ComparisonofthefrontFroudenumber( Fr N )ofcertainselectedcases having =1.25 withthepredictionsofBenjaminFroudenumber( Fr B )andtheFroude numberbasedonno-returnow( Fr NR )proposedin x 2.2 69

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t Fr 5 10 15 20 25 30 35 0.8 1 1.2 1.4 1.6 1.8 B1Fr B Fr NR (a) ~ t Fr 5 10 15 20 25 30 35 0.8 1 1.2 1.4 1.6 1.8 B3Fr B Fr NR (b) ~ Figure2-18:ComparisonofthefrontFroudenumber( Fr N )ofcertainselectedcases having =0.90 withthepredictionsofBenjaminFroudenumber( Fr B )andtheFroude numberbasedonno-returnow( Fr NR )proposedin x 2.2 70

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CHAPTER3 DYNAMICSOFCOMPLETETURBULENCESUPPRESSIONINTURBIDITY CURRENTSDRIVENBYMONODISPERSESUSPENSIONSOFSEDIMENTS 3.1Background Horizontaldensitydifferencesinauidundertheactionof gravitationaleld resultinowsknownasgravitycurrents[ 3 39 ].Turbiditycurrentsarespecialcasesof gravitycurrentswheresuspendedsedimentsareresponsibl eforthedensitydifference. Turbiditycurrentsarepredominantlyoceancurrentsowin galongtheoceanoor.They canbetriggeredbyvariousmechanisms,butessentiallyatu rbiditycurrentinvolves sedimentsowingdowntheslopingoceanoordraggingthesu rroundingwaterwithit. Theresultingowistypicallyturbulent,whichinturnkeep sthesedimentsinsuspension todrivetheow.Turbiditycurrentsareknowntoreachspeed softheorderof 5 m = s in submarinecanyons[ 50 ]andtravelhundredsofkilometers[ 71 ].Thisabilitytoreach highspeedsandtravellongdistancesmaketurbiditycurren tsanimportantsediment transportmechanisminthesubmarineworld[ 49 ]. Suspendedsedimentsplayacrucialroleinbothsustaininga ndkillingturbidity currents.Theincreaseddensityduetothesuspendedsedime ntsdrivestheow, andthegeneratedturbulenceprovidestheenergyrequiredt okeepthesedimentsin suspension.Thedistributionofsuspendedsedimentsinatu rbiditycurrentisgoverned bybalancebetweendownwarduxofsedimentsduetosettling andupwarduxof sedimentsduetoturbulentmixing.Thisexchangeleadstoas tablestraticationof sedimentsintheow.Thisstraticationsuppressesturbul encebylimitingtheexchange ofmassandmomentuminthebed-normaldirectionandinextre mecasescanlead tocompleteshutdownofturbulence.Insuchextremecases,a bsenceofturbulence leadstoheavydepositionofsedimentswhichresultsincomp letecessationoftheow [ 20 83 ].Thisdualroleofsedimentsmakestheowinterestingandc omplextoanalyze. Thischapterdissectsthesetworolesusingdirectnumerica lsimulations(DNS)and attemptstoexplainhowsuspendedsedimentsaffectowturb ulence. 71

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Inanactualturbiditycurrenttherearesedimentsofvariou ssizes[ 65 ]andtheow itselfishighlyturbulentwithalargeReynoldsnumberof O (10 6 ) ormore.Sufciently largesedimentsdonotgetsuspendedintotheow,butaretra nsportedalongthebed asbedloadthroughrollingandoccasionalsaltation.Onthe otherhand,washload sedimentsaresonethattheyeffectivelydonotsettleandr emainwell-mixedinthe ow.Theeffectofwashloadistomaketheuidheavierthanit ssurrounding.Inthis study,onlyintermediatesizeofsedimentswillbeconsider edthatsettletowardsthebed andcanbere-entrainedintosuspensionbyturbulence.Thep articularlimitwherethe rateofdepositionisexactlybalancedbytherateofresuspe nsionisconsidered,and thusthetotalamountofsuspendedsedimentsisconservedin theow.Inliteraturethis limitwherethecurrenthaszeronetuxofsedimentstotheoc eanbedisidentiedas bypassorauto-suspensionmode.Physicallythismodeimpli esthatthedrivingpotential ofsuchacurrentisalwaysconserved.Thisregimeofsedimen ttransportisthelimiting casebetweenthedepositionalmode,forwhichthenetamount ofsedimentsdecay overtimeordistanceofspreadingleadingtoultimatecessi onofthecurrent,andthe resuspensionmode,forwhichthenetamountofsedimentsinc reasesalongwiththe intensityofthecurrent[ 64 66 ]. DNSofaeld-scaleturbiditycurrentisnotpossibleevenwi ththecurrentstateof-the-arthigh-performancecomputers.Inordertogainfu ndamentalunderstanding oftheeffectsofsuspendedsedimentsonturbulence,simula tionsmustberestrictedto lowandmoderateReynoldsnumberturbiditycurrents.Theun derstandinggainedfrom suchsimulationscanthenbeextrapolatedandusedinvariou smodelsofeldscale turbiditycurrents.Recentinvestigationsofturbulenttu rbiditycurrentsusingthe”turbidity currentwitharoof”(TCR)model[ 15 19 ]demonstratethatturbulencecanbecompletely dampedwhenthesettlingvelocityofsedimentsexceedacert ainthresholdvalue. ThischapterextendstheworkbyCantero etal. [ 15 19 ]inasignicantwaytoobtain acomprehensiveunderstandingofthemechanismoftotaltur bulencesuppression. 72

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Theearliertheoreticalmodel(TCR)employedanarticialr igidtopboundary,which somewhatcomplicateditsdirectapplicabilitytoactualtu rbiditycurrents.Asaresult, evenwhenstraticationsuppressedturbulenceclosetothe bottomboundary,turbulence continuedtogenerateneartherigidtopboundaryanddiffus eddownwardtothebottom boundary,thusinuencingturbulencesuppressionbystrat ication.Thisfeatureaffected identicationofthesuppressionmechanismandpreventedc learinterpretationof theresults.Hereamodelofequilibriumturbiditycurrents thatusesanstress-free topboundaryisimplemented,whichlimitsturbulencegener ationtoonlythebottom boundary.Thepresentapproachallowscompletesuppressio nofturbulenceoverthe entirelayerofuid.However,asintheTCRmodel,theslowst reamwisedevelopmentof thecurrentisignoredbynotallowingentrainmentofambien tuid. Themodelpresentedinthischapterpreservestheessential featuresofturbidity currents,i.e.,theowisentirelydrivenbythesuspendeds edimentsandthesettling ofsedimentsself-stratiestheow.Thesefeaturesarerep resentativeofanactual turbiditycurrentinthebypassmode.Thepresentmodelhasb eenusedtoseparatethe drivingandthestraticationeffectsofthesuspendedsedi mentsbyindividuallyturning onandofftheseeffects.Themostintriguingaspectofthere sultsistheabruptnatureof totalturbulencesuppressionwithincreaseinthesettling velocityofsediments(aproxy forthesizeofsediments).Moreprecisely,belowathreshol dvalueofsettlingvelocity theturbiditycurrentremainsvigorouslyturbulentandthe levelofturbulenceisonly moderatelyinuencedbythesuspendedsediments.However, oncethethresholdvalue iscrossed,anabruptchangeinbehavioroccursandturbulen ceistotallyextinguished. Thisabrupttransitionisakintotheonsetofinstabilityin alaminarow,butthekey differenceisthatthenatureofreverse-transitiontolami narconditionsisinherently non-linear.Thischapterfurtherexploresthistransition bothfromastatisticalpointof viewandalsofromamechanisticpointofviewintermsofhowt heauto-generationof turbulentvorticalstructuresiscompletelydisruptedabo vethethreshold. 73

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Themathematicalmodelofturbiditycurrentsemployedinth isstudyservesthe purposeofdevelopingfundamentalunderstandingofintera ctionofsuspendedsedimentsandturbulence.Existenceofacriticalsettlingvelo cityforagiven Re andchannel slope haspracticalimplications.Thisobservationmeansthaton lyacertainsizeof sedimentscanbekeptinsuspensioninaowofgivenintensit y.Theknowledgewe gainbringsusclosertounderstandingthetransitionoftur biditycurrentsfrombeing neterosional(self-accelerating)tonetdepositional.Th evalidationofthissimple,yet effectivemathematicalmodelisdifcultaslaboratoryexp erimentscannotrecreatesuch conditionsandtheeldobservationsaretoocomplextobeex actlyrelevant.However, someindirectindicationsofturbulencesuppressionexist .ExperimentsofSequeiros etal. [ 77 ]showtransitionbetweenneterosionaltonetdepositional ow.Insuchatransitionthenetdepositionalowwilleventuallyleadtocomp leteshutdownofturbulence andtheow.However,inlaboratorythiseventualstatecann otbeobservedowingtothe sizeconstrainsoftheume.Talling etal. [ 83 ]reportedeldobservationsthatsudden turbulencesuppressioncanoccurleadingtoabruptlossofc arryingcapacityoftheow leadingtoheavysedimentation. 3.2ProblemFormulation Theturbiditycurrentismodeledasaninclinedchannelwith aslopeof withrespecttothehorizontal.Theowinthechannelisdrivenbyam ono-dispersesuspension ofsedimentswhichdragtheuidinthestreamwisedirection andalsosettledownstratifyingtheow(seeFigure 3-1 ).Thesuspensionisassumedtobedilutesothatcollision betweensedimentparticlesandrheologyeffectscanbenegl ected,settlingvelocityis independentofconcentration,andBoussinesqapproximati oncanbeemployed.Under 74

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thesecircumstancestheowisgovernedbythefollowingequ ations @ u @ t + u r u = 1 w r p + r 2 u + w w g (3–1) r u =0, (3–2) @ c @ t +( u + V ) r c = Dr 2 c (3–3) where u = f u v w g isthevelocityvector, p isthepressure, w isthedensityofwater, isthekinematicviscosityofwater, c isthevolumetricconcentrationofsediments, = w (1+ Rc ) isthedensityofthemixturewith R =( s w ) = w and s the densityofthesediments, g = f g x ,0, g z g istheaccelerationduetogravitywith g x and g z thecomponentsalongthestreamwiseandbed-normaldirecti onsofthechannel, V = f V x ,0, V z g istheterminalsettlingvelocityofanisolatedsedimentpa rticleina quiescentambient,and D isthediffusivityofthesediments.Byusingtheseequation s itisassumedthattheinertialeffectsofthesedimentsareo fsecondordercomparedto settlingeffects[ 17 21 28 ]. Sedimentparticlesareassumedtobenon-cohesiveandlarge enoughthattheir Brownianmotioncanbeignored.Itisnowwellestablishedth atevensuchlargeparticleseffectivelydiffuseduetolongrangehydrodynamicint eractionsmediatedbyparticle numberdensityuctuations[ 58 76 ].Thus D istakentobetheeffectiveconstantdiffusivityofthesediments.Thediffusivetermin( 3–3 )alsoprovidesamechanismto resuspendsedimentsfromthebed[ 30 ]. Thechannelisassumedtobeperiodicinthestreamwiseandsp anwisedirections. Theheightofthesimulationdomainis h .Thelengthofthesimulationdomain(streamwisedirection)is L x =4 h andthewidth(spanwisedirection)is L y =4 h = 3 .Along thestreamwiseandspanwisedirectionsperiodicboundaryc onditionsareimposedfor allvariables.Thetopboundaryofthechannelissmoothwith anostressboundary condition,whilethebottomboundaryrepresentsthebedwhe reano-slipconditionis 75

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imposed.Thetopandbottomboundaryconditionsforconcent rationstatethatsettling exactlybalancesresuspension.Thus,forthetopandbottom boundariestheimposed boundaryconditionsare u =0 at z =0 (3–4) @ u @ z =0, @ v @ z =0 and w =0 at z = h (3–5) cV z = D @ c @ z at z =0 and z = h (3–6) Animportantconsequenceofassuming( 3–6 )isthatthenetvolumeconcentration ofsedimentswithinthedomainisconserved.Thisallowsfor thestatisticallystationary stateoftheturbulentow.3.2.1MeanFlowEquations Theproblemdenedin x 3.2 hasafullydevelopedstatisticallystationarystate.The meanowequationsobtainedbyaveragingthemomentumandco ncentrationequations overturbulenceread d 2 u dz 2 d dz ( u 0 w 0 )+ R cg x =0, (3–7) 1 w d p dz + d dz ( w 0 2 )+ R cg z =0, and(3–8) D d 2 c dz 2 + V z d c dz d dz ( c 0 w 0 )=0. (3–9) Here,themeanvariablesarerepresentedbyanoverbar( )andtheuctuationsfrom themeanarerepresentedbyaprime( 0 ).Averagingisperformedovertimeandover thetwohomogeneousdirections x and y .Sincethereisnogravitationalacceleration alongthespanwisedirection( y ),themeanowisassumedtwo-dimensionalwith v =0 .Furthermore,owingtotheincompressibilityoftheuidan dtotheno-penetration conditionatthetopandbottomboundaries, w =0 .Observethatpressureappearsonly 76

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in( 3–8 ),fromwhich p = p + w w 0 2 + w Rg z Z z 0 c ( ) d (3–10) where p isthemeandynamicpressurethatremainsaftersubtracting thehydrostatic pressureduetosuspendedsediments,anditsonlyroleistoe nsuretheincompressibilitycondition.From( 3–7 )thebottomshearstress b canbeexpressedas b w = Rg x Z h 0 c ( z ) dz (3–11) sincethestressatthetopboundaryiszeroandtheReynoldss tress( u 0 w 0 )iszeroatthe topandbottomboundaries.Basedon( 3–11 )avelocityscaleisdenedas u 2 = b w = Rg x c ( v ) h (3–12) where c ( v ) = 1 h Z h 0 c ( z ) dz (3–13) isthevolumeaveragedsedimentconcentration.Itisappare ntthatfor V = j V j =0 thesedimentconcentrationwillremainwellmixedwithouta nyverticalgradient,i.e. d c = dz =0 .Inthislimittheproblemreducestoachannelowdrivenbya uniformbody forcegivenby w Rg x c ( v ) FollowingtheworkbyGeyer[ 32 ],turbulentsedimentuxescanbeapproximatedas w 0 c 0 = D t d c dz (3–14) where D t = u z (1 z = H ) istheturbulentdiffusivityand istheconstantinthevelocity log-law( u + =(1 = )log( z + )+ B ).( 3–9 )canbeintegratedtotheRouseprole c c b = ( H z ) = z ( H b ) = b Ro (3–15) where Ro istheRousenumbergivenby Ro = V z = u b =0.01 H and c b isthevalueof concentrationat z = b 77

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3.2.2DimensionlessEquations Thefollowingscalesareemployedtodenedimensionlessva riables: u forvelocity, h forlength, h = u fortime, c ( v ) forconcentrationofsediments,and w u 2 forpressure. Thedimensionlessgoverningequationsare @ ~ u @ ~ t + ~ u ~ r ~ u = ~ r ~ p + 1 Re ~ r 2 ~ u +~ c e g (3–16) ~ r ~ u =0, (3–17) @ ~ c @ ~ t +( ~ u + ~ V ) ~ r ~ c = 1 Re Sc ~ r 2 ~ c (3–18) where e g = f 1,0, 1 = tan g and tan = g x = g z .Dimensionlessvariablesarerepresented by( ~ ).Thedimensionlessnumbersin( 3–16 )-( 3–18 )aretheReynoldsnumber( Re )and theSchmidtnumber( Sc ),whicharedenedas Re = u h and Sc = D (3–19) Theothertwoparametersinvolvedinthemodelarethenon-di mensionalsedimentsettlingvelocity j ~ V j = ~ V andtheslopeofthechannel tan .Notethattheabove mathematicalmodel,with( 3–6 )asconcentrationboundarycondition,considersthesedimentationandresuspensionuxestobalanceeachotherloc allyandinstantaneously. Thisindicatesthatsedimentsthatsettleonthebedareinst antlyresuspendedbackinto theow.Thisstrongassumptionimpliesthattheowisinthe bypassmodeforallthe casestobesimulated,evenaswevarythesettlingvelocity. Notethatinrealturbidity currentsinordertostrictlyremaininthebypassmode,anin creaseinsettlingvelocity mustbeaccompaniedbyacorrespondingincreaseinthedrivi ngforce. Typicallytheinclinationofthecontinentalslopeontheoc eanoorrangesfrom 1 to 10 [ 70 ],hencetheslopeofthechannelinallthesimulationsisxe dat =5 and 1 = tan =11.43 .Forthecaseofaeldturbiditycurrentofheight h =20 mrunning onaslopeof =5 withameanvolumeconcentration c ( v ) =0.005 ofsandparticles inwater( R =1.65 and =10 6 m 2 /s),thedenitionin( 3–12 )gives u =0.38 m/s. 78

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Thecorresponding V forsedimentdiameters 70 mand 120 mare V =0.004 m/sand V =0.01 m/s[ 65 ],withcorresponding ~ V z =1 10 2 and ~ V z =2.6 10 2 ,respectively. ThecorrespondingReynoldsnumber( Re =7.5 10 6 )istoolargeandcannotbestudied withDNS.Thepresentsimulationsemploy Re =180 whichresultsinamatureturbulent ow.FollowingMucha&Brenner[ 58 ]andSegre etal. [ 76 ]aroughapproximationfor sedimentdiffusivityis D 10 aV ,where a istheradiusofthesedimentparticles.Thus, fortheaboveexample D 1.5 6.1 10 6 m 2 = s .Itshouldbenotedthat D willdepend onthelocalconcentration,concentrationgradientandshe arstress,butwesuppress suchdependenciesinourpresentmodel.Basedonthendings ofNecker etal. [ 59 ] andCantero etal. [ 18 ]thatthesimulationresultsofturbiditycurrentsareinse nsitiveto theprecisevaluesof Sc aslongasitis O (1) ,thepresentsimulationsemploy Sc =1 Table 3-1 showsthelistofcasesstudied. Thebodyforceterminequation( 3–16 )canberedenedas ~ c e g =(~ c av +~ c )( ^ e x ^ e z tan ), (3–20) where ~ c av istheinstantaneousconcentrationaveragedoverplanespa ralleltothebed, ~ c isthecorrespondinguctuationfromtheaverageconcentra tioneldand ^ e x and ^ e z areunitvectorsinthe ~ x and ~ y directionrespectively.Thetermsintheaboveequation canbegroupedasfollows ~ c e g =(~ c av +~ c ) ^ e x ~ c av tan ^ e z ~ c tan ^ e z (3–21) Thesecondtermontherightintheaboveequationcanbecombi nedwiththepressure gradientterminthegoverningequation( 3–16 ),andtherearrangedpressureeldis givenby ^ p =~ p 1 tan Z ~ z 0 ~ c av ( ) d (3–22) 79

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Thepresentnumericalsimulationssetthebed-normalpress uregradienttobezeroat thetopandbottomboundaries,andtheabovedenitionofpre ssureisessentialtoavoid theformationofaspuriouspressureboundarylayerinthedo main. Thebulkvelocityoftheowisdenedasfollows ~ u b = Z 1 0 ~ u (~ z ) d ~ z (3–23) BulkReynoldsnumbercanbedenedas Re b = u b h = .Thecorrespondingvaluesof bulkvelocity ~ u b andbulkReynoldsnumber Re b forallthecasesarelistedinTable 3-1 For Re =180 bulkReynoldsnumberisoftheorderof3500andtheunstrati edowis turbulent. 3.3NumericalMethod Thedimensionlessgoverningequations( 3–16 )-( 3–18 )aresolvedusingadealiased pseudo-spectralcode[ 24 ].Fourierexpansionsareemployedinthedirectionstangen tial tothebed( x y ),whileaChebyshevexpansionisusedinthedirectionnorma l tothebed( z ).Asplittingmethodisusedtosolvethemomentumequationa ndthe incompressibilitycondition.Alowstoragemixedthird-or derRunge-KuttaandCrankNicolsonschemeisusedfortemporaldiscretizationofadve ctionanddiffusionterms. Thisschemeiscarriedoutinthreestageswithpressurecorr ectionattheendofeach stage.Detailsontheimplementationofthisschemecanbefo undintheworkby Cortese&Balachandar[ 26 ].Thegridresolutionis ( N x N y N z )=(96,96,97) 3.4LaminarSolution Thelaminarsolutionofthegoverningequationsisofacadem icinterestonly. Notethatthelaminarowwillnotberealizedsinceitisnotp ossibleforthenitesizedsedimentstostayinsuspensionwithoutthepresenceo fturbulentmixingand turbulentsedimentresuspensionfromthebed.Thelaminars olutionis,however,useful inestablishingtotalsuppressionofturbulence. 80

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Thelaminarowisassumedtobeonedimensional,steadyandu niform.This meansthat ~ v =0 ~ w =0 @=@ ~ t =0 @=@ ~ x =0 and @=@ ~ y =0 .Inthiscasethegoverning equations( 3–16 )-( 3–18 )simplifyto d 2 ~ u d ~ z 2 = Re ~ c and ~ V z d ~ c d ~ z = 1 Re Sc d 2 ~ c d ~ z 2 (3–24) Integrating( 3–24 )withboundaryconditions( 3–4 )-( 3–6 )yields ~ c = A exp( ~ V z Re Sc ~ z ), (3–25) and ~ u = A ~ V 2 z Re Sc 2 h exp( ~ V z Re Sc ~ z ) Re ~ V z Sc ~ z exp( ~ V z Re Sc )+1 i (3–26) Theconstant A cannotbedeterminedusingtheboundaryconditions.Theamo unt ofsedimentspresentintheowatsteadystatedependsonthe initialcondition.Thus, thesystemisclosedwiththefollowingcriterion Z 1 0 ~ cd ~ z =1, (3–27) whichgives A = ~ V z Re Sc 1 exp( ~ V z Re Sc ) (3–28) Thelimitingcaseof ~ V =0 leadstoauniformconcentrationproleandaparabolic velocityprole ~ u = Re 2 (~ z 2 2~ z ). (3–29) Intheotherlimitingcaseofverylargesettlingvelocity,i .e. ~ V !1 ,thesediments settletothebottomandtheconcentrationproleisgivenby aDiracdeltafunction locatedat ~ z 0 + .Itcanbeshownfrom( 3–26 )that ~ u max 0 as ~ V z !1 ,andtherefore theowceasestoexist.Figure 3-2 showsthelaminarvelocityandconcentrationproles for ~ V =0.0265 1 = tan =11.43 Re =180 and Sc =1 81

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3.5Results Asmentionedbefore,inturbiditycurrentssuspendedsedim entsplayadualrole ofbothdrivingandstratifyingtheow.Underextremesitua tions,straticationcan leadtocompleteturbulencesuppression.Atthisstate,the owlosesitsabilityto keepsedimentsinsuspensionanditceasestoexistduetocom pletesedimentation. Previousresearchhashighlightedtheexistenceofacritic alsettlingvelocitybeyond whichcompleteturbulencesuppressionoccurs[ 15 19 ],buttheunderlyingmechanisms remaintobeexplored. Table 3-1 listsallthedifferentsimulationsanalyzedinthischapte rtoaddress themechanismofturbulencedampinginturbiditycurrents. Sixdifferentcasesof stratiedowsareconsidered.Cases1to6correspondtosim ulationswherethe settlingvelocityofsedimentsisincreasedfrom ~ V =1 10 2 to ~ V =2.65 10 2 Case 0 ,with ~ V =0 ,isthereferencecase,sinceitssolutioncorrespondstoun stratied turbulentchannelowdrivenbyauniformbodyforce.Asshow ninFigure 3-2 (a),the concentrationproleforcase0isuniform.Thecorrespondi ngvelocityproleisshown inFigure 3-2 (b),andinFigure 3-3 inwallunitsas u + = u = u vs z + = zu = ,wherevery goodagreementisobservedwiththelawofthewall( u + = z + )for z + < 5 andwiththe loglaw( u + =1 = 0.41log( z + )+5.5 )for z + > 30 .Table 3-1 alsoreportscasesS1and S2,whichcorrespondtoowswherethestraticationeffect ofsuspendedsedimentsis turnedoffbysuppressingrelevanttermsinthemomentumequ ation. Theresultspresentedhererepresentlongtermasymptotics tationarystates(statisticallysteadystates)oftheow.Twodifferentinitial conditionswereconsideredin thisstudytoaddressthesensitivityofthestationarystat etotheinitialconditions:a)unstratiedturbulentchannelowwithuniformconcentratio nofsediments(thestationary stateofcase0),andb)aninstantaneousvelocityandconcen trationeldsofanearby turbulentcasetakenduringitsstationarystate(forexamp le,aninstantaneousoweld ofthestationarystateofcase2isusedasinitialcondition forcase3).Severalofthe 82

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casesinTable 3-1 weresimulatedwiththesetwoinitialconditions.Itwasobs ervedthat theapproachtothestationarystatefromtheinitialcondit ionisoscillatory.Quantities suchasnear-bedhorizontallyaveragedconcentrationofse dimentsevolvefromthe initialconditiontothenalstationaryvaluedisplayingo vershootsandundershoots.The farthertheinitialconditionisfromthestationaryvalue, thelargertheseovershootsand undershootsare.Furthermore,itisobservedthattheoscil latorybehaviorofthesolution hasanimpactonthenalstateonlyforthecaseswherethesed imentsettlingvelocity isjustbelowthecriticalvalueforcompleteturbulencesup pression(onlycase5inthe presentinvestigation).Inthissense,thenalasymptotic statecanbetakentobeinsensitivetothepreciseinitialconditionemployed.Theresul tspresentedhereforcase5 thatfallsinthissmallsensitiverangewereobtainedbymin imizingtheinitialovershootof thesolution.Theoscillatorybehaviorofthesolutionatea rlytimeswillnotbediscussed here.3.5.1MeanValuesandTurbulentFluxesinStratiedFlows ThemeanconcentrationprolesofselectedcasesgiveninTa ble 3-1 areshownin Figure 3-2 (a).Asthesettlingvelocityofsedimentsincreases,there sultingconcentration prolenolongerremainsuniformandincreasinglydeviates fromthereferenceprole ofcase0.Theresultingnon-uniformmeanconcentrationofs edimentsisskewed towardsthebed,whichaffectstheowintwoways.First,the streamwisedrivingforce, representedbythersttermontherighthandsideof( 3–21 ),isalsocorrespondingly skewedtowardsthebed.Second,thestablestraticationin ducedbyconcentration gradientstendstodampenturbulenceandthiseffectiscapt uredbythelasttwoterms ontherighthandsideof( 3–21 ).Asitwillbediscussedbelow,theeffectofskewed drivingforceisnotasstrongandthestraticationeffecti sthedominantmechanism responsibleforcompleteturbulencesuppression. Figure 3-2 (a)showsaclearchangeofregimefromcase5tocase6.Whilec ase5 showsawell-mixedconcentrationprole,theconcentratio nproleforcase6isnearly 83

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zerointhetoppartofthechannelandpresentslargevaluesi nthenear-bedregion. CorrespondingmeanvelocityprolesareshowninFigure 3-2 (b),andtheyalsoshowa clearchangeofregimefromcase5tocase6.Whilethevelocit yproleforcase5shows asmalldeviationfromtheturbulentvelocityproleofcase 0,thevelocityproleforcase 6showslessmixingandresemblesalaminarprole.Intheins etofFigure 3-2 (a)mean concentrationprolesofcase1and5arecomparedwiththeco rrespondingRouse proles.Rouseprole( 3–15 )isanapproximationfortheconcentrationproleachieved bycharacterizingtheturbulentmixingtermin( 3–9 )byeddydiffusivity.Rouseprolesof cases1and5areevaluatedusingthemodiedvonKarmanconst ant whichisgiven inTable 3-1 .Thisquantiestheeffectofstraticationontheturbulen tmixinginthe channel.FromtheFigureitisclearthatRouseprolesoverpredicttheturbulentmixing instratiedowsleadingtofullerconcentrationproles. Figure 3-3 presentsthevelocity prolesinwallunitsonlog-linearscales.Alsoincludedin thisgureisthelogarithmic lawofthewall u + =1 = log( z + )+ B forcases0and5withadjustedconstants.Thebest tvaluesof and B forallcasesarelistedinTable 3-1 .Thevelocityprolesforcases0 to5showadeniteturbulentnaturewithadenitelogarithm icregionfor z + > 30 .The deviationfromthefullyturbulentlogarithmiclaw(case0) isonlyintermsoftheconstant values.Thischangeintheconstantscanbeattributedtothe dampingofturbulence becauseofstratication[ 15 ].Case6,ontheotherhand,doesnotdisplayalogarithmic region.Figures 3-2 (a)and 3-2 (b)showalsothelaminarsolutioncomputedforthesame setofparametersusedforcase6.Almostperfectagreementi sobservedbetweencase 6andthecorrespondinglaminarsolution. Thenatureofthechangeofowregimecanbefurtherexplored inFigure 3-4 wherethemeanvelocityatthetopboundary( ~ u t )andthemeanconcentrationatthebed ( ~ c b )areplottedasafunctionofsettlingvelocityofsediments .Belowthecriticalvalue, increaseinthesettlingvelocityofsedimentsresultsinmo destincreaseofthenetow rate,asindicatedbytheincreaseofthemeanvelocityatthe topstress-freeboundary. 84

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Aroundthecriticalsettlingvelocitythereisanabruptinc reasein ~ u t signalingthechange ofowregime.Asimilarbehaviorcanalsobeobservedinthem eanconcentration ofsedimentsatthebed,whichincreasesslowlyforbelow-cr iticalvaluesofsettling velocity.Acrossthecriticalvalue,anabruptincreaseint hebottomconcentrationcanbe observed. Thebalancebetweenthesettlinguxofsediments( ~ V z ~ c )andtheturbulentuxof sediments( ~ w 0 ~ c 0 )canbeobtainedfrom( 3–9 )as ~ V z ~ c = ~ c 0 ~ w 0 1 ReSc d ~ c d ~ z (3–30) Increaseofthesettlingvelocityincreasesthesettlingu xofsediments.Inordertokeep thesedimentswellmixedintheow( d ~ c = d ~ z 0 ),theremustbeaproportionalincrease intheturbulentuxofsediments.Inotherwords,inawell-m ixedchanneltheratioof turbulentuxtosettlinguxofsedimentsapproaches1.The relativeimportanceof turbulentuxofsedimentscanbeseeninFigure 3-5 (a),whichshowsthevariationof theratioofturbulentuxtosettlinguxinthebed-normald irection.FromFigure 3-5 (a) itisevidentthattheincreaseinturbulentuxisnotpropor tionaltosettlinguxforallthe casesshown.Thisratioshowsadecreasingtrendwithincrea singsettlingvelocityof sediments,butthedecreaseisnotverysensitiveforcases1 to5.Also,thedecreaseis notuniformthroughoutthechannel,itismorepronouncedne arthetopboundarythan nearthebed.Thisasymmetryisduetoreductionofturbulent transportfromnearthe bed,whereturbulenceproductionoccurs,totheupperparto fthechannel.Forcase6 theturbulentuxofsedimentsiszero.Thistrendshowsthee ffectofstraticationbutit cannotexplainthesuddentransitionseenatthecriticalse ttlingvelocity. Figure 3-5 (b)showsprolesoftheReynoldsstress ~ u 0 ~ w 0 .Forcases1to5thesmall reductioninReynoldsstresswithincreaseinsettlingvelo cityofsedimentssuggeststhat theeffectofstraticationisweak.However,withfurtheri ncreaseinsettlingvelocityof sedimentsabovethecriticalvalue,adramaticshut-offiss eenintheReynoldsstress. 85

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ThecorrespondingDNSresultsofKim etal. [ 47 ]forpurechannelowarealsoshownin Figure 3-5 (b).Theperfectagreementwiththeresultsforcase0serves asvalidationof thesimulationprocedureandtheresolutionemployedhere. Turbulentintensityorrootmeansquare(rms)velocityuct uations( ~ u rms = p ~ u 0 2 )are showninFigure 3-6 .Thevariationofallrmsvelocityuctuationsisverysmall forcases 1to5.Forcase6 ~ w rms and ~ v rms arezero,whilethestreamwisevelocityuctuations ~ v rms arenitebutsmall.Thus,forcase6thestraticationeffec tsaresuccessfulin completelysuppressingthewall-normalandspanwiseveloc ityuctuationsandthereby allwall-normalandspanwisemomentumandmasstransport.H owever,somesheet-like streamwisevelocityuctuationspersistcontributingto ~ u rms .Again,thecorresponding DNSresultsofKim etal. [ 47 ]forunstratiedchannelowarealsoshowninFigure 3-6 whichareinperfectagreementwiththeresultsforcase0. Theseresultsleadtotheviewpointthatthereexistsacriti calvalueforsettling velocityofsedimentsandevenasmallincreasebeyondthisv aluewillcauseasudden changeintheowregime.Itwillbeshowninthefollowingsec tionsthattheowfor cases1( ~ V s =0.01 )to5( ~ V s =0.026 )remainsturbulent,whilecase6( ~ V s =0.0265 ) undergoescompleteturbulencesuppression.3.5.2IsolatedEffectsofNon-UniformStreamwiseForcing Oneoftheeffectsofsettlingsedimentsistofocusthedrivi ngforceclosetothe bed.Inordertounderstandtheisolatedeffectofthenon-un iformdrivingforceonow turbulence,somesimulationswereperformedwherethestra ticationeffectontheow isturnedoff.Thisisachievedbyneglectingthebodyforcet erm(thelasttwoterms in( 3–21 ))inthebed-normal( z -direction)momentumequation.Thesesimulations arelistedinTable 3-1 ascasesS1andS2.Thesettlingvelocityofsedimentsincase S1isexactlythesameascase3andthereforetheirresultsca nbecomparedto understandtheeffectofstraticationonturbulentows.A swillbeshowninthissection, theabruptsuppressionofturbulenceintheowdoesnotoccu rwithouttheinuenceof 86

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stratication.Inordertoclearlydemonstratethisfeatur ecaseS2considerssediments withlargesettlingvelocityofnearlyfourtimesthecritic alsettlingvelocityforcomplete turbulencesuppressionidentiedintheprevioussection. InFigure 3-7 (a)and 3-7 (b)themeanconcentrationandmeanvelocityprolesfor thecasesS1andS2arecomparedwiththoseofcases0and3.Cas e0corresponds toauniformdrivingforceandservesasthereferencecase.A scanbeexpected,with increasingsettlingvelocitythemeanconcentrationprol esalsoskewtowardthebedfor casesS1andS2.Theskewnessoftheseprolesis,however,sm allerthanthatofthe prolesinFigure 3-2 (a).Sincethetotalvolumeaveragedconcentrationofsedim entsis constrainedtobeaconstant,thetotalstreamwisedrivingf orceremainsthesameinall cases,butthestrongbed-normalvariationinthestreamwis edrivingforceforcaseS2is reectedinitsconcentrationprole.Concentrationprol esofcasesS1andS2arealso comparedwiththecorrespondingRouseproles( 3–15 )intheinsetofFigure 3-7 (a). TheRouseprolesshowhighergradientsascomparedtotheco rrespondingproles obtainedfromtheDNS.ThissuggeststhatRouseprolesunde rpredicttheamountof turbulentmixinginthesecases. Theisolatedeffectoftheskeweddrivingforceonthemeantu rbulentvelocityprole isshowninFigure 3-7 (b).InallthecaseslistedinTable 3-1 thedimensionlessvelocity gradientatthebottomboundaryis Re bythedenitionof u (see( 3–12 )).Asseenin Figure 3-7 (b)forcasesS1andS2,theeffectofskewedforcingistomake thevelocity prolefullerbyreducingthevelocitygradientawayfromth ebed.Thecorresponding bulkvelocityforcasesS1andS2slightlyincreaseswithset tlingvelocity(seeTable 3-1 ). Ontheotherhand,themeanstreamwisevelocityatthetop(lo cationofthemaximum value)slightlydecreases.Nevertheless,theisolatedeff ectofskeweddrivingforceon themeanvelocityprolesisnegligiblewhencomparedtothe effectofstraticationseen forcase3inFigure 3-7 (b)andforallcasesshowninFigure 3-2 (b). 87

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Asmentionedpreviously,inordertokeepthesedimentswell mixedinthechannel theratioofturbulentuxtosettlinguxofsedimentsmusta pproachone.Figure 3-8 (a) showstheratioofturbulentuxtosettlinguxofsediments forcasesS1,S2and3.It canbeobservedinthisgurethattheratioislargerforcase sS1andS2thanforcase3, andveryclosetoone.Specically,forcaseS2theratioprac ticallyreachesunityclose tothecenterofthechannel.Theratiodecreasesandreaches zeroattheboundaries bydenitionofturbulentuxes.Similarbehaviorcanbesee ninFigure 3-8 (b)where thestreamwisermsvelocityuctuationsareshown.Thedamp ingeffectisnegligible forcaseS1.Adecreaseofabout 10% canbeobservedinthepeakvalueforcaseS2. Thiseffect,however,canalsobeconsideredtobeminorwhen comparedtocase6in Figure 3-6 (a).Similarbehaviorsareobservedforspanwiseandbed-no rmalrmsvelocity uctuationsandarenotshownhere. Theseresultsshowclearlythatturbulentmixingisfullyac tiveforcasesS1andS2, thusreinforcingtheideathattheisolatedeffectofskewed drivingforceonturbulence dampingisnegligibleascomparedtotheeffectofstratica tion.Thefocusingofthe drivingforceclosetothebedinducesamodestreductionint heturbulentintensities, buttheturbulentnatureandmixingcapabilityoftheowpre vail.Withsignicantlylarger settlingvelocitiesthanthatofcaseS2,asthedrivingforc emovesveryclosetothe bed,wheretheviscouseffectsdominate,totalsuppression ofturbulencecouldstillbe achievedevenintheabsenceofstraticationeffects.3.5.3TurbulentKineticEnergyBalance Thedimensionlessturbulentkineticenergy(TKE)equation forthepresentstatisticallystationaryandhorizontallyhomogeneousowcanbeex pressedas ~ P ~ + d d ~ z 1 Re d ~ k d ~ z ~ w 0 ~ p 0 + 1 2 ~ u 0 i ~ u 0 i # = ~ F x + ~ F z tan (3–31) 88

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wheretheTKE ~ k ,theTKEproduction ~ P ,theTKEdissipation ~ ,andtheturbulentuxof sediments ~ F i areexpressedas ~ k = 1 2 ~ u 0 i ~ u 0 i ~ P = ~ u 0 i ~ u 0 j @ ~ u i @ ~ x j ,~ = 1 Re @ ~ u 0 i @ ~ x j @ ~ u 0 i @ ~ x j ~ F i = ~ u 0 i ~ c 0 (3–32) Figures 3-9 (a)and 3-9 (b)showprolesofTKEproductionanddissipationfor selectedcasesinTable 3-1 .Productionhasapronouncedpeakclosetothebed.For case0thepeakisat ~ z =0.067 whichcorrespondsto z + 12 ,anditsdimensionless valueis 40 ,whichcorrespondwelltoreportedresultsforunstratied channelow[ 47 ]. Theinuenceofnon-zerosettlingvelocityonthelocationo fproductionpeakissmall: thepeaktendstoslightlyshiftawayfromthebed.Alsothepe akvalueofproduction isseentodecreasewithincreaseinsettlingvelocityofsed iments.Thereis 13% reductioninproductionpeakfromcase0tocase5.Tworegion sofoppositetrends canbeobservedinFigure 3-9 (a).Intheregionbetweenthebedand ~ z 0.1 thereis dampingofTKEproductionwithincreaseinsettlingvelocit yofsediments.Beyondthis location( ~ z > 0.1 )riseintheTKEproductionisobservedwithincreaseinsett lingvelocity ofsediments.Theinuenceofstraticationcausedbysuspe ndedsedimentsontheTKE dissipationisnotaspronouncedasforproduction. ShowninFigures 3-9 (a)and 3-9 (b)aretheresultsforcase6,whichcorresponds tocurrentsdrivenbysedimentswithsupercriticalsettlin gvelocity.BothTKEproduction anddissipationareidenticallyzerothroughoutthechanne l.Alsoplottedinthesegures aretheresultsforcaseS2correspondingtothelargersettl ingvelocitywithstratication effectsturnedoff.Consistentwiththeotherturbulentqua ntitiessomereductionin bothTKEproductionanddissipationisobserved,buttheow stillremainsvigorously turbulent. ThecontributionofthesuspendedsedimentstotheTKEbudge t(righthandside termsin( 3–31 ))isplottedinFigure 3-10 (a).Suspendedsedimentsactasasinkand dampTKE.ThisamountofTKEisexpendedinkeepingthesedime ntssuspended. 89

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Figure 3-10 (a)showsthatTKEdamping,inducedduetosediments,peaksn earthe bedataboutthelocationwheretheTKEproductionpeaks,but thepeakisbroaderas thedecayofTKEdampingawayfromthepeakintothechannelis slowerthantheTKE production.ThisbehaviorcanbeclearlyseeninFigure 3-10 (b)wheretheratioofTKE dampingtoTKEproductionisshown.Theratioisonlyafewper centintheregionwhere boththeproductionanddampingpeak,butlargerawayfromth epeakwhereproduction becomesverysmall. TheglobalbalanceofTKEintheowcanbeobtainedbyintegra ting( 3–31 )inthe bed-normaldirection ~ P ~ E + 1 Re 8<: d ~ k d ~ z # 10 + tan 9=; = + ~ V z tan (3–33) wherewehaveusedthefactthat ~ F z = ~ w 0 ~ c 0 = ~ V z ~ c +1 = ( Re Sc ) d ~ c = d ~ z (referto( 3–30 )), and ~ P = Z 1 0 ~ P d ~ z ~ E = Z 1 0 ~ d ~ z = Z 1 0 ~ u 0 ~ c 0 d ~ z = ~ c b ~ c t Sc (3–34) Here ~ c t and ~ c b arethemeanconcentrationonthetopboundaryandbed,respe ctively. Thersttwotermsonthelefthandsideof( 3–33 )arethebulkTKEproductionand thebulkTKEdissipation.Thetwotermsontherighthandside of( 3–33 )arethebulk sediment-induceddampingofTKEandrepresentthetotalamo untofTKEspentto keepthesedimentssuspendedandwellmixedinthebed-norma ldirection.Theterm arisesfromtheterm ~ F x in( 3–31 ).Theseconddampingtermisthemanifestationof the ( ~ F z = tan ) termin( 3–31 ).Thelasttermonthelefthandsideof( 3–33 )iscomposed oftwoeffects:bulkdiffusionofTKEandbulkhydrodynamicd iffusionofsediments.For largeReynoldsnumbersthistermbecomesnegligibleascomp aredtotheothertermsin thebudget. ThetermsinvolvedintheglobalbalanceofTKEareshowninTa ble 3-2 .Asseen fromthetable,therearetwotermswhichcontributetoTKEpr oduction.Thesetermsare 90

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theTKEproduction ~ P ,andthebulkhydrodynamicsediment-sedimentandsediment -bed interactions ( = ( Re tan )) .Althoughtheintroductionof ( = ( Re tan )) intothebulkTKE equationisthroughadampingterm ~ F z ,fromitscontributiontothebalanceitcanbe thoughtasamechanismthatsupportsTKE.AtlargeReynoldsn umbers,however,this termbecomesnegligible.Table 3-2 showsthatthesediment-induceddampingterms amounttoonly 2 6% oftheTKEproduction.Observethatforcase5thesedimentinduceddampingofTKEis 6.3% ,whileforcase6,withasettlingvelocityonly 2% larger thancase5,thesediment-induceddampingofTKEiscomplete Theterm isabout30-40%of ( ~ V z = tan ) overtherangeofsettlingvelocities considered.Assumingthatthescalingof with ( ~ V z = tan ) remainvalidoverawide rangeofReynoldsnumbersofpracticalinterestthentheove rallsuspended-sedimentinducedTKEdampingcanbeexpressedassomefactorof ( ~ V z = tan ) .Thus,the entiredampingprocesscanberepresentedbytheparametric combinationofslope ofthechannel( )andsettlingvelocityofthesediments( ~ V z )andthecombination takestheform ~ V z = tan Recently,[ 23 ]havesuggestedthatthereexistsauniversal criterionfortotalturbulencesuppression.Theauthorsha veidentiedthattheextent ofturbulencedampingcanbequantiedbasedontheparametr icgroupingofsettling velocityofsediments( V z )andslopeofthechannel( )whichcombinetogive ~ V z = tan Therefore,atanygiven Re thereexistsacriticalvalueof ~ V z = tan beyondwhich completeturbulencesuppressionoccurs.Thisalsomeansth atforsedimentsofcertain settlingvelocitythereexistsacriticalvalueforchannel slopebelowwhichcomplete turbulencesuppressionoccurs.[ 20 ]and[ 83 ]haveshownthatachangeintheslope ofthechannelcanresultincompletesuppressionofturbule nceleadingtomassive sedimentation. Itisveryintriguingthatcompleteturbulencesuppression couldbeabruptlyproducedbythesmallvaluesofsediment-inducedTKEdampingsh owninTable 3-2 .What 91

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causesthecollapseofturbulenceandhowthedampingtermsi nitiatetheprocessof completeturbulencesuppressionarethecorequestions.3.5.4EnergySpectra Inordertoexplorethedependenceofsuspendedsedimentsan dtheirdominant straticationeffectonowscales,the x and y energyspectraofthestreamwise, spanwiseandbed-normalcomponentsofvelocityareconside red.Comparingthe energyspectraofcase0andcase5(referencecaseandlarges tsettlingvelocitybefore completeturbulencesuppression,respectively)givesani deatowhatlevelthedifferent scalesofturbulenceareinuencedbythesuspendedsedimen ts.Figure 3-11 presents theenergyspectraofallthecomponentsofvelocityatthree differentdistancesaway fromthebed: z + 12 (bufferlayer), z + 40 (logarithmiclayer)and z + 150 (away formthebed).Topframescorrespondto x -spectraandbottomframescorrespondto y -spectra.Itisworthnoticingthatallthespectrashowseve raldecadesofenergydecay indicatingadequatespatialresolutionoftheentirerange ofturbulentscales. Closetothebed(bufferandlogarithmiclayers)the x -spectraofstreamwisevelocity ( E uu ( k x ) )forcases0and5virtuallyoverlap,suggestingverylittle inuenceofsuspended sediments.Incontrast,the x -spectraofspanwiseandbed-normalcomponentsof velocity( E vv ( k x ) and E ww ( k x ) )aredampedoverallwavenumbersintheseregions.The levelofdampingissomewhatlargerfor k x < 7 .The y -spectrashowsubtledifferences withrespecttothe x -spectrainthebufferandlogarithmiclayers.Forlowwaven umbers ( k y < 10 ), E vv ( k y ) and E ww ( k y ) displaysimilarinuencebysuspendedsedimentswhilea slightdampingisseenfor ~ E uu ( k y ) .Forhighwavenumbers( k y > 10 ),negligibledamping isobservedforallthreeenergyspectra. Awayfromthebed( z + > 150 ),theinuenceofsuspendedsedimentsonthe energyspectraismoresubstantial.The x -spectraand y -spectraofallthreevelocity componentsshowsignicantreductionintheenergycontent forcase5.Thegeneral trendobservedintheregionsclosetothebedprevails.That is,theeffectofsuspended 92

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sedimentsismorepronouncedforthespanwiseandbed-norma lvelocitycomponents thanforthestreamwisecomponentandthedecayismoreprono uncedatlowerwave numbersthanathigherwavenumbers. Thefollowingdeductionscanbemadefromtheenergyspectra .Firstandforemost, straticationaffectsthelargeandmoderatescaleeddystr ucturesoftheowmorethan thesmall(Kolmogorov)scaleeddies.Also,energycontento fthespanwiseandbednormalvelocityuctuationsaredampedmuchmorethanthest reamwisecomponent, suggestingthatstraticationhasastrongereffectonstre amwiseorientedvortical structures.Sinceturbulenceproductionpredominantlyoc cursintheregionofthebuffer layer,itcanbearguedthatwhilestraticationhasadirect inuenceonturbulence productionatthelargeandintermediatescalemotionsinth enear-bedregion,italso hasastrongeffectonturbulencetransporttotheupperregi onsoftheow,which resultsinsubstantialreductioninthespectraofallthree velocitycomponentsinthe regionawayfromthebed.Theprocessofcompleteturbulence suppressionstarts withdampingofthelargeandintermediatescaleeddiesinth enear-bedregion,where thesourceofsustainingturbulencelies.Bydampingsuched dies,theturbulence generationmechanismsareoverpoweredandeventuallylead totheobservedabrupt turbulencesuppression.Section 3.5.7 willfocusonturbulentowstructurestosupport thishypothesisandshowthatthemechanismresponsiblefor completeturbulence suppressionisthedampingofnear-bedstreamwiseoriented eddies. 3.5.5ReynoldsStressBalance Thepuzzlingaspectoftheobservedcompleteturbulencesup pressionisitsabrupt onsetwithincreasingsettlingvelocityofsediments.Infa ct,aswasshowninTable 3-2 justpriortoonsetofcompleteturbulencesuppression,the sediment-induceddamping ofTKEaccountsforonlyafewpercentofTKEproduction.Sinc eReynoldsstress ~ u 0 ~ w 0 playsacrucialroleinTKEproduction,thissectionfurther probesintohowReynolds 93

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stressbalanceanddistributionareinuencedbyincreasin gthesettlingvelocityof sediments. Forthepresentproblem,thestationarystateoftheReynold sstressbalancein dimensionlessformcanbeexpressedas ~ P rey ~ 13 + @ ~ u 0 ~ w 0 ~ w 0 @ ~ z + ~ 13 1 Re @ 2 ~ u 0 ~ w 0 @ ~ z 2 = ~ F x tan + ~ F z (3–35) where ~ P rey = ~ w 0 ~ w 0 @ ~ u @ ~ z ~ 13 = 2 @ ~ u 0 1 @ ~ x j @ ~ u 0 3 @ ~ x j ~ 13 = ~ w 0 @ ~ p 0 @ ~ x + ~ u 0 @ ~ p 0 @ ~ z (3–36) Thersttermonthelefthandsideisproduction,thesecondt ermisdissipation,and thelastthreetermsaretheturbulent,pressureandviscous transport.Thetermsonthe righthandsidearethesediment-induceddampingofReynold sstress.Thesedamping termsarisesoutoftheuctuationsinconcentrationandvel ocityelds. Figure 3-12 showstheReynoldsstressproductionandsediment-induced damping forselectedcasesfromTable 3-1 .Reynoldsstressproductionisnotstronglyinuenced bysettlingvelocityofsediments.Theproductionpeakrema insnearlythesame. Interestinglyenough,Reynoldsstressproductionslightl yincreasesawayfromthepeak withincreasingsettlingvelocityofsediments.Thiscanbe explainedbyobservingthat productiondependsonboththermsbed-normalvelocityuct uations ~ w rms andthe meanvelocitygradient @ ~ u =@ ~ z .While ~ w rms decreaseswithincreasingsedimentsettling velocity,thecorrespondingmeanvelocitygradientincrea sestocompensate.Incontrast, thesediment-induceddampingofReynoldsstressshowssign icantincreasewith increasingsettlingvelocityofsediments. TheglobalbalanceofReynoldsstressinthechannelisobtai nedbyintegratingthe Reynoldsstressequation( 3–35 )inthebed-normaldirection Z 1 0 ~ 13 ~ 13 d ~ z + 1 Re ( d ~ u 0 ~ w 0 d ~ z 10 + ) = tan + ~ V z (3–37) 94

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where = Z 1 0 ~ P rey d ~ z Hereagain,( 3–30 )hasbeenusedtore-write ~ F z or ~ c 0 ~ w 0 as ~ V z ~ c +1 = ( Re Sc ) @ ~ c =@ z .The termsinvolvedinthebulkReynoldsstressbalancearetabul atedinTable 3-3 .Clearly, themostimportantsediment-induceddampingtermis = tan .Observethat ~ V z is about 2% of = tan forallcasesconsidered.LikeintheTKEbalance, isaproduction term.Therefore,thesediment-induceddampingofReynolds stressisrepresented by = tan + ~ V z .UnlikeintheTKEbalance,herethesediment-induceddampi ngof Reynoldsstressforcase5isabout30%ofproduction.Thefra ctionaldampingof Reynoldsstressbysedimentsismuchstrongerthanthedampi ngofTKE.Thus,the pathwaytoturbulencesuppressionisperhapsthroughmodul ationofReynoldsstress producingturbulentevents.3.5.6ReynoldsStressEvents Recentresearchefforts[ 10 14 90 91 ]havebeguntoshedlightonhowturbulencesustainsitselfeveninaperfectlysmoothwalledchan nel.Oneoftheproposed mechanismreliesontheauto-generationofvorticalstruct uresandtheirarrangementas streamwisealignedpackets[ 90 91 ].Hairpinandotherquasi-streamwisevorticesina turbulentwalllayergeneratesecondary,tertiaryandaddi tionalvorticalstructuresover timeonlywhentheiramplitudeisaboveacertainthreshold. Theimplicationisthatifthe strengthofthevorticalstructuresfallsbelowacertainth resholdtheylosetheirabilityto generatethenextgenerationofvorticalstructuresandthu sfailtosustainturbulence. Thislineofargumentispursuedinthissectiontoexploreth eeffectofsuspended sedimentsonthetypicalReynoldsstressproducingevents. Forthedifferentcases considered,atdifferentlocationsfromthebed,aquadrant analysisofthestreamwise andbed-normalvelocityuctuationsisperformedtoobtain astatisticalpictureofthe Reynoldsstressdistribution.Theprobability P ( )= Pr f ~ u 0 ~ w 0 = g thatthelocal 95

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instantaneousReynoldsstresstakesavalue iscomputedfromthe ~ u 0 ~ w 0 scatterplot. Traditionallythe ~ u 0 ~ w 0 scatterplotisdividedintofourquadrants,ofwhichquadra nt2(Q2)andquadrant-4(Q4)contributetonegativeReynoldss tressandhavebeen recognizedtobethemostsignicant.AQ2eventcorresponds to ~ u 0 < 0 and ~ w 0 > 0 whileaQ4eventcorrespondsto ~ u 0 > 0 and ~ w 0 < 0 .Q4eventwillbringthehigher momentumuidfromthebulkowclosetothebed,whileQ2even twillcauselow momentumuidpresentclosetothebedtobeejectedintotheb ulkow.Vertical mixingduetoQ2eventsleadtoresuspensionofsettlingsedi ments.Also,Q2events correspondtoquasi-streamwiseandhairpinvorticesinthe owwhichareresponsible forsustainingturbulence.ThereforenegativeReynoldsst resscontributionsfromQ2and Q4eventsareseparatedincomputingtheprobability. Figure 3-13 shows j P ( ) j versusReynoldsstressvalue( )forthesecondquadrant eventsat ~ z + =18 forcases0and5.Resultsforotherbed-normallocationsand for othercasesaresimilarandthereforenotshown.TheReynold sstressatwhich j P ( ) j reachesapeakvalueisdenotedas m ,whichcanbeinterpretedastheturbulent eventthatmakesthelargestcontributiontothetimeandhor izontalaveragedmean Reynoldsstress.Forcase0contributiontothemeanReynold sstressismaximizedfor m = 1.26 andthecorrespondingvalueoftheturbulentevent(uctuat ingvelocity) is ~ u 0m =( 2.42,0.0,0.521) .Theconditionallyaveragedoweld(conditionaleddy) correspondingtothisReynoldsstresseventisahairpinvor tex[see 1 ].Sothequestion ofturbulencesustainmentcanbeposedintermsoftheabilit yforthisReynoldsstress maximizinghairpinvortextoproduceandpopulatethesubse quentgenerationsof vorticalstructures[ 2 ].Intheclearuidcase(case0)theturbulentvorticalstru cturesare self-sustainingandtheturbulencegenerationcontinuesu ninterrupted. Forcase5thepeakcontributiontoReynoldsstressoccursat alowervalueof m = 0.68 andthecorrespondingturbulentvelocityeventis ~ u 0m =( 2.10,0.0,0.323) AsitisseeninFigure 3-5 (b),themeanReynoldsstressfor ~ z + =18 (peaklocation) 96

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decreasesbyonlyabout17%fromcase0tocase5.However,the strengthofthe Reynoldsstressmaximizingevent m hasdecreasedtoabout46%.Moreimportantly, thisdecreaseinReynoldsstresscomesaboutbya38%reducti oninthebed-normal velocityuctuations. Insteadof j P ( ) j onecanalsolookattheReynoldsstressinquadrant2that maximizes P ( ) .Suchaneventcanbedenotedas p andinterpretedasthemost likelyReynoldsstressevent(asopposedtoonethatmaximiz escontributiontomean Reynoldsstress).ThemostlikelyReynoldsstresseventcom putedinthismannerfor case0isgivenby p = 0.167 ~ u 0p =( 0.505,0.00,0.331) .Thecorrespondingvalues forcase5are p = 0.167 ~ u 0p =( 0.728,0.00,0.229) .ThemostlikelyReynoldsstress eventdoesnotchangewithsettlingvelocity,buttheturbul entvelocityvectorchanges. Fromcase0tocase5thereis44%increaseinstreamwiseveloc ityuctuations,and adecreaseof33%inthebed-normalvelocityuctuations.Th us,increaseinsettling velocityofsedimentsresultsinreorientationofthemostl ikelyQ2Reynoldsstressevent vectorwhilekeepingtheintensityoftheeventconstant. Zhou etal. [ 91 ]consideredconditionallyaveragedvorticalstructuresi nunstratied channelowobtainedfordifferentlevelsofturbulentReyn oldsstresseventsgiven by 2 m .Astheconstant variedfrom0.25to3.0,theintensityoftheinitialvortica l structureincreased.Theyobservedthatonlyfor greaterthan 2.0 theconditional vorticalstructurewasabletoauto-generatefuturegenera tionofhairpinandquasilongitudinalvortexstructures.Initialvorticalstructu resofstrength lessthan 2.0 were unabletoauto-generatenewvorticalstructure.FromFigur e 3-13 (b)theprobabilityof Reynoldsstresseventsofstrengthatleastfourtimes m canbecomputedas P ( < 5.04) .Forcase0thisprobabilityis4%at z + 18 anditissufcienttospawnthe nextgenerationofvorticalstructuresandmaintainturbul enceintheow.Forcase5the probability P ( < 5.04 )hasdecreasedto2%.Notethatalthough m forcase5islower 97

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than m forcase0,herethethresholdReynoldsstressforregenerat ionismaintainedat 5.04 Itisimportanttomentionthattheauto-generationcriteri ongivenbyZhou etal. [ 91 ] wasforunstratiedsinglephasechannelowandhenceitcan notbedirectlyextended tostratiedmulti-phaseows.Asdiscussedabove,theeffe ctofincreasedsettling velocityofsedimentsisnotonlytodecrease m ,butalsotoreorienttheeventvector ~ u 0m tobeincreasinglyparalleltothebed.Astheeventvectora ttensitisprobablethateven higherinitialvortexstrengthwillberequiredtospawnthe nextgenerationofstructures. Thusthevalueof islikelytobedifferentandmuchgreaterthan 2 forauto-generation incaseofowswithsuspendedsediments.TheresultsofZhou etal. [ 91 ]illustratethe dependenceofthethresholdamplitude(forregenerationof vorticalstructures)onthe characteroftheeventandtherebyontheinitialvortexstru cture.Itcanthereforebe conjecturedthatwhensettlingvelocityofsedimentsisinc reasedabovethethreshold, thechangeinthespatialstructureofthevorticalstructur escombinedwiththereduction intheirstrength,togetherprecludesustainedregenerati onofturbulentvorticalstructures andultimatelytheowlosesallturbulence.Timeresolvedm ovies(tobediscussed in x 3.5.7 )whichshowturbulentvorticalstructuresexistingincase 5andhowthese structuresevolveincase6,supportthisviewpoint.3.5.7Near-BedCoherentStructuresandConcentrationFluc tuations Withincreasing ~ V thesettlinguxofsedimentsincreases.Forstationarysta te thesettlinguxmustbebalancedbythecombinationofupwar ddiffusiveuxand turbulentux(mixing)ofsediments.Inaturbulentowthed iffusiveuxdominates onlyintheregionclosetothebed.Thus,awayfromthisregio n,anincreaseinsettling uxdemandsaconcomitantincreaseintheturbulentux ~ w 0 ~ c 0 .FromFigure 3-6 (b)itis evidentthatthelevelofverticalvelocityuctuationsinf actdecreasewith ~ V andthusan increaseintheturbulentuxofsedimentsrequiresastrong increaseintheuctuations ofsedimentconcentrationanditscorrelationwiththevelo cityuctuations. 98

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Figure 3-4 showsthevariationinboththemeanconcentration( ~ c b )andrmsconcentrationuctuations( ~ c rms b )atthebedwithincreasingsettlingvelocityofsediments. Asexpectedthemeanconcentrationatthebedgraduallyincr easeswithincreasein settlingvelocity,however,thermsconcentrationuctuat ionsexplosivelyincreasesasthe criticalsettlingvelocityisapproached.Thermsvalueinc reasesbyafactorof5asthe settlingvelocityofsedimentsincreasesfrom 0.01 to 0.026 .Forcase1( ~ V =0.01 )the rmsvalueisabout 3% ofthemeanconcentrationanditincreasesto 12.5% forcase5 ( ~ V =0.026 ).Theexplosivegrowthinconcentrationuctuationsindic ateslocalsegregationofsedimentsintoregionsofmuchhigherconcentrati onthanthemeanandother regionsofmuchlowerconcentrationthanthemean.Furtherm ore,inordertomaintain thestationarystatebyincreasingtheturbulentupwardux ofsediments,theregions ofhighconcentrationmustbecorrelatedwithregionsofupw ard(positive)bed-normal velocityuctuation.Theexistenceofsuchregionsoflarge sedimentconcentrations wouldrequiretheowtolocallyspendalargeamountofenerg yinordertoachieve therequiredupwardbed-normaltransport.Thiscanleadtoa nabruptshutdownof theauto-generationmechanismofhairpinvortexstructure sintheowwhichcausesa completecollapseoftheowturbulence. Figures 3-14 and 3-15 showplotsofstreamwisevelocity( ~ u 0 ),bed-normalvelocity ( ~ w 0 )andconcentration( ~ c 0 )uctuationsforcases1and5inthe ~ x ~ y (streamwisespanwise)planeat z + 12 .Thislocation z + 12 isaroundtheregionwhere maximumturbulentkineticenergyproductionandmaximumRe ynoldsstressareseen. Theplotsofconcentrationandstreamwisevelocityuctuat ionsshowlongstreamwisealignedstreakystructuresinbothcases.Itcanbeobserved that ~ u 0 and ~ c 0 distributions arenegativelycorrelated.Inotherwords,thesedimentcon centrationishigher(lower) alongthelow(high)speedstreaks.Thebed-normalvelocity uctuationsarealsowell correlatedwiththestreamwisevelocityandconcentration uctuations.Butsince ~ w 0 does notshowalongstreakystructure,thecorrelationisnotaso bvious. 99

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Fromcase1tocase5thereisaslightreductionintheminimum andmaximum valuesof ~ u 0 and ~ w 0 ,whilethereissubstantialincreaseintheminimumandmaxi mum valuesof ~ c 0 .Comparisonofthetwocasesrevealcertaininterestingfea tures.Inthecontourplotsofcase1,thelowandhighspeedsteaksandtheregi onoflargebed-normal velocityuctuations,whichareindicativeoflocalquasistreamwiseandhairpinvortical structures,aremoredenselypopulatedandaredistributed moreorlessuniformlyover thehorizontalplane.Withincreaseinsettlingvelocityth edensityoflowandhighspeed streaksisreducedandasaresulttheuniformityoftheirspa tialdistributionappearsto belost.Intheconcentrationcontours,incase1theregiono fpositivelargevaluesare longandthinandspreadovertheentireplane,whileincase5 theseregionsarelong, thick,andmoreintenseandtheyarenowspatiallylocalized .Thissuggestsareduction inthedensityofturbulentvorticalstructures. Figures 3-16 3-17 and 3-18 showiso-surfacesofswirlingstrengthforcases0,5 and6.Swirlingstrengthisdenedastheimaginarypartofth ecomplexeigenvaluesof thelocalvelocitygradienttensorandhasbeenshowntobeve ryeffectiveinextracting vorticalregionsinturbulentows[ 25 91 ].Thesegurespresentthevorticalstructures oftheturbulentowataselectedtimeinstant.Incases0and 5,theowremainsturbulentandthustheturbulentstructuresseeninFigures 3-16 and 3-17 arerepresentative ofwhatareseenatothertimesaswell.Case0showsadensedis tributionofhairpin-like andinclinedquasi-streamwisevorticesintheow.Withinc reasingsettlingvelocity,the straticationoftheowincreasesandthestructuresbecom esparsecreatingpockets intheowcompletelydevoidofvorticalstructures.Case5c orrespondstotheclosest turbulentcasetothecriticalsettlingvelocity.InFigure 3-17 substantialdecreaseinthe densityofthestructuresisseenbuttheowremainsturbule nt.Thisimpliesthatsome oftheexistingstructuresintheowareintenseenoughtosp awnthenextgeneration ofstructurestosustainturbulence.Case6correspondstoa settlingvelocityslightly 100

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greaterthanthatincase5( 2% )andabovecriticality.Atthissettlingvelocitycompleteturbulencesuppressionisobserved.Figure 3-18 showstheturbulentstructures incase6ataninstantwhentheowisstillintheprocessoftu rbulencesuppression. Notethatcase6wasstartedwithaturbulentinitialconditi ontakenfromcase5andthus startedwithvorticalstructuressimilartothoseshowninF igure 3-17 .However,dueto theincreasedsettlingvelocity,auto-generationprocess wasdisruptedandashorttime later,asshowninFigure 3-18 ,theowhaslostmostofitsturbulentstructures.When continued,eventhevorticalstructuresseeninFigure 3-18 werenotabletospawnthe nextgenerationofvorticalstructuresandtheowlostalli tsturbulentuctuations 3.5.8MechanisticViewofTurbulenceSuppression Fromtheturbulentstatisticsandthepictureofinstantane ousturbulentstructures presentedintheprevioussectionsthefollowingsimplesce narioisproposed,which leadstoseeminglysuddenandabruptlossofturbulence.For thecaseofclearuid auto-generationandturbulencesustainmentmechanismsar ewellunderstood.Inthis limitofzerosettlingvelocity,thesedimentsareperfectl ywellmixedatalltimeswithzero concentrationuctuations.AsdiscussedinZhou etal. [ 91 ]theturbulentwalllayeris populatedwithstreamwisealignedhairpinvortexpacketsa ndasthesepacketstravel downstream,betweentheirquasi-streamwiselegstheycoop erativelypumpthenearbedlowmomentumuidintotheow.Theconsequenceofthisco operativeactionofthe hairpinpacketsarethelowspeedstreaks,whicharemuchlon gerthantheindividual hairpinvortices.Thesestreaksprovidetheenvironmentfo rsubsequentformationofthe nextgenerationofhairpinpackets. Nowconsidersuddenlyintroducingasettlingvelocityfort hesediments(thisis similartostartingasimulationofnon-zero ~ V withcase0astheinitialvelocityeld).Now sedimentsareresuspendedonlyinregionsofsufcientupwa rd(positive)bed-normal velocityuctuations,whiletheysettledowninregionsofd ownwarduidmotion.This createsincreasedconcentrationuctuations.Inparticul ar,thecooperativeupward 101

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pumpingofuidbythestreamwisealignedhairpinvortexpac ketsprovidestheperfectenvironmentfortheupwardtransportofsediments.Iti sintheseregionswhere increasedsedimentconcentrationareexpected.Thus,asth ehairpinpacketstravel downstream,similartothelongstreaksoflowspeeduid,th eyleavebehindwellcorrelatedlongstreaksofhighsedimentconcentration.Thus, subsequentgenerationof hairpinvorticesarebeingformedinregionsofsteadilyinc reasingsedimentconcentration. Asseeninthequadrantanalysisofthevelocityuctuations ,theincreasedconcentrationofsedimentatthelowspeedstreaksaltersthechara cterofthenewlyformed vorticalstructures.Furthermore,theadditionalbuoyanc yeffectposedbythesediments, canbeexpectedtoincreasethethresholdamplitudeforthef ormationofnextgeneration ofvorticalstructures.Thus,increasingsettlingvelocit yofsedimentsleadstoadecrease inthedensityofvorticalstructures,whoseeffectcanalso beseeninthereductionof thekeyturbulencestatistics.Itcanthenbeconjecturedth atonceathresholdsettling velocityisexceeded,thelocalaccumulationofsedimentsa tthelow-speedstreaksdo notpermitfurthergenerationofvorticalstructures.With outtheauto-generationprocess active,theturbulencevertimefullydecays. 3.6Deductions Diluteturbiditycurrentsdrivenbymono-dispersesuspens ionofsedimentsinbypass modearemodeledasaninclinedchannelowdrivenbysuspend edsediments.The suspendedsedimentsundertheinuenceofgravitydrivethe owinthechanneland simultaneouslysettletowardsthebed.Theinteractionofs edimentsandturbulencelead to:(a)skewingofthestreamwisedrivingforcetowardstheb edand(b)stablestraticationthatdampsbed-normalmomentumandmasstransport.S everalsimulationsof thismodelwerecarriedouttounderstandthetwoeffectsand theirroleinturbulence suppression.SimulationsS1andS2considerowswherethes traticationeffectin thebed-normalmomentumequationisturnedoff.Fromtheser esultsitisshownthat 102

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straticationisthedominanteffectwhichbringsabouttot alturbulencesuppression. Skewingthedrivingforcetowardsthebedcausesturbulence dampingbutthiseffectis modestand,evenforverylargesettlingvelocity,withoutt hestraticationeffectinthe bed-normalmomentumequationtheowremainsturbulent. Straticationisthemanifestationofthebalancebetweent urbulentmixingandthe settlingofthesediments.Atthecriticalsettlingvelocit y,owturbulenceisnolonger abletokeepthesedimentsinsuspensionandtheowinthecha nneltendstowardsthe laminarsolution.Forthepresentsimulations,with Re =180 and =5 ,themean velocityandmeanconcentrationprolesclearlyshowthatt hereissuddentransitionin thenatureoftheowfromcase5( ~ V =0.026 )tocase6( ~ V =0.0265 ),andthusthe criticalsettlingvelocityfortotalturbulencesuppressi onisintherange (0.026,0.0265) Theowcanthenbedividedintotworegimes,i.e.,turbulenc e-stratiedowsand turbulence-suppressedows,andthelatterisrealizedfor settlingvelocitieslargerthan thecriticalsettlingvelocity.Thissuddentransitionisf ascinatingandtheabruptchange isseeninseveralturbulentstatistics.Thisprocessissim ilartotheonsetofinstability, butthedifferenceisitshighlynon-linearbehavior.Analy sisofturbulentkineticenergy equation,Reynoldsstresstransportequationandonedimen sionalspanwiseand streamwiseenergyspectrasuggestthattotalturbulencesu ppressionmaybebrought aboutbydampingofReynoldsstressproducingturbulentvor ticalstructuresclosetothe bed. Theterm 1 = tan inthegoverningequationcanbeidentiedastheRichardson number Ri ,whichistheratioofpotentialenergyduetobuoyancyeffec ttokinetic energyoftheow.Inthepresentproblemthestraticatione ffectduetosedimentsis quantiedby R c ( v ) g z H ,whilethekineticenergyofthecurrentcanbeexpressedas u 2 = R c ( v ) g x H .Thus,theirratioyields Ri = g z = g x =1 = tan .Inanexternallydriven owonlythebuoyancyeffectdependsonmeansedimentconcen tration,andthus Ri willbeaproxyfor c ( v ) .However,inthepresentcasetheowisdrivenbythesuspend ed 103

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sedimentsandasaresult c ( v ) appearsinboththenumeratorandthedenominator, andRichardsonnumber Ri issimplyrelatedtothebedslope.Theentiredamping processcanberepresentedbytheparametriccombinationof slopeofthechannel( ) andsettlingvelocityofthesediments( ~ V z )andthecombinationtakestheform ~ V z = tan ( Ri ~ V z )[ 23 ]. IntheworkofZhou etal. [ 91 ],itwasestablishedthatturbulenceinanunstratied channelowsustainsitselfbyauto-generationofvortical structures.Theyshowed thatonlyhairpinvorticalstructureswhosestrengthisgre aterthanathresholdvalue arecapableofspawningthenextgenerationofstructures.T hischapterproposes anexplanationfortheabruptextinctionofturbulenceinth eproblemstudiedusing statisticalanalysisoftheQ2Reynoldsstresseventsfordi fferentcasesandusingthe auto-generationcriteriongivenbyZhou etal. [ 91 ].Fromtheprobabilitydensityofthese Q2eventsitisshownthatwithincreasingsettlingvelocity theintensityofQ2events decreaseandtheeventvectorattens.Thesetwoeffectscau sespatialmodulations intheturbulenthairpinvortices:theirintensityreduces andtheirspatialdistribution becomessparse.Asthesettlingvelocityincreasesbeyondt hecriticalvalue,the thresholdvorticalstrengthbecomeslargerandtheexistin gowstructuresareincapable ofauto-generating.Theowthusevolvestocompletesuppre ssionofturbulence. 104

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Table3-1:Listofsimulations. ~ V isthedimensionlesssettlingvelocityofthesediments. ~ u b isthebulkvelocityoftheow. Re b istheReynoldsnumberbasedonbulkvelocity. ~ c b and ~ c t arethedimensionlessmeanvolumeconcentrationofsedimen tsatthe bedandthetopboundary. and B arethebesttconstantsforthevelocitylog-law ( u + =1 = log( z + )+ B ).Valuesof and B forcase6arenotshownasthereiscomplete turbulencesuppressionandthelog-lawvelocityproleisa bsent. case ~ V ~ u b Re b ~ c b ~ c t B 00.015.5227941.001.000.4105.5010.0116.1329031.220.820.3484.4020.0217.0930761.550.580.2721.8230.02518.0632511.830.440.2410.8040.025518.1532661.880.430.2380.8050.02618.3333001.930.420.2370.8060.026527.3249004.780.043-S10.02515.6228111.580.670.4105.8S20.115.6328137.630.130.4807.6 Table3-2:Turbulentkineticenergybudgetofcaseslistedi nTable 3-1 case ~ k ~ P ~ E ~ V z = tan = ( Re tan ) tan = ~ V z ( ~ V z = tan + ) = ~ P 11.716.346.040.03580.1140.02540.310.02421.626.546.270.07720.2280.06190.340.04631.586.576.230.10400.2850.08770.360.05941.646.546.200.11000.2910.09180.380.06151.716.596.230.11700.2960.09580.400.063 Table3-3:ReynoldsstressbudgetofcaseslistedinTable 3-1 case ~ V z = tan = Re ~ V z tan = ( ~ V z + = tan )/ 14.240.010.4120.002220.0240.09924.530.020.8800.005430.0220.2034.570.0251.1860.007670.0210.26544.610.02551.2540.008030.0200.2854.680.0261.3300.008400.0200.29 105

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Figure3-1:Schematicrepresentationofthemodelofaturbi ditycurrent.Themodel preservesthemostessentialfeaturesofturbiditycurrent s,i.e.theowisentirelydriven bythesuspendedsedimentsandthesettlingofsedimentssel f-stratiestheow. c z 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 case0case1case2case5case6 (a) LaminarSolution ~~ u z 0 10 20 30 40 0 0.2 0.4 0.6 0.8 1 (b) ~~ cz0.511.52 0 0.2 0.4 0.6 0.8 1 case1case5 ~~RouseProfile RouseProfile Figure3-2:(a)Meanconcentrationand(b)Meanvelocitypro lesofvariouscases listedinTable 3-1 .Thelaminarsolutioniscomputedfor ~ V =0.0265 1 = tan =11.43 Re =180 and Sc =1 (samesetofparametersforcase6).Theinsetin(a)compares meanconcentrationprolesforcases1andcase5withtheirR ouseproles.Referto ( 3–15 )fortheexpressionforRouseproles.Itshouldbenotedtha t intheexpression ofRouseprolesisnotthevonKarmanconstantbutthemodie dconstantbasedon bestttothevelocitylog-law.RefertoTable 3-1 forvaluesof 106

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z + u + 10 1 10 0 10 1 10 2 0 10 20 case0case1case5case6 u + =z + LogLaw Figure3-3:Meanvelocityprolesinwallunits.Thetwologlawsshowncorrespondto besttvaluespresentedinTable 3-1 forcases0and5. V u t c b c rms,b 0 0 0.01 0.01 0.02 0.02 0.03 0.03 15 20 25 30 35 40 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.05 0.1 0.15 0.2 0.25 c b c rms,b u t ~~ ~ ~~ ~ ~ Figure3-4:Variationintopmeanvelocity ~ u t ,meanconcentrationofsedimentsatthe bed ~ c b andrmsuctuationsofconcentrationofsedimentsatthebed ~ c rms b withsettling velocity.Thereisclearchangeofowregimeaboutthecriti calsettlingvelocitywhich liesbetweenthecorrespondingvaluesforcases5and6. 107

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w'c'/V z c z 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ~ ~ ~~~ (a) u'w' z 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 case0case1case2case5case6Kimet.al. (b) ~ ~~(1987) Figure3-5:Prolesof(a)theratioofsedimentsturbulent uxtosettlingux ( ~ w 0 ~ c 0 = ~ V z ~ c )and(b)Reynoldsstress( ~ u 0 ~ w 0 ).ResultsfromKim etal. [ 47 ]arealso plottedtovalidatecase0. 108

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u rms z 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 case0case1case2case5case6Kimet.al. (a) ~~(1987) w rms z 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (b) ~~ v rms z 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 (c) ~~ Figure3-6:Prolesofrmsvelocityuctuations.(a) ~ u rms .(b) ~ w rms .(c) ~ v rms .CorrespondingprolesfromKim etal. [ 47 ]arealsopresentedasvalidationforcase0. 109

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u z 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 (b) ~~ c z 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 case0case3caseS1caseS2 (a) ~~ cz02468 0 0.2 0.4 0.6 0.8 1 caseS1RouseprofilecaseS2Rouseprofile ~~ Figure3-7:(a)Meanconcentrationand(b)Meanvelocitypro lesofCasesS1andS2. Theinsetin(a)showstheRouseprolesforcasesS1andS2.Re ferto( 3–15 )forthe expressionofRouseprole. intheexpressionofRouseproleisnotthevonKarman constantbutthemodiedconstantbasedonbestttothevelo citylog-law.Referto Table 3-1 forvaluesof 110

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w'c'/V z c z 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ~ ~ ~ ~~(a) u rms z 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 case0case3caseS1caseS2 (b) ~~ Figure3-8:Prolesof(a)ratioofsedimentsturbulentuxt osettlinguxand(b)streamwisermsvelocityuctuations.CasesS1andS2donotconside rstraticationeffects. 111

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z 0 10 20 30 40 0 0.2 0.4 0.6 0.8 1 case0case1case2case5case6caseS2 (a) ~~ e z 30 20 10 0 0 0.2 0.4 0.6 0.8 1 (b) ~~ Figure3-9:Prolesof(a)turbulentkineticenergyshearpr oductionand(b)turbulent kineticenergydissipation F x +F z /tan q z 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 1 (a) ~ ~~ (F x +F z /tan q )/ z 0 0.1 0.2 0.3 0.4 0 0.2 0.4 0.6 0.8 1 case1case2case5case6 (b) ~ ~~~ Figure3-10:Prolesof(a)sediment-inducedTKEdampingan d(b)theratioof sediment-inducedTKEdampingtoTKEproduction 112

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k x 10 0 10 1 10 2 10 5 10 4 10 3 10 2 10 1 10 0 z + ~12 k x 10 0 10 1 10 2 10 5 10 4 10 3 10 2 10 1 10 0 z + ~40 k x 10 0 10 1 10 2 10 5 10 4 10 3 10 2 10 1 10 0 z + ~150 k y 10 0 10 1 10 2 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0 E uu case0 E vv case0 E ww case0 E uu case5 E vv case5 E ww case5 z + ~12 k y 10 0 10 1 10 2 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0 z + ~40 k y 10 0 10 1 10 2 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0 z + ~150 Figure3-11:Energyspectraat z + 12 (bufferlayer), z + 40 (logarithmiclayer)and z + 150 (awayformthebed).Topframescorrespondto x -directionspectra,andbottom framescorrespondto y -directionspectra 113

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P rey z 0 5 10 15 0 0.2 0.4 0.6 0.8 1 case1case2case5case6 ~~(a) F x /tan q +F z z 1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 ~~(b) ~ Figure3-12:Prolesof(a)Reynoldsstressproductionand( b)sediment-induced Reynoldsstressdamping 114

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x x P( x ) 20 15 10 5 0 0 0.01 0.02 0.03 0.04 0.05 0.06 case0case5 x m,case0 =1.26 x m,case5 =0.68 Q2 (a) x P( x ) 6 4 2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Q2 (b) 1.5 1 0.5 0 0 0.05 0.1 0.15 0.2 0.25 0.3 3 2 1 0 0 0.02 0.04 0.06 7 6 5 4 0 0.002 0.004 0.006 0.008 0.01 Figure3-13:Prolesof(a)weightedprobabilitydensityfu nctionofQ2eventsand(b) probabilitydensityfunctionofQ2events.Thebed-normall ocationis z + 18 m isthe maximizingReynoldsstressQ2eventanditsvalueforcase0a ndcase5isshownin (a).Theinsetin(a)highlightsthechangeinthemaximizing ReynoldsstressQ2event fromcase0tocase5.Theinsetsin(b)showstheregionaround themostprobableQ2 event( p ),whichdoesnotchangefromcase0tocase5,andthebottompa rtofthe curves. 115

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Figure3-14:Contoursof(a) ~ u 0 ,(b) ~ w 0 and(c) ~ c 0 forcase1at z + 12 116

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Figure3-15:Contoursof(a) ~ u 0 ,(b) ~ w 0 and(c) ~ c 0 forcase5at z + 12 117

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Figure3-16:Iso-surfacesof ci forcase0.Theiso-surfacesare ci =22 118

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Figure3-17:Iso-surfacesof ci forcase5.Theiso-surfacesare ci =22 119

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Figure3-18:Iso-surfacesof ci forcase6.Theiso-surfacesare ci =22 120

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CHAPTER4 TOWARDSAUNIVERSALCRITERIAFORTURBULENCESUPPRESSIONIN DILUTE TURBIDITYCURRENTSWITHNON-COHESIVESEDIMENTS 4.1Background Turbiditycurrentsderivetheirmotionfromtheexcessdens ityimposedbysuspendedsediments.Theincreaseddensityofthesediment-la denuidcreatesahorizontalpressuregradientthatdrivestheow.Iftheowissu fcientlyintenseturbulence issustained.Flowturbulenceiscrucialsincetheenhanced shearstressatthebed enablesresuspension.Ifresuspensiondominatesoverdepo sitionthecurrentcould self-accelerate[ 66 ].Ifdepositiondominates,theintensityoftheowwilldec reaseand willeventuallyextinguish[ 20 83 ].Snowavalanches,pyroclasticowsandduststorms arealsoexamplesofthesetypeofows. Thegoverningequationsofagranularnon-cohesiveturbidi tycurrentowingdown aslope canbenon-dimesionalizedwith H = currentheightasthelengthscaleand shearvelocity u = p b = f asthevelocityscale,where b isthebottomshearstress and f = uiddensity.Inthegoverningequations,threedimensionl essparameters canbedened:Reynoldsnumber( Re ),Richardsonnumber( Ri ),anddimensionless sedimentsettlingvelocity( ~ V z ) Re = u H Ri = g z RC v H u 2 and ~ V z = V u cos (4–1) Here R =( p f ) = f p = sedimentsdensity, = kinematicviscositywhichistaken asconstant, V = settlingvelocityofanisolatedsedimentparticle, C v = meanvolume concentrationofsuspendedsediments,and g z = g cos = bed-normalcomponent ofgravity.Aglobalbalanceofthedrivingforceandthebott omshearstressyields u 2 = RC v Hg x ,where g x = g sin = isthebed-tangentialcomponentofgravity. Re representstheratioofinertialtoviscousforcesandthusm easurestheowintensity. Straticationeffectsareproportionaltothegradientofd ensity.Byexpressingtheabove bulkRichardsonnumberas Ri = g z R ( C v = H ) = ( u = H ) 2 itcanbeseenthatitisaglobal 121

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equivalentofgradientRichardsonnumber. Ri representstheratiobetweenbuoyancy andinertialforcesandthusservesasameasureofstratica tioneffectsinmomentum transfer.Inaturbiditycurrentthesuspendedsedimentsbo thdriveandstratifytheow andthustheRichardsonnumberbecomes Ri =1 = tan .Thedensitygradientsare measuredby ~ V z = Z where Z istheRousenumberand isthevonKarmanconstant. Forthecaseofaeldturbiditycurrentofheight H =20 mrunningonaslope =5 with ameanvolumeconcentration C v =0.005 of 110 msandsedimentsinwater( R =1.65 and =10 6 m 2 /s),theabovedenitionsgive Re =7.5 10 6 Ri =11.43 and ~ V z =2.3 10 2 Thischapterpresentsevidencetoshowthataturbiditycurr entofxed Re remains turbulentprovidedtheproduct Ri ~ V z isbelowathresholdvalue.Ifeither Ri or ~ V z isincreasedtogobeyondthethreshold,turbulenceiscompl etelyextinguishedand theoweventuallydissipatesassedimentssettleonthebed .Asimpletheoretical argumentispresentedtoobtain Ri ~ V z asthecriticalparameter.Thisproductarises naturallyasthedimensionlessmeasureofturbulentkineti cenergy(TKE)dissipation duetostablestratication.Itisobserved,however,thatt hedampingofturbulenceis weakuntilthethreshold Ri ~ V z isreached,beyondwhichturbulenceisabruptlyand completelydamped.Turbulencedampingcanalsobeachieved inowsladenwith cohesivesedimentsowingtotheirrheologicaleffects[ 6 7 82 ].Oneofthenovelties ofthisstudyistoshowthatcompleteturbulencesuppressio ncanalsobeachieved inturbiditycurrentsdrivenbygranularnon-cohesivesedi mentsowingtostratication effects.Pastresearchonstraticationhasmostlyfocused ondensitystratication[ 5 see].Incontrastthesettlingtendencyofthesedimentsske wsthestraticationeffects towardsthebedinducingabreakofsymmetryintheow[ 15 19 ]. Numericalresultsandlaboratoryobservations[ 77 ]indicatethethreshold Ri ~ V z tobebetween0.25to0.5forlowtomoderate Re .Incomparison,eldoccurrences ofturbiditycurrentsareat Re ordersofmagnitudelarger[ 37 62 88 ].Recentresults 122

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on Re scalingofTKEproductionanddissipation[ 51 ]areemployedinthischapterto proposeasimilarscalingforbuoyancy-inducedturbulence damping,whichyieldsa logarithmicdependenceforthreshold Ri ~ V z on Re 4.2Methods Following[ 15 ],asimpliedmodelforthebodyofaturbiditycurrentiscon sidered asaninclinedchannelinwhichtheowisdrivenbygranularn on-cohesivesediments ofuniformsize,nitesettlingvelocityandnegligibleine rtia.Theowisassumeddilute (volumeaverageconcentration C v 1% )sothatinteractionbetweensedimentparticles andrheologyeffectscanbeneglected,Boussinesqapproxim ationcanbeemployed (densityofthecurrentisslightlyhigherthantheambient uidandthereforeitisacknowledgedonlyviathebodyforceterminthegoverningequa tion),andsettlingvelocity isindependentofconcentration.Thesettlingvelocitycan becomputedemploying, forexample,Dietrich'sformula[see 65 ,pg.41].Employingscales H u and C v the dimensionlesssetofequationsthatdescribestheoware[ 15 ] @ ~ u @ ~ t +~ u r ~ u = r ~ p + 1 Re r 2 ~ u +~ c e x Ri ~ c e z r ~ u =0, (4–2) @ ~ c @ ~ t + ~ u + ~ V x e x ~ V z e z r ~ c = 1 Re Sc r 2 ~ c (4–3) where ~ u = dimensionlessuidvelocity, ~ c = dimensionlessvolumetricconcentration ofsediments, ~ p = dimensionlesspressure, Sc = = D = Schmidtnumberwith D the diffusivityofsedimentsemployedtomodelresuspensionfr omthebed, ~ V x = V sin = u and e x and e z areunitvectorsinthestreamwiseandbed-normaldirection s.These equationsaresolvedusingade-aliasedpseudospectralcod eonagrid N x =96 N y = 96 N z =97 inthestreamwise,spanwiseandbed-normaldirections.Per iodicboundary conditionareemployedinthestreamwiseandspanwisedirec tions.Forvelocity,no-slip hasbeenimposedatthebottomboundary,whilefree-slip(se tAinTable 4-1 )andno-slip (setBinTable 4-1 )havebeenemployedforthetopboundary.Theinitialcondit ionsfor 123

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thesimulationsareturbulentchannelowvelocityeldand awell-mixedconcentration eld. Thedynamicsofsedimentresuspensionisverycomplex[ 30 56 75 ]andstillnot wellunderstood.IntheDNS,sedimentresuspensionisincor poratedbymeansofthe diffusiontermontherighthandof( 4–3 ).Thisdiffusioninnon-cohesivesedimentsarise fromtheirlongrangehydrodynamicinteraction[ 76 ].Theboundaryconditionimposed forconcentrationiszerototaluxatthetopandbottomboun daries,thusconserving thetotalamountofsuspendedsedimentsandallowingtheow toreachsteadystate. Whenturbulenceisdamped,thesedimentsaresequesteredin athin,near-bedzone, whosethicknesscanbeexpectedtogrowduetodepositionsin cethebasalshearstress isnotsufcienttoresuspendsediments.However,whenturb ulenceismaintained, thethicknessofthislayerremainsxed.Thedevelopmentof high-concentrationbasal layersisnotaccountedbythismodel.Moredetailsonthemat hematicalmodelandits implementationcanbefoundinCantero etal. [ 15 16 22 ]. ItisworthmentioningthatDNSprovidescompletethree-dim ensionaltimedependentinformationofthevelocityandconcentratione lds,andnoturbulence modelingisrequired.Meanvariablesarecomputedasaverag esoverplanesparallel tothebedandovertime.DerivedvariablessuchasTKEanddis sipationaredirectly computedfromtheirdenitions,withnoassumptions. 4.3EvidenceofTotalTurbulenceSuppression Thissectionpresentsevidencefortheabruptchangeinowb ehavior.Results fromtwosetsofsimulationslistedinTable 4-1 arepresented.Therangeof ~ V z and Ri employedinthesimulationsareoftheorderofvaluespresen tedinsection 4.1 for atypicaleldturbiditycurrents,withtheexceptionof Re =180 .Although Re inthe simulationsismuchsmallerthanineldcases,theresultin gturbiditycurrentsarefully turbulentwhenstraticationeffectsareweak.Inthisrega rds,oneofthemainobjectives 124

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istoshedlightontheReynoldsscalingofstraticationeff ectsonturbiditycurrentsto allowextrapolationofpresentDNSresults. Figure 4-1 showsthewall-normalvariationofthemeanconcentrationp roles.The resultsforsetAshowanabruptchangeinbehavioras ~ V z increasesfrom 0.0255 to 0.0275 .For ~ V z 0.0255 theowremainsturbulentandthesedimentsconcentration remainsnearlyuniform,exceptclosetothetopandbottombo undarieswherethe sedimentdiffusiondominatesoverturbulentmixingtobala ncethesettlingux.For ~ V z =0.0275 theconcentrationproleshowsanexponentialdecayconsis tentwiththe laminarsolution.Thebehaviorsfor ~ V z =0.02125 and 0.023 ofsetBarequitedifferent andconsistentwiththeabruptsuppressionofturbulenceat some ~ V z inbetween. Withincreasing ~ V z ,thesedimentconcentrationnearthebottomboundaryincre ases contributingtostrongerstablestraticationeffects. Theabruptnatureofturbulencesuppressioncanbebetterap preciatedinFigure 4-2 wheretheTKEandTKEproduction(seeprecisedenitionin( 4–6 )ofsection 4.4 ) prolesareplottedforallcasesofsetA.Whencomparedtopu rechannelow(case A-1),theinuenceofincreasingsedimentsettlingonTKEis quitesmallfor ~ V z 0.0255 Anabruptchangecanbeclearlyseenas ~ V z increasesto 0.0275 .ForcaseA-4both TKEandTKEproductionarecompletelydampedacrosstheenti reuidlayer.The resultsofsetBalsoshowsuchanabruptchange,however,not asclearowingto continuedTKEproductionatthetopboundary.Thecritical ~ V z fortotalturbulence suppressioniscomparableforthetwocongurations,illus tratingrelativeinsensitivity todetailsawayfromthebottomboundary.Figure 4-2 isfor Ri =11.4 andsimilar thresholdbehaviorisobservedforother Ri .Alsoforaxednon-dimensionalsediment settlingvelocitythereexistsacritical Ri beyondwhichtheowturbulenceiscompletely suppressed. Directobservationoftotalturbulencesuppressionandcom pletesettlingofsedimentsishardtoachieveinlaboratoryumes.Recentexperi ments[ 77 ],however, 125

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havedocumentedself-acceleratingowstotransitionandb ecomedepositionalwith increasingsedimentssize,providingsupporttothethresh oldbehavior. 4.4TheoreticalConsiderations Theparametricgroupingcontrollingtheobservedsharptra nsitiontocomplete turbulencesuppressioncanbederivedfromthefollowingan alysis.Considerasteady diluteturbiditycurrentwithonlynon-cohesivesediments inequilibriumwhereresuspensionanddepositionatthebedareinbalance,andambientui dentrainmentatthetop interfaceisnegligible[ 15 ].Then,thegoverningequationsofthemeanoware: d d z u 0 w 0 + d u d z + cRg x =0, d d z c 0 w 0 cV z = D d 2 c d z 2 (4–4) P + d d z d k d z w 0 p 0 c + 1 2 u 0 i u 0 i # + c 0 u 0 Rg x c 0 w 0 Rg z =0. (4–5) Here u ( = u 1 )and w ( = u 3 ) = streamwiseandbed-normalowvelocity, p = pressure, c = sedimentsconcentration,barsandprimerepresentmeanand perturbations,and k = 1 2 u 0 i u 0 i P = u 0 w 0 d u d z = @ u 0 i @ x i @ u 0 i @ x i (4–6) aretheTKE,TKEproductionandTKEdissipation.Inthelastt ermof( 4–4 )left,thesuspendedsedimentsdrivethemeanowandtheforcingisdictat edbytheconcentration prole c ( z ) .In( 4–5 )thelasttwotermsembodythedampinginuenceofsuspended sediments. Equations( 4–4 )canbeintegratedto u 2 = RC v Hg x and c 0 w 0 = cV z + d c d z (4–7) where C v =1 = H R H 0 c d z Employing( 4–7 ),( 4–5 )canbeintegratedtoitsdimensionlessform ~ P ~ E + 1 Re d ~ k d ~ z 10 + Ri Sc = Ri ~ V z + (4–8) 126

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where ~ P = 1 u 3 H Z 0 P d z ~ E = 1 u 3 H Z 0 d z = 1 HC v u H Z 0 c 0 u 0 d z and = c b c t C v (4–9) arethedimensionlessbed-normal-integratedTKEproducti onanddissipation, isa componentofTKEdamping,and c t and c b arethetopandbottomwallconcentrations. Here,tildemeansdimensionlessvariables. Equation( 4–8 )dictatestheglobalTKEbalanceandthetwotermsontherigh t accountfortheTKEconsumptiontomaintainsedimentsinsus pension.WhentheavailabilityofTKEtomaintainthesedimentsinsuspensionisnot enough,netsedimentation occursandtheowextinguisheseventually.Inthischapter Ri ~ V z isproposedtocapture totalTKEconsumptiontomaintainsedimentsinsuspension, since isobservedto be 30% of Ri ~ V z inallthecasesconsidered.Therefore,theconditionfortu rbulence suppressionis Ri ~ V z > K c ,where K c isthecriticalvaluebeyondwhichsuspended sedimentscompletelyextinguishturbulence. 4.5ReynoldsNumberScaling ForthetwosetsofsimulationsinTable 4-1 K c iscomputedtorangefrom 0.25 to 0.313 .Correspondingresultsforthehigher Re =400 yield K c intherange 0.35 to 0.39 [ 20 ],thusexhibitingaweak Re dependence.Itisofinteresttoscaletheseresultsto verylarge Re ofrelevancetoeldscaleturbiditycurrents. AsdisplayedinFigure 4-2 ,aslongastheowremainsturbulent,turbulence statisticsshowsonlyaweakdependenceon ~ V z .Thisisconsistentwiththefactthatfor caseA-3 ~ P =6.6 and Ri ~ V z + =0.397 ,i.e.sediment-induceddissipationisonly 6% of ~ P .The Re scalingfor K c isthussoughtfromthe Re dependenceofTKEinapure turbulentchannelowwithoutsediments.Inthislimit ~ P canbeexactlyexpressedas [ 89 ]: ~ P = Re ( g (~ z ) ~ z )( 1 g (~ z ) ) where g (~ z )= ~ u 0 ~ w 0 +~ z (4–10) 127

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Recentinvestigations[ 51 52 ]haveshownpeakTKEproductiontoscalelinearlyas Re butthelocationofthepeaktoscaleas Re 1 .Toobtainanestimateof ~ P theempirical function[ 63 ] g (~ z )= 2 arctan 0.82 Re ~ z 1 exp Re ~ z 7.8 2 (4–11) isused.Integrating( 4–10 ) ~ P 2.46ln ( Re ) 5.97 Figure 4-3 showstheabovescalingalongwiththesimulationresultsfo rturbulent channelow[ 40 ],andresultsfromtwoofourturbiditycurrentsimulations :(i) Re =180 Ri =11.4 ~ V z =0.015 and(ii) Re =400 Ri =2.5 ~ V z =0.175 .WhileTKEdissipation balancesproductionintheabsenceofsediments(as Re !1 ),theirdifferenceissmall inaturbiditycurrentandisbalancedby Ri ~ V z + Basedontheresultsabove,alogarithmicscalingof K c ispostulated.Laboratory andelddataforturbiditycurrentsreportedinTable 4-2 isusedtovalidatethishypothesis.Thelaboratoryexperimentsby[ 77 ]wereperformedina15m-longumewitha 5 % slope,andcorrespondtoacontinuouscurrentloadedwithno n-cohesiveplastic particlesrangingfrom20to200 m.Theevolutionofparticleconcentrationandthe speedofthecurrentatdownstreamlocationsweremonitored tocategorizethecurrent intoselfaccelerating,depositionalandauto-suspension .Thevaluesfor[ 88 ]correspond toobservationsofturbiditycurrentsintheMonterreyCany on,westofCalifornia,USA. VelocitymeasurementswereperformedwithacousticDopple rcurrentprolers(ADCP), andsedimentconcentrationshavebeenestimatedtobelower than0.05.Thesediments canbeconsiderednesand. Itmustberecognized,however,thattheaboveeldobservat ionsandlaboratory experimentswerenotconductedwiththegoalofestablishin gthethresholdvalueof K c Asaresult,someoftheinformationneededinthecalculatio nof K c wasnotmeasured orreported,inwhichcaseweestimateaplausiblerangeofva luesandcomputethe correspondingrangeof K c inTable 4-2 .Also,thelimitingequilibriumstatebetween self-acceleratinganddepositionalow(calledauto-susp ensionmode)istakenhereto 128

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bethecriticalstagebeforeturbulencesuppression.Exper imentalandeldobservations thusinterpretedareshowninFigure 4-4 wherevaluesof K c areplottedagainst Re Thisgurealsoincludesthesimulationresultsfor Re =180 and 400 .Thebesttis Ri ~ V z crit = K c ( Re ) 0.041ln( Re )+0.11 Theexperimentalandelddatashowadeniteupwardtrend,b utexhibitconsiderablescatter.Thus,thescalingfor K c isintendedasaguideline.Clearly,additional higher Re simulationsandexperimentswithafocusonobservingturbu lencesuppressionbystraticationareneeded. AtlowReynoldsnumbersadditional Re dependencemayarise.Ascanbeseen from( 4–8 ),thediffusivetermisnegligibleforlarge Re ,butmayrepresentasignicant sinkofTKEatlower Re .Thus,low Re simulationsareinadequatetoobtainasymptotic scaling.Forpurechannelow Re & 500 wasrequiredfortheviscouseffecttobe negligibleintheproduction-dissipationbalance[ 52 ]. 4.6Deductions Indiluteturbiditycurrents,non-cohesivesuspendedsedi mentshavetwoeffects: a)todrivetheowandsupplyenergyforturbulence,andb)to createstabledensity straticationandsuppressturbulence.Thesetwocompetin geffectsarecharacterizedby threeparameters: Re Ri and ~ V z .Acriticalvalueof Ri ~ V z isobservedtoexist,beyond whichturbulenceiscompletelyextinguishedbystraticat ion.FromtheTKEequation theparametricgrouping Ri ~ V z istheoreticallyobtained,whichcanbeinterpreted astheenergyspenttokeepsedimentsinsuspension.Simulat ionsshowthatthe transitionisabruptandhappenswhenonlyabout6%ofthebed -normal-integrated TKEproductionisconsumedtomaintainsedimentsinsuspens ion.Forvaluesof Ri ~ V z lowerthancriticalitytheeffectofsuspendedsedimentson turbulenceremainssmall, butonceexceededturbulenceiscompletelyturnedoff.Inac hannelowthewallnormal-integratedTKEproductionanddissipationscaleas ln( Re ) ,basedonwhicha logarithmicdependenceforcritical Ri ~ V z withincreasing Re isproposed.Laboratory 129

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andeldobservations,alongwithpresentresultsareusedt oobtainabesttforthis Re dependence.Ithasbeenshownthatturbulencedampingcanbe achievedbyrheology effectinturbiditycurrentcarryingsufcientamountofco hesivesediments.Thischapter showsthatcompleteturbulencesuppressioncanalsobeachi evedduetostratication effectsingranularnon-cohesiveturbiditycurrents. 130

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Table4-1:Listofdirectnumericalsimulations(DNS).SetA :free-sliptopwall.SetB: no-sliptopwall. Case Re Ri ~ V z State A-118011.40.0TurbulentA-218011.40.01TurbulentA-318011.40.0255TurbulentA-418011.40.0275TurbulencesuppressedB-118011.40.02125TurbulentB-218011.40.023Turbulencesuppressed Table4-2:Listofexperimentalandeldobservationsusedi nFigure 4-4 .Currenthalfheighthasbeenusedaslengthscale.Currentheight H expressedinm,sediments diameter d expressedin m,andsettlingvelocity V z expressedinmm/s. and y :Test 11,datafor x =4.8 mandfor x =14.3 m,respectively,andsettlingvelocityreferstothe concentration-weightedvalues. Case HC v SdRV z Re K c [ 77 ] 0.337.8 10 2 5.0 10 2 20-2000.31.2 7.2 10 3 0.56 [ 77 ] y 0.368.5 10 2 5.0 10 2 20-2000.31.1 8.6 10 3 0.48 [ 88 ] 60 100 < 5.0 10 2 > 10 2 1001.657.5 7.0 10 7 0.54-1.07 c/C v z/H 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 A1A2A3A4B1B2 Figure4-1:MeanconcentrationprolesofcaseslistedinTa ble 4-1 131

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k/u 2 z/H 0 2 4 0 0.2 0.4 0.6 0.8 1 A1A2A3A4 (a) H/u 3 z/H 0 20 40 0 0.2 0.4 0.6 0.8 1 (b) Figure4-2:Bed-normalprolesof(a):dimensionlessturbu lentkineticenergy( k = u 2 );(b): dimensionlessturbulentkineticenergyproduction( P H = u 3 ).Resultsforcaseslistedin Table 4-1 .Seedenitionin( 4–6 ).Theturbulentkineticenergymeasurestheturbulence intensityoftheow,andtheturbulentkineticenergyprodu ctionmeasurestheabilityof theowtocontinuecreatingturbulence. X X X X Re t P, e 10 2 10 3 10 4 4 6 8 10 12 14 16 18 JimenezJimenezComteBellotDNSPresentstudyDNSPresentstudyOurEquation X P e e P P ~ ~ ~ ~ ~~ ~ Figure4-3:Evolutionofbed-normal-integratedturbulent kineticenergyproductionand dissipationwithReynoldsnumber.DatalabeledasJimenezt akenfrom[ 40 ],anddata labeledComte-Bellottakenfrom[ 51 ]. 132

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Re t K c10 2 10 3 10 4 10 5 10 6 10 7 10 8 0 0.5 1 1.5 ABSequeiros(2009)Xu(2004)Trendline Figure4-4:Evolutionof K c with Re .DatapointsareobtainedfromourDNScases,experimentsandeldobservations.Thedataforthisplotissh owninTable 4-2 .Atrend lineisttedthroughscattereddatapointstoshowthelogar ithmicdependenceof K c on Re 133

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CHAPTER5 ANALYSISOFTURBULENCESUPPRESSIONINTURBIDITYCURRENTSD RIVEN BYBI-DISPERSESUSPENSIONSOFSEDIMENTS 5.1Background Turbiditycurrentscanbeconceptualizedassubmarinerive rsandareknowtobe oneofthemainmechanismfortransportingsedimentsinsusp ensiontodeepocean [ 31 ].Similarowsareabundantingeophysical,environmental andindustrialprocesses. Snowavalanches,duststorms,pyroclasticows,lavaows, leakageofpoisonousuid intheenvironmentanddambreaksarejustsomeexamples[ 81 ].Turbiditycurrents canbeextremelyenergetic,erosive,andcancarrylargeamo untsofsuspended sedimentsandpropagatelongdistances[ 50 71 83 ].Recurringoccurrencesofturbidity currentsareknowntocarveoutdifferenttopographiesonth eoceanoor.Someof thehugesubmarinecanyonsaretestimonyoftheerosivenatu reofsuchcurrents[ 50 ]. Furthermore,thehydrocarbonreservoirsformedinsidethe oceanoorovergeological timescalesarefromthedepositedsedimentstransportedby mechanismsliketurbidity currents.Thereforeturbiditycurrentsareanimportantca tegoryofsubmarineowsand animprovedunderstandingoftheirdynamicsisessential. Ingeneral,aturbiditycurrentfreelyinteractswiththeoc eanoorbyexchanging sedimentsbymeansofdeposition(duetosettling)andresus pension(duetoerosion). Similarly,turbiditycurrentsinteractwiththesurroundi ngbyentrainingambientwater atthetopinterface.Thenatureoftheseinteractionsareco ntrolledbyfactorslike theamountofsedimentload,propertiesofthesuspendedsed iments,stateofthe oceanoor,andambientconditions.Inotherwords,thesefa ctorsplayacrucialrolein determiningthemechanismsresponsibleforentrainmentof sedimentsfromtheocean oorandkeepingtheminsuspension.Here,werestrictourat tentiontodiluteturbidity currents(volumetricconcentrationofsuspendedsediment sisusuallylessthat 1 %)so thatturbulenceintheowisthesolemechanismresponsible forentrainingsediments fromthebedandkeepingtheminsuspension. 134

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Indiluteturbiditycurrentssuspendedsedimentsplayadua lrole.Ononehand, theydrivetheowwhichgeneratesturbulencenecessarytoe ntrainsedimentfrom thebedandtomixtheminthebodyofthecurrent.Ontheotherh and,thesettling tendencyofsedimentsleadstodensitystraticationwhich stronglyinteractswiththe owturbulencediminishingtheabilityoftheturbiditycur renttoerodesedimentsfrom thebedandtokeeptheminsuspension.Previousresearch[ 61 64 ]analyzedsuch currentsandclassiedthemintothreestatesbasedontheir abilitytoretainsediments. Whenthecurrentisincapableofretainingsedimentsinsusp ension,itprogressively losesthedrivingpotentialandeventuallyceasestoexist. Currentsinsuchstateare calleddepositional.Ontheotherhand,currentsthathaven etentrainmentofsediments fromtheoceanooraretermedtobeinthestateofself-accel eration.Becauseofnet entrainmentofsediments,thecurrentfurtherintensiess ettingupareinforcingcycle andhencethenameself-accelerating[ 66 67 77 ].Autosuspensionorbypassisthe limitingstatebetweenthedepositionalandself-accelera tionstates[ 64 66 ].Herethe currentisenergeticenoughsothatitexactlymaintainsthe sedimentload,i.e.,the drivingpotentialisalwaysconserved. Theinterplayofturbulenceandsuspendedsedimenthasbeen studiedinCantero etal. [ 15 ]bymeansofdirectnumericalsimulations(DNS)ofasimpli edmodelfor continuousdiluteturbiditycurrents.Theymodeledthesec urrentsasinclinedchannel owdrivenbytheexcessdensityimposedbyamono-dispersed suspension.Themodel assumedtheowtobeinbypassmodesoastoreachastatistica llysteadystate. Cantero etal. [ 15 ]reportedcompleteturbulencesuppressionwhenthesettli ngvelocity ofsedimentswasgreaterthanacriticalvalue.Completetur bulencesuppressionimplies thattheowhaslostthecapacitytokeepthesedimentsinsus pension.Suchaow willbedepositionalandwillloseeventuallyallthesuspen dedsediments.InChapter 3 thecompleteturbulencesuppressionprocesswasanalyzedi ngreaterdetailanda mechanismthatcausescompleteturbulencesuppressionint heowwasproposed. 135

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Chapter 3 proposesthatthestablestraticationinthecurrentcreat espocketsof owwithintensesedimentconcentrationuctuationsthata recorrelatedwithvarious turbulentstructureslikehairpinandquasi-streamwisevo rticesintheow.Pocketsof largesedimentconcentrationuctuationsimpliesthatthe owstructureslocallyneed tospendmoreenergytokeepthesedimentsfromsettling.Thi sleadstodampingin theintensity(strength)andspatialre-orientationofthe structures.Itisexpectedthat thisdampingprocessintensieswithincreaseinthesettli ngvelocityofsedimentsand eventuallyreachesapointwherethestructureslosetheira uto-regenerationabilitywhich culminatestocompleteturbulencesuppression. Thismechanismofcompleteturbulencesuppressionisparam eterizedinChapter 4 .Inthechapteritissuggestedthatturbulencesuppression canbequantiedbythe parametricgrouping Ri V z = u whichalsorepresentstheamountofenergyspentby turbulencetokeepthesettlingsedimentsinsuspension.He re Ri istheRichardson numberwhichsimpliedto 1 = tan orslopeofthechannel, V z isthesettlingvelocityof thesuspendedsedimentsinthedirectionnormaltothebed,a nd u istheshearvelocity oftheow.Chapter 4 proposesthatcompleteturbulencesuppressionwilloccuri na owcorrespondingtoasupercriticalvalueof Ri V z = u .Furthermore,itisnotedthat thecriticalvaluewilldependontheReynoldsnumberofthe ow( Re ).Limitedeld observations,DNSandexperimentssuggestalogarithmicde pendenceon Re Thecriticalvalueofsettlingvelocityreportedby[ 15 ]is V z = u 0.025 fora Ri 11.5 .Incontrast,experimentalevidenceontransportofsuspen dedsedimentbyturbidity currents[ 30 ]atasimilarvalueof Ri 12.5 reportsedimentsthatareinsuspension evenforvaluesof V z = u aslargeas0.5[ 30 ,seeFigure11in].Clearly,thepredictionof [ 15 ]underestimatesthelargersizeofsedimentthancanbecarr iedinsuspensionby turbiditycurrents.Thisunderestimationcannotbeexplai nedbytheReynoldscorrection proposedinChapter 4 136

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Inthischapterweextendtheworkof[ 15 ]andtheworkpresentedinChapter 3 toturbiditycurrentsdrivenbybi-dispersesuspension,an daddressthestratication effectsinturbiditycurrentsandtheircarryingcapacityi ntermsofsedimentsizes. TheturbulencesuppressioncriteriaproposedinChapter 4 isextendedtobi-disperse suspensionsandshowbetteragreementwithexperimentalda taby[ 30 ]. 5.2ProblemFormulation Thissectiondescribesthemathematicalmodelfordilutetu rbiditycurrentsdriven bybi-dispersesuspensionofsediment.Thismathematicalm odelissimilartothemonodispersemodelusedinCantero etal. [ 15 ]andChapter 3 ,wheretheturbiditycurrents aremodeledasinclinedchannelowsdrivenbymono-dispers esuspensionofsediment. SeeFigure 3-1 foraschematicrepresentationofthemathematicalmodel.H erewe assumethatthecurrentisdrivenbyexcessdensityimposedb ybi-dispersesuspension ofsediment.Furthermore,thebi-dispersesuspensionisas sumedtobediluteenough sothatthesedimentparticle-particlecollisionsareunim portantandrheologyeffects canbeneglected.Equilibrium-Eulerianmodelisusedtodes cribethedynamicsofthe ow(seeFerry&Balachandar[ 28 ]fordetailsonEquilibrium-Eulerianmodel).Moreover, asaconsequenceofdilutesuspension,Boussinesqapproxim ationisalsoemployed. Followingarethedimensionalgoverningequationsforsuch aow: @ u @ t + u r u = 1 w r p + r 2 u +( R c c c + R f c f ) e g (5–1) r u =0, (5–2) @ c m @ t +( u + V m ) r c m = D m r 2 c m (5–3) where u = f u v w g isthevelocityvector, p isthepressure, w isthedensityofwater, isthekinematicviscosityofwaterand e g = f g x ,0, g z g istheaccelerationdueto gravitywith g x and g z beingcomponentsinthestreamwiseandbednormaldirection s. Thelasttermin( 5–1 )quantiesthebodyforceduetothebi-dispersesuspension and 137

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thesubscripts' c 'and' f 'representpropertiesassociatedwithcoarse(large)and ne (small)sedimentparticlesinthebi-dispersesuspension. Therefore c c and c f arethe sedimentconcentrationofcoarseandnesedimentparticle sintheowrespectively. Thevariables R c and R f arethemeasureofspecicgravityofthecoarseandne sedimentparticlesinsuspension.Ingeneraltheycanbewri ttenas R m =( sm w ) = w where'm'canbe' c 'forcoarseor' f 'fornesedimentparticles.Thecontinuousphase (water)satisesthecontinuityequation( 5–2 ),whilethetwodispersedphasesare representedbytheircorrespondingtransportequationwhi chhasthegeneralformas shownby( 5–3 ). V m = f V mx ,0, V mz g isthesettlingvelocityofsedimentparticles oftype' m '(asbefore' m 'becomes' c 'forcoarseand' f 'fornesedimentparticlesin suspension)inquiescentambient.Sedimentparticlesarea ssumedtobenon-cohesive and D m isthediffusivityofsedimentparticlesoftype'm'.Heredi ffusivityisassumedto beisotropicandconstantforbothsedimentsizes,i.e. D c D f D .Itshouldalsobe notedthatdiffusivityisnotduetothebrownianmotionofse dimentparticles,butdueto longrangehydrodynamicforcesmediatedthroughthecontin uousphaseduetorandom uctuationsintheparticlenumberdensity[ 58 76 ].Thisdiffusivetermalsoactsasa mechanismtoerodeorentrainsedimentparticlesfromthebe d[ 30 ].Theformofthe convectiontermintheconcentrationtransporttermimplie sthatthesedimentparticles havesmallresponsetimeandtheirinertiaisofsecondorder importanceascompared totheirsettling[ 15 17 28 ]. Theheightofthechannelis L z = h anditsstreamwiseandspanwiselengthsare L x =4 h and L y =4 h = 3 ,respectively.Thechannelisassumedtobeperiodicinthe streamwiseandspanwisedirections.No-slipboundarycond itionisimposedonthe bottomboundaryorthebed,andthetopboundaryimposesno-s tresscondition.For eachsedimenttypethebottomandtopboundaryofthechannel imposeszeronetux ofsedimentconcentrationbyenforcingthelocalsettling uxofsedimentstobalance thelocalsedimentconcentrationgradient.Physicallythi smeansthatsedimentsthat 138

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settleontheboundaryareinstantlyresuspendedintotheo w.Thisallowstheowto conservethetotalsedimentloadandreachastatisticallys teadystate.Notethatthis boundaryconditionisappliedtobothconcentrationequati ons.Thereforetotalvolume concentrationofeachsedimentsizeisconservedandasares ultthetotalvolume concentrationofthebi-dispersesuspensionisconserved. Alltheaboveboundary conditionscanbemathematicallyexpressedasfollows u =0 at z =0, (5–4) @ u @ z =0, @ v @ z =0 and w =0 at z = h (5–5) c m V mz = D @ c m @ z at z =0 and z = h (5–6) 5.3DimensionlessEquations Theboundaryconditionsimposedonthegoverningequations ensurethatthe owreachesastatisticallysteadystate.Theforcebalance atthefullydeveloped statisticallysteadystatewillbeusedtodenerelevantsc alestogetthedimensionless equations.Themeanowequationsforfullydevelopedstati sticallysteadyoware obtainedbyaveragingthemomentumandconcentrationequat ionsoverturbulence. Averagingisdonespatiallyinthetwohomogenousdirection si.e.streamwiseand spanwisedirectionsandovertime.Theaveragequantitiesa rerepresentedby ( ) and thecorrespondinguctuationsarerepresentedby ( 0 ) .Followingarethemeanow equations d 2 u dz 2 d dz ( u 0 w 0 )+ ( R c c c + R f c f ) g x =0, (5–7) 1 w d p dz + d dz ( w 0 2 )+ ( R c c c + R f c f ) g z =0, and(5–8) D d 2 c m dz 2 V zm d c m dz d dz ( c 0 m w 0 )=0. (5–9) 139

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Meanmomentumequationinthespanwisedirectionistrivial asforcinginspanwise directionisabsent( g y =0 ).Thisgives v =0 .Similarly,meanvelocityinthebed-normal directioniszero( w =0 )duetocontinuityandnopenetrationboundaryconditionon the bedandtopboundary.Noticethatpressureappearsonlyin( 5–8 )(bed-normaldirection) andintegrating( 5–8 )givesthefollowingformforit, p = p + w w 0 2 + w g z Z z 0 ( R c c c ( )+ R f c f ( ) ) d (5–10) Similarlyintegrating( 5–7 )inthebed-normaldirectiongivesthesteadystatebalance betweenthefrictiononthebedandtopboundaryandthedrivi ngforceduetobuoyancy effectimposedbysuspendedsedimentparticles.Sincethet opboundaryisno-stress, drivingforceisbalancedbystressonthebedanditisasshow nbelow b w = g x h R c C ( v ) c + R f C ( v ) f = Rg x hC ( v ) (5–11) where C ( v ) c = 1 h Z h 0 c c dz C ( v ) f = 1 h Z h 0 c f dz (5–12) C ( v ) = C ( v ) c + C ( v ) f and R = R c C ( v ) c + R f C ( v ) f C ( v ) (5–13) NotethattheReynoldsstress u 0 w 0 termdropsoutasitisexactlyzeroontheboundaries inthebednormaldirection.Althoughtheformulationiskep tgeneral,butastraight forwardsimplicationwouldbetoassumethatallthesedime nttypeshavesimilar density,i.e. sc sf s ,whichmeansthat R c R f R .Frictionvelocity( u )canbe denedbasedon( 5–11 ) u 2 = b w = Rg x hC ( v ) (5–14) Fromabovevariousscalesrelevanttotheproblemcanbeden ed. u isselectedas velocityscale,Lengthscaleis' h ',i.e.heightofthechannel,timescaleis h = u ,pressure scaleis w u 2 andconcentrationscaleis C ( v ) .Usingthesescales,governingequations 140

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( 5–1 )-( 5–3 )reducetothefollowingdimensionlessform @ ~ u @ ~ t + ~ u r ~ u = r ~ p + 1 Re r 2 ~ u +( ~ R c ~ c c + ~ R f ~ c f ) e (5–15) r ~ u =0, (5–16) @ ~ c m @ ~ t +( ~ u + ~ V m ) r ~ c m = 1 Re Sc r 2 ~ c m (5–17) Intheabove, (~ ) representsdimensionlessvariables, ~ R c = R c = R ~ R f = R f = R and e = f 1,0, g z = g x = 1 = tan g isthedimensionlessgravityvector.Reynoldsnumber ( Re )andSchmidtnumber( Sc )aretwodimensionlessparametersgoverningtheow andtheirformsareasshownhere Re = u h and Sc = (5–18) Including Re and Sc ,thereare 8 parametersthatfullydenetheabovesetofgoverning equations.Theseparametersare Re Sc ,channelinclination( ),settlingvelocityand densityofcoarseandnesedimentparticles,i.e. ~ R c ~ V c ~ R f and ~ V f ),totalvolumetric concentrationofeithercoarseornesedimentparticles(s incesumoftotalvolumetric concentrationofcoarseandnesedimentparticlesis 1 ).Notethatthetotalvolumetric concentrationofaparticularsedimentcanbewrittenas r m = C ( v ) m C ( v ) sothat r c + r f =1. (5–19) Simpliedbi-dispersemodel. Toprobetheeffectsofbi-dispersesuspension onowturbulenceasimpliedbi-dispersemodelcanbepropo sed.Firstofall,letus assumethatthesedimentparticleshavenearlythesamedens ity,i.e., ~ R f ~ R c 1 Furthermore,alimitingcaseofnesedimentparticlesisco nsideredwheretheparticles areassumedtobesonethattheyhavenegligiblesettlingve locity( ~ V f 0 ).Thismakes thetransportequationfornesedimentparticlestriviala ndthereforedropsoutofthe governingequations.See( 5–9 ),with V f 0 thesedimentconcentrationwillremain 141

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uniforminthebed-normaldirection.Thustheresultingbidispersemodelwillbea 5 parameterproblemwherethenalsetofparametersare Re Sc ~ V c and r f .Thenal formofthegoverningequationsareasshownbelow @ ~ u @ ~ t + ~ u r ~ u = r ~ p + 1 Re r 2 ~ u +(~ c c + r f ) e (5–20) r ~ u =0, (5–21) @ ~ c c @ ~ t +( ~ u + ~ V c ) r ~ c c = 1 Re Sc r 2 ~ c c (5–22) Henceforththismodelwillbereferredtoasthebi-disperse modelandthewordsimpliedwillbedropped.Inthismodel,theinclinationofchanne lisxedat( =5 ).This valueisselectedasitlieswellwithintherangeofinclinat ionobservedforthecontinental slopeontheoceanoor(itsrangeis 1 to 10 ,seePinet[ 70 ]).Also =5 wasthe inclinationusedinthepreviousDNSsimulations[ 15 23 ]andthemono-dispersesimulationspresentedinChapter 3 .Usingthesamevalueheremakesthecomparisonof thebi-dispersesimulationswiththemmorerelevant.Schmi dtnumber, Sc ,comesabout fromthediffusionterminthesedimenttransportequation( see( 5–3 )).Thisdiffusion termisusedasamodeltoapproximatethesedimenterosionpr ocessformtheocean oor(see[ 30 ]wheretheyhavedenedentrainmentfunctionbasedontheco ncentration gradients).Furthermore,theexistenceofsuchadiffusion termfornitesizesediment particles(Brownianmotionisnegligible)isnotcompletel yunphysical.Recentobservationsshowthatevennitesizeparticlescandiffuseduet olongrangehydrodynamic forcesmediatedthroughthecontinuousphaseduetorandom uctuationsintheparticle (sediment)numberdistribution[ 76 ],[ 58 ].Thusthesedimentparticlediffusiontermis inplacetomimicalltheaboveprocess.Ideally, Sc willbeanisotropicduetoitsdependenceonlocalsedimentconcentrationandlocalshear. Sc mayalsohaveaparametric dependenceonthesedimentsettlingvelocity(sedimentsiz e).Inthepresentmathematicalmodel, Sc isassumedtobeisotropicandindependentofthesettlingve locityof 142

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sedimentsanditissetto 1 forallthesimulations.PreviousstudieslikeCantero etal. [ 15 ],Shringarpure etal. [ 79 ]haveusedsimilarvalues,whilestudieslikeBonometti& Balachandar[ 11 ],Cantero etal. [ 21 ],Necker etal. [ 59 ]haveshownthateffectof Sc is negligibleaslongasitis O (1) inmagnitude. Inrealturbiditycurrents Re islargeandusuallyofthe O (10 4 ) ormore.DNSof suchhigh Re isimpossibleevenwiththecurrentstate-of-the-artsuper computers. Here,Reynoldsnumberisxedat Re =180 whichresultsinmatureturbulencefor apurechannelow.Thisvalueisalsoselectedsoastoaidthe comparisonofthese simulationswiththepreviousresearch[ 15 23 79 ]. Thisbringsustothequestionofselectingappropriatevalu esforthetworemaining parameters ~ V c and r f .Clearly,thedimensionlessbulkdrivingforceisdividedi nto 2 components: r f and r c =1 r f .Since ~ V f =0 ,thesesedimentsremainwellmixedin thechannelwithuniform(mean)concentrationof c f = r f .Thereforethebodyforcedue tonesedimentsisequivalenttoconstantanduniformstrea mwisepressuregradient thatdrivestheow.Notethatnesedimentswillhavenostra ticationorturbulence dampingeffects.Whiletheremainingportionofthebulkdri ving, r c ,isduetosettling sedimentswithvelocity ~ V c .Thecoarsesedimentparticleswillskewthedrivingforce closetothebedandstratifytheowinthebed-normaldirect ion.Thereforethecoarse sediment( r c )issolelyresponsibleforinducingturbulencedampingeff ectsontheow. Frompreviousresearch[ 15 ]andthemono-dispersesimulationspresentedinChapter 3 ,itisclearthatfor ~ V c < ~ V critical mono thestraticationeffectdueto r c willneverbe sufcienttocausetotalturbulencesuppression.Thusthea boveproblemisinteresting andnon-trivialonlywhen ~ V c > ~ V critical mono andthereexistacriticalvalueof r f below whichthestraticationeffectduetothecoarsesedimentsb ecomesstrongenoughto causecompleteturbulencesuppression.Usingtheabovemat hematicalmodelDNSare carriedoutunder 3 sets.Ineachset ~ V c (whichisgreaterthan ~ V critical mono =0.026 for Re =180 ,seeTable 3-1 )isheldxedand r f isvariedtoobtainitscriticalvaluebelow 143

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whichthereiscompleteturbulencesuppressionintheow. ~ V c =0.025,0.035 and 0.05 insetA,BandC,respectively.Thedetailsofallthesimulat ionsaregiveninTable 5-1 Numericalmethod: Thedimensionlessgoverningequations( 5–20 )-( 5–22 )are solvedusingadealiasedpseudo-spectralcode[ 24 ].TheowvariablesareapproximatedbyFourierexpansionsinthedirectiontangentialto thebed( ~ x ~ y )andby Chebychevexpansionsinthebed-normaldirection( ~ z ).Momentumequationalongwith theincompressibilitycriteriaissolvedbyasplittingmet hod.Alow-storagemixedthird orderRunge-KuttaandCrank-Nicolsonschemeisusedfortem poraldiscretizationof advectionanddiffusionterms.Thisschemeiscarriedoutin threestageswithpressure correctionattheendofeachstage.RefertoCortese&Balach andar[ 26 ]forcomplete detailsontheimplementationofthescheme.Thegridresolu tionof( N x N y N z )= ( 96,96,97 )isusedanditisfoundtobesufcientfortheReynoldsnumbe rselectedin thisstudy[ 15 79 ]. 5.4LaminarSolution Thesimpliedbi-dispersemodelwherethenesedimentshav enegligiblesettling velocityadmitsalaminarsolution.Itneedstobeemphasize dthatthislaminarstatewill notexistinrealows.Itispurelyamanifestationofthemat hematicalmodelproposed here.Inrealowsstrongstraticationcangreatlydamptur bulenceandaffectthe sedimentcarryingcapacityoftheow.Insuchcases,theow willprogressivelydeposit sedimentsonthebedandeventuallyceasetoexist.Thelamin arsolutionofthismodel isanindicationofthatstatebeingreached.Whentheabovem athematicalmodeladmits thelaminarsolutionasastablestate,itimpliesthatstrat icationisstrongenoughthatit completelysuppressesturbulenceintheow. Startingfromthegoverningequations( 5–20 )-( 5–22 )andwiththeassumptionthat owislaminar,steadyandfullydevelopedinthehomogeneou sdirections( ~ x and ~ y ),the 144

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equationsreducetothefollowingform d 2 ~ u d ~ z 2 = Re ( r f +~ c c ) (5–23) ~ V cz d ~ c c d ~ z = 1 Re Sc d 2 ~ c c d ~ z 2 (5–24) Integrating( 5–24 )andusingtheboundaryconditions( 5–6 )givestheexpressionfor concentrationofcoarsesedimentsinthechannelas ~ c c = A exp( ~ V cz Re Sc ~ z ). (5–25) Theintegrationconstant A cannotbecomputedusingtheboundaryconditions.Itsform isdeterminedsuchthatthetotalvolumetricconcentration ofcoarsesedimentsinthe channelis r c =1 r f .Applyingthisclosureweget A = (1 r f ) ~ V cz Re Sc 1 exp( V cz Re Sc ) (5–26) Nowusingtheaboveexpressionforconcentration,( 5–23 )canbesolvedtogivethe followingexpressionforstreamwisevelocity ~ u = Re r f ~ z 2 2 ~ z 1 r f ~ V cz Sc 1 exp( ~ V cz Re Sc ) h exp( ~ V cz Re Sc ~ z ) + Re Sc ~ V cz ~ z exp( ~ V cz Re Sc ) 1 i (5–27) Theaboveexpression( 5–27 )clearlyshowssuperpositionoftwoeffects:rstterm isthetypicalparabolicproleduetoconstantpressuregra dientowandthesecond termisthecontributionduetosettlingsediments.The r f =1 limitgivesamonodispersesuspensionwithsedimentparticleshavingneglig iblesettlingvelocity.Forthis casewerecoverthevelocityproleforuniformpressuregra dientdrivenchannelow. Similarly, r f =0 limitalsogivesamono-dispersesuspensionbutwithcoarse sediment particleshavingnon-negligiblesettlingvelocity.Forth iscasewerecovertheexactsame 145

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expressionforstreamwisevelocitythatwasobtainedfromt helaminarsolutionofmonodispersemodelinChapter 3 (see( 3–26 )).Figures 5-1 (a)and(b)showsthelaminar velocityandtotalsedimentconcentration( r f +~ c c )prolesfor Re =180 r f =0.235 Sc =1 and j ~ V c j =0.035 5.5Results Inturbiditycurrentstheextentofstraticationdependso ntherelativemagnitudeof thesettlingtendencyofsedimentsandturbulentmixingint heow.Sincethesuspended sedimentsarealsoresponsiblefordrivingtheow,thereis tightinterplaybetween straticationandturbulencemodulatedbythesizeofsedim entparticles.PreviousDNS ofturbiditycurrentsusingthemono-dispersemodelhavesh ownthatonlysuspensions ofsub-criticalsedimentsizescandrivetheow[ 15 23 79 ].Webelievethatasimilar conclusioncanbemadeforturbiditycurrentsdrivenbybi-d ispersesuspensionscan bemade.Ingeneral,thestateofacurrentwithbi-disperses uspensiondependson thecombinedeffectofturbulenceandstraticationeffect sofindividualsedimentsizes inthesuspension.Forexample,thevolumetricconcentrati onofeachsedimentsize willimposebodyforceonthecurrent.Thestreamwisecompon entwillcontribute towardsgeneratingturbulence,whilethebed-normalcompo nentwillcontributetowards stratifyingtheow.Therefore,thestateofthecurrentwil ldependonthecombinedeffect ofbothsedimentsizes,i.e.,ifthenetstraticationiswea kthecurrentremainsactive andowissustained,otherwise,strongstraticationcanl eadtocompleteturbulence suppressionandtheowwillceasetoexist.Thisistheunder lyingprinciplethatgives turbiditycurrentstheabilitytotransportlargesizesedi ments.Inotherwords,itcanbe saidthatturbiditycurrentstransportlargesedimentswhe ntheyareapartofafavorable suspensioncomposition. Thebi-disperseformulationispresentedin x 5.2 .Differentsuspensionsareconsideredunder3setsA,BandC.Inthesesets(A,BandC)thesettli ngvelocityofcoarse 146

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sedimentsisxed( 0.0275 0.035 and 0.05 )andtheproportionofcoarsetonesedimentsisvaried.Ineachsetwestartwithasuspensioncontai ningalotofnesediments ( r f 1 )whichkeepsthestraticationeffectstominimum.Insubse quentcases,the quantityofcoarsesedimentsisincreasedattheexpenseof nesediments.Thusweincreasethestraticationeffectintheowwhilekeepingthe totalsedimentloadconstant. Eventuallyacriticalsuspensioncompositionisreachedso thatevenasmallincrease intheamountofcoarsesedimentswillresultincompletetur bulencesuppression.The criticalsuspensionliesbetweencases6Aand7AforsetA,6B and7BforsetBand6C and7CforsetC.Completedetailsofallthesimulationsarel istedinTable 5-1 Thisanalysisisintendedtoshowtheeffectofbi-disperses uspensiononthe owandtheexistenceofacriticalcompositionthatbringsa boutcompleteturbulence suppression.Figure 5-1 (a)presentsthemeansedimentconcentration( ~ c c + r f )proles inthebed-normaldirectionforcases1B,3B,6B,7Band r f =1 r f =1 impliesturbidity currentdrivenbyauniformbodyforce(oramono-dispersesu spensionofsediments withzerosettlingvelocity,seecase0fromTable 3-1 ). r f =1 isusedasareferenceto comparethestraticationeffectsofdifferentbi-dispers esuspensions.Fromcase1Bto 7B,theamountofcoarsesedimentsinthesuspensionisprogr essivelyincreasedatthe expenseofnesediments.Thismeansthatthestratication effectineachsubsequent casewillbegreaterthanitspredecessor.Thisisexactlywh atweseeinFigure 5-1 (a). Themeansedimentconcentrationatthebedincreasesfrom 1.232 forcase1Bto 5.036 forcase7B.Similarlytheconcentrationgradientneartheb edincreasesfromcase1Bto case7B.Observethatfromcase6Btocase7Bthereisasuddenc hangeinthemean concentrationprole.Incase7Bmostofthesedimentsareno wconcentratedclose tothebedindicatinglossinthemixingabilityoftheow.Al soshowninthegureare squaresymbolsthatrepresentthelaminarsolutioncorresp ondingtotheparameters ofcase7B.Clearlythemeanconcentrationproleobtainedf romtheDNSisinvery 147

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goodagreementwiththelaminarsolution.Thissuggeststha tcompleteturbulence suppressionhasoccurredincase7B. MeanvelocityprolesshowninFigure 5-1 (b)alsotellasimilarstory.Fromcase1B to7Basthestraticationincreases,itinhibitsthevertic alexchangeofmomentumand asaconsequence,themeanstreamwisevelocityincreasesin thechannel.Itcanbe inferredfromthegureandalsoformtheDNSdatathatthebul kstreamwisevelocity(or averagemeanstreamwisevelocityorowrate)increasesfro mcase1Bto7B.Similarto themeanconcentrationproles,asubstantialchangeinthe ~ u prolesisseenfromcase 6Bto7B.Thereisanabruptjumpinthebulkvelocityandthema ximumstreamwise velocityfromcase6Btocase7B.Furthermore,the ~ u proleofcase7Bisinverygood agreementwithitscorrespondinglaminarsolution.Theref ore,itisevidentthatcomplete turbulencesuppressionhasindeedoccurred. Figure 5-2 (a)presentstheReynoldsstress( ~ u 0 ~ w 0 )prolesofdifferentcasesfrom setB.Reynoldsstressmodulationsareanindicationofturb ulencesuppressionand henceitrepresentthestraticationeffectontheow.Reyn oldsstressprolesshow slightdampingas r f decreasesfrom 0.75 forcase1Bto 0.24 forcase6B.Beyondcase 6B,evenaslightdropin r f to 0.235 (case7B)causesabruptandcompletesuppression ofReynoldsstressintheow.Figure 5-2 (b)showstheprolesofratioofReynolds ux( ~ w 0 ~ c 0 )tosettlingux ~ V cz d ~ c c fordifferentcasesfromsetB.SinceReynoldsuxis ameasureofturbulentmixinginthebed-normaldirection,t heratioofReynoldsuxto settlinguxrepresentsthebed-normalmixingabilityofth eow.SimilartoReynolds stressproles,weobservethatfromcase1Btocase6Bonlysl ightdampingisseen inthemixingabilityoftheow.But,beyondcase6Btherecom pletelossofthemixing abilityoftheow. SimilarabruptsuppressiontrendsareseeninFigure 5-3 (a),(b)and(c)forthe streamwise,bed-normalandspanwisermsvelocityproleso fsomeofthecasesfrom setB.Insummary,alloftheaboveturbulencestatisticsrev ealthatcompleteturbulence 148

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suppressionoccurswhen r f isreducedfrom 0.24 to 0.235 .Anaiveextensiontorealturbiditycurrentwouldimplythatat Re =180 andslope =5 ,totransportsediments withsettlingvelocityof 0.035 inbi-dispersesuspension,thenesediments( j ~ V f j 0.0 ) needtomakeupatleast24%ofthetotalsedimentload.Simila robservationscanbe madefromtheothertwosets,AandC.5.5.1CompleteTurbulenceSuppressionCriteriaforBidisp reseSuspensions Chapter 4 proposesthatturbulencedampingcanbequantiedbythepar ametric grouping Ri ~ V z foraturbiditycurrentdrivenbyamono-dispersesuspensio n.Inaddition, itwasalsoshownthattheabruptandcompleteturbulencesup pressionoccurswhenthe parametricgrouping Ri ~ V z increasesbeyondacriticalvalue.Chapter 4 alsoproposes thatthecriticalvaluefor Ri ~ V z hasalogarithmicdependenceon Re .Inthissection, fromtheturbulentkineticenergy(TKE)equationforabi-di spersesuspension,itis shownthatasimilarparametricgroupingcanbegiventoquan tifyturbulencedamping. Furthermore,itisalsoshownthatacriticalvalueforthisp arametricgroupingexistthatis similartothepreviousmono-dispersecase. Turbulentkineticenergyequationforturbiditycurrentsd rivenbybi-dispersesuspensionofsedimentswhentheowisinstatisticallysteadysta teisgivenbelow ~ P ~ d d ~ z ~ w 0 ~ p 0 + ~ k ~ w 0 1 Re d ~ k d ~ z # = ~ u 0 ~ c 0 c + Ri ~ w 0 ~ c 0 c (5–28) whereturbulentkineticenergy(TKE) ~ k ,TKEproduction ~ P andTKEdissipation ~ are expressedas ~ k = ~ u 0 i ~ u 0 i 2 ~ P = ~ u 0 ~ w 0 d ~ u d ~ z and ~ = 1 Re @ ~ u 0 i @ ~ x j @ ~ u 0 i @ ~ x j (5–29) TheTKEequation( 5–28 )cannowbeintegratedinthebed-normaldirection( ~ z )tothe gettheglobalbalance. ~ P ~ E + 1 Re 8<: d ~ k d ~ z # 10 + Ri c 9=; = c + Ri ~ V cz r c (5–30) 149

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where ~ P = Z ~h 0 ~ P d ~ z ~ E = Z ~ h 0 ~ d ~ z c = Z ~h 0 ~ u 0 ~ c 0 c d ~ z c = ~ c cb ~ c ct Sc and Ri = 1 tan (5–31) Termsontherighthandsideof( 5–30 )quantifyturbulencedampingduesuspended sediments. Ri c isthebulkhydrodynamicinteractionsofsedimentparticle sbetween themselvesandwiththechannelboundaries.Asexpectedthe turbulencedamping effectsareonlyduetothecoarsesediments.Table 5-2 presentstheTKEbudget ofcasesthatareclosesttothecriticalturbulencedamping limitforeachofthesets A,BandC.AlsopresentedinthetableistheTKEbudgetofcase 5from 3-1 .Case 5representsthecriticalturbulencedampinglimitforcurr entswithmonodisperse suspensionofsediment.NoticethattheTKEbudgetofallthe casesshowstriking agreement.ThismeansthatatthecriticalstatethebulkTKE production( ~ P ),bulkTKE dissipation( ~ E )andbulkTKEdampingterms c and Ri ~ V cz r c inthechannelseemsto beinsensitivetothecompositionofthesuspension.Furthe rmore,aneffectivesettling velocity ~ V e = r c ~ V cz canbedenedforthebi-dispersecases6A,6B,6Candthisval ue closelymatcheswiththemonodispersecase5.Fromtheseobs ervationsitcanbe concludedthatthecriteriaforcompleteturbulencesuppre ssionforcurrentsdrivenby monodispersesuspensionsholdsforcurrentsdrivenbybi-d ispersesuspensions.An implicationofthisisthatthescalingrelationsforthetur bulencesuppressioncriteria developedinCantero etal. [ 23 ](showninChapter 4 )shouldholdforcurrentsdriven bybi-dispersesedimentsuspensions.Itcanalsobeconject uredthatthisturbulence suppressioncriteria[ 23 ]canbeextendeduniversallyforcurrentsdrivenbyagenera l poly-dispersesedimentsuspension(seeAppendix C ).Mathematicallythiscanbe writtenas Ri ~ V z j crit Ri r c ~ V cz j crit =0.041ln( Re )+0.11 (5–32) 150

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5.5.2TurbulenceStatisticsofFlowswithNearCriticalTur bulenceDamping TheturbulencesuppressioncriteriawhichwasbasedontheT KEbudgetofcurrents withmonodispersesuspensionholdsforcurrentswithbi-di spersesuspensions.Since thecriteriaisbasedonthebulkow,itwillbeinterestingt ocomparethelocalmeanand otherhigherorderturbulencestatisticstounderstandthe dynamicsoftheowinmore detail.Notethatthecasesthatareclosetothecriticaltur bulencedampinglimitwillbe referredtoasnear-criticalcaseshenceforth. Figure 5-4 (a)presentsthemeansedimentconcentration( ~ c c + r f )prolesofall near-criticalcases(case6A,6B,6Candcase5)inthebed-no rmaldirection.Allthe meansedimentconcentrationprolesshownin 5-4 (a)areclose,butnotexactlyon topofeachother.Thedifferencesintheprolesareapparen tespeciallyclosetothe bedandnearthetopboundaryofthechannel.Nearthebed,the meansediment concentrationanditsgradientincreaseswithincreaseint hesettlingvelocityofcoarse sedimentparticles.Thereforehigherconcentrationandst rongerstraticationmay beobservedfornear-criticalcasesthatcontainsedimentp articleswithadifferent settlingvelocity.Inthecentralregionofthechannel,the prolesofnear-criticalcases donotoverlap,butthegradientsintheproleseemtobecomp arable.Closetothetop boundarymeansedimentconcentrationisobservedtoincrea sewithincreaseinthe settlingvelocityofcoarsesediments.Thisbehavior,alth oughcounterintuitive,manifests fromtheinteractioncoarseandnesedimentparticles.Inc reaseinthesettlingvelocity ofcoarsesedimentparticlesresultsinsubstantialincrea seinthestraticationeffect, andtocounterthiseffectmoreamountofnesedimentsarere quiredtokeepthe turbulenceactive.Thereforetheincreaseinsedimentconc entrationatthetopboundary ispredominantlyduetothenesedimentsor r f .Forexample,incase6Cand6A r f are 0.469 and 0.04 ,respectivelywhichmeansthatvolumetricconcentrationo fcoarse sedimentatthetopboundary( ~ c ct )forcase6Cand6Ais 0.124 and 0.375 respectively. 151

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Thetop( ~ c ct + r f )andbottom( ~ c cb + r f )meanconcentrationvaluesofallthenearcriticalcasesarelistedinTable 5-1 .Noticethateventhoughthemeanconcentrationat thebedandatthetopboundarydonotmatch,thedifferencein theirvaluesisclosely maintained.ThisdifferenceshowsupintheTKEbudgetasapa rtoftheterm c (referto ( 5–31 )).Theentireformofthistermis Ri c = Re anditquantiesthebulkhydrodynamic interactionsofsedimentparticlesbetweenthemselvesand withthechannelboundaries. FromTKEbudgetweseethatthistermremainsnearlyconstant forallthenear-critical cases. Figure 5-4 (b)presentsthemeanstreamwisevelocity( ~ u )prolesforallthenearcriticalcases.Alltheprolesarerightontopofeachother showingextremelygood agreementdespitethedifferencesobservedinthemeanconc entrationproles.The subtlevariationsinthemeansedimentconcentrationwillc orrespondtovariationsin thebodyforceexertedbythesedimentsuspensioninthestre amwiseandbed-normal direction.Thefactthatthishasnegligibleeffectontheme anstreamwisevelocityproles isdenitelyintriguing.Table 5-1 listthebulkvelocity( ~ u b )andthemaximumstreamwise velocity( ~ u t )inthechannel.Observethat ~ u b forallthenear-criticalcases6A,6B,6C andevencase5fromShringarpure etal. [ 79 ]arecloselyboundaroundavalueof 18.41 .Similarly ~ u t ,maximumstreamwisemeanvelocityinthechannelisinveryg ood agreementforallthenear-criticalcases.Thereforeitcan beconcludedthatatthenearcriticalstraticationlimit( Re xed)themeanstreamwisevelocityproleisinsensitiveto thespeciccompositionofthesedimentsuspension. Figure 5-5 presentsthestreamwise,bed-normalandspanwiserootmean square (rms)velocityprolesofallthenear-criticalcases.Noti cethatalltheprolesare comparable,andtheagreementisespeciallygoodfor ~ u rms prolesofcases6A,6Band 6C.The ~ u rms proleofcase5matcheswellwiththeprolesofnear-critic alcasesclose tothebed.Slightoffsetinthe ~ u rms proleofcase5fromothercasesisseenawayfrom thebed.Incaseof ~ w rms allthefourprolesareverycloselyboundwhichsuggeststh at 152

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theextentofdampinginthebed-normaldirectionissimilar inallthefournear-critical cases.Figure 5-6 (a)presentstheReynoldsstressprolesofallthenear-cri ticalcases. Allthecasesarenearlyontopofeachotherwhichsuggeststh atthelocalReynolds stressisnegligiblyaffectedbythecompositionofthesusp ension.Thisobservation isespeciallytrueclosetothebed.Awayfromthebed,inthet ophalfofthechannel, Reynoldsstressprolesareslightlysensitivetowardsthe sizeofcoarsesedimentsin thesuspension.Thissensitivitymaybeduetothefactthatt hecases6A,6B,6Cand case5arenotthetruecriticalstatesbutonlyclosetoit.Fo rcasesthatmaylieeven closertothetruecriticalstate,itispossiblethatReynol dsstressproleswillbenearly ontopofeachothereverywhereinthechannel. Anotherimportantmeasureoftheowisitsmixingability.F igure 5-6 (b)presents theprolesoftheratioofReynoldsux( ~ w 0 ~ c 0 c )tosettlingux( ~ V cz ~ c c ).Thisratioisthe measureofmixingabilityoftheow;highertheratiobetter isthemixingability.Inthe upperhalfofthechannelweseethattheratioincreaseswith increaseinthesettling velocityofcoarsesedimentparticles.Thisapparentincre aseinthemixingabilityof theowispredominantlyduetothelackofcoarsesedimentsi ntheupperhalfofthe channel(see ~ c t intable 5-1 ).Thislackofsedimentsreducesthesettlingux(seegure 5-7 (a)).OntheotherhandReynoldsuxisdampedtoalesserexte nt.Thisincreases theratio ~ w 0 ~ c 0 c / ~ V cz ~ c c withincreaseinthesettlingvelocityofthecoarsesedimen ts.Inthe lowerhalfofthechannelitseemsthatthemixingabilityoft heowremainsnearlysame forallthecases.Figure 5-7 (a)showstheprolesofsettlinguxassociatedwithallthe nearcriticalcases.Clearlyinthelowerhalfofthechannel thesettlinguxincreases withincreaseinthesettlingvelocityofthecoarsesedimen ts.Thusinordertomaintain theratio ~ w 0 ~ c 0 c / ~ V cz ~ c c same,therehastobeaproportionalincreasein ~ w 0 ~ c 0 c associated withtheow.Figure 5-7 (b)showstheReynoldsux( ~ w 0 ~ c 0 c )prolesofallthenearcritical cases.Observethat ~ w 0 ~ c 0 c increasessubstantiallyfromcase6Atocase6Cnearthebed sothatthemixingabilityoftheowinthelowerhalfofthech annelremainssame. 153

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WeknowthatReynoldsuxisresponsibleforkeepingthesett lingsedimentsin suspensionandthereforeisanimportantquantitythatrepr esentsthemixingabilityof theow.Reynoldsuxistheresultofcorrelationbetweenu ctuationsinbed-normal velocity( ~ w 0 )andsedimentconcentration( ~ c 0 c ).Figure 5-5 (b)showsthat ~ w rms isinsensitive tothecompositionofthesuspensioninthelowerhalfofthec hannel,thatmeansthe modulations(increase)seenintheReynoldsuxprolesare duetotheconcentration uctuations.Figure 5-8 presentsthe ~ c crms prolesofallthenear-criticalcases.As expected ~ c crms dependsonthesettlingvelocityofcoarsesedimentsintheb i-disperse suspension.Thetrendsin ~ c crms prolesconformtothoseseenfor ~ w 0 ~ c 0 c .Closetothebed ~ c crms tendstoincreasewithincreaseinthesettlingvelocityofc oarsesediments,while awayfromthebedthetrendisexactlyopposite. WehaveseenthattheTKEbudgetofnear-criticalcases6A,6B ,6Cand5is insensitivetothecompositionofthesuspensiondrivingth eturbiditycurrent.Here wewillcomparethebed-normalprolesofTKEproduction,di ssipationanddamping fordifferentnear-criticalcases.Figure 5-9 presentstheTKEproduction( ~ P )andTKE dissipation( ~ )prolesofallthenear-criticalcases(referto( 5–28 )fortheformsof ~ P and ~ ).NoticethatalltheTKEproductionprolesarerightontop ofeachother.Thismeans thatTKEproductionassociatedwiththeowforallthenearc riticalcasesmatches locallyinsidethechannel.Animplicationofthisisthatth ebulkTKEproductionfor alltheabovecaseswillbesimilar.SimilarlyTKEdissipati onprolesareobservedto berightontopofeachother.Slightsensitivityinthedissi pationprolesisseenvery closetothebed,butthismaybeduetothefactthatthesecase sarenottruecritical states,merelynear-critical.Figure 5-10 showstheprolesofTKEdampingtermsfor allthenear-criticalcases(referto( 5–28 )fortheformofdampingterms).FromtheTKE budgetitwasobservedthatthebulkTKEdampingforcases6A, 6B,6Candcase5are close.Thismeanseventhoughthetotalenergyspentbytheo wtokeepsediments insuspensionissimilar,theTKEdampingprolesrevealtha ttheenergyspentlocally 154

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toretainthesedimentsdependsonthecompositionofthesus pension.Thisdifference inthelocalTKEdampingisbecausetheamountofenergyspent locallydependson thelocalsedimentconcentrationanditsgradient.Inother words,theenergyspent byTKEtokeepthesedimentsinsuspensiondirectlydependso nthesettlinguxand thestraticationeffectimposedbytheconcentrationgrad ient.Fromthesettlingux prolesandthemeanconcentrationprolesshowningures 5-7 5-4 (a)itisclearthat theenergyspentbytheowlocallywillbedifferentforthev ariousnear-criticalcases consideredhere.5.5.3TurbulenceStatisticsofFlowswithSimilarTurbulen ceDamping Previoussection x 5.5.2 presentedthestatisticallysteadyowstatisticsofvario us near-criticalcases.Themeanandotherturbulencestatist icsassociatedwiththe continuousphase(water)wereobservedtobeinsensitiveto thecompositionofthe sedimentsuspension.Thisobservationsuggeststhatinstr atiedturbulentturbidity currents,theextentofturbulencedampingisthesoleparam eterthatcontrolsthe statisticsofcontinuousphase.Inotherwords,wecanconje cturethattheparametric combination Ri ~ V e willcontroltheowpropertiesassociatedwiththecontinu ous phase.Hereforthesakeofbrevity,wewillrefertoalltheca sesinwhichbulkTKE dampingisfarlessthanthecriticallimitassub-criticalc ases. Considerthesub-criticalcases3A,3B,2Candcase2fromTab le 3-1 thathave turbulencedampingofaround 0.22 .CompletedetailsofthesecasesarelistedinTable 5-1 .Figure 5-11 (a)and(b)presentsthemeansedimentconcentrationandmea n streamwisevelocityoftheaboveselectedsub-criticalcas es.Asobservedinthenear criticalcases,themeansedimentconcentrationprolesar enotonthetopofeachother. Thedifferencesinthesettlingvelocityofcoarsesediment smanifestsasincreasein themeansedimentconcentrationandconcentrationgradien tnearthebed.Despite suchdifferencesinthemeansedimentconcentrationprole s,themeanstreamwise velocityprolesarenearlyontopofeachother.Otherprope rtiesassociatedwiththe 155

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owlikethebulkvelocityandthemaximummeanstreamwiseve locityshowverygood agreement(seetable 3-1 ). Figure 5-12 presentsthestreamwise,bed-normalandspanwisermsveloc ityof allthesub-criticalcases.Noticethatthermsvelocitypro leslieveryclosetoeach other.SimilarlyFigure 5-13 (a)presentsthecorrespondingReynoldsuxprolesof thesub-criticalcases.Wendallofthemtobeinverygoodag reement.Thisbehavior suggestsnegligibledependenceonthecompositionofthesu spensionaslongasthe turbulencedampinginthecasesiscomparable.Figure 5-13 showstheprolesofthe ratioofReynoldsux( ~ w 0 ~ c 0 )tosettlingux( ~ V cz ~ c c )forthesub-criticalcases.Thetrends seenintheseprolesareconsistentwiththoseseenfornear criticalcases(referto gure 5-6 (b)).Figure 5-14 presentstheTKEproductionandTKEdissipationprolesof thesub-criticalcases.Extremelygoodagreementisseenfo ralltheproles.Figure 5-15 showstheTKEdampingprolesofthesecases.Asexpected,be causeofthedifference inthesettlingvelocityofthecoarsesedimentstheenergys pentbyTKElocallyto retainthesedimentsinsuspensionisdifferent.Noticetha tthetrendsseenintheTKE dampingprolesasafunctionofsettlingvelocityofcoarse sedimentsiscomparableto thatobservedforthenearcriticalcases(refertogure 5-10 ).Fromthecomparisonof variousturbulencestatisticsandTKEbudget,itisapparen tthattheowpropertiesof thecontinuousphasewillbenegligiblyaffectedbythecomp ositionofthesuspensionif thebulkTKEdampingremainscomparable.SincebulkTKEdamp ingisquantiedby theparametricgrouping Ri ~ V e ,thisprovidesanelegantsimplicationforcurrentswith bidisperseorpolydispersesuspensionstocomparablemono dispersesuspension. 5.6CarryingCapacityofaTurbidityCurrent Sofartheanalysiswasfromtheperspectiveofstraticatio neffectssuppressing turbulenceintheow.Chapter 3 analyzedtheturbulencesuppressionprocessina turbiditycurrentdrivenbymono-dispersesuspensionsofs ediments.InChapter 4 a criteriatoquantifyturbulencesuppressioninmono-dispe rsecurrentswasproposed. 156

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Thischapterextendedthepreviousanalysisofmono-disper sesuspensionandshowed thattheturbulencesuppressioncriteriaholdsevenforcur rentsdrivenbybi-disperse sedimentsuspension.Furthermore,thischapteralsoshows thatthelocalbehaviorof owisonlyafunctionoftheparametricgrouping Ri ~ V cz r c (seeFigures 5-4 5-15 ).As aconsequenceoftheaboveobservations,theturbulencesup pressioncriteriacanbe interpretedasacarryingcapacitycriteriaforthecurrent .Inotherwords,foragiven owintensity( Re xed)andsizeofcoarsesediments j ~ V c j thereisalwaysamaximum amountofcoarsesedimentsthattheowcancarry(keepinsus pensionor r c max ).From ( 5–32 ), r c max canbewriteas r c max = 0.041ln( Re )+0.11 Ri ~ V cz (5–33) Notethattheaboverelationisbecausethenesedimentpart icleshavenegligible settlingvelocity.Whenthenesedimentshavenitesettli ngvelocitytheaboveequation willbemodiedasshownhere r c max = 0.041ln( Re )+0.11 Ri r f ~ V fz Ri ~ V cz (5–34) Thisisausefulinterpretationofturbulencesuppressionc riteriaandcoupledwiththe Reynoldsnumberscalingitcanbeusedtoestimatethesedime nttransportcapacityof currentsateldscale.5.6.1ExperimentalEvidence Inthissectionwecontrastourpredictionofturbulencesup pressionwiththeexperimentalobservationofGarcia&Parker[ 30 ].TheexperimentsinGarcia&Parker[ 30 ] consistedofasalinedensitycurrentthatoverrananerodib lebed,resuspendingsedimentandthusresultingina(mixeddensity-)turbiditycurr enttransportingsuspended sediment.Garcia&Parker[ 30 ]reportedthattheresultingturbiditycurrenttransporte d suspendedsedimentwithsettlingvelocitiesashighas j ~ V c j 0.3 forsedimentparticles ofsize 180 m (seeFigure11in[ 30 ]).Theexperimentswereperformedinachannel 157

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withaslopeof S =0.08 resultingin Ri =1 = S =12.5 ,andunderconditionssuch that u 1.3 cm/s(seeTable2inGarcia&Parker[ 30 ]).Theseexperimentswereat Re 400 .NotethatforthisReynoldsnumberestimation, h 3 cm.Thisvalueof h representsapproximatelythemaximumheightabovethebedw heresuspendedsedimentconcentrationdropstozero(seeFigure7inGarcia&Par ker[ 30 ]).Thisdenition of h isusedbecauseitrelateswelltothemonoandbi-dispersemo delpresentedhere. Clearlythepredictionforthelimiting j ~ V j accordingtothemono-dispersecriteriagiven inChapter 4 x 4.4 and x 4.5 forexperimentsinGarcia&Parker[ 30 ]doesnotmatchwell. Formono-dispersesuspensionscritical j ~ V is 0.028 at Re =400 whilethedensity currentwasabletocarrysedimentparticleshavingsettlin gvelocityashighas 0.3 .The bi-dispersemodelpresentedinthischapterhastobeusedto reconciletheprediction madebythemono-dispersecriteriawiththeexperimentalob servationsinGarcia& Parker[ 30 ]. Thebi-dispersemodelisquiteamenabletotheexperimentsd onebyGarcia& Parker[ 30 ].Thenesedimentparticlesinthebi-dispersemodelhaven egligiblesettling velocityandwillcorrespondtothepropertiesofdissolved solutethatdrivesthedensity currentintheexperimentsofGarcia&Parker[ 30 ].Furthermore,thecoarsesediments inthebi-dispersemodelwilltakethepropertiesofthesedi mentparticleslyingonthe bedinGarcia&Parker[ 30 ].Toapplythecarryingcapacityorturbulencesuppression criteriatotheexperimentsinGarcia&Parker[ 30 ],weneedanestimateoftheratioof volumetricconcentrationofdissolvedsolute( C ( v ) f )tothesuspendedsedimentparticles ( C ( v ) c ).Intermsofthebi-dispersemodel,weneedtodetermine r c =r f appropriatefor experimentsinGarcia&Parker[ 30 ].Thisestimatecanbeobtainedfromthenetdriving forcetermasshownhere Bodyforceduetocoarsesediment Bodyforceduetonesediment = R c C ( v ) c R f C ( v ) f j model = RC ( v ) c o j exp (5–35) 158

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Assuming R c R f inthebi-dispersemodelgives r c r f j model = RC ( c ) c o (5–36) where o =( so w ) = w so isthedensityofthesalineunderow, w isthedensityof theclearwater, R isthemeasureofspecicgravityofsedimentparticles R = SG 1 and C ( v ) c isthevolumetricconcentrationofsedimentparticles.The sequantitiesare estimatedfromtheexperimentalresultsgiveninGarcia&Pa rker[ 30 ]forsediment particlesofsize 180 m o 0.020 (seecasesB,CandDinTable1inGarcia&Parker [ 30 ]), R =0.35 (coalparticles,thevalueofSGisgiveninGarcia&Parker[ 30 ])and C ( v ) c =0.003 (approximatevalueobtainedfromFigure7thatcorresponds tovolumetric concentrationat400cmdownstreamlocation).Fromtheseva lues r c =r f isestimatedto be r c r f 0.06 and r c 0.057 (5–37) Thereforethelimiting ~ V z fortheexperimentsinGarcia&Parker[ 30 ],accordingtoour bidispersecriteriais ~ V z h 0.041ln ( Re ) +0.11 i = ( r c Ri ) 0.5 .[ 30 ]showssediments insuspensionwith j ~ V c j 0.3 ,andthusthebidispersepredictionisquitegood.Afurther interpretationofthisresultsisthat[ 30 ]turbiditycurrentswasnotatcapacityandcould havecarriedevenalargerfractionofsediments. Carryingcapacitycriteriaasapredictiontool. Theabovecomparisonestablishesthatturbiditycurrentstransportinglargesizesed imentparticlescanbepredicted usingasimplebi-dispersemodel.Infactthemainadvantage ofthismodelisthatit canpredicttheamountoflargesizesedimentparticlescarr iedbythecurrent.Followingsectiondescribeshowthecarryingcapacitycriteriaca nbeusedtoestimatethe amountofcoarsesedimentparcelscarriedbythecurrent.Co nsiderthataturbidity currentdrivenbymono-dispersesuspensionofnesediment ( j ~ V f j 0 )isowingon topofabedwithaconstantslope.Letthecurrentencountera nerodibleportionofthe bedcontainingcoarsemono-dispersesedimentparticles.A sthecurrentpropagates 159

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overthebeditstartsentrainingsedimentparticlesintoth eowanditseffectivedensity increases.Thiswillleadtoareenforcingmechanismthatco ntinuestoentrainparticles untilthecarryingcapacityofthecurrentisreached.Using thebi-dispersemodelwecan predicttheamountofsedimentparticlesthatneedstobeent rainedtoreachthecarrying capacity.Wecanwritethefollowingrelationfortheveloci tyscalefromthebi-disperse modeldescribedabove u 2 = RHC ( v ) f g x + RHC ( v ) c g x (5–38) anddecomposetheReynoldsnumberoftheowintocontributi onduetoneand coarsesedimentsas Re 2 = Re 2 f + Re 2 c = Re 2 f 1 r c (5–39) Pluggingthisrelationinscalingrelationgivenin( 5–32 )gives Ri ~ V cz r c =0.041ln( Re f ) 0.041 2 ln(1 r c )+0.11. (5–40) Replacingthesecondnaturallogtermbyseriesexpansionwe getthefollowing Ri ~ V cz r c =0.041ln( Re f )+ 0.041 2 1 X n =1 r n c n +0.11. (5–41) Thisequationissolvediterativelyuntilthesolutionconv ergestoavaluefor r c 5.7Deductions Inthischapteramathematicalmodelforcontinuousturbidi tycurrentsdrivenby bi-dispersesuspensionofsedimentisproposed.Inthemode l,nesedimentparticles havenegligiblesettlingvelocityandhencetheyimposenos traticationeffect.They areonlyresponsiblefordrivingtheowinthestreamwisedi rectionbyimposingexcess density(uniformbodyforce).Ontheotherhand,coarsesedi mentparticleshavealarge settlingvelocityandbythemselvesimposestrongturbulen cedampingeffectonthe ow.Thereforetheasymptoticstateoftheowdependsonthe combinedeffectofthese twosediments.Ifthenetturbulencedampingeffectduetola rgesedimentparticles dominatesthentheowwillundergocompleteturbulencesup pressionandloseits 160

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abilitytoretainsettlingsediments.Throughsimulations itisevidentthatforacertainsize ofcoarsesedimentparticlesthereisaminimumquantityof nesedimentsneededto keeptheowalivei.e.tokeepturbulenceactive.Thisisthe underlyingmechanismby whichrealturbiditycurrentsderivetheirabilitytotrans portlarge/heavysedimentsfor longdistances. InChapter 3 theprocessofturbulencesuppressioninturbiditycurrent sdrivenby mono-dispersesuspensionofsedimentwasidentied.Inthe followingchapter(Chapter 4 )itwasshownthatturbulencesuppressionprocessorturbul encedampingcanbe quantiedbytheparametricgrouping Ri ~ V z .Thisparametricgroupingrepresentsthe amountofTKEspentbytheowtokeepthesettlingsedimentpa rticlesinsuspension. Inthischapter,themono-dispersemodelwasextendedtoinc ludebi-dispersesuspensions.Usingthismodelvarioussimulationswerecarriedou tunderthreesets(A,Band C)toanalyzetheeffectofbi-dispersityonturbulencesupp ression.Ineachsimulation set,thesettlingvelocityofcoarsesedimentparticleswas xedalongwith Re =180 Sc =1 and Ri =11.4 andtheproportionofneandcoarsesedimentparticlesalte red todeterminethecriticalcompositionthatbringsabouttot alturbulencesuppression. SetA,BandCcorrespondtocoarsesedimentparticleswithse ttlingvelocity 0.0275 0.035 and 0.05 ,respectively.Theminimumamountofnesedimentsnecessa ryto keepturbulenceactiveinsetA,BandCwasobservedtobe r f =0.04,0.24,0.469 respectively.Furthermore,fromtheTKEbudgetofallthene ar-criticalcases(6A,6Band 6C)completeturbulencesuppressionwasobservedtooccurw henbulkTKEdamping reachesavalueof0.30.Thisobservationagreeswellwithth ecriticalbulkTKEdamping forcurrentswithmonodispersesuspensionwhichwasobserv edtobearound 0.294 in Chapter 3 .Basedonthisobservationtheuniversalturbulencesuppre ssioncriteriaproposedinChapter 4 wasextendedtoincludebi-dispersesuspensions.Theparam etric groupingthatquantiesturbulencedampinginbi-disperse suspensiontakestheform Ri r c ~ V cz 161

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InthisChapter,thelocalbehavioroftheowdrivenbydiffe rentbi-dispersesuspensionsbutwithnearlysamebulkTKEdampingorvalueforparam etricgrouping Ri r c ~ V cz werecompared.Theprolesofmeansedimentconcentrationa ndotherstatisticsrelatedtosedimentconcentrationwereobservedtobeslightl ydifferent.Thesedifferences weremanifestationsofthesettlingvelocityofcoarsesedi mentparticles.Thismeantthat eventhoughtheowgloballyspentsimilaramountofenergyi nkeepingthesediment particlesinsuspensions,thelocalstatisticalbehavioro fpropertiesrelatedtosediment concentrationweredependentonthesettlingvelocity.How ever,surprisingly,theproles ofturbulencestatisticsofthecontinuousphase(water)li kethemeanstreamwisevelocity,rmsvelocity,Reynoldsstress,etc.,werenearlyontop ofeachother.Fromthiswe canconcludethatturbulencedampingisaverysubtleproces s.Itdoesnotdrastically alterthestatisticalbehavioroftheowbutmaychangethes tructureofturbulencedrastically.Thefactthatthestatisticalprolesshowanabrup tshutdownofturbulenceisan indicationofthis. Anotherimportantinsightgainedthroughthesesimulation sistheinterpretation ofcompleteturbulencesuppressionassedimentcarryingca pacityofthecurrent. Thefactthataminimumamountofnesedimentparticlesaren ecessarytotransport coarsesedimentparticleshasdirectimplicationonthesed imentcarryingcapacityof thecurrent.Thepreviousstatementcanbeeasilyrephrased tohighlightthesediment carryingcapacityofthecurrent.Itcanbesaidthatforagiv enamountofnesediment particlesinthecurrent,thecurrentwillbealwayslimited byamaximumamountof coarsesedimentparticlesthatitcansupportinsuspension .Thisinterpretationofthe bi-dispersemodelwasusedtoanalyzetheexperimentalresu ltsofGarcia&Parker [ 30 ].Fromsimpleestimatesoftheexperimentaldata,bi-dispe rsemodelpredictions reconcilesverywellwiththeexperiments.Inotherwordsth eerosionofsediment particlesofsize 180 m andthecorrespondingconcentrationlevelsattainedinthe currentwerewithinthesedimentcarryingcapacitypredict ionsofthebi-dispersemodel. 162

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Sincetheexperimentswerenotintendedtomeasurethecarry ingcapacityofthe current,detailedcomparisonofthebi-dispersemodelcoul dnotbedone. 163

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Table5-1:Listofsimulations: r f referstothevolumefractionofnesedimentsinthe mixtureand r c =1 r f isthevolumefractionofcoarsesediments. ~ u b isthebulkstreamwisevelocity, ~ u t isthemaximummeanstreamwisevelocity, ~ c cb + r f and ~ c ct + r f are thetotalconcentrationofsedimentsatthebedandatthetop boundaryofthechannel, respectively.TheabbreviationCTSstandsforcompletetur bulencesuppression.Also presentedinthetablearetwocases,case2and5,fromTable 3-1 case j ~ V c j r f r c ~ u b ~ u t ~ c cb + r f ~ c ct + r f State 1A0.02750.750.2515.9118.961.1670.90Turbulent2A0.02750.500.5016.5920.071.3710.776Turbulent3A0.02750.300.7017.1921.071.5720.644Turbulent4A0.02750.1250.87517.8822.151.7990.499Turbulent5A0.02750.060.9418.2322.651.9110.437Turbulent6A0.02750.040.9618.4122.911.9510.415Turbulent 1B0.0350.750.2516.0619.221.2320.881Turbulent2B0.0350.500.5016.8820.601.5400.717Turbulent3B0.0350.450.5517.1921.081.6090.676Turbulent4B0.0350.300.7017.7822.281.8800.546Turbulent5B0.0350.260.7418.3522.761.9880.507Turbulent6B0.0350.240.7618.4422.902.0270.4877Turbulent7B0.0350.2350.76530.6339.965.0360.244CTS 1C0.050.750.2516.4819.841.4020.8422Turbulent2C0.050.600.4017.3221.181.7410.720Turbulent3C0.050.550.4517.7521.771.8950.673Turbulent4C0.050.500.5018.1522.402.0650.622Turbulent5C0.050.4750.52518.4422.812.1680.597Turbulent6C0.050.4690.53118.5122.902.1980.593Turbulent7C0.050.460.5435.3849.335.3020.466CTS 20.020.01.017.0921.051.5540.578Turbulent50.0260.01.018.3322.771.930.42Turbulent 164

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Table5-2:Turbulentkineticenergybudgetforallthecriti calcasesfromthepresent studyandthecriticalcase(case5)fromTable 3-1 case j ~ V c j r f ~ P ~ E c Ri r c V cz Ri c = Re r c V cz 6A0.02750.046.5346.1790.1130.3020.09780.02646B0.0350.246.5986.2370.1170.3040.09770.0266 6C0.050.4696.5166.1620.1130.3030.10190.0266 50.0261.06.5906.2300.1170.2960.09580.0260 Table5-3:Turbulentkineticenergybudgetofsub-critical casesfromthepresentstudy andtheintermediatecase(case2)fromTable 3-1 case j ~ V c j r f ~ P ~ E c Ri r c V cz Ri c = Re r c V cz 3A0.02750.306.5766.3020.01400.2190.06890.01923B0.0350.456.5806.3040.01400.2190.05920.0192 2C0.050.606.5836.2960.01430.2280.06480.0199 20.021.06.5446.2650.01450.2280.06200.0199 c c + g f z 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 g f =1.0 case1Bcase3Bcase6Bcase7BLaminar ~~(a) Solution u z 0 10 20 30 40 0 0.2 0.4 0.6 0.8 1 ~~(b) Figure5-1:(a)Meanconcentrationand(b)meanstreamwisev elocityprolesofcases fromsetB.RefertoTable 5-1 fordetails. 165

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u'w' z 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 g f =1.0 case1Bcase3Bcase6Bcase7B (a) ~~~ w'c'/V cz c z 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ~ ~~~ c (b)~ Figure5-2:(a)Reynoldsstressand(b)Reynoldsuxproles ofsomeofthecasesfrom setB.RefertoTable 5-1 fordetails. 166

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u rms z 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 g f =1.0 case1Bcase3Bcase6Bcase7B (a)~~ w rms z 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (b) ~~ v rms z 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 (c) ~~ Figure5-3:(a) ~ u rms ,(b) ~ w rms and(c) ~ v rms prolesofsomeofthecasesfromsetB.Refer toTable 5-1 fordetails. 167

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u z 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 ~~(b) c c + g f z 0.4 0.8 1.2 1.6 2 0 0.2 0.4 0.6 0.8 1 case6Acase6Bcase6Ccase5 ~~(a) Figure5-4:(a)Meansedimentconcentration( ~ c c + r f )and(b)meanstreamwisevelocity( ~ u )prolesofcriticalcasesfromeachsetandthecriticalcas eofmono-disperse simulationsfromTable 3-1 .RefertoTable 5-1 forthedetails. 168

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u rms z 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 case6Acase6Bcase6Ccase5 ~~(a) w rms z 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ~~(b) v rms z 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 ~~(c) Figure5-5:(a) ~ u rms ,(b) ~ w rms and(c) ~ v rms prolesofcriticalcasesfromeachsetandthe criticalcaseofmono-dispersesimulationsfromTable 3-1 .RefertoTable 5-1 forthe details. 169

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u'w' z 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 case6Acase6Bcase6Ccase5 ~ ~~(a) w'c'/V cz c c z 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ~~(b) ~~ ~ c Figure5-6:(a)Reynoldsstressand(b)ratioofReynoldsux tosettlinguxprolesof criticalcasesfromeachsetandthecriticalcaseofmono-di spersesimulationsfrom Table 3-1 .RefertoTable 5-1 forthedetails. 170

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V cz c c z 0 0.02 0.04 0.06 0.08 0.1 0 0.2 0.4 0.6 0.8 1 case6Acase6Bcase6Ccase5 ~~~ (a) w'c' z 0.03 0.02 0.01 0 0 0.2 0.4 0.6 0.8 1 ~~ ~ (b) Figure5-7:(a)Settlingux ~ V cz ~ c c and(b)Reynoldsux ~ w 0 ~ c 0 prolesofcriticalcases fromeachsetandthecriticalcaseofmono-dispersesimulat ionsfromTable 3-1 .Refer toTable 5-1 forthedetails. 171

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c crms z 0.1 0.15 0.2 0.25 0.3 0 0.2 0.4 0.6 0.8 1 case6Acase6Bcase6Ccase5 ~~ Figure5-8: ~ c crms prolesofnear-criticalcasesfromeachsetandthecritica lcaseof mono-dispersesimulationsfromTable 3-1 .RefertoTable 5-1 forthedetails. 172

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P z 0 10 20 30 40 0 0.2 0.4 0.6 0.8 1 case6Acase6Bcase6Ccase5 ~(a) ~ e z 20 10 0 0.2 0.4 0.6 0.8 1~(b) ~ Figure5-9:Turbulentkineticenergy(a)Productionand(b) Dissipationprolesofnearcriticalcasesfromeachsetandthecriticalcaseofmono-di spersesimulationsfrom Table 3-1 .RefertoTable 5-1 forthedetails. 173

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u'c'+Ri t w'c' z 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 case6Acase6Bcase6Ccase5 ~~~ ~~ c c Figure5-10:Prolesofturbulentkineticenergydampingof criticalcasesfromeachset andthecriticalcaseofmono-dispersesimulationsfromTab le 3-1 .RefertoTable 5-1 for thedetails. 174

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u z 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 ~ (b)~ c c + g f z 0.4 0.8 1.2 1.6 2 0 0.2 0.4 0.6 0.8 1 case3Acase3Bcase2Ccase2 ~~ (a) Figure5-11:(a)Meansedimentconcentration( ~ c c + r f )and(b)meanstreamwisevelocity( ~ u )prolesofcasesfromeachsetwhichhavesimilarturbulenc edampingeffects andthecorrespondingcaseofmono-dispersesimulationsfr omTable 3-1 .Turbulence dampingisgivenbytheparametricgrouping Ri r c ~ V cz andthevalueofthisparametric groupingfortheabovecasesis 0.22 .RefertoTables 5-1 5-3 forthedetails. 175

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u rms z 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 case3Acase3Bcase2Ccase2 ~~(a) w rms z 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ~~(b) v rms z 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 ~~(c) Figure5-12:(a) ~ u rms ,(b) ~ w rms and(c) ~ v rms prolesofcasesfromeachsetwhichhave similarturbulencedampingeffectsandthecorrespondingc aseofmono-dispersesimulationfromTable 3-1 .Turbulencedampingisgivenbytheparametricgrouping Ri r c ~ V cz andthevalueofthisparametricgroupingfortheabovecases is 0.22 .RefertoTables 5-1 5-3 forthedetails. 176

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u'w' z 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 case3Acase3Bcase2Ccase2 ~ ~~(a) w'c'/V cz c c z 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ~ ~~(b) c ~ ~ Figure5-13:(a)Reynoldsstressand(b)ratioofReynoldsu xtosettlinguxprolesof casesfromeachsetwhichhavesimilarturbulencedampingef fectsandthecorrespondingcaseofmono-dispersesimulationfromTable 3-1 .Turbulencedampingisgivenby theparametricgrouping Ri r c ~ V cz andthevalueofthisparametricgroupingfortheabove casesis 0.22 .RefertoTable 5-1 5-3 forthedetails. 177

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P z 0 10 20 30 40 0 0.2 0.4 0.6 0.8 1 case3Acase3Bcase2Ccase2 ~~(a) e z 30 20 10 0 0 0.2 0.4 0.6 0.8 1 ~~(b) Figure5-14:Turbulentkineticenergy(a)Productionand(b )Dissipationprolesof casesfromeachsetwhichhavesimilarturbulencedampingef fectsandthecorrespondingcaseofmono-dispersesimulationfromTable 3-1 .Turbulencedampingisgivenby theparametricgrouping Ri r c ~ V cz andthevalueofthisparametricgroupingfortheabove casesis 0.22 .RefertoTable 5-1 5-3 forthedetails. 178

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u'c'+Ri t w'c' z 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 1 case3Acase3Bcase2Ccase2 ~~~~~ Figure5-15:Prolesofturbulentkineticenergydampingfo rcasesfromeachsetwhich havesimilarturbulencedampingeffectsandthecorrespond ingcaseofmono-disperse simulationfromTable 3-1 .Turbulencedampingisgivenbytheparametricgrouping Ri r c ~ V cz andthevalueofthisparametricgroupingfortheabovecases is 0.22 .Referto Table 5-1 5-3 forthedetails. 179

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CHAPTER6 HYBRIDOPENMP-MPIPSEUDO-SPECTRALCODE:VALIDATION&OPTI MIZATION Allthesimulationresultspresentedsofarwereobtainedfr omapseudo-spectral code(completedetailsofthealgorithmcanbefoundinCorte se&Balachandar[ 26 ]) implementedforsharedmemoryarchitectures.Insharedmem oryarchitectures(SMA) multipleprocessors(CPUs)havedirect(logical)accessto alargeglobalmemorybank. WithoutgettingintothedetailsofvariousSMA,inverysimp listicterms,itisclearthat thesucharchitectureswillhaveadvantagesfromthepointo fviewofdatasharingand communicationbetweentasks.Furthermore,programmingfo rsucharchitecturesis easyastheentirememoryislogicallyaccessibletoallthep rocessors.However,such architectureslackscalabilitybetweenmemoryandCPU.The communicationtrafc onsharedmemory-processor(CPU)pathincreasesgeometric allyasmoreCPUsare added.Furthermore,sincetheentirememoryislogicallyav ailabletoalltheprocessors, changesinthememoryaffectedbyoneprocessorsarevisible toallotherprocessors (onSMAswherecachecoherencyisimplemented).Thisimpose srestrictiononthe programmingaspectandsynchronizationconstructshaveto beincorporatedtoensure integrityofthedata.AnotherissuewithSMAsisthattheybe comeextremelyexpensive todesignandproduceforlargenumberofCPUs. Ontheotherhand,therearedistributedmemoryarchitectur esorsystemswhere individualprocessors(CPUs)donothavedirectaccesstoal lthememoryavailablein thesystem.Eachprocessorhasitsownlocalmemoryandthere isnocrossmapping betweenthememoryaddressesofdifferentprocessors.Asac onsequencethereis nocachecoherencyinthesystem.Inordertoaccessdatathat liesonthememory thatislocaltosomeotherprocessors,theprogrammerhasto explicitlyimplement themethodandtimeofcommunicationforthisdatatransfer. Herethecommunication methodologyandprotocoldependsonthearchitectureofthe processorsandthe networkfabric.Becauseofthedistributednature,memoryi nthesystemlinearlyscales 180

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withnumberofprocessors.Infact,withoff-the-shelfproc essorsandnetworkhardware, veryconvenientlyandinacosteffectiveway,extremelymas sivedistributedsystems canbeassembled.IncomparisonwiththeSMAs,distributeds ystemssaveonthe overheadsthatSMAsincurduetointerferenceinthememorya ccessbymultiple processorsandensuringcachecoherencyinthesystem.Buto ntheotherhand, thecommunicationoverthenetworkinadistributedsystemi smuchslowerthanthe localmemoryaccessinaSMA.Insummary,eventhoughdistrib utedsystemsprovide greatexibilityandscalability,theprogrammingnolonge rremainstrivial.Toperform simulationsondistributedmemoryarchitectures,theprog rammerhastoexplicitly implementallthecommunicationbetweentheprocessorsand makesurethatthe algorithmremainsscalableandefcient. Today,thelargestcomputersemployahybridarchitecture, i.e.,theyhavemany smallSMAsconnectedoverthenetwork.Thisleveragesthead vantagesofboth systemsandprovidesamassiveparallelizationcapability .Howdoesthishelpusin preformingDNSofhigherReynoldsnumberows?Andwhatcapa bilitiesdoweneed toperformsuchstate-of-the-artsimulations?Basedonver yroughestimates,tocarry outapureturbulentchannelowsimulationat Re =4000 willrequireatleast O (10 10 ) gridpoints.Toreachstatisticallysteadystate,thenumbe rofdataaccesscallsand numberofoperationstobeperformedareenormous.Suchasim ulationwilltrulyneed supercomputerswithpetascalerating.SMAarchitecturesd onotscaletosuchlevels andhencemostofthepresentpetascalesystemsarebasedona hybridarchitecture. Theimplicationofthisis,wewillneedtodistributetheent irecomputationaldomain intopiecessothateachpieceliesonthelocalmemoryofvari ousnodesonthesystem (entiresystemismadeupofnodesconnectedbyanetworkfabr icandeachnodecan bethoughttobeaSMA).Inordertomarchintime,eachnodewil lsolveitspieceof thedomainwhichmayrequireaccessinginformationthatlie sonothernodesonthe network.Thecomputercodewillbeimplementedsuchthatthi sinformationisfetched 181

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asandwhenneededsothattimemarchingalgorithmcancontin ue.Suchtypeof computingparadigmiscalledSingleProgramMultipleData, i.e.,thesameprogramruns oneachnodebutitishandlingdifferentdata. OurcollaboratorDr.JustinR.Davisfromthedepartmentofc ivilandcoastal engineeringhasimplementedanewhybridshared/distribut edmemoryparallelcode forfullyturbulentoscillatoryows.ThiscodeusesOpenMP andMPIdirectivesto implementaparallelized,highlyaccuratepseudo-spectra lscheme.Dataisstoredlocally andtransferredbetweencomputationalnodessuchthatcomp utationslikeFFTs,x,y andzderivativesarecalculatedcompletelyin-processor. Thecodeiscongurableat compiletimetobesetupforvariousmodesofoperation(seri al,OpenMP,MPIand OpenMP+MPI),differentFFTlibraries(DFTIandFFTW3),per sistentandnon-persistent MPIcommunication,etc.Thecodeisalsocapableofevaluati ngvariouscomputational methodsatruntimetopickthemostoptimumalgorithmsuitab leforthegivendomain sizeandcomputerresources. Followingisabriefdescriptionoftheoriginalversionoft hecode.Thede-aliased pseudospectralschemeemployslowstorageR-K3methodfort imeintegration,Fourier expansionsin x and y directionsandChebyshevpolynomialsin z direction.Itusesprojectionmethodtogetintermediatevelocitybysolvingthea dvection-diffusionequation (orHelmholtzequation)andthenenforcesincompressibili tyviathepressurecorrection step(Poisson'spressureequation).Apartfromthe 3 componentsofvelocity,there isanadditionalscalarequationfortheconcentrationeld thatneedstobesolved. Inthecode,thetimeintegrationstepisaccomplishedbygen ericmodularstepsas showninFigure 6-1 .Eachblockrepresentsagenericsubroutinethatperformss pecic tasks.Firsttwoblockscomputetheadvectionanddiffusion termsandstorethemin thecomputationalspacei.e.( z kx ky )space.Notethat( z kx ky )spaceimpliesthat x and y directionsaretransformedintowavenumberspace.Assembl eRHSblockin theowchartrepresentsgroupofscalingandadditionopera tionsthataredoneon 182

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theadvectionanddiffusiontermstogettheresidueorRHSfo rthetimeintegration step.TheHelmholtzsolverblockisresponsibleforadvanci ngtheowvariableslike velocityorthescalareldintimetothenextRK3stage.Atth eendoftheRK3stage pressurecorrectionstep(Poissonsolverblock)isimpleme ntedandtheintermediate velocityeldobtainedaftertheHelmholtzsolveriscorrec ted(velocitycorrectionblock). Figure 6-1 alsopresentsbriefsummaryofoperationscarriedoutbyvar iousblocksin thealgorithm.Thisgivesanat-a-glancerepresentationof theextentofcomputations andcommunicationsinvolvedinthescheme.Noticethatallt hecommunicationislimited totheadvectionblock.Aftertheadvectionblock,thereisa bsoluteparallelizationofthe scheme.Thismeansthattheindividualprocessescanindepe ndentlycontinuecomputingthediffusionterms,HelmholtzsolverandthePoisso nsolverontheirlocalpiece ofdata.Thisisanimportantfeatureofthecodeandthebasis ofthislieswiththeway data/domainisdistributedontheprocesses. HereIwillbrieysummarizethestrategyemployedinthecod etodistributedata ontheprocessesandtransposeitontheywhenrequiredbyth ealgorithm.Inallthe transposeoperationsanddistributionalgorithmsdescrib edhere,thereisoneimportant featureincorporatedinthem.Thisassumptionstatesthatf oragivenorientationof data,therstindexisalwayskeptlocalor”inprocessor”an dtheothertwoindicesare distributed.Furthermore,computationsaresetupsuchtha ttheyare”inprocessor”.As aconsequenceofthisstrategy,whenevercomputationsarer equiredtobedoneon 2 nd or 3 rd index,ontheytransposeofdatasetisnecessary.Thesetra nsposeoperations willrequirecommunicationoverdistributedsystemtoreor ientthedataset.Figure 6-2 showstheowchartofdatatransposeoperationsimplemente dinthecode.There arethreeorientationsinwhichdataexist:Realspace( y z x ),maincomputational space( z kx ky )andintermediatespace( x z ky ).Forwardroutewilltakethedata fromrealspacetomaincomputationalspaceviatheintermed iatespace.Thiswill involve 2 forwardFFTsand 2 communicationcallsthatimplementtransposeoperations 183

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(seetheFigure 6-2 fortheorderinwhichtheseoperationsaredone).Similarly ,in thereverseroute,datainthemaincomputationalspaceistr ansformedbacktoreal space.Evenherethereare 2 backwardFFTsand 2 communicationcallsthatimplement transposeoperations.Atthispointitisfairtoraisethequ estion,whydoweneedsuch amechanism?Inthealgorithm,itisconvenienttocomputeth enonlineartermsinN-S equationsorinthescalarconcentrationequationinrealsp ace,whilethetimeintegration methodalongwithallothercomputationsareefcientwhend oneinthecomputational space.Thereforeitisimportantthatwehavetransposemech anismsthatcantransform datafromrealspacetocomputationalspaceandviceversa. 6.1ModifyingGoverningEquationsandRelatedParameters TheoriginalversionofthecodedevelopedbyDr.JustinR.Da viswastostudythe effectofsedimentsuspensioninoscillatingboundarylaye rows(seeOzdemir etal. [ 60 ]forthecompletesetofgoverningequations).Heretheowi sexternallydrivenby anoscillatingpressuregradient,whilethesuspendedsedi mentssettledownunderthe inuenceofgravitytostratifytheow.Theproblemofprese ntinterestdiffersfromthe oscillatoryowsetup.Firstofall,thechannelisinclined withrespecttohorizontaland thereisnostreamwisepressuregradienttodrivetheow.Th esuspendedsediments intheinclinedchannelareresponsiblefordrivingtheowi nthestreamwisedirection andatthesametimetheytendtosettledowntostratifytheo weldinthebednormal direction.Thedifferencesinthetwoproblemstatementsma nifestthroughthedenition ofthedimensionlessparametersthatcontroltheow.Follo wingchangesweremadeto ensurethattherequiredgoverningequationsweresolved. Makethechannelinclined. Tomakethechannelinclined,asimpletrickisemployed.Thereferenceframeisattachedandalignedwiththe bottomwallofthechannel. Thismeansthatforaninclinedchannel,thegravityvectorh astwocomponents; g y in thestreamiwsedirection(notethat' y 'isthestreamwisedirectionand' x 'isthespanwise 184

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direction)and g z inthebed(bottomwall)normaldirection.Inthecodethecha nnelinclinationisfedasaninputatruntimeandthatsetsupthecorrec tcomponentsofgravityto computethebodyforcesinstreamwiseandbednormaldirecti onmomentumequations. InterpretationofRichardsonnumber. Richardsonnumberistheratioofthe effectofbuoyancyinthebednormaldirectiontotheinertia oftheow.Inotherwords, Richardsonnumberisameasureofstraticationintheow.I noscillatingpressuregradientdrivenboundarylayerows,Richardsonnumberisapro xyforthetotalvolumetric concentrationofsedimentsintheow.Whileinthepresentp roblem,astheowisboth drivenandstratiedbysediments,Richardsonnumberisame asureofinclinationofthe channelwithrespecttothehorizontal.Inotherwords,Rich ardsonnumberrepresents howstrongisthebuoyancyeffectinbednormaldirectiontot hetotalvolumetricbody forcecausedbysedimentsuspensioninthestreamwisedirec tion.Suitablechanges aremadesothatbothinterpretationsoftheRichardsonnumb ercanbespeciedinthe code. Normalizedtotalvolumetricsedimentconcentration. Inthepresentproblem denition,totalvolumetricconcentrationofsedimentsis notanindependentparameter. ItisindirectlyspeciedbytheReynoldsnumberoftheow(s eeChapter 3 x 3.2 formore details).Furthermore,inthegoverningequations,concen trationofsedimentsisnormalizedbytotalvolumetricconcentrationofsediments.He nce,anyinitialconditionfor concentrationeldhastobenormalizedbythetotalvolumet ricsedimentconcentration beforestartofthesimulations.Thisfeaturehasnowbeenim plementedinthecode. Thesechangesensurethattheappropriategoverningequati onsaresolvedinthe code.Thenextstepistovalidatetheimplementation. 6.2Validation 6.2.1RecoverLaminarSolution Oneofthemostbasicandpreliminaryvalidationstepistosi mulatelaminarchannel ow.Hereweconsiderowbetweentwoparallelwallsorplate s(seeFigure 3-1 )driven 185

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byaconstantpressuregradient.Inthistestcase Re =180 anddimensionlesschannel heightis1.Thelengthofthechannelinstreamwiseandspanw isedirectionis 4 and 4 = 3 andthedimensionlesspressuregradientissetto1.Periodi cboundaryconditions areemployesinstreamwiseandspanwisedirections.Bottom wallisnoslipwhilethetop wallisno-stressboundary.Thelaminarsolutionforthiso wis ~ u = Re 2 ~ z 2 2~ z (6–1) Figure 6-4 (a)showsthevelocityproleobtainedfromthesimulationa ndtheanalytical solution.AlsoshowninFigure 6-4 (b)istheproleofpercentageerrorinstreamwise velocitycomputedbythesimulationasafunctionofdistanc efromthebottomwall. Percentageerrorisaround 0.1% throughoutthechannelwhichshowsthatthecodeis abletorecoverthelaminarsolutionofapurechannelow.Ne xt,wewillconsidercase 6fromTable 3-1 andverifythattheequationforscalareldissolvedcorrec tlyalong withtheNavier-Stokesequationstorecoverthecorrectlam inarsolution.Theanalytical solutionforthiscaseisgivenby 3–25 and 3–26 inChaper 3 .Figure 6-5 showsthe streamwisevelocityandsedimentconcentrationprolesob tainedfromthesimulation andthecorrespondinganalyticalsolutionproles.Alsosh owninFigure 6-6 arethe percentagerelativeerrorsinthestreamwisevelocityands edimentconcentration.Notice thatthetotalrelativeerrorinbothstreamwisevelocityan dsedimentconcentration remainsaround 0.1% 6.3FullyTurbulentSolution Thepresentproblemofinterestisintheturbulentregimean dtousethiscodeto simulatesuchowsextensivevalidationoffullyturbulent testcasesisrequired.Because thetestcasesareturbulentows,thereisnoanalyticalsol utionavailabletovalidatethe simulation.Furthermore,wecannotperformgridpointbygr idpointcomparisonwiththe simulationresultsoftheoldpseudo-spectralcode.Valida tionwillbedonebymatching thestatisticallystationaryoweldsfromboththeruns. 186

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Thepresentversionofthecodeimplementsanelaboratealgo rithmthatcomputes upto4thorder1-pointstatisticsandaccumulatestheminti me.Thisimplementation willbecapableofcomputingmostoftheimportanttermsinRe ynoldsstresstransport equationandallthetermsintheturbulentkineticenergyeq uation.Thetestsperformed herewillnotonlyvalidatethealgorithm,theimplementati onofthecodethatcomputes thestatisticalaveragesaswell.Firsttestcaseistosimul atefullyturbulentchannel owat Re =180 drivenbyaconstantpressuregradient.Thiscasewillcorre spond tocase 0 inTable 3-1 .Figure 6-7 (a)showsthemeanstreamwisevelocityprolesof case 0 obtainedusingoldsharedmemoryspectralcodeandthenewHy bridcode.Also showninFigure 6-7 (b)isthepercentagerelativeerrorinthemeanstreamwisev elocity obtainedformthehybridcode.SimilarlyFigure 6-8 showsthermsvelocityprolesand Figure 6-9 showsthecorrespondingpercentagerelativeerrorintherm svelocitiesthat areobtainedfromthehybridcode.Figure 6-10 (a)showstheReynoldsstressprole obtainedfromthetwosimulationsandFigure 6-10 (b)showsthepercentagerelative errorintheReynoldsstressprole.Itisevidentthatthest atisticsobtainedfromthenew Hybridcodeareincloseagreementwiththatoftheoldspectr alcodesimulation.Next, consideraturbulentchannelowdrivenbymono-dispersesu spensionofsediment. TheReynoldsnumberoftheowis Re =180 andthedimensionlessterminalsettling velocityofthesedimentsinquiescentambientis 0.02 .Thiscasereferstocase 2 in Table 3-1 .ThiscaseissimulatedusingthenewHybridcodeandstatist icallysteady meanstreamwisevelocityandsedimentconcentrationprol esarecomparedwiththe oldsimulationinFigure 6-11 .Correspondingpercentagerelativeerrorinthestreamwis e meanvelocityandsedimentconcentrationareshowninFigur e 6-12 .Similarly,Figures 6-13 and 6-15 showtheRMSvelocities.Thepercentagerelativeerrorinth ermsvalues areshowninFigures 6-14 and 6-16 .Furthermore,ReynoldsstressandReynoldsux prolesfromthetwosimulationsandcomparedinFigure 6-17 andtheirpercentage relativeerrorsareshowninFigure 6-18 .Alltheseguresclearlyshowthatthestatistics 187

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obtainedformthehybridcodeareincloseagreementwiththe previoussharedmemory code. 6.4Optimization TheabovetestcaseshavesufcientlyvalidatedthenewHybr idcode.Thenext logicalstepsaretobenchmarktheperformanceofthecode,i dentifybottlenecks anddetermineitsscalability.Thisexerciseisextremelyi mportant;itwillleadtoan optimizedcodethatiscapableofefcientlysimulatinghig hReynoldsnumberowson massivelyparallelcomputingenvironments.Theoptimizat ionprocesscanbedivided intotwoaspectsofthecode:computationalandcommunicati on.Bothofthemaretobe tweakedandtunedtoderivethebestperformanceoutoftheco de.Thecomputational aspectofthecodecanbebrokendownintobasicoperationsli kescaling,matrixsum, matrix-matrixelementbyelementmultiplication,matrixm ultiplicationandFourier transform.Intheoptimizationprocess,alternatemethods areimplementedforeachof theseoperationsandaretimedtodeterminethefastestorth emostefcientmethod. Followingisabriefsummaryoftheeffortsthatwentintothe optimizationprocessof computationalaspectsofthecode.Simplebackoftheenvelo peanalysisrevealsthat themostcomputationallyintensiveoperationsarematrixm ultiplicationandFourier transforms.6.4.1OptimizeMatrixMultiplicationOperation Letsrstconsidermatrixmultiplicationoperation.Actua ltasksinthehybridcode likerstandsecond z derivative,Helmholtzsolver,Poissonsolveretcaremadeu pof manymatrixmultiplicationoperations.Suchtasksinvolve atwodimensionaldifferentiationmatrixoratwodimensionalEigenvectormatrixoperat ingrepeatedlyonthedata matrix.Thereareapproximately O ( N 4 ) oatingpointoperationsinvolvedinthesetasks where N isthenumberofgridpointsin z (bednormal)direction.Clearlythetotaltime takenforthesetasksscaleasthe4thpowerofnumberofgridp ointsin z direction.This meansthatthesetaskswillbecomputationallyquiteintens iveforhighReynoldsnumber 188

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owswhichrequireextremelyneresolutionin z direction.Furthermore,anysavings accomplishedintheoptimizationprocesscansubstantiall yimprovethescalabilityofthe codeforhighReynoldsnumberows.Followingisasummaryof improvementsmadeto thematrixmultiplicationoperationsthatresultedinsubs tantialsavings. Declarematrixmultiplicationoperatorsasrealvariables Inallthetasksthat involvematrixmultiplicationoperation,thedatathatisp rocessediscomplexwhilethe multiplicationoperatorslikeChebyshevdifferentiation matrixandEigenvectormatrix arepurelyreal.Intheoriginalversion,boththeoperatorm atrixandthedatamatrixare declaredascomplexvariables.Thisleadstoredundantoper ationswheretheimaginary part(zero)oftheelementsintheoperatormatrixmultiplie sthecomplexdataelements. Anewmethodisimplementedthatmakesthematrixmultiplica tionoperationdevoid ofsuchredundancies.Inthismethodtheoperatormatrixisd enedtoberealandit operatesovertherealandimaginarypartsofthedatamatrix separately. UseofBLASlibrary. BasicLinearAlgebraSubprogramsisahighlyoptimized implementationoflow-levelkernelsubroutinesforbasicl inearalgebraoperations likematrixmultiplication.BLASdriverroutinesarespeci callyimplementedforhigh performancearchitectureandallthehardwarevendorslike IntelandAMDprovide optimizedBLASroutinesfortheirhardware.Theoptimizedm atrixmultiplicationmethod implementedinthenewversionusesBLASmatrixmultiplicat ionroutine”DGEMM”. ParallelizationusingopenMPthreads. Themultiplicationoperationisparallelized evenfurtherbyspawningmultipleopenMPthreadssothateac hthreadpreformsapart ofthematrixmultiplicationoperation.Inotherwords,the BLASdriverroutinecallsare madeinsideeachopenMPthreadsthatoperatesoverdifferen tpiecesofthedatamatrix. LoadbalancingoveropenMPthreads. ThedatathatislocaltotheMPIprocess isunwrappedandredistributedintopiecessothateachopen MPthreadhasequal workload.Afterthecomputationiscomplete,processeddat aisre-conguredintoits originaldistributionformat.LoadBalancingensuresthat computationsareequally 189

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distributedandthereforetherearenobottlenecksthatcan reducetheefciencyofthe code. Theabovefeaturesareimplementedasapartofnewmethodint hesubroutines thatinvolvematrixmultiplication.Thismethodisatleast 5 timesfasterthantheoriginal method(seetable 6-2 ). 6.4.2OptimizeFourierTransformOperation The3DdataorthedomainisdistributedovertheMPIprocesse ssuchthatthe rstindexislocalandtheothertwoindicesaredistributed .Asaconsequence,Fourier transformsarecomputedinonedirectionatatime.Duringth eFouriertransform,each stringofdatacorrespondingtotherstindexistransforme dandthisoperationisiterated overthesecondandthirdindicesthatarelocaltothe(MPI)p rocess.ThisFourier transformoperationcanbeimplementedintwoways.Thesimp lestwayistopasseach stringofdatatotheFFTlibraryroutinefrominsidethenest edloopsthatgoesoverall thelocalvaluesofthesecondandthirdindices.Letscallth ismethod1.Anotherway istocalltheFFTlibraryroutineoncesuchthattheroutinec omputesFFTovermany stringsofdata.Letscallthismethod2.Intheoriginalvers ionofthecode,method1is seriali.e.theFFTlibraryroutineusestheserialversiono fthelibrary.Whilemethod2 isparallelastheFFTlibraryroutineusedalltheopenMPthr eadsavailabletotheMPI processtocomputeFouriertransformsonmultiplestringso fdata.Inthesemethods, thecodecanuseeitherintel'sFouriertransformlibrary”D FTI”orFastFourierTransform oftheWestlibrary”FFTW3.X”.Preliminaryanalysisshowth atboththelibrariesare equallyefcient. AftertimingtheFouriertransformmethodsforboththelibr ariesitwasfoundthat parallelversionormethod2wasslower/inefcientthanthe serialversionormethod 1fortheworkloadsizesrelevanttoourproblem.Thismeantt hattherewasscopefor implementingabettermethodthatprovidesmoreexibility andcontrolovertheFourier 190

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transformoperations.Followingisabriefsummaryofthefe aturesthathavebeen incorporatedintheoptimizedFouriertransformmethods. SingleandmanyFFTtransforms. Twomethodsareimplementedbasedonhow manytimestheFFTlibraryroutinesarecalled.Inmethod1,t heFFTlibraryroutineis calledforeachstringofdatainanestedloopthatgoesovert he(distributed)second andthirdindiceslocaltotheMPIprocess.Thekeydifferenc ebetweenthisandthe methodintheoriginalversionisthatthenewermethodispar allel,i.e.theentirework loadissharedoveralltheopenMPthreadsavailabletotheMP Iprocess.Thismeans thatapartofthenestedloopresidedonaopenMPthreadandma nysuchthreadswork simultaneouslytotransformthedata.Hereweusetheserial versionoftheFFTlibrary butsimultaneouslycallmanyinstancesofthelibrarythato perateondifferentdatato accomplishthetransformfaster.Inmethod2,theFFTlibrar yroutinesarepassedmany stringsofdatasothattherearefarfewercallsmadetotheli brary.Also,likemethod1, manyinstancesofthelibraryroutinerunsimultaneouslyon severalopenMPthreads thatarespawnedbytheMPIprocess.Notethatthesetwometho dscanuseboththe librariesbutatpresentonlyFFTWlibraryisused. LoadbalancingacrossopenMPthreads. Sinceboththeversionsareparallel, i.e.,openMPthreadsarespawnedtoloopoversecondandthir dindex,itisimportant thateachopenMPthreadsharesnearlyequalworkloadsothat themethodisefcient andscalable.Dataisrstunwrappedandnearlyequalpieces arefedtotheopenMP threadstocomputetheFouriertransforms.Afterthetransf ormation,thedataiswrapped backintoitsoriginaldistributionscheme.6.4.3OptimizeCommunication CommunicationalgorithmisanimportantaspectoftheHybri dcode.Asexplained earlier,communicationisnecessarytotransposedatasoth atallthecomputationscan bekept”inprocessor”.Bottlenecksandinefcienciesinth ecommunicationschemecan severelyaffecttheperformanceandscalabilityofthecode .Thereforeitisimportant 191

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toanalyzethecommunicationalgorithmindetailandblendi twiththecomputational schemesothatanyredundanttransposeoperationscanbeeli minated.Following aresomeofthecrucialimprovementsincorporatedinthecod ethatwillimproveits performanceandscalability. Memorybottlenecks. Thetransposeoperationsareaccomplishedbymapping theorientationofdatasetacrosstheMPIprocessesandsett ingupnecessarycommunicationderivativestotransfertheinformationoverthen etworkfabric.Thiswillinvolve settingupanelaboratemachineryduringinitializationpr ocessthatstoresthemapping ofthedatasetorientationacrossalltheMPIprocessesandb etweenvarioustranspose operations.Thismachinerywillbeusedlaterinthemainalg orithmtocarryoutthecommunicationprocess.Thereforeitisnecessarythatthememo ryusagebythismachinery isoptimumanddoesnotcripplethescalabilityofthecode.C onsidertheowchartof transposeoperationsshowninFigures 6-2 and 6-3 .Thesetransposeoperationscanbe generalizedintotwocategories:rstcategoryiswherethe rstandthesecondindexis swappedandsecondcategoryiswheretherstandthelastind exisswapped.Inthe formercategory,thecommunicationisalmostalwaysrestri ctedtofewneighboringMPI processes.Whileinthelatercategory,thecommunicationi swithalmostalltheMPI processestakingpartinthesimulation.Thisdifferenceha stwoimportantimplications. Firstiswithrespecttothenumberofcommunicationmessage sthatneedstobepassed toaccomplishthetransposeandsecondiswithrespecttothe extentofmachinery requiredtoaccomplishthetransposeoperations.Intheori ginalversionofthecode,the initializationprocessthatsetsupthemachineryisnotsca lableandexible.Itfailsto takeintoaccountthefactthatcertaintransposeoperation s(rstcategorytranspose)will requirecommunicationwithfarfewerMPIprocesses.Thisca usessubstantialmemory bottlenecksasthecodeisscaleduptosolvebiggergridsize problems.Inthenew version,theinitializationprocessismadeintelligentan dlocaltotheMPIprocess.Itsets upthemachineryinaleanandefcientwaybyrequestingmemo rythatisjustsufcient 192

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forcarryingoutcommunicationsrelevanttotheMPIprocess .Thischangehasresolved thememorybottleneckassociatedwiththecommunicationma ppingprocessofthe code.Forexample,foraproblemsizeof 256 x 256 x 257 on 16 MPIprocessorsthetotal memoryusedbyeachMPIprocesswhensimulatedusingtheoldv ersionwas 852.8 MBandusingthenewversionwas 154.96 MB.Thisamountsto 80 %savinginmemory utilization. Suppresscommunicationofzerosorredundantdata. Inthecode,thetime integrationstepisaccomplishedin( z kx ky )space,i.e., z directionislocalandforward transformed x and y directionsaredistributedovertheMPIprocesses.Ontheot her hand,theadvectiontermsthatarenecessarytocomputether esidueforthetime integrationsteparecomputedin( yl z xl )space.Theadvectiontermsarede-aliased byinterpolatingtheoriginaldataoveranerresolution.L etter' l 'standsforenlarged gridresolution.Aswemoveineitherforwardorreversedire ctionofthetranspose operation,”convert( yl z xl )to( z kx ky )”,datathatisredundantorzeroiscarriedin thecommunicationcalls.Forexample,considerthetranspo seoperation( xl z kyl )to ( kyl z xl ). kyl dataisobtainedbypaddingzeroafter ky elements.Thisisdonetoget globallyinterpolatedvaluesatthenergridresolutionin therealspace.Therefore,the zeropaddeddataneednotbecommunicatedduringthetranspo seoperation.Thiswill resultinsmallercommunicationmessagesand/orfewernumb erofmessagestobe exchangedbetweentheMPIprocesses.Similarargumentscan bemadefor( kyl z xl ) to( xl z kyl )transpose.Bothtransformsandthecorrespondingmachine ryismodiedto implementtheabovefeature.Table 6-2 showsthatbyincorporatingthis,transposetime wasreducedby 20 27 %(seeTable 6-2 forcompletedetailsofthetestcase). Integratetransposeandcomputationtasks. Asshownearlier,communication processesarerestrictedtotheadvectionblockofthealgor ithm.Thisfeatureisdueto thefactthatadvectiontermsneedtobecomputedinthereale nlargedspace( yl z xl ) andthetimesteppingschemeisimplementedinthecomputati onalspace( z kx ky ).By 193

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integratingthetransposeandcomputationaltasksinthead vectionblock,itispossible tominimizethetransposecallsnecessarytocomputetheadv ectionterms.Following isabriefdescriptionofthisprocess.Advectiontermscanb ewritteninconservativeor non-conservativeformasshownbelow @ ~ u i ~ u j @ ~ x j or ~ u j @ ~ u i @ ~ x j (6–2) Theseformsareusedalternately,i.e.,eventimestepsusec onservativeformandodd timestepsusenonconservativeform.Theowchartforthetw omethodsareshown inFigures 6-19 and 6-20 .Figure 6-21 hasthekeyorthedescriptionofthesymbols usedintheowchart.Noticethatboththemethodsaregeneri c,i.e.,theycancompute advectiontermsforanycomponentofvelocityorthescalar eld.Wejustneedto passthevariablecontainingthevelocitycomponentorthes calareldasargument tothemethodandanothervariabletostoretheadvectionter m.Table 6-1 showsthe numberoftransposecallsinvolvedincomputingadvectiont ermsusingboththese forms.Analyzingthealgorithmscloselyrevealsthatthere arepossiblesavinginterms ofthenumberoftransposeoperationsnecessarytocomputet headvectionterms.In ordertoachievethis,theprocessofcomputingtheadvectio ntermsforallthethree velocitydirectionshavetobeintegrated.Thisintegratio nprocesswillallowsomeofthe variablestobereusedandeliminateredundanttransposeop erations.Forexample, whilecomputingtheadvectiontermsinitsconservativefor m,wecanmakeuseofthe symmetrypropertyofthenonlinearterms,i.e., ~ u i ~ u j issameas ~ u j ~ u i .Thereforeafter computingthetransposefrom( yl z xl )to( z kx ky )space,variableslike ~ u ~ v ~ u ~ w and ~ v ~ w arestoredsothattheycanbereusedtocomputetheadvection termsforother directions.Similarly,incaseofnonconservativeformofa dvectionterms,fewtricks canbeusedasfollows.All y derivativescanbeefcientlycomputedfromvariablesin realspacebycarryingoutforwardFFT,multiplyingFFTcoef cientsbycorresponding wavenumber( i k y )andthencarryingoutbackwardFFT.Thisgives @ u =@ y @ v =@ y and 194

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@ w =@ y in( yl z xl )spaceandavoids6transposeoperations.Similarlyxderiv atives canbecomputedbycallingonly1transposeoperation.Thusw ecanavoid9transpose operationsbyimplementingthisintegrationprocess.Tofo rmulateanewalgorithm withintegratedadvectiontermsisnottrivial.Theowchar toftheactualalgorithm implementedinthenewversionofthecodeisshowninFigures 6-22 and 6-23 andthe resultingsavingsaresummarizedinTable 6-1 6.5TimingtheNewOptimizedHybridCode Alloftheaboveoptimizationfeaturesareimplementedinth enewversionofthe code.Testcaseshavebeensetupthatrunusingtheoldversio nandthenewversionof thecodeandalltheimportanttasksaretimed.Furthermore, theserunsareperformed onthesamemachinesoronmachinesthathaveexactsamearchi tecture.Thisensures thatthesavingareduetheimplementationofanoptimizedal gorithminthecode.Itmust alsobeemphasizedthattheserunsarepurelyintendedtomea suretheperformanceof thecodeandhencethesimulationresultswillnotbepresent edhere. Thesesimulationsarerunon”HIPERGATOR”machineofHighPe rformance ComputingcenterattheUniversityofFlorida.Thesimulati onsusedagridsizeof 256 x 256 x 257 pointsin x y z directionandthechannelinclinationissetto =5 .The oweldisarbitrarilyinitializedbyarandomnumbergener atorthatcantakeavalue between 1 to 1 .Pressuregradientinthechannelisturnedoff.Sedimentsu spension isintroducedinthechannelwiththesedimentparticlesett lingvelocityof 0.035 .Both simulationsareperformedusing 64 cores( 1 node)suchthatthereare 8 MPIprocesses and 8 openMPthreadsperMPIprocess. 100 iterationsortimestepsarecomputedin bothcasessoastoaccumulatetheexecutiontimeofvarioust asksforbetterstatistics. Theexecutiontimeofthesesimulationsandtheirtasksissu mmarizedinTable 6-2 Thecolumns”Oldcode”and”Newcode”givetheaverageexecut iontime(in seconds)ofaparticulartaskinthesimulation.”%saving”c olumntellshowmuchfaster istheexecutionofaparticulartaskinthenewversionascom paredtotheoldversionof 195

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thecode.Thelastcolumnweightstheaverageexecutiontime oftaskswiththenumber oftimestheyarecalledinthesimulationtodeterminetheir shareinthetotalsavings achievedbythenewcode.Thisrepresentstheimpactofoptim izationofaparticular taskontheoverallsimulation.Now,considerthe”zderivat ive”task,thetableshowsthat thistaskinthenewcodeis 80 %fasterthanitsimplementationintheold.Furthermore, theoptimizationofthistaskhasasubstantialimpactonmak ingthenewcodefaster. Thetimesavingsduetothisoptimizationamountstoabout 43 %ofthetotalsavings fortheentiresimulation.OthercomputationaltaskslikeF FTs,diffusion,divergence, Helmholtzsolver,Poissonsolvershow 50 to 70 fasterexecutiontime.Mostofthesetask (individually)contributeabout 2 to 7 %ofthetotalsavingsfortheentiresimulation.The communicationprocesseslikesenddatafrom( kyl z xl )to( xl z kyl )andviceversa,are 17 to 27 %fasterandtheircontributiontowardstotalsavingsisabo ut 2% .Thesavings achievedfortaskshigherleveltaskslikeconvert( yl z xl )to( z kx ky ),Advectionterms, Advancetimearemainlyduetothesavingsachievedatthelow erleveltaskslikesend ( kyl z xl )to( xl z kyl ), z derivative,Helmholtzsolver,Poissonsolveretcthatmake themup.Forexample,convert( yl z xl )to( z kx ky )taskismadeupofForwardFFT inyl,ForwardFFTinxl,Send( kyl z xl )to( xl z kyl )andsend( kxl z kyl )to( z kx ky ) andthereforethesavingincoverttaskismainlyduetothesa vinginthebasictasks puttogether.Thetask”Mainloop”inthetablecontainsthei mplementationofRK3 integrationstepwhichmeansthateachcallto”Mainloop”wi lladvancethesimulationby 1timestep.Thisisthetaskatthehighestlevelandtherefor ewillcontainallthetasks showninthetable.Theaveragetimetakenforexecutionofon ecalltotask”Mainloop” representhowfastisthecode.Thistestclearlyshowsthatt heoptimizationprocesshas madethenewerversionofthecodeatleast 60 %fasterthantheoldcode. 196

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Table6-1:Summaryoftransposeoperationsintheoldandthe newoptimizedversion ofthehybridcode.Notethattheadvectiontermsforthescal arequationemployconservativeformonly.Hence,inthetable,forconservativef ormthenumberoftranspose operationsalsoincludesthescalarequation. AdvectionblockOldversionNewversion TransposeConsNon-consConsNon-cons ( z kx ky )to( kxl z kyl )41246 ( xl z kyl )to( kyl z xl )41249 ( kxl z kyl )to( z kx ky )12393 ( kyl z xl )to( xl z kyl )12393 Table6-2:Comparisonoftheexecutiontimeofvarioustasks intheoldandnewversionoftheopenMP-MPIhybridcode.Timeisinseconds.Perce ntsavingsachievedfor individualtasksisgivenin%savingscolumn.%Totalsaving columnrepresentsthe proportionofindividualtasksascomparedtothetotalsavi ngsaccomplishedbythenew code. TaskOldcodeNewcode%saving%totalsaving ForwardFFTin xl 0.0490.01177.553.60 BackwardFFTin xl 0.050.01276.003.36 ForwardFFTin yl 0.0820.01976.835.97 BackwardFFTin yl 0.0810.02075.316.55 z 'derivative0.3750.06682.4043.00 Diffusion0.3840.07480.7316.98 Divergence0.400.08179.804.04 SolveHelmholtz0.2670.12553.187.18 SolvePoisson0.2540.12351.571.66 Send( kyl z xl )to( xl z kyl )0.0950.06927.372.46 Send( xl z kyl )to( kyl z xl )0.0860.07117.441.33 Convert( yl z xl )to( z kx ky )0.2650.14644.9117.29 Advectionterms4.0361.85154.1427.61 Advancetime30.6611.2863.2381.67 Mainloop37.5913.8563.16100 197

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Figure6-1:Blockdiagramrepresentationofthehybridopen MP-MPIpseudospectral code 198

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Figure6-2:Blockdiagramrepresentationoftheconvertope rationthattransformsdata betweenrealspaceandcomputationalspace Figure6-3:Blockdiagramrepresentationoftheconvertope rationthattransformsdata betweenenlargedrealspace(forde-aliasing)andcomputat ionalspace 199

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Figure6-4:Velocityproleoflaminarchannelowdrivenby constantpressuregradient (seecase0fromTable 3-1 )thatisobtainedfromthehybridcodesimulationandthe analyticalsolution.Frame(b)showstherelativeerrorint heproles. Figure6-5:Laminar(a)velocityand(b)concentrationpro leofchannelowdrivenby monodispersesuspension(seecase6fromTable 3-1 )obtainedfromhybridcodeand theanalyticalsolution 200

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Figure6-6:(a)Percentagerelativeerrorinthestreamwise velocityand(b)sediment concentrationobtainedfromthesimulationofchannelowd rivenbymono-disperse suspensionofsediment(seecase6fromTable 3-1 ) u z 0 5 10 15 0 0.2 0.4 0.6 0.8 1 Hybrid codeold code ~~ (a) % error u z 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 Hybrid code ~~ (b) Figure6-7:(a)Meanstreamwisevelocityprolesofturbule ntchannelow(seecase0 fromTable 3-1 )obtainedfromthehybridcodeandoldsharedmemorycode.(b )shows thepercentagerelativeerrorinthemeanstreamwiseveloci typroleobtainedfromhybridcode. 201

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u rms z 0 1 2 3 0 0.2 0.4 0.6 0.8 1 Hybrid codeold code ~~ (a) w rms z 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1~~ (b) v rms z 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1~~ (c) Figure6-8:(a) ~ u rms ,(b) ~ w rms and(c) ~ v rms prolesofturbulentchannelowobtainedfrom thehybridcodeandoldsharedmemorycode % error u rms z 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 Hybrid code ~~ (a) % error w rms z 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1~~ (b) % error v rms z 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1~~ (c) Figure6-9:Prolesofpercentagerelativeerrorin(a) ~ u rms ,(b) ~ w rms and ~ v rms ofturbulent channelowsimulationusingthenewhybridcode 202

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u'w' z 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 Hybrid codeold code ~~~(a) % error u'w' z 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 ~ ~~(b) Figure6-10:Prolesof(a)Reynoldsstress(~ u 0 ~ w 0 )forturbulencechannelowobtainedfromhybridcodeandoldsharedmemorycode.(b)Perce ntagerelativeerrorin Reynoldsstressobtainedfromthehybridcode. 203

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u z 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 Hybrid codeold code (a) c z 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 (b) Figure6-11:(a)Meanstreamwisevelocityprolesand(b)Me ansedimentconcentrationprolesobtainedfromthesimulationofturbulentc hannelowdrivenbymono dispersesuspensionofsediment(seecase6fromTable 3-1 )usingthenewhybridcode andoldsharedmemorycode. % error u z 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 (a) % error c z 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 1 (b) Figure6-12:Percentagerelativeerrorin(a)Meanstreamwi sevelocityand(b)Mean sedimentconcentrationofturbulentchannelowthatisdri venbymonodispersesuspensionofsediment(seecase6fromTable 3-1 )andsimulatedusingthehybridcode. 204

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u rms z 0 1 2 3 0 0.2 0.4 0.6 0.8 1 ~~(a) w rms z 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Hybrid codeold code ~~(b) Figure6-13:(a) ~ u rms and(b) ~ w rms prolesofturbulentchannelowwhichisdrivenby monodispersesuspensionofsediment(seecase6fromTable 3-1 )andsimulatedusing thehybridcodeandoldsharedmemorycode. % error u rms z 1 2 0 0.2 0.4 0.6 0.8 1 ~~(a) % error w rms z 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 ~~(b) Figure6-14:Percentagerelativeerrorin(a) ~ u rms and(b) ~ w rms ofturbulentchannelow whichisdrivenbymonodispersesuspensionofsediment(see case6fromTable 3-1 ) andsimulatedusingthehybridcode. 205

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v rms z 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 ~~(a) c rms z 0 0.05 0.1 0.15 0 0.2 0.4 0.6 0.8 1 Hybrid codeold code ~~(b) Figure6-15:(a) ~ v rms and(b) ~ c rms prolesofturbulentchannelowwhichisdrivenby monodispersesuspensionofsediment(seecase6fromTable 3-1 )andsimulatedusing thehybridcodeandoldsharedmemorycode. % error v rms z 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 ~~(a) % error c rms z 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ~~(b) Figure6-16:Percentagerelativeerrorin(a) ~ v rms and(b) ~ c rms ofturbulentchannelow whichisdrivenbymonodispersesuspensionofsediment(see case6fromTable 3-1 ) andsimulatedusingthehybridcode. 206

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u'w' z 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 Hybrid codeold code ~~~ (a) w'c' z 0 0.005 0.01 0.015 0.02 0 0.2 0.4 0.6 0.8 1 ~~~ (b) Figure6-17:(a)Reynoldsstress( ~ u 0 ~ w 0 )and(b)Reynoldsux( ~ w 0 ~ c 0 )prolesofturbulentchannelowwhichisdrivenbymonodispersesuspension ofsediment(seecase6 fromTable 3-1 )andsimulatedusingthehybridcodeandoldsharedmemoryco de. u'w' z 0 1 2 3 0 0.2 0.4 0.6 0.8 1 ~~~ (a) w'c' z 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 ~~~ (b) Figure6-18:Percentagerelativeerrorin(a)Reynoldsstre ss( ~ u 0 ~ w 0 )and(b)Reynolds ux( ~ w 0 ~ c 0 )ofturbulentchannelowwhichisdrivenbymonodispersesu spensionof sediment(seecase6fromTable 3-1 )andsimulatedusingthehybridcode. 207

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Figure6-19:Thesequenceofoperationsinvolvedincomputi ngtheadvectiontermsin theconservativeform. Figure6-20:Thesequenceofoperationsinvolvedincomputi ngtheadvectiontermsin thenonconservativeform. 208

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Figure6-21:Descriptionofthesymbolsusedinalltheowch artsshowninthischapter 209

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210

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Figure6-22:Anoptimizedmethodofcomputingtheadvection termsintheconservative form.Thismethodensuresthatadvectiontermsarecomputed withminimumnumber oftransposeoperations.Thismethodspansoverthreeframe s(a),(b)and(c).Forthe descriptionofthesymbolsrefertoFigure 6-21 211

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212

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Figure6-23:Anoptimizedmethodofcomputingtheadvection termsinthenonconservativeform.Thismethodensuresthatadvectionterm sarecomputedwithminimumnumberoftransposeoperations.The”NCblock”seeninf rames(a)and(b)isa genericsubroutinethattakesinanyprimaryvariablein( z kx ky )spaceandtransforms itinto( yl z xl )space.Furthermore,duringthetransformation,italsoco mputesthex andyderivativeofthevariablein( yl z xl )space.Thedescriptionofthisblockisshown inFigure 6-24 213

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Figure6-24:Flowchartshowingthesequenceofoperationsi ntheNCblockthatisused inFigure 6-23 (a)and(b). 214

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CHAPTER7 CONCLUSIONSANDFUTUREWORK 7.1Conclusions Gravitycurrentsthataredrivenbyactivescalareldsandg eneratedbycontinuous dischargeintotheambientexhibitcomplexphysics.Suchpr oblemsposeaformidable challengeindevelopingaccuratepredictivemodels.Typic ally,thecomplexowis brokendownintosimplermodels(problems)thatprobespeci caspectsoftheow. Studyingsuchmodelproblemsprovideimportantinsightson thebehaviorofthe actualow.Theseinsightsarecrucialastheybecomethegui dingprinciplesbehind developing(better)modelsthatarebasedonfundamentalun derstandingoftheow. Inthisdissertation,threeaspectsoftheaboveproblemare probed:1)Howdoesthe frontconditionbehaveasafunctionofinowdischargeandt heroleofambientow directiononthefrontcondition?2)Straticationeffects ofdispersedphaseonthe turbulentmixingintheow?3)Sedimentcarryingcapacityo fthecurrentanditsability totransportlargesedimentsizes. Chapter 2 dealswiththerstaspectmentionedabove.Herethefrontco nditionof aconservativecurrent,generatedbycontinuousdischarge intotheambient,isderived fromcontrolvolumeanalysis(conservationofmomentumand energy)aboutthefront ofthecurrent.Inthisanalysis,theambientowdirectioni skeptunconstrained.Anew parameter r isintroducedthatspeciesthefractionofambientuidtha towsinthe oppositedirectionofcurrent.Thecontrolvolumeanalysis givesageneralexpressionfor thefrontFroudenumberwhichisafunctionofcurrentdepthr atio” a ”andtheambient owparameter r (see( 2–5 )).Thisexpressionclearlyshowsthattheambientow directionhasasubstantialeffectonthefrontFroudecondi tion.When r =0 orunder fullreturnow(ambientandcurrentowintheoppositedire ction,seeBenjamin[ 9 ]or ( 2–4 ))thefrontFroudeconditionisastrongdecreasingfunctio nofdepthratio(front Froudeconditiondropsbyafactorof2from p 2 to 1 = p 2 asthedepthratioincreases 215

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from0to0.5).Ontheotherhand,when r =1 orundernoreturnow(ambient andcurrentowareinthesamedirection,see( 2–8 ))frontFroudeconditionremains nearlyconstantthroughouttherangeofdepthratio.Thesei nferencesdrawnfromthe theoreticalconsiderationsarevalidatedbycomparingthe simulationresultsofthetwo extremecases: r =0 and r =1 .Thismeansthatambientowdirectioncangreatly inuencethefrontFroudeconditionofthecurrent.Asacons equence,appropriatefront Froudenumbermustbespeciedthattakesintoaccounttheam bientowdirection whenitisusedasaninputforvariousmodelslikeshallowwat erequations,boxmodels, etc.Chapter 2 alsopresentsthesimulationresultsofgravitycurrentsge neratedby variableinowintheno-returnconguration.Twovaluesfo rtheinowparameter are selected: =1.25 thatrepresentswaxinginow(rateofinowincreaseswitht ime) and =0.90 thatrepresentswaninginow(rateofinowdecreaseswitht ime).Front conditionofthesevariableinowsimulationswerecompare dwiththefrontcondition ofconstantinowsimulationsandtheoreticalfrontFroude numberexpressions Fr B and Fr NR .Suchacomparisonrevealedthattheoreticalexpressionsd evelopedfor steady,inviscidgravitycurrentfrontscanbeappliedtoin viscid,unsteadyfrontsasjump conditionsacrossthedensitydiscontinuity(frontofthec urrent).Wealsondthatthe frontconditionofvariableinow(unsteady)gravitycurre ntsarecomparabletothefront conditionofcontinuousinow(steady)currentsforsimila rdepthratios.Furthermore, thefrontconditionisobservedtobeweaklydependentonthe valueof andthe dependencewasobservedtobenon-uniformovertherangeofd epthratios. Chapter 3 addressesthesecondaspectoftheow,i.e.,theeffectofst raticationon owturbulence.Chapter 3 considersbodyofaconstantinownon-conservativedilute gravitycurrent,inparticularaturbiditycurrent.Inatur biditycurrenttheinteractionof settlingsedimentsandturbulenceleadsto:(a)skewingoft hestreamwisedrivingforce towardsthebedand(b)stablestraticationthatdampsbednormalmomentumand masstransport.Boththeseeffectshaveaninuenceonthetu rbulenceproductionand 216

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turbulencedampingintheow.Infactinsuchtypeofowsund erstandingtheturbulencedampingprocessisextremelyimportantasowturbule nceisthesolemechanism responsibleforkeepingthesettlingsedimentsinsuspensi on.Withoutturbulence,the owwillprogressivelylooseallthesedimentsandwillceas etoexist.Inthischaptera simplemodelforturbiditycurrentsdrivenbymono-dispers esuspensionofsediment isproposed.Severalsimulationsareperformedat Re =180 andwithsuspensions containingdifferentsizeofmono-dispersesedimentparti cles.Thesesimulationsreveal thatskewingthedrivingforcetowardsthebedcausesturbul encedampingbutthiseffect ismodestevenforsuspensionscontainingsedimentparticl eswithverylargesettlingvelocity.Thismeansthatstraticationeffectisthedominan teffectresponsibleforcausing extremesuppressionofturbulenceintheow.Furthermore, thisstudyobservedthatthe processofcompleteturbulencesuppressionwassudden,i.e .,theturbulencestatistics oftheowshowedminimaldampingeffectswhenthesediments ettlingvelocity(aproxy forsedimentsize)waslessthanacriticalvalue,butassoon asthesettlingvelocitywas increasedevenbyasmallamountbeyondthecriticalvalueit ledtocompleteshutdown ofturbulence.Thissuddentransitionisfascinatingandth eabruptchangeisseeninseveralturbulentstatistics.Analysisofturbulentkinetice nergyequation,Reynoldsstress transportequationandonedimensionalspanwiseandstream wiseenergyspectrasuggestthattotalturbulencesuppressionmaybebroughtabout bydampingofReynolds stressproducingturbulentvorticalstructuresclosetoth ebed.Thischapterproposed anexplanationfortheabruptextinctionofturbulenceusin gstatisticalanalysisoftheQ2 Reynoldsstresseventsforvariouscasesandtheauto-gener ationcriterionforturbulent vorticalstructuresclosetothebed(wall)givenbyZhou etal. [ 91 ].Fromtheprobability densityoftheseQ2eventsitisshownthatwithincreasingse dimentsettlingvelocitythe intensityofQ2eventsdecreaseandtheeventvectorattens .Thesetwoeffectscause spatialmodulationsintheturbulenthairpinandquasi-str eamwisevortices:theirintensity reducesandtheirspatialdistributionbecomessparse.Ast hesettlingvelocityincreases 217

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beyondthecriticalvalue,thethresholdvorticalstrength necessaryforauto-generation increasesandtheexistingowstructuresbecomeincapable ofauto-generating.The owthusevolvestocompletesuppressionofturbulence.Thr oughthisstudy,thedynamicsofsuppressionofturbulenceinturbiditycurrentshasb eenidentied.Furthermore,it canbeconjecturedthatturbiditycurrentsbecomedepositi onalandloosesedimentsto theoceanoordueamechanismsimilartotheoneproposedher e. Havingidentiedtheturbulencesuppressionprocessintur biditycurrents,itwillbe usefultocharacterizethisprocessusingtheparametersof theowsothatitcanbe usedtomakepredictionsaboutthestateoftheow.Chapter 4 considerstheglobal TKEbudgetofthemathematicalmodelusedtosimulateturbid itycurrentsinChapter 3 andidentiesthattheturbulencesuppressionmechanismca nbecharacterized bythreeparameters: Re Ri and ~ V z .Theparametricgrouping Ri ~ V z isidentiedas itnaturallyarisesfromtheTKEequationastheenergyspent byturbulencetokeep thesedimentsinsuspension.Throughsimulationspresente dinChapter 3 and 4 ,it wasclearthatacriticalvalueof Ri ~ V z exist.Increasingthevalueof Ri ~ V z beyondits criticalvalueresultsincompleteturbulencesuppression .Thesimulationsshowthat thetransitionisabruptandhappenswhenonlyabout 6% ofthebed-normal-integrated TKEproductionisconsumedtomaintainsedimentsinsuspens ion.Inachannelowthe wall-normal-integratedTKEproductionanddissipationsc aleas ln( Re ) ,basedonwhich alogarithmicdependenceforcritical Ri ~ V z withincreasing Re isproposed.Laboratory andeldobservations,alongwithpresentresultsareusedt oobtainabesttforthis Re dependence. Chapter 5 extendsthemathematicalmodelproposedinChapter 3 toincludebidispersesuspensionofsediment.Fromthisasimpliedmode lisformulatedsothat thene(small)sedimentparticleshavenegligiblesettlin gvelocity.Thissimplication reducestheeffectofnesedimentparticlestobelikeaunif ormconstantstreamwise pressuregradient(nostraticationeffect).Ontheotherh andthecoarsesediment 218

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particleshavealargesettlingvelocitysothattheyimpose astrongstraticationeffect ontheow.Thisisthesimplestmodelthatcanbeformulatedt odemonstratethe abilityofturbiditycurrentstotransportlargesedimentp articlesoverlongdistances.In otherwordstheaboveproblemreducestodeterminingthemin imumamountofne sedimentsnecessarytoensurethattheturbulenceintheow remainsactive.Inorder toaidcomparisonwithsimulationresultsofmono-disperse modelpresentedinprevious chapters, Re =180 Sc =1 and Ri =11.4 .Thebi-dispersemodelintroducestwo moreparameters r f (fractionalamountofnesedimentparticlesinthesuspens ion)and ~ V c (settlingvelocityofcoarsesedimentparticles).Simulat ionsarecarriedoutin3sets: A,BandCwhere ~ V c isxedat 0.0275,0.035 and 0.05 respectively.Foreachsetthe minimumamountofnesedimentsnecessarytokeepturbulenc eactiveisfoundtobe r f =0.04,0.24 and 0.469 respectively.Theseobservationsdemonstratethatevenla rge sedimentparticlescanbetransportediftheoverallsedime ntsuspensionisfavorablefor sustainingturbulence. AnotheraspectthatisprobedinChapter 5 istheapplicabilityofturbulencesuppressioncriteriatoturbiditycurrentsdrivenbybi-dispe rsesuspensionofsediments. FromtheTKEbudgetofallthenear-criticalcases(6A,6Band 6C),completeturbulence suppressionisobservedtooccurwhenbulkTKEdampingreach esavalueof0.30. ThisobservationagreeswellwiththecriticalbulkTKEdamp ingforcurrentswithmono dispersesuspension.Thismeansthattheparametricgroupi ngthatquantiescomplete turbulencesuppressionwilltaketheform Ri r c ~ V cz anditwillhavethesamecritical valueastheoneobservedformono-dispersesuspension.Fro mthisitfollowsthatthe logarithmicdependenceofthecriticalvaluefortheparame tricgroupingon Re willhold forbi-dispersesuspensions. Theturbulencesuppressioncriteriaforbi-dispersesuspe nsionscanbeeasily recastasthecarryingcapacitycriteriaforturbiditycurr ents.Thefactthataminimum amountofnesedimentparticlesarenecessarytotransport coarsesedimentparticles 219

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hasdirectimplicationonthesedimentcarryingcapacityof thecurrent.Whenareal turbiditycurrentencountersanerodiblebed(assumingtha ttheowisenergeticto entrainsediments),themaximumamountofsedimentparticl esthatcanbeentrained fromthebedwilldependontheexistingcompositionofthese dimentsuspension.This isapowerfulinterpretationandcanbeusedtopredictthema ximumsedimentcarrying capacityofaturbiditycurrent.Toprobethevalidityofthi shypothesis,theexperimental resultsofGarcia&Parker[ 30 ]wereanalyzedusingthesimplebi-dispersemodel presentedinthischapter.Fromthesimpleestimatesofexpe rimentaldataappliedto thebi-dispersemodelyieldedpredictionsthatreconciled verywellwiththeexperiments. Inotherwordstheerosionofsedimentparticlesofsize 180 m andthecorresponding concentrationlevelsattainedinthecurrentwerewithinth esedimentcarryingcapacity predictionsofthebi-dispersemodel.Sincetheexperiment swerenotintendedto measurethecarryingcapacityofthecurrent,detailedcomp arisonofthebi-disperse modelcouldnotbedone. Finally,inChapter 6 thevalidationandoptimizationofthehybridopenMP-MPI pseudospectralcodeisdiscussed.ThenewhybridopenMP-MP Icodeisimplemented toleveragethemassivelyparallelarchitectureofthelate stsupercomputerssothathigh delitysimulationsofparticulategravitycurrentscanbe performedathigherReynolds numbers.Thevalidationprocesshasestablishedthecorrec tnessofthecodeand theoptimizationprocesshassubstantiallyspeduptheexec utiontimeofthecode. Approximately 63 %speedupwasattainedafteroptimization.Inconclusion,t hrough Chapters 2 3 4 5 allthethreeaspectsofcontinuousparticulategravitycur rentswere analyzed.Moreovertheanalysishasbeenabletobringoutim portantinsightsthatwill helpthedevelopmentofbetterpredictivemodelsforsuchco mplexows. 7.2FutureWork Inthepresentworkwehaveaddressedfewfundamentalaspect sofparticulate gravitycurrentsbyconsideringsimpliedmathematicalmo dels.Thishasclearly 220

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helpedusinidentifyingtheunderlyingmechanismsthatpla yacrucialroleinthe evolutionofthesecurrents.Nowwiththisstrongfoundatio nweneedtodevelopdetailed mathematicalmodelsthatcanincorporate,inanincrementa lmanner,themanyfeatures ofparticulategravitycurrents.Followingaresomeofthea venuesthatwillfollowfrom thisstudy Simulationofgravitycurrentsgeneratedbyvariableinow athigherReynolds number. PerformhighReynoldsnumbersimulationsof3Dvariablein owgravity currentsanddetermineitseffectonthefrontFroudecondit ion.3Dsimulationswill beabletocapturethemixingassociatedatthefrontingreat detail.Also,perform simulationsofnon-conservativecurrents.Suchsimulatio nswilladdextraparameters likedispersedphaseparticlediameterandinertiaofthepa rticletothemodel.Thiswill beaninterestingstudythatwillhelpusunderstandtherole ofinertialparticlesonthe frontconditionofnon-conservative,variableinowcurre nts. Simulationsofpoly-disperseturbiditycurrents. Innaturethedispersedphase isnevermonoorabi-dispersesuspension.Infact,itisalwa ysacontinuousdistribution ofparticlesizes.Torepresentcontinuousparticledistri butionposesagreatchallenge. Wecannotuseinnitelymanydiscretesedimentsizestorepr esentthedistribution.This makestheproblemuntenableforthepresentcomputerresour ces.Thebestoptionhere istodirectlysolvetheprobabilitydistributionfunction fortheparticlesuspension.The evolutionequationthatrepresentssuchsystemsiscalledt hePopulationBalanceEquation(PBE).Thereareseveralwaysdocumentedintheliterat uretosolvecoupledPBE andN-Sequations.Oneofthemostsuitablemethodsforoutpr oblemisQuadrature MethodofMoments(QMOM).Thismethodusesquadraturetorep resenttheprobability densityfunctionintermsofitsmomentsandsolvesthePBEby computingtheevolution ofmomentsinstead.Thismethodiscomputationallyfarmore cheaperandefcientthan solvingmanyconcentrationequationsfordiscretesedimen tsizes. 221

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Morerealisticmodelforresuspensionux. Atpresenttheresuspensionuxis notafunctionofthelocalshearstressonthewall.Furtherm ore,thereisinstantaneous andlocalbalancebetweentheresuspensionuxandthesettl inguxatthewall.This isanoverconstrainedboundaryconditions.Wecanimplemen tanewresuspension uxformulationthatisdenedlocallybasedonthelocalshe arstress.Thisimproved resuspensionuxwillallowustoanalyzeitseffectonthetu rbulencesuppressionorthe carryingcapacitycriteriaforturbiditycurrents.Furthe rmore,sucharesuspensionmodel canalsohelpusinunderstandingtheprocessofbedmorpholo gy. 222

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APPENDIXA ANOTHERDERIVATIONOF FR r Fig. A-1 (a)isaschematicdescriptionofthecurrentinthelaborato ryframe.Supposeweswitchtoaframemovingwithspeed u F =(1 r ) aU totheright.Inthisframe thecurrentmoveswithspeed U T = U u F ,andtheambientisstationary;seeFig. A-1 (b). Forthetransformedcurrentwithspeed U T ,theobserverinthemovingframe“sees” aclassicalBenjaminproblem,andtherefore U T = Fr B ( a )( g 0 h ) 1 = 2 (A–1) Transformingbackintothelaboratoryframe,weobtain U ( g 0 h ) 1 = 2 = 1 1 (1 r ) a Fr B ( a )= Fr r ( a ). (A–2) Itcanbeshownthattheenergydissipationisnotaffectedby theframetransformation. 223

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FigureA-1:Thecurrentsystem(a)inthelaboratoryframe;( b)inaframemovingwith u F = Ua (1 r ) totheright. 224

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APPENDIXB EVALUATIONOFDEPTHRATIOFOR =1 CASE H and q (dimensional)aregiven.Inthe =1 case, h = h N and u = u N are constants. Indimensionalform: u N h N = Fr r ( a ) g 0 1 = 2 h 3 = 2 N = q (B–1) Consequently h N H 3 Fr 2 ( a )= q 2 g 0 H 3 (B–2) TheRHSisgivenforaparticularsystem,andtheLHSisasimpl eexpressionfor a = h N = H .Inparticular,fortheNRcase,weobtain a 3 2 a 1 a 2 = q 2 g 0 H 3 (B–3) Thesolutionprovides a .Actually,thisprovidesthecompletedescriptionofthis ow, becausenow h N = aH isdetermined,and u N = q = h N Indimensionlessform,thepreviousequationreads a 3 2 a 1 a 2 = Fr 2 in ~ H 3 (B–4) ThispredictionistestedinTable 2-1 .TheagreementwiththeNSsimulationsis,in general,good. 225

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APPENDIXC MATHEMATICALMODELOFTURBIDITYCURRENTSDRIVENBYPOLY-DI SPERSE SUSPENSIONOFSEDIMENTS Appendix Thisappendixdescribestheformulationofamodelforturbi ditycurrentsdrivenby poly-dispersesuspensionofsediment.Asbefore(see 3 x 3.2 ),thebodyofaturbidity currentismodeledasaninclinedchannelowthatisdrivens olelybytheexcess densityimposedbythesedimentsuspension.RefertoFigure 3-1 foraschematicofthe model.Hereweconsiderapoly-dispersesuspensionthatcan bedescribedbyasetof discretesedimentsizes.Tokeeptheformulationgeneral,l etssaythereare'N'discrete sizesthatmakeupthepoly-dispersesuspension.Weassumet hatthesuspension isdilutesothatthecollisionsbetweensedimentparticles andrheologyeffectscan beneglected.Boussinesqassumptionsareemployedowingto dilutesuspension assumption.Equilibrium-Eulerianformulationisusedtod escribethedynamicsofthis problem.Withtheseassumptions,thegoverningequationsi nthedimensionalformcan bewrittenasshowbelow @ u @ t + u r u = 1 w r p + r 2 u + N X m =1 ( R m c m ) e g (C–1) r u =0, (C–2) @ c m @ t +( u + V m ) r c m = Dr 2 c m (C–3) ( C–1 )isthecontinuousphase(water)momentumequationwhere u = f u v w g isthevelocityvector, w isthedensityofwater, p isthepressure, isthekinematic viscosityofwater.Notethatthelasttermontherighthands ideof( C–1 )isthetotal effectofortheexcessdensityimposedbythepoly-disperse suspensionontheow. Theterminsidethesummationistheexcessdensity(bodyfor ce)contributionof sedimentoftype”m”ontheow.Thevariablesinsidethesumm ationare c m isthe 226

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concentrationofsediment, R m =( sm w ) = w isthemeasureofexcessdensityof sedimentand sm istheactualdensityofsediment.Theeffectofthisexcessd ensity indifferentdirectionsiscontrolledbythegravityvector e g = f g x ,0, g z g where g x and g z arethecomponentsofgravityinthestreamwiseandbed(wall )normaldirection. Thecontinuousphasesatisesthecontinuityequation( C–2 ).Whilethe' N 'discrete dispersedphasesaregovernedbycorrespondingscalartran sportequations.The generalformofthisequationisasshownin( C–3 ). IntheEquilibrium-Eulerianformulation,weassumethatth einertialeffectsofall sedimentsizesarenegligiblecomparedtotheirsettlingte ndency.Asaconsequence, thevelocityofthesedimentparticlesisgivenbysuperposi tionofthelocaluidvelocityandthesettlingvelocityofanisolatedsedimentpar ticleinquiescentambient. Mathematicallyitcanbewrittenas V d = u + V m ,where V m = f V mx ,0, V mz g isthe settlingvelocityofsedimentparticleoftype'm'inquiesc entambient.Itisnecessaryto emphasizeatthispointthatthesedimentparticlesarelarg eenoughthatthediffusivity in( C–3 )isnotduetoBrownianmotion.Thiseffectivediffusionisd uetothelongrange hydrodynamicinteractionscausedbyrandomuctuationsin theparticlenumberdensity. Therefore, D istakenasaneffectiveconstantdiffusivityofsedimentpa rticles.Also implicitinthisisthefactthatwehaveusedthesamediffusi vityforallsedimentsizes. Anotheraspectofthediffusivetermsisthatitprovidesame chanisminthemodelto implementresuspensionofsedimentparticlesatthebed. Theaboveequationsaresolvedwiththefollowingboundaryc onditions.Weemploy periodicboundaryconditionsinthestreamwiseandspanwis edirections.Thelength ofthedomaininthesedirectionareappropriatelyspecied .Theheightofthechannel issetto h .Foruidvelocitythebottomwalloffersnoslipcondition, whilethetopwall isnostress.Fortheconcentrationeld,foreachsediments ize,thetopandbottom boundaryconditionsstatethatsettlingexactlybalancesr esuspension.Mathematically 227

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theseboundaryconditionscanbewrittenasfollows u =0 at z =0 (C–4) @ u @ z =0, @ v @ z =0 and w =0 at z = h (C–5) c m V mz = D @ c m @ z at z =0 and z = h (C–6) Specifying( C–6 )asaboundaryconditionimpliesthatthesettlinguxisexa ctlybalancedbyresuspensionux(diffusiveuxhere)locallyandi nstantaneously.Asa consequencetheabovemodelconservestheamountofsedimen tsinsuspensionatall times. C.1DimensionlessEquations:SelectionofAppropriateSca les Theimposedboundaryconditionsforconcentrationeldens uresthatthetotal amountofsedimentintheowisalwaysconserved.Thisdrive stheturbulentowto afullydevelopedstatisticallystationarystate.Theglob alforcebalanceatthisstate isusedtogettherelevantscalesfortheproblem.Herewepre sentthemeanow equationsofthefullydevelopedstatisticallysteadystat e.Toobtainmeanproperties, averagingisdonespatiallyinthetwohomogeneousdirectio ns,i.estreamwiseand spanwiseandtheaveragesareaccumulatedovertime.Theave ragequantitiesare representedby ( ) andthecorrespondinguctuationsarerepresentedby ( 0 ) .Following arethemeanowequations d 2 u dz 2 d dz ( u 0 w 0 )+ N X m =1 ( R m c m ) g x =0, (C–7) 1 w d p dz + d dz ( w 0 2 )+ N X m =1 ( R m c m ) g z =0, and(C–8) d 2 c m dz 2 V zm d c m dz d dz ( c 0 m w 0 )=0. (C–9) 228

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( C–7 )and( C–8 )aremeanmomentumequationsinstreamwiseandwallnormal direction.Themeanmomentumequationinthespanwisedirec tionistrivialasthere isnoforcinginthatdirection(notethat g y =0 ).Furthermore,itcanbeshownthat themeanvelocityinthebed-normaldirectioniszero( w =0 )duetocontinuityandno penetrationboundaryconditiononthebedandthetopbounda ry.Noticethatpressure appearsonlyin( C–8 )(bed-normaldirection)andintegrating( C–8 )givesthefollowing formforit, p = p + w w 0 2 + w N X m =1 R m g z Z z 0 c m ( ) d (C–10) Similarlyintegrating( C–7 )inthebed-normaldirectiongivesthesteadystatebalance betweenthefrictiononthebedandthetopboundaryandthedr ivingforcedueto buoyancyeffectimposedbysuspendedsediments.Sincethet opboundaryisno-stress, drivingforceisbalancedbystressonthebedanditisasshow nbelow b w = g x h N X m =1 R m C ( v ) m = Rg x hC ( v ) (C–11) where C ( v ) m = 1 h Z h 0 c m dz and C ( v ) = N X m =1 C ( v ) m (C–12) NotethattheReynoldsstress u 0 w 0 termdropsoutasitisexactlyzeroontheboundaries inthebednormaldirection.Also,itisassumedthatallthes edimenttypeshavesimilar density,i.e. sm s ,whichmeansthat R m R .Frictionvelocity( u )canbedened basedon( C–11 )andisusedasthevelocityscale. u 2 = b w = Rg x hC ( v ) (C–13) Lengthscaleis' h ',i.e.heightofthechannel,timescaleis h = u ,pressurescaleis w u 2 andconcentrationscaleis C ( v ) .Usingthesescales,governingequations( C–1 )-( C–3 ) reducetothefollowingdimensionlessform 229

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@ ~ u @ ~ t + ~ u r ~ u = r ~ p + 1 Re r 2 ~ u + N X m =1 ~ c m e (C–14) r ~ u =0, (C–15) @ ~ c m @ ~ t +( ~ u + ~ V m ) r ~ c m = 1 Re Sc r 2 ~ c m (C–16) Intheabove, (~ ) representsdimensionlessvariablesand e = f 1,0, g z = g x = 1 = tan g isthedimensionlessgravityvector.Reynoldsnumber( Re )andSchmidtnumber( Sc ) aretwodimensionlessparametersgoverningtheowandthei rformsareasshownhere Re = u h and Sc = (C–17) Including Re and Sc ,thereare 2 N +2 parametersthatfullydenetheabovesetof governingequations.Theseparametersare Re Sc ,channelinclination( ),settlingvelocityof'N'discretesedimenttypes( ~ V m )andtotalvolumetricconcentrationofindividual sedimenttypes( N 1 parameters).Wecandenethetotalvolumetricconcentrati onof aparticularsedimenttypeas r m = C ( v ) m C ( v ) sothat N X m =1 r m =1. (C–18) Insummary,tosimulateaowwithpolydispersesuspensionc ontaining' N 'discrete sedimentsizesmeanstosolve N +4 (3momentumequations,1continuityequationfor thecontinuousphaseandNtransportequationsforNdiscret esedimentsizes)PDEs simultaneously.Clearly,forlargeNtheproblembecomesco mputationallyexpensive,if notintractable. C.2CompleteTurbulenceSuppressionCriteria Inthissectionwewillshowthataparametricgroupingsimil artothemono-disperse modelcanbegiventoquantifyturbulencedampinginturbidi tycurrentsdrivenbypolydispersesuspension.Webelievethattherewillexistacrit icalvalueforthisparametric 230

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groupingbeyondwhichcompleteturbulencesuppressionwil loccur.Furthermore,weexpectthecriticalvaluetobesimilartotheoneobservedform ono-dispersesuspensions. Turbulentkineticenergy(TKE)equationforaturbiditycur rentdrivenbypolydispersesuspensionofsedimentandinthefullydevelopeds tatisticallysteadyregimeis givenbelow ~ P ~ d d ~ z ~ w 0 ~ p 0 + ~ k ~ w 0 1 Re d ~ k d ~ z # = N X m =1 ~ u 0 ~ c 0 m + N X m =1 ~ w 0 ~ c 0 m tan (C–19) whereturbulentkineticenergy(TKE) ~ k ,TKEproduction ~ P andTKEdissipation ~ are expressedas ~ k = ~ u 0 i ~ u 0 i 2 ~ P = ~ u 0 ~ w 0 d ~ u d ~ z and ~ = 1 Re @ ~ u 0 i @ ~ x j @ ~ u 0 i @ ~ x j (C–20) TheaboveTKEequation( C–19 )canbeintegratedinthebed-normaldirection( ~ z )tothe gettheglobalbalance. ~ P ~ E + 1 Re 8<: d ~ k d ~ z # ~h0 + Ri N X m =1 m 9=; = N X m =1 m + Ri N X m =1 ~ V mz r m (C–21) where ~ P = Z ~h 0 ~ P d ~ z ~ E = Z ~ h 0 ~ d ~ z m = Z ~h 0 ~ u 0 ~ c 0 m d ~ z m = ~ c mb ~ c mt Sc and Ri = 1 tan (C–22) Intheabove,terms m and Ri ~ V mz r m aremanifestationsoftheturbulentmixingterm ~ w 0 ~ c 0 m intheTKEequation( C–19 ).Thisturbulentmixingtermcanbeequivalently expressedbyintegrating( C–9 )whichgivestheextratermsintheglobalTKEbalance. ~ c mb and ~ c mt arethevolumetricconcentrationofsedimentparticlesoft ype' m 'atthe bottomandthetopboundariesofthechannel.In( C–19 ),thetwosummationsonthe rightquantifytheamountofenergyspentbyTKEtokeepthese dimentsinsuspension. Thersttermonthelefthandsideoftheequationisthetotal TKEproductionandthe 231

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secondtermisthetotalTKEdissipation.Thetwotermsinsid ethebracketsonthe leftsideoftheTKEequation( C–19 )representTKEdiffusionandbulkhydrodynamics sediment-sedimentandsediment-bedinteractions.Theset wotermsareusually negligibleathighReynoldsnumbersandtheTKEbudgetreduc estoabalancebetween TKEproduction,dissipationanddamping.TheglobalTKEbal anceequationforpolydispersesuspensionisquietsimilartothemono-disperses uspensions(refertoequation ( 3–33 )inChapter 3 and( 4–8 )inChapter 4 ).Webelievethattheobservationspresented inChapter 4 canbeextendedtocurrentsdrivenbypoly-dispersesuspens ionswhich meansthattheparametricgroupingthatcharacterizesturb ulencedampingwillbe Ri N X m =1 ~ V mz r m (C–23) Inotherwords,apolydispersesimulationcanbeconsidered asamono-disperse suspensionwithaneffectivesettlingvelocityoreffectiv esedimentsizewhichcanbe givenas ~ V e = N X m =1 ~ V mz r m (C–24) 232

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BIOGRAPHICALSKETCH MrugeshSurendraShringarpurewasborninThane,Maharasht ra,IndiaonDecember4,1985.HegraduatedfromSEShighschoolin2003andj oinedK.J.Somiaya collegeatMumbaiUniversitytostudyMechanicalengineeri ng.InJune2007,hegraduatedwithBachelorofEngineeringdegree.Aftergraduatio n,hewasrecruitedby Larsen & ToubroLtd.,anengineeringandconstructionconglomerate ,toworkasa graduateengineerintheirprecisiontoolsdivision.InAug ust2008,afterworkingfor LarsenandToubroLtd.forayear,hejoinedUniversityofFlo ridaatGainesvillefor M.S.inAerospaceEngineering.HegraduatedfromUniversit yofFloridawithaPh.D.in MechanicalEngineeringinDecember2013. 240